SET007 Axioms: SET007+189.ax


%------------------------------------------------------------------------------
% File     : SET007+189 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Reduction Relations
% 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    : rewrite1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  124 (   2 unit)
%            Number of atoms       :  622 (  50 equality)
%            Maximal formula depth :   21 (   7 average)
%            Number of connectives :  529 (  31 ~  ;  15  |; 219  &)
%                                         (  43 <=>; 221 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   40 (   1 propositional; 0-3 arity)
%            Number of functors    :   26 (   8 constant; 0-2 arity)
%            Number of variables   :  326 (   0 singleton; 310 !;  16 ?)
%            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(cc1_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m1_rewrite1(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(cc2_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v3_rewrite1(A) )
     => ( v1_relat_1(A)
        & v2_relat_2(A)
        & v1_rewrite1(A) ) ) )).

fof(cc3_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_relat_2(A)
        & v1_rewrite1(A) )
     => ( v1_relat_1(A)
        & v3_rewrite1(A) ) ) )).

fof(cc4_rewrite1,axiom,(
    ! [A] :
      ( ( v1_xboole_0(A)
        & v1_relat_1(A) )
     => ( v1_relat_1(A)
        & v2_relat_2(A)
        & v1_rewrite1(A)
        & v2_rewrite1(A)
        & v3_rewrite1(A) ) ) )).

fof(rc1_rewrite1,axiom,(
    ? [A] :
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v2_relat_2(A)
      & v1_rewrite1(A)
      & v2_rewrite1(A)
      & v3_rewrite1(A) ) )).

fof(cc5_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v3_rewrite1(A) )
     => ( v1_relat_1(A)
        & v2_rewrite1(A) ) ) )).

fof(cc6_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v8_rewrite1(A) )
     => ( v1_relat_1(A)
        & v7_rewrite1(A) ) ) )).

fof(cc7_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v7_rewrite1(A) )
     => ( v1_relat_1(A)
        & v8_rewrite1(A)
        & v9_rewrite1(A) ) ) )).

fof(cc8_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v6_rewrite1(A) )
     => ( v1_relat_1(A)
        & v7_rewrite1(A) ) ) )).

fof(cc9_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v8_rewrite1(A) )
     => ( v1_relat_1(A)
        & v5_rewrite1(A) ) ) )).

fof(cc10_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v5_rewrite1(A) )
     => ( v1_relat_1(A)
        & v4_rewrite1(A) ) ) )).

fof(cc11_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_rewrite1(A)
        & v4_rewrite1(A) )
     => ( v1_relat_1(A)
        & v8_rewrite1(A) ) ) )).

fof(cc12_rewrite1,axiom,(
    ! [A] :
      ( ( v1_xboole_0(A)
        & v1_relat_1(A) )
     => ( v1_relat_1(A)
        & v4_rewrite1(A)
        & v5_rewrite1(A)
        & v6_rewrite1(A)
        & v7_rewrite1(A)
        & v8_rewrite1(A)
        & v9_rewrite1(A) ) ) )).

fof(rc2_rewrite1,axiom,(
    ? [A] :
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v2_relat_2(A)
      & v1_rewrite1(A)
      & v2_rewrite1(A)
      & v3_rewrite1(A)
      & v4_rewrite1(A)
      & v5_rewrite1(A)
      & v6_rewrite1(A)
      & v7_rewrite1(A)
      & v8_rewrite1(A)
      & v9_rewrite1(A) ) )).

fof(cc13_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v3_rewrite1(A)
        & v9_rewrite1(A) )
     => ( v1_relat_1(A)
        & v4_rewrite1(A)
        & v5_rewrite1(A)
        & v7_rewrite1(A)
        & v8_rewrite1(A)
        & v9_rewrite1(A) ) ) )).

fof(cc14_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v10_rewrite1(A) )
     => ( v1_relat_1(A)
        & v2_relat_2(A)
        & v1_rewrite1(A)
        & v2_rewrite1(A)
        & v3_rewrite1(A)
        & v4_rewrite1(A)
        & v5_rewrite1(A)
        & v7_rewrite1(A)
        & v8_rewrite1(A)
        & v9_rewrite1(A) ) ) )).

fof(cc15_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v3_rewrite1(A)
        & v7_rewrite1(A) )
     => ( v1_relat_1(A)
        & v10_rewrite1(A) ) ) )).

fof(rc3_rewrite1,axiom,(
    ? [A] :
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v2_relat_2(A)
      & v1_rewrite1(A)
      & v2_rewrite1(A)
      & v3_rewrite1(A)
      & v4_rewrite1(A)
      & v5_rewrite1(A)
      & v6_rewrite1(A)
      & v7_rewrite1(A)
      & v8_rewrite1(A)
      & v9_rewrite1(A)
      & v10_rewrite1(A) ) )).

fof(rc4_rewrite1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v2_relat_2(A)
      & v1_rewrite1(A)
      & v2_rewrite1(A)
      & v3_rewrite1(A)
      & v4_rewrite1(A)
      & v5_rewrite1(A)
      & v7_rewrite1(A)
      & v8_rewrite1(A)
      & v9_rewrite1(A)
      & v10_rewrite1(A) ) )).

fof(d1_rewrite1,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) )
             => ( ( ( A = k1_xboole_0
                    | B = k1_xboole_0 )
                 => ( C = k1_rewrite1(A,B)
                  <=> C = k7_finseq_1(A,B) ) )
                & ~ ( A != k1_xboole_0
                    & B != k1_xboole_0
                    & ~ ( C = k1_rewrite1(A,B)
                      <=> ? [D] :
                            ( m2_subset_1(D,k1_numbers,k5_numbers)
                            & ? [E] :
                                ( v1_relat_1(E)
                                & v1_funct_1(E)
                                & v1_finseq_1(E)
                                & k3_finseq_1(A) = k1_nat_1(D,np__1)
                                & E = k7_relat_1(A,k2_finseq_1(D))
                                & C = k7_finseq_1(E,B) ) ) ) ) ) ) ) ) )).

fof(t1_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( k1_rewrite1(k1_xboole_0,A) = A
        & k1_rewrite1(A,k1_xboole_0) = A ) ) )).

fof(t2_rewrite1,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] :
              ( A != k1_xboole_0
             => k1_rewrite1(k7_finseq_1(B,k9_finseq_1(C)),A) = k7_finseq_1(B,A) ) ) ) )).

fof(t3_rewrite1,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,D] : k1_rewrite1(k7_finseq_1(A,k9_finseq_1(C)),k7_finseq_1(k9_finseq_1(D),B)) = k7_finseq_1(k7_finseq_1(A,k9_finseq_1(D)),B) ) ) )).

fof(t4_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( A != k1_xboole_0
         => k1_rewrite1(k9_finseq_1(B),A) = A ) ) )).

fof(t5_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ~ ( A != k1_xboole_0
          & ! [B,C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ~ ( A = k7_finseq_1(k9_finseq_1(B),C)
                  & k3_finseq_1(A) = k1_nat_1(k3_finseq_1(C),np__1) ) ) ) ) )).

fof(d2_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( m1_rewrite1(B,A)
          <=> ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(C,k4_finseq_1(B))
                      & r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(B)) )
                   => r2_hidden(k4_tarski(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))),A) ) ) ) ) ) ) )).

fof(t6_rewrite1,axiom,(
    $true )).

fof(t7_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] : m1_rewrite1(k9_finseq_1(B),A) ) )).

fof(t8_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r2_hidden(k4_tarski(B,C),A)
         => m1_rewrite1(k10_finseq_1(B,C),A) ) ) )).

fof(t9_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m1_rewrite1(B,A)
         => ! [C] :
              ( m1_rewrite1(C,A)
             => ( k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1)
               => m1_rewrite1(k1_rewrite1(B,C),A) ) ) ) ) )).

fof(t10_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m1_rewrite1(B,A)
         => m1_rewrite1(k3_finseq_5(B),k4_relat_1(A)) ) ) )).

fof(t11_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( r1_tarski(A,B)
           => ! [C] :
                ( m1_rewrite1(C,A)
               => m1_rewrite1(C,B) ) ) ) ) )).

fof(d3_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
        <=> ? [D] :
              ( m1_rewrite1(D,A)
              & k1_funct_1(D,np__1) = B
              & k1_funct_1(D,k3_finseq_1(D)) = C ) ) ) )).

fof(d4_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r2_rewrite1(A,B,C)
        <=> r1_rewrite1(k2_xboole_0(A,k4_relat_1(A)),B,C) ) ) )).

fof(t12_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
        <=> ? [D] :
              ( v1_relat_1(D)
              & v1_funct_1(D)
              & v1_finseq_1(D)
              & ~ r1_xreal_0(k3_finseq_1(D),np__0)
              & k1_funct_1(D,np__1) = B
              & k1_funct_1(D,k3_finseq_1(D)) = C
              & ! [E] :
                  ( m2_subset_1(E,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(E,k4_finseq_1(D))
                      & r2_hidden(k1_nat_1(E,np__1),k4_finseq_1(D)) )
                   => r2_hidden(k4_tarski(k1_funct_1(D,E),k1_funct_1(D,k1_nat_1(E,np__1))),A) ) ) ) ) ) )).

fof(t13_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] : r1_rewrite1(A,B,B) ) )).

fof(t14_rewrite1,axiom,(
    ! [A,B] :
      ( r1_rewrite1(k1_xboole_0,A,B)
     => A = B ) )).

fof(t15_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
         => ( r2_hidden(B,k3_relat_1(A))
            | B = C ) ) ) )).

fof(t16_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r2_hidden(k4_tarski(B,C),A)
         => r1_rewrite1(A,B,C) ) ) )).

fof(t17_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C,D] :
          ( ( r1_rewrite1(A,B,C)
            & r1_rewrite1(A,C,D) )
         => r1_rewrite1(A,B,D) ) ) )).

fof(t18_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m1_rewrite1(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,k4_finseq_1(B))
                      & r1_xreal_0(C,D) )
                   => r1_rewrite1(A,k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) )).

fof(t19_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
         => ( B = C
            | ( r2_hidden(B,k3_relat_1(A))
              & r2_hidden(C,k3_relat_1(A)) ) ) ) ) )).

fof(t20_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
         => ( r2_hidden(B,k3_relat_1(A))
          <=> r2_hidden(C,k3_relat_1(A)) ) ) ) )).

fof(t21_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
        <=> ( B = C
            | r2_hidden(k4_tarski(B,C),k17_finseq_1(A)) ) ) ) )).

fof(t22_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
        <=> r1_rewrite1(k17_finseq_1(A),B,C) ) ) )).

fof(t23_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( r1_tarski(A,B)
           => ! [C,D] :
                ( r1_rewrite1(A,C,D)
               => r1_rewrite1(B,C,D) ) ) ) ) )).

fof(t24_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C,D] :
          ( r1_rewrite1(A,C,D)
        <=> r1_rewrite1(k2_xboole_0(A,k6_relat_1(B)),C,D) ) ) )).

fof(t25_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
         => r1_rewrite1(k4_relat_1(A),C,B) ) ) )).

fof(t26_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
         => ( r2_rewrite1(A,B,C)
            & r2_rewrite1(A,C,B) ) ) ) )).

fof(t27_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] : r2_rewrite1(A,B,B) ) )).

fof(t28_rewrite1,axiom,(
    ! [A,B] :
      ( r2_rewrite1(k1_xboole_0,A,B)
     => A = B ) )).

fof(t29_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r2_rewrite1(A,B,C)
         => ( r2_hidden(B,k3_relat_1(A))
            | B = C ) ) ) )).

fof(t30_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r2_hidden(k4_tarski(B,C),A)
         => r2_rewrite1(A,B,C) ) ) )).

fof(t31_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C,D] :
          ( ( r2_rewrite1(A,B,C)
            & r2_rewrite1(A,C,D) )
         => r2_rewrite1(A,B,D) ) ) )).

fof(t32_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r2_rewrite1(A,B,C)
         => r2_rewrite1(A,C,B) ) ) )).

fof(t33_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r2_rewrite1(A,B,C)
         => ( B = C
            | ( r2_hidden(B,k3_relat_1(A))
              & r2_hidden(C,k3_relat_1(A)) ) ) ) ) )).

fof(d5_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r3_rewrite1(A,B)
        <=> ! [C] : ~ r2_hidden(k4_tarski(B,C),A) ) ) )).

fof(t34_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( ( r3_rewrite1(A,B)
            & r1_rewrite1(A,B,C) )
         => B = C ) ) )).

fof(t35_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ~ r2_hidden(B,k3_relat_1(A))
         => r3_rewrite1(A,B) ) ) )).

fof(d6_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r4_rewrite1(A,B,C)
        <=> ( r3_rewrite1(A,C)
            & r1_rewrite1(A,B,C) ) ) ) )).

fof(d7_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r5_rewrite1(A,B,C)
        <=> ? [D] :
              ( r1_rewrite1(A,B,D)
              & r1_rewrite1(A,C,D) ) ) ) )).

fof(d8_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r6_rewrite1(A,B,C)
        <=> ? [D] :
              ( r1_rewrite1(A,D,B)
              & r1_rewrite1(A,D,C) ) ) ) )).

fof(d9_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r7_rewrite1(A,B,C)
        <=> ? [D] :
              ( ( r2_hidden(k4_tarski(B,D),A)
                | B = D )
              & ( r2_hidden(k4_tarski(C,D),A)
                | C = D ) ) ) ) )).

fof(d10_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r8_rewrite1(A,B,C)
        <=> ? [D] :
              ( ( r2_hidden(k4_tarski(D,B),A)
                | B = D )
              & ( r2_hidden(k4_tarski(D,C),A)
                | C = D ) ) ) ) )).

fof(t36_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ~ r2_hidden(B,k3_relat_1(A))
         => r4_rewrite1(A,B,B) ) ) )).

fof(t37_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r1_rewrite1(A,B,C)
         => ( r5_rewrite1(A,B,C)
            & r6_rewrite1(A,B,C)
            & r5_rewrite1(A,C,B)
            & r6_rewrite1(A,C,B) ) ) ) )).

fof(t38_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( ( r5_rewrite1(A,B,C)
            | r6_rewrite1(A,B,C) )
         => r2_rewrite1(A,B,C) ) ) )).

fof(t39_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r5_rewrite1(A,B,B)
          & r6_rewrite1(A,B,B) ) ) )).

fof(t40_rewrite1,axiom,(
    ! [A,B] :
      ( ( r5_rewrite1(k1_xboole_0,A,B)
        | r6_rewrite1(k1_xboole_0,A,B) )
     => A = B ) )).

fof(t41_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r5_rewrite1(A,B,C)
         => r5_rewrite1(A,C,B) ) ) )).

fof(t42_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r6_rewrite1(A,B,C)
         => r6_rewrite1(A,C,B) ) ) )).

fof(t43_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C,D] :
          ( ( ( r1_rewrite1(A,B,C)
              & r5_rewrite1(A,C,D) )
            | ( r5_rewrite1(A,B,C)
              & r1_rewrite1(A,D,C) ) )
         => r5_rewrite1(A,B,D) ) ) )).

fof(t44_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C,D] :
          ( ( ( r1_rewrite1(A,C,B)
              & r6_rewrite1(A,C,D) )
            | ( r6_rewrite1(A,B,C)
              & r1_rewrite1(A,C,D) ) )
         => r6_rewrite1(A,B,D) ) ) )).

fof(t45_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r7_rewrite1(A,B,C)
         => r5_rewrite1(A,B,C) ) ) )).

fof(t46_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r8_rewrite1(A,B,C)
         => r6_rewrite1(A,B,C) ) ) )).

fof(d11_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r9_rewrite1(A,B)
        <=> ? [C] : r4_rewrite1(A,B,C) ) ) )).

fof(t47_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ~ r2_hidden(B,k3_relat_1(A))
         => r9_rewrite1(A,B) ) ) )).

fof(d12_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ( r9_rewrite1(A,B)
            & ! [C,D] :
                ( ( r4_rewrite1(A,B,C)
                  & r4_rewrite1(A,B,D) )
               => C = D ) )
         => ! [C] :
              ( C = k2_rewrite1(A,B)
            <=> r4_rewrite1(A,B,C) ) ) ) )).

fof(d13_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_rewrite1(A)
      <=> v1_wellord1(k4_relat_1(A)) ) ) )).

fof(d14_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v2_rewrite1(A)
      <=> ! [B] :
            ( r2_hidden(B,k3_relat_1(A))
           => r9_rewrite1(A,B) ) ) ) )).

fof(d15_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v3_rewrite1(A)
      <=> ! [B] :
            ( m1_pboole(B,k5_numbers)
           => ~ ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => r2_hidden(k4_tarski(k1_funct_1(B,C),k1_funct_1(B,k1_nat_1(C,np__1))),A) ) ) ) ) )).

fof(d16_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_rewrite1(A)
      <=> ! [B] :
            ~ ( r1_tarski(B,k3_relat_1(A))
              & B != k1_xboole_0
              & ! [C] :
                  ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ~ ( r2_hidden(D,B)
                          & C != D
                          & r2_hidden(k4_tarski(C,D),A) ) ) ) ) ) )).

fof(t48_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_rewrite1(A) )
     => ! [B] :
          ( v1_relat_1(B)
         => ( r1_tarski(B,A)
           => v1_rewrite1(B) ) ) ) )).

fof(d17_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( r10_rewrite1(A,B)
          <=> ! [C,D,E] :
                ~ ( r2_hidden(k4_tarski(C,D),A)
                  & r2_hidden(k4_tarski(C,E),B)
                  & ! [F] :
                      ~ ( r1_rewrite1(B,D,F)
                        & r1_rewrite1(A,E,F) ) ) ) ) ) )).

fof(d18_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( r11_rewrite1(A,B)
          <=> ! [C,D,E] :
                ~ ( r1_rewrite1(A,C,D)
                  & r1_rewrite1(B,C,E)
                  & ! [F] :
                      ~ ( r1_rewrite1(B,D,F)
                        & r1_rewrite1(A,E,F) ) ) ) ) ) )).

fof(t49_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( r11_rewrite1(A,B)
           => r10_rewrite1(A,B) ) ) ) )).

fof(d19_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v4_rewrite1(A)
      <=> ! [B,C] :
            ( ( r3_rewrite1(A,B)
              & r3_rewrite1(A,C)
              & r2_rewrite1(A,B,C) )
           => B = C ) ) ) )).

fof(d20_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v5_rewrite1(A)
      <=> ! [B,C] :
            ( ( r3_rewrite1(A,B)
              & r2_rewrite1(A,B,C) )
           => r1_rewrite1(A,C,B) ) ) ) )).

fof(d21_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v6_rewrite1(A)
      <=> ! [B,C,D] :
            ( ( r2_hidden(k4_tarski(B,C),A)
              & r2_hidden(k4_tarski(B,D),A) )
           => r7_rewrite1(A,C,D) ) ) ) )).

fof(d22_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_rewrite1(A)
      <=> ! [B,C] :
            ( r6_rewrite1(A,B,C)
           => r5_rewrite1(A,B,C) ) ) ) )).

fof(d23_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v8_rewrite1(A)
      <=> ! [B,C] :
            ( r2_rewrite1(A,B,C)
           => r5_rewrite1(A,B,C) ) ) ) )).

fof(d24_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v9_rewrite1(A)
      <=> ! [B,C,D] :
            ( ( r2_hidden(k4_tarski(B,C),A)
              & r2_hidden(k4_tarski(B,D),A) )
           => r5_rewrite1(A,C,D) ) ) ) )).

fof(t50_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v6_rewrite1(A)
       => ! [B,C,D] :
            ( ( r1_rewrite1(A,B,C)
              & r2_hidden(k4_tarski(B,D),A) )
           => r5_rewrite1(A,C,D) ) ) ) )).

fof(t51_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_rewrite1(A)
      <=> r11_rewrite1(A,A) ) ) )).

fof(t52_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_rewrite1(A)
      <=> ! [B,C,D] :
            ( ( r1_rewrite1(A,B,C)
              & r2_hidden(k4_tarski(B,D),A) )
           => r5_rewrite1(A,C,D) ) ) ) )).

fof(t53_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v9_rewrite1(A)
      <=> r10_rewrite1(A,A) ) ) )).

fof(t54_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v4_rewrite1(A) )
     => ! [B,C,D] :
          ( ( r4_rewrite1(A,B,C)
            & r4_rewrite1(A,B,D) )
         => C = D ) ) )).

fof(t55_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_rewrite1(A)
        & v4_rewrite1(A) )
     => ! [B] : r4_rewrite1(A,B,k2_rewrite1(A,B)) ) )).

fof(t56_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v2_rewrite1(A)
        & v4_rewrite1(A) )
     => ! [B,C] :
          ( r2_rewrite1(A,B,C)
         => k2_rewrite1(A,B) = k2_rewrite1(A,C) ) ) )).

fof(d25_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v10_rewrite1(A)
      <=> ( v7_rewrite1(A)
          & v3_rewrite1(A) ) ) ) )).

fof(t57_rewrite1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v8_rewrite1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v8_rewrite1(B) )
         => ( r11_rewrite1(A,B)
           => v8_rewrite1(k2_xboole_0(A,B)) ) ) ) )).

fof(t58_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_rewrite1(A)
      <=> v9_rewrite1(k17_finseq_1(A)) ) ) )).

fof(t59_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_rewrite1(A)
      <=> v6_rewrite1(k17_finseq_1(A)) ) ) )).

fof(d26_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( r12_rewrite1(A,B)
          <=> ! [C,D] :
                ( r2_rewrite1(A,C,D)
              <=> r2_rewrite1(B,C,D) ) ) ) ) )).

fof(d27_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r13_rewrite1(A,B,C)
        <=> ( r8_rewrite1(A,B,C)
            & ~ r5_rewrite1(A,B,C) ) ) ) )).

fof(t60_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B,C] :
          ( r13_rewrite1(A,B,C)
         => r2_rewrite1(A,B,C) ) ) )).

fof(t61_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( ! [B,C] : ~ r13_rewrite1(A,B,C)
       => v9_rewrite1(A) ) ) )).

fof(t62_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( ! [C,D] :
                ( r2_hidden(k4_tarski(C,D),B)
               => r13_rewrite1(A,C,D) )
           => r12_rewrite1(A,k2_xboole_0(A,B)) ) ) ) )).

fof(t63_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ? [B] :
          ( v1_relat_1(B)
          & v10_rewrite1(B)
          & r1_tarski(k3_relat_1(B),k3_relat_1(A))
          & ! [C,D] :
              ( r2_rewrite1(A,C,D)
            <=> r5_rewrite1(B,C,D) ) ) ) )).

fof(d28_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v10_rewrite1(B) )
         => ( m2_rewrite1(B,A)
          <=> ! [C,D] :
                ( r2_rewrite1(A,C,D)
              <=> r5_rewrite1(B,C,D) ) ) ) ) )).

fof(t64_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m2_rewrite1(B,A)
         => r12_rewrite1(A,B) ) ) )).

fof(t65_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v10_rewrite1(B) )
         => ( r12_rewrite1(A,B)
           => m2_rewrite1(B,A) ) ) ) )).

fof(t66_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m2_rewrite1(B,A)
         => ! [C,D] :
              ( r2_rewrite1(A,C,D)
            <=> k2_rewrite1(B,C) = k2_rewrite1(B,D) ) ) ) )).

fof(s1_rewrite1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r2_hidden(A,k4_finseq_1(f1_s1_rewrite1))
              & r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(f1_s1_rewrite1)) )
           => p1_s1_rewrite1(k1_funct_1(f1_s1_rewrite1,A),k1_funct_1(f1_s1_rewrite1,k1_nat_1(A,np__1))) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r2_hidden(A,k4_finseq_1(f2_s1_rewrite1))
              & r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(f2_s1_rewrite1)) )
           => p1_s1_rewrite1(k1_funct_1(f2_s1_rewrite1,A),k1_funct_1(f2_s1_rewrite1,k1_nat_1(A,np__1))) ) )
      & ~ r1_xreal_0(k3_finseq_1(f1_s1_rewrite1),np__0)
      & ~ r1_xreal_0(k3_finseq_1(f2_s1_rewrite1),np__0)
      & k1_funct_1(f1_s1_rewrite1,k3_finseq_1(f1_s1_rewrite1)) = k1_funct_1(f2_s1_rewrite1,np__1) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r2_hidden(A,k4_finseq_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1)))
            & r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1))) )
         => ! [B,C] :
              ( ( B = k1_funct_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1),A)
                & C = k1_funct_1(k1_rewrite1(f1_s1_rewrite1,f2_s1_rewrite1),k1_nat_1(A,np__1)) )
             => p1_s1_rewrite1(B,C) ) ) ) )).

fof(s2_rewrite1,axiom,
    ( ( v1_rewrite1(f1_s2_rewrite1)
      & ! [A] :
          ( ! [B] :
              ( r2_hidden(k4_tarski(A,B),f1_s2_rewrite1)
             => ( A = B
                | p1_s2_rewrite1(B) ) )
         => p1_s2_rewrite1(A) ) )
   => ! [A] :
        ( r2_hidden(A,k3_relat_1(f1_s2_rewrite1))
       => p1_s2_rewrite1(A) ) )).

fof(dt_m1_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m1_rewrite1(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) ) ) ) )).

fof(existence_m1_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ? [B] : m1_rewrite1(B,A) ) )).

fof(dt_m2_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( m2_rewrite1(B,A)
         => ( v1_relat_1(B)
            & v10_rewrite1(B) ) ) ) )).

fof(existence_m2_rewrite1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ? [B] : m2_rewrite1(B,A) ) )).

fof(symmetry_r10_rewrite1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => ( r10_rewrite1(A,B)
       => r10_rewrite1(B,A) ) ) )).

fof(symmetry_r11_rewrite1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => ( r11_rewrite1(A,B)
       => r11_rewrite1(B,A) ) ) )).

fof(symmetry_r12_rewrite1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => ( r12_rewrite1(A,B)
       => r12_rewrite1(B,A) ) ) )).

fof(dt_k1_rewrite1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( v1_relat_1(k1_rewrite1(A,B))
        & v1_funct_1(k1_rewrite1(A,B))
        & v1_finseq_1(k1_rewrite1(A,B)) ) ) )).

fof(dt_k2_rewrite1,axiom,(
    $true )).
%------------------------------------------------------------------------------