SET007 Axioms: SET007+65.ax


%------------------------------------------------------------------------------
% File     : SET007+65 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Recursive Definitions
% 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    : recdef_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   0 unit)
%            Number of atoms       :  435 ( 154 equality)
%            Maximal formula depth :   19 (  12 average)
%            Number of connectives :  428 (  18 ~  ;  32  |; 246  &)
%                                         (   0 <=>; 132 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   0 propositional; 1-3 arity)
%            Number of functors    :   77 (  61 constant; 0-4 arity)
%            Number of variables   :  171 (   0 singleton; 127 !;  44 ?)
%            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(s1_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
            ? [C] : p1_s1_recdef_1(A,B,C) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B,C,D] :
              ( ( p1_s1_recdef_1(A,B,C)
                & p1_s1_recdef_1(A,B,D) )
             => C = D ) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = k5_numbers
        & k1_funct_1(A,np__0) = f1_s1_recdef_1
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s1_recdef_1(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) )).

fof(s2_recdef_1,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ! [B] :
            ( m1_subset_1(B,f1_s2_recdef_1)
           => ? [C] :
                ( m1_subset_1(C,f1_s2_recdef_1)
                & p1_s2_recdef_1(A,B,C) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,f1_s2_recdef_1)
        & m2_relset_1(A,k5_numbers,f1_s2_recdef_1)
        & k8_funct_2(k5_numbers,f1_s2_recdef_1,A,np__0) = f2_s2_recdef_1
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s2_recdef_1(B,k8_funct_2(k5_numbers,f1_s2_recdef_1,A,B),k8_funct_2(k5_numbers,f1_s2_recdef_1,A,k1_nat_1(B,np__1))) ) ) )).

fof(s3_recdef_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = k5_numbers
      & k1_funct_1(A,np__0) = f1_s3_recdef_1
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_funct_1(A,k1_nat_1(B,np__1)) = f2_s3_recdef_1(B,k1_funct_1(A,B)) ) ) )).

fof(s4_recdef_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,f1_s4_recdef_1)
      & m2_relset_1(A,k5_numbers,f1_s4_recdef_1)
      & k8_funct_2(k5_numbers,f1_s4_recdef_1,A,np__0) = f2_s4_recdef_1
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,f1_s4_recdef_1,A,k1_nat_1(B,np__1)) = f3_s4_recdef_1(B,k8_funct_2(k5_numbers,f1_s4_recdef_1,A,B)) ) ) )).

fof(s5_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f2_s5_recdef_1,A)
              | ! [B] :
                ? [C] : p1_s5_recdef_1(A,B,C) ) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f2_s5_recdef_1,A)
              | ! [B,C,D] :
                  ( ( p1_s5_recdef_1(A,B,C)
                    & p1_s5_recdef_1(A,B,D) )
                 => C = D ) ) ) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & k3_finseq_1(A) = f2_s5_recdef_1
        & ( k1_funct_1(A,np__1) = f1_s5_recdef_1
          | f2_s5_recdef_1 = np__0 )
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => ( r1_xreal_0(f2_s5_recdef_1,B)
                | p1_s5_recdef_1(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ) )).

fof(s6_recdef_1,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r1_xreal_0(np__1,A)
         => ( r1_xreal_0(f3_s6_recdef_1,A)
            | ! [B] :
                ( m1_subset_1(B,f1_s6_recdef_1)
               => ? [C] :
                    ( m1_subset_1(C,f1_s6_recdef_1)
                    & p1_s6_recdef_1(A,B,C) ) ) ) ) )
   => ? [A] :
        ( m2_finseq_1(A,f1_s6_recdef_1)
        & k3_finseq_1(A) = f3_s6_recdef_1
        & ( k1_funct_1(A,np__1) = f2_s6_recdef_1
          | f3_s6_recdef_1 = np__0 )
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => ( r1_xreal_0(f3_s6_recdef_1,B)
                | p1_s6_recdef_1(B,k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ) )).

fof(s7_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ~ ( r1_xreal_0(np__1,A)
                & ~ r1_xreal_0(k3_finseq_1(f1_s7_recdef_1),A)
                & ! [C] : ~ p1_s7_recdef_1(k1_funct_1(f1_s7_recdef_1,k1_nat_1(A,np__1)),B,C) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B,C,D,E] :
              ( ( r1_xreal_0(np__1,A)
                & E = k1_funct_1(f1_s7_recdef_1,k1_nat_1(A,np__1))
                & p1_s7_recdef_1(E,B,C)
                & p1_s7_recdef_1(E,B,D) )
             => ( r1_xreal_0(k3_finseq_1(f1_s7_recdef_1),A)
                | C = D ) ) ) )
   => ? [A,B] :
        ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & A = k1_funct_1(B,k3_finseq_1(B))
        & k3_finseq_1(B) = k3_finseq_1(f1_s7_recdef_1)
        & k1_funct_1(B,np__1) = k1_funct_1(f1_s7_recdef_1,np__1)
        & ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,C)
             => ( r1_xreal_0(k3_finseq_1(f1_s7_recdef_1),C)
                | p1_s7_recdef_1(k1_funct_1(f1_s7_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) ) ) ) )).

fof(s8_recdef_1,axiom,(
    ? [A,B] :
      ( v1_relat_1(B)
      & v1_funct_1(B)
      & v1_finseq_1(B)
      & A = k1_funct_1(B,k3_finseq_1(B))
      & k3_finseq_1(B) = k3_finseq_1(f1_s8_recdef_1)
      & k1_funct_1(B,np__1) = k1_funct_1(f1_s8_recdef_1,np__1)
      & ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,C)
           => ( r1_xreal_0(k3_finseq_1(f1_s8_recdef_1),C)
              | k1_funct_1(B,k1_nat_1(C,np__1)) = f2_s8_recdef_1(k1_funct_1(f1_s8_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C)) ) ) ) ) )).

fof(s9_recdef_1,axiom,
    ( ( k1_relat_1(f2_s9_recdef_1) = k5_numbers
      & k1_funct_1(f2_s9_recdef_1,np__0) = f1_s9_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => p1_s9_recdef_1(A,k1_funct_1(f2_s9_recdef_1,A),k1_funct_1(f2_s9_recdef_1,k1_nat_1(A,np__1))) )
      & k1_relat_1(f3_s9_recdef_1) = k5_numbers
      & k1_funct_1(f3_s9_recdef_1,np__0) = f1_s9_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => p1_s9_recdef_1(A,k1_funct_1(f3_s9_recdef_1,A),k1_funct_1(f3_s9_recdef_1,k1_nat_1(A,np__1))) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B,C,D] :
              ( ( p1_s9_recdef_1(A,B,C)
                & p1_s9_recdef_1(A,B,D) )
             => C = D ) ) )
   => f2_s9_recdef_1 = f3_s9_recdef_1 )).

fof(s10_recdef_1,axiom,
    ( ( k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,np__0) = f2_s10_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => p1_s10_recdef_1(A,k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,A),k8_funct_2(k5_numbers,f1_s10_recdef_1,f3_s10_recdef_1,k1_nat_1(A,np__1))) )
      & k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,np__0) = f2_s10_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => p1_s10_recdef_1(A,k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,A),k8_funct_2(k5_numbers,f1_s10_recdef_1,f4_s10_recdef_1,k1_nat_1(A,np__1))) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m1_subset_1(B,f1_s10_recdef_1)
             => ! [C] :
                  ( m1_subset_1(C,f1_s10_recdef_1)
                 => ! [D] :
                      ( m1_subset_1(D,f1_s10_recdef_1)
                     => ( ( p1_s10_recdef_1(A,B,C)
                          & p1_s10_recdef_1(A,B,D) )
                       => C = D ) ) ) ) ) )
   => f3_s10_recdef_1 = f4_s10_recdef_1 )).

fof(s11_recdef_1,axiom,
    ( ( k1_relat_1(f3_s11_recdef_1) = k5_numbers
      & k1_funct_1(f3_s11_recdef_1,np__0) = f1_s11_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => k1_funct_1(f3_s11_recdef_1,k1_nat_1(A,np__1)) = f2_s11_recdef_1(A,k1_funct_1(f3_s11_recdef_1,A)) )
      & k1_relat_1(f4_s11_recdef_1) = k5_numbers
      & k1_funct_1(f4_s11_recdef_1,np__0) = f1_s11_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => k1_funct_1(f4_s11_recdef_1,k1_nat_1(A,np__1)) = f2_s11_recdef_1(A,k1_funct_1(f4_s11_recdef_1,A)) ) )
   => f3_s11_recdef_1 = f4_s11_recdef_1 )).

fof(s12_recdef_1,axiom,
    ( ( k8_funct_2(k5_numbers,f1_s12_recdef_1,f4_s12_recdef_1,np__0) = f2_s12_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,f1_s12_recdef_1,f4_s12_recdef_1,k1_nat_1(A,np__1)) = f3_s12_recdef_1(A,k8_funct_2(k5_numbers,f1_s12_recdef_1,f4_s12_recdef_1,A)) )
      & k8_funct_2(k5_numbers,f1_s12_recdef_1,f5_s12_recdef_1,np__0) = f2_s12_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,f1_s12_recdef_1,f5_s12_recdef_1,k1_nat_1(A,np__1)) = f3_s12_recdef_1(A,k8_funct_2(k5_numbers,f1_s12_recdef_1,f5_s12_recdef_1,A)) ) )
   => f4_s12_recdef_1 = f5_s12_recdef_1 )).

fof(s13_recdef_1,axiom,
    ( ( k8_funct_2(k5_numbers,k1_numbers,f3_s13_recdef_1,np__0) = f1_s13_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,k1_numbers,f3_s13_recdef_1,k1_nat_1(A,np__1)) = f2_s13_recdef_1(A,k8_funct_2(k5_numbers,k1_numbers,f3_s13_recdef_1,A)) )
      & k8_funct_2(k5_numbers,k1_numbers,f4_s13_recdef_1,np__0) = f1_s13_recdef_1
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,k1_numbers,f4_s13_recdef_1,k1_nat_1(A,np__1)) = f2_s13_recdef_1(A,k8_funct_2(k5_numbers,k1_numbers,f4_s13_recdef_1,A)) ) )
   => f3_s13_recdef_1 = f4_s13_recdef_1 )).

fof(s14_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f2_s14_recdef_1,A)
              | ! [B,C,D] :
                  ( ( p1_s14_recdef_1(A,B,C)
                    & p1_s14_recdef_1(A,B,D) )
                 => C = D ) ) ) )
      & k3_finseq_1(f3_s14_recdef_1) = f2_s14_recdef_1
      & ( k1_funct_1(f3_s14_recdef_1,np__1) = f1_s14_recdef_1
        | f2_s14_recdef_1 = np__0 )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f2_s14_recdef_1,A)
              | p1_s14_recdef_1(A,k1_funct_1(f3_s14_recdef_1,A),k1_funct_1(f3_s14_recdef_1,k1_nat_1(A,np__1))) ) ) )
      & k3_finseq_1(f4_s14_recdef_1) = f2_s14_recdef_1
      & ( k1_funct_1(f4_s14_recdef_1,np__1) = f1_s14_recdef_1
        | f2_s14_recdef_1 = np__0 )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f2_s14_recdef_1,A)
              | p1_s14_recdef_1(A,k1_funct_1(f4_s14_recdef_1,A),k1_funct_1(f4_s14_recdef_1,k1_nat_1(A,np__1))) ) ) ) )
   => f3_s14_recdef_1 = f4_s14_recdef_1 )).

fof(s15_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f3_s15_recdef_1,A)
              | ! [B] :
                  ( m1_subset_1(B,f1_s15_recdef_1)
                 => ! [C] :
                      ( m1_subset_1(C,f1_s15_recdef_1)
                     => ! [D] :
                          ( m1_subset_1(D,f1_s15_recdef_1)
                         => ( ( p1_s15_recdef_1(A,B,C)
                              & p1_s15_recdef_1(A,B,D) )
                           => C = D ) ) ) ) ) ) )
      & k3_finseq_1(f4_s15_recdef_1) = f3_s15_recdef_1
      & ( k1_funct_1(f4_s15_recdef_1,np__1) = f2_s15_recdef_1
        | f3_s15_recdef_1 = np__0 )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f3_s15_recdef_1,A)
              | p1_s15_recdef_1(A,k1_funct_1(f4_s15_recdef_1,A),k1_funct_1(f4_s15_recdef_1,k1_nat_1(A,np__1))) ) ) )
      & k3_finseq_1(f5_s15_recdef_1) = f3_s15_recdef_1
      & ( k1_funct_1(f5_s15_recdef_1,np__1) = f2_s15_recdef_1
        | f3_s15_recdef_1 = np__0 )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => ( r1_xreal_0(f3_s15_recdef_1,A)
              | p1_s15_recdef_1(A,k1_funct_1(f5_s15_recdef_1,A),k1_funct_1(f5_s15_recdef_1,k1_nat_1(A,np__1))) ) ) ) )
   => f4_s15_recdef_1 = f5_s15_recdef_1 )).

fof(s16_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B,C,D,E] :
              ( ( r1_xreal_0(np__1,A)
                & E = k1_funct_1(f1_s16_recdef_1,k1_nat_1(A,np__1))
                & p1_s16_recdef_1(E,B,C)
                & p1_s16_recdef_1(E,B,D) )
             => ( r1_xreal_0(k3_finseq_1(f1_s16_recdef_1),A)
                | C = D ) ) )
      & ? [A] :
          ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A)
          & f2_s16_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
          & k3_finseq_1(A) = k3_finseq_1(f1_s16_recdef_1)
          & k1_funct_1(A,np__1) = k1_funct_1(f1_s16_recdef_1,np__1)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__1,B)
               => ( r1_xreal_0(k3_finseq_1(f1_s16_recdef_1),B)
                  | p1_s16_recdef_1(k1_funct_1(f1_s16_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) )
      & ? [A] :
          ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A)
          & f3_s16_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
          & k3_finseq_1(A) = k3_finseq_1(f1_s16_recdef_1)
          & k1_funct_1(A,np__1) = k1_funct_1(f1_s16_recdef_1,np__1)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__1,B)
               => ( r1_xreal_0(k3_finseq_1(f1_s16_recdef_1),B)
                  | p1_s16_recdef_1(k1_funct_1(f1_s16_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1))) ) ) ) ) )
   => f2_s16_recdef_1 = f3_s16_recdef_1 )).

fof(s17_recdef_1,axiom,
    ( ( ? [A] :
          ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A)
          & f3_s17_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
          & k3_finseq_1(A) = k3_finseq_1(f1_s17_recdef_1)
          & k1_funct_1(A,np__1) = k1_funct_1(f1_s17_recdef_1,np__1)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__1,B)
               => ( r1_xreal_0(k3_finseq_1(f1_s17_recdef_1),B)
                  | k1_funct_1(A,k1_nat_1(B,np__1)) = f2_s17_recdef_1(k1_funct_1(f1_s17_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B)) ) ) ) )
      & ? [A] :
          ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A)
          & f4_s17_recdef_1 = k1_funct_1(A,k3_finseq_1(A))
          & k3_finseq_1(A) = k3_finseq_1(f1_s17_recdef_1)
          & k1_funct_1(A,np__1) = k1_funct_1(f1_s17_recdef_1,np__1)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__1,B)
               => ( r1_xreal_0(k3_finseq_1(f1_s17_recdef_1),B)
                  | k1_funct_1(A,k1_nat_1(B,np__1)) = f2_s17_recdef_1(k1_funct_1(f1_s17_recdef_1,k1_nat_1(B,np__1)),k1_funct_1(A,B)) ) ) ) ) )
   => f3_s17_recdef_1 = f4_s17_recdef_1 )).

fof(s18_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
            ? [C] : p1_s18_recdef_1(A,B,C) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B,C,D] :
              ( ( p1_s18_recdef_1(A,B,C)
                & p1_s18_recdef_1(A,B,D) )
             => C = D ) ) )
   => ( ? [A,B] :
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & A = k1_funct_1(B,f2_s18_recdef_1)
          & k1_relat_1(B) = k5_numbers
          & k1_funct_1(B,np__0) = f1_s18_recdef_1
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => p1_s18_recdef_1(C,k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) )
      & ! [A,B] :
          ~ ( ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & A = k1_funct_1(C,f2_s18_recdef_1)
                & k1_relat_1(C) = k5_numbers
                & k1_funct_1(C,np__0) = f1_s18_recdef_1
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => p1_s18_recdef_1(D,k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) )
            & ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & B = k1_funct_1(C,f2_s18_recdef_1)
                & k1_relat_1(C) = k5_numbers
                & k1_funct_1(C,np__0) = f1_s18_recdef_1
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => p1_s18_recdef_1(D,k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) )
            & A != B ) ) )).

fof(s19_recdef_1,axiom,
    ( ? [A,B] :
        ( v1_relat_1(B)
        & v1_funct_1(B)
        & A = k1_funct_1(B,f2_s19_recdef_1)
        & k1_relat_1(B) = k5_numbers
        & k1_funct_1(B,np__0) = f1_s19_recdef_1
        & ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => k1_funct_1(B,k1_nat_1(C,np__1)) = f3_s19_recdef_1(C,k1_funct_1(B,C)) ) )
    & ! [A,B] :
        ~ ( ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & A = k1_funct_1(C,f2_s19_recdef_1)
              & k1_relat_1(C) = k5_numbers
              & k1_funct_1(C,np__0) = f1_s19_recdef_1
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_funct_1(C,k1_nat_1(D,np__1)) = f3_s19_recdef_1(D,k1_funct_1(C,D)) ) )
          & ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & B = k1_funct_1(C,f2_s19_recdef_1)
              & k1_relat_1(C) = k5_numbers
              & k1_funct_1(C,np__0) = f1_s19_recdef_1
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_funct_1(C,k1_nat_1(D,np__1)) = f3_s19_recdef_1(D,k1_funct_1(C,D)) ) )
          & A != B ) )).

fof(s20_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m1_subset_1(B,f1_s20_recdef_1)
             => ? [C] :
                  ( m1_subset_1(C,f1_s20_recdef_1)
                  & p1_s20_recdef_1(A,B,C) ) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m1_subset_1(B,f1_s20_recdef_1)
             => ! [C] :
                  ( m1_subset_1(C,f1_s20_recdef_1)
                 => ! [D] :
                      ( m1_subset_1(D,f1_s20_recdef_1)
                     => ( ( p1_s20_recdef_1(A,B,C)
                          & p1_s20_recdef_1(A,B,D) )
                       => C = D ) ) ) ) ) )
   => ( ? [A] :
          ( m1_subset_1(A,f1_s20_recdef_1)
          & ? [B] :
              ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,f1_s20_recdef_1)
              & m2_relset_1(B,k5_numbers,f1_s20_recdef_1)
              & A = k8_funct_2(k5_numbers,f1_s20_recdef_1,B,f3_s20_recdef_1)
              & k8_funct_2(k5_numbers,f1_s20_recdef_1,B,np__0) = f2_s20_recdef_1
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => p1_s20_recdef_1(C,k8_funct_2(k5_numbers,f1_s20_recdef_1,B,C),k8_funct_2(k5_numbers,f1_s20_recdef_1,B,k1_nat_1(C,np__1))) ) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s20_recdef_1)
         => ! [B] :
              ( m1_subset_1(B,f1_s20_recdef_1)
             => ~ ( ? [C] :
                      ( v1_funct_1(C)
                      & v1_funct_2(C,k5_numbers,f1_s20_recdef_1)
                      & m2_relset_1(C,k5_numbers,f1_s20_recdef_1)
                      & A = k8_funct_2(k5_numbers,f1_s20_recdef_1,C,f3_s20_recdef_1)
                      & k8_funct_2(k5_numbers,f1_s20_recdef_1,C,np__0) = f2_s20_recdef_1
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => p1_s20_recdef_1(D,k8_funct_2(k5_numbers,f1_s20_recdef_1,C,D),k8_funct_2(k5_numbers,f1_s20_recdef_1,C,k1_nat_1(D,np__1))) ) )
                  & ? [C] :
                      ( v1_funct_1(C)
                      & v1_funct_2(C,k5_numbers,f1_s20_recdef_1)
                      & m2_relset_1(C,k5_numbers,f1_s20_recdef_1)
                      & B = k8_funct_2(k5_numbers,f1_s20_recdef_1,C,f3_s20_recdef_1)
                      & k8_funct_2(k5_numbers,f1_s20_recdef_1,C,np__0) = f2_s20_recdef_1
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => p1_s20_recdef_1(D,k8_funct_2(k5_numbers,f1_s20_recdef_1,C,D),k8_funct_2(k5_numbers,f1_s20_recdef_1,C,k1_nat_1(D,np__1))) ) )
                  & A != B ) ) ) ) )).

fof(s21_recdef_1,axiom,
    ( ? [A] :
        ( m1_subset_1(A,f1_s21_recdef_1)
        & ? [B] :
            ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,f1_s21_recdef_1)
            & m2_relset_1(B,k5_numbers,f1_s21_recdef_1)
            & A = k8_funct_2(k5_numbers,f1_s21_recdef_1,B,f3_s21_recdef_1)
            & k8_funct_2(k5_numbers,f1_s21_recdef_1,B,np__0) = f2_s21_recdef_1
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k8_funct_2(k5_numbers,f1_s21_recdef_1,B,k1_nat_1(C,np__1)) = f4_s21_recdef_1(C,k8_funct_2(k5_numbers,f1_s21_recdef_1,B,C)) ) ) )
    & ! [A] :
        ( m1_subset_1(A,f1_s21_recdef_1)
       => ! [B] :
            ( m1_subset_1(B,f1_s21_recdef_1)
           => ~ ( ? [C] :
                    ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,f1_s21_recdef_1)
                    & m2_relset_1(C,k5_numbers,f1_s21_recdef_1)
                    & A = k8_funct_2(k5_numbers,f1_s21_recdef_1,C,f3_s21_recdef_1)
                    & k8_funct_2(k5_numbers,f1_s21_recdef_1,C,np__0) = f2_s21_recdef_1
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => k8_funct_2(k5_numbers,f1_s21_recdef_1,C,k1_nat_1(D,np__1)) = f4_s21_recdef_1(D,k8_funct_2(k5_numbers,f1_s21_recdef_1,C,D)) ) )
                & ? [C] :
                    ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,f1_s21_recdef_1)
                    & m2_relset_1(C,k5_numbers,f1_s21_recdef_1)
                    & B = k8_funct_2(k5_numbers,f1_s21_recdef_1,C,f3_s21_recdef_1)
                    & k8_funct_2(k5_numbers,f1_s21_recdef_1,C,np__0) = f2_s21_recdef_1
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => k8_funct_2(k5_numbers,f1_s21_recdef_1,C,k1_nat_1(D,np__1)) = f4_s21_recdef_1(D,k8_funct_2(k5_numbers,f1_s21_recdef_1,C,D)) ) )
                & A != B ) ) ) )).

fof(s22_recdef_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ~ ( r1_xreal_0(np__1,A)
                & ~ r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),A)
                & ! [C] : ~ p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(A,np__1)),B,C) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B,C,D,E] :
              ( ( r1_xreal_0(np__1,A)
                & E = k1_funct_1(f1_s22_recdef_1,k1_nat_1(A,np__1))
                & p1_s22_recdef_1(E,B,C)
                & p1_s22_recdef_1(E,B,D) )
             => ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),A)
                | C = D ) ) ) )
   => ( ? [A,B] :
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & v1_finseq_1(B)
          & A = k1_funct_1(B,k3_finseq_1(B))
          & k3_finseq_1(B) = k3_finseq_1(f1_s22_recdef_1)
          & k1_funct_1(B,np__1) = k1_funct_1(f1_s22_recdef_1,np__1)
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__1,C)
               => ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),C)
                  | p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))) ) ) ) )
      & ! [A,B] :
          ~ ( ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C)
                & A = k1_funct_1(C,k3_finseq_1(C))
                & k3_finseq_1(C) = k3_finseq_1(f1_s22_recdef_1)
                & k1_funct_1(C,np__1) = k1_funct_1(f1_s22_recdef_1,np__1)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(np__1,D)
                     => ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),D)
                        | p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) ) ) )
            & ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C)
                & B = k1_funct_1(C,k3_finseq_1(C))
                & k3_finseq_1(C) = k3_finseq_1(f1_s22_recdef_1)
                & k1_funct_1(C,np__1) = k1_funct_1(f1_s22_recdef_1,np__1)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(np__1,D)
                     => ( r1_xreal_0(k3_finseq_1(f1_s22_recdef_1),D)
                        | p1_s22_recdef_1(k1_funct_1(f1_s22_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D),k1_funct_1(C,k1_nat_1(D,np__1))) ) ) ) )
            & A != B ) ) )).

fof(s23_recdef_1,axiom,
    ( ? [A,B] :
        ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B)
        & A = k1_funct_1(B,k3_finseq_1(B))
        & k3_finseq_1(B) = k3_finseq_1(f1_s23_recdef_1)
        & k1_funct_1(B,np__1) = k1_funct_1(f1_s23_recdef_1,np__1)
        & ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,C)
             => ( r1_xreal_0(k3_finseq_1(f1_s23_recdef_1),C)
                | k1_funct_1(B,k1_nat_1(C,np__1)) = f2_s23_recdef_1(k1_funct_1(f1_s23_recdef_1,k1_nat_1(C,np__1)),k1_funct_1(B,C)) ) ) ) )
    & ! [A,B] :
        ~ ( ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & v1_finseq_1(C)
              & A = k1_funct_1(C,k3_finseq_1(C))
              & k3_finseq_1(C) = k3_finseq_1(f1_s23_recdef_1)
              & k1_funct_1(C,np__1) = k1_funct_1(f1_s23_recdef_1,np__1)
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__1,D)
                   => ( r1_xreal_0(k3_finseq_1(f1_s23_recdef_1),D)
                      | k1_funct_1(C,k1_nat_1(D,np__1)) = f2_s23_recdef_1(k1_funct_1(f1_s23_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D)) ) ) ) )
          & ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & v1_finseq_1(C)
              & B = k1_funct_1(C,k3_finseq_1(C))
              & k3_finseq_1(C) = k3_finseq_1(f1_s23_recdef_1)
              & k1_funct_1(C,np__1) = k1_funct_1(f1_s23_recdef_1,np__1)
              & ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(np__1,D)
                   => ( r1_xreal_0(k3_finseq_1(f1_s23_recdef_1),D)
                      | k1_funct_1(C,k1_nat_1(D,np__1)) = f2_s23_recdef_1(k1_funct_1(f1_s23_recdef_1,k1_nat_1(D,np__1)),k1_funct_1(C,D)) ) ) ) )
          & A != B ) )).

fof(dt_k1_recdef_1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & m1_relset_1(B,A,k5_numbers)
        & m1_subset_1(C,A) )
     => m2_subset_1(k1_recdef_1(A,B,C),k1_numbers,k5_numbers) ) )).

fof(redefinition_k1_recdef_1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & m1_relset_1(B,A,k5_numbers)
        & m1_subset_1(C,A) )
     => k1_recdef_1(A,B,C) = k1_funct_1(B,C) ) )).
%------------------------------------------------------------------------------