SET007 Axioms: SET007+204.ax


%------------------------------------------------------------------------------
% File     : SET007+204 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Graphs
% 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    : graph_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   97 (  10 unit)
%            Number of atoms       :  652 (  69 equality)
%            Maximal formula depth :   25 (   7 average)
%            Number of connectives :  571 (  16 ~  ;   4  |; 329  &)
%                                         (  29 <=>; 193 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   38 (   1 propositional; 0-4 arity)
%            Number of functors    :   27 (   5 constant; 0-4 arity)
%            Number of variables   :  215 (   0 singleton; 190 !;  25 ?)
%            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(rc1_graph_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v1_graph_1(A) ) )).

fof(rc2_graph_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v1_graph_1(A)
      & v2_graph_1(A) ) )).

fof(rc3_graph_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v7_graph_1(A) ) )).

fof(rc4_graph_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v2_graph_1(A)
      & v3_graph_1(A)
      & v4_graph_1(A)
      & v5_graph_1(A)
      & v6_graph_1(A)
      & v7_graph_1(A) ) )).

fof(rc5_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v8_graph_1(B,A) ) ) )).

fof(rc6_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B) ) ) )).

fof(rc7_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v8_graph_1(B,A) ) ) )).

fof(rc8_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v10_graph_1(B,A) ) ) )).

fof(rc9_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v8_graph_1(B,A)
          & v10_graph_1(B,A) ) ) )).

fof(rc10_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m3_graph_1(B,A)
          & v1_graph_1(B)
          & v2_graph_1(B) ) ) )).

fof(d1_graph_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( v2_graph_1(A)
      <=> ~ v1_xboole_0(u1_graph_1(A)) ) ) )).

fof(d2_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
              & r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
           => ! [C] :
                ( ( v1_graph_1(C)
                  & v2_graph_1(C)
                  & l1_graph_1(C) )
               => ( C = k1_graph_1(A,B)
                <=> ( u1_graph_1(C) = k2_xboole_0(u1_graph_1(A),u1_graph_1(B))
                    & u2_graph_1(C) = k2_xboole_0(u2_graph_1(A),u2_graph_1(B))
                    & ! [D] :
                        ( r2_hidden(D,u2_graph_1(A))
                       => ( k1_funct_1(u3_graph_1(C),D) = k1_funct_1(u3_graph_1(A),D)
                          & k1_funct_1(u4_graph_1(C),D) = k1_funct_1(u4_graph_1(A),D) ) )
                    & ! [D] :
                        ( r2_hidden(D,u2_graph_1(B))
                       => ( k1_funct_1(u3_graph_1(C),D) = k1_funct_1(u3_graph_1(B),D)
                          & k1_funct_1(u4_graph_1(C),D) = k1_funct_1(u4_graph_1(B),D) ) ) ) ) ) ) ) ) )).

fof(d3_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( r1_graph_1(A,B,C)
              <=> ( r1_partfun1(u4_graph_1(B),u4_graph_1(C))
                  & r1_partfun1(u3_graph_1(B),u3_graph_1(C))
                  & g1_graph_1(u1_graph_1(A),u2_graph_1(A),u3_graph_1(A),u4_graph_1(A)) = k1_graph_1(B,C) ) ) ) ) ) )).

fof(d4_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v3_graph_1(A)
      <=> ! [B,C] :
            ( ( r2_hidden(B,u2_graph_1(A))
              & r2_hidden(C,u2_graph_1(A))
              & k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u3_graph_1(A),C)
              & k1_funct_1(u4_graph_1(A),B) = k1_funct_1(u4_graph_1(A),C) )
           => B = C ) ) ) )).

fof(d5_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v4_graph_1(A)
      <=> ! [B,C] :
            ( ( r2_hidden(B,u2_graph_1(A))
              & r2_hidden(C,u2_graph_1(A)) )
           => ( ( ~ ( k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u3_graph_1(A),C)
                    & k1_funct_1(u4_graph_1(A),B) = k1_funct_1(u4_graph_1(A),C) )
                & ~ ( k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u4_graph_1(A),C)
                    & k1_funct_1(u3_graph_1(A),C) = k1_funct_1(u4_graph_1(A),B) ) )
              | B = C ) ) ) ) )).

fof(d6_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v5_graph_1(A)
      <=> ! [B] :
            ~ ( r2_hidden(B,u2_graph_1(A))
              & k1_funct_1(u3_graph_1(A),B) = k1_funct_1(u4_graph_1(A),B) ) ) ) )).

fof(d7_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v6_graph_1(A)
      <=> ! [B] :
            ( ( v2_graph_1(B)
              & l1_graph_1(B) )
           => ! [C] :
                ( ( v2_graph_1(C)
                  & l1_graph_1(C) )
               => ~ ( r1_xboole_0(u1_graph_1(B),u1_graph_1(C))
                    & r1_graph_1(A,B,C) ) ) ) ) ) )).

fof(d8_graph_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( v7_graph_1(A)
      <=> ( v1_finset_1(u1_graph_1(A))
          & v1_finset_1(u2_graph_1(A)) ) ) ) )).

fof(d9_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_graph_1(A))
             => ! [D] :
                  ( r2_graph_1(A,B,C,D)
                <=> ( ( k1_funct_1(u3_graph_1(A),D) = B
                      & k1_funct_1(u4_graph_1(A),D) = C )
                    | ( k1_funct_1(u3_graph_1(A),D) = C
                      & k1_funct_1(u4_graph_1(A),D) = B ) ) ) ) ) ) )).

fof(d10_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_graph_1(A))
             => ( r3_graph_1(A,B,C)
              <=> ? [D] :
                    ( r2_hidden(D,u2_graph_1(A))
                    & r2_graph_1(A,B,C,D) ) ) ) ) ) )).

fof(d11_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( m1_graph_1(B,A)
          <=> ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(np__1,C)
                      & r1_xreal_0(C,k3_finseq_1(B)) )
                   => r2_hidden(k1_funct_1(B,C),u2_graph_1(A)) ) )
              & ? [C] :
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v1_finseq_1(C)
                  & k3_finseq_1(C) = k1_nat_1(k3_finseq_1(B),np__1)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k3_finseq_1(C)) )
                       => r2_hidden(k1_funct_1(C,D),u1_graph_1(A)) ) )
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k3_finseq_1(B))
                          & ! [E] :
                              ( m1_subset_1(E,u1_graph_1(A))
                             => ! [F] :
                                  ( m1_subset_1(F,u1_graph_1(A))
                                 => ~ ( E = k1_funct_1(C,D)
                                      & F = k1_funct_1(C,k1_nat_1(D,np__1))
                                      & r2_graph_1(A,E,F,k1_funct_1(B,D)) ) ) ) ) ) ) ) ) ) ) )).

fof(d12_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
         => ( v8_graph_1(B,A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(np__1,C)
                 => ( r1_xreal_0(k3_finseq_1(B),C)
                    | k1_funct_1(u3_graph_1(A),k1_funct_1(B,k1_nat_1(C,np__1))) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,C)) ) ) ) ) ) ) )).

fof(d13_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
         => ( v2_funct_1(B)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ ( r1_xreal_0(np__1,C)
                        & ~ r1_xreal_0(D,C)
                        & r1_xreal_0(D,k3_finseq_1(B))
                        & k1_funct_1(B,C) = k1_funct_1(B,D) ) ) ) ) ) ) )).

fof(d14_graph_1,axiom,(
    $true )).

fof(d15_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_funct_1(B)
            & m2_graph_1(B,A) )
         => ( v10_graph_1(B,A)
          <=> ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C)
                & k3_finseq_1(C) = k1_nat_1(k3_finseq_1(B),np__1)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ( r1_xreal_0(np__1,D)
                        & r1_xreal_0(D,k3_finseq_1(C)) )
                     => r2_hidden(k1_funct_1(C,D),u1_graph_1(A)) ) )
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ ( r1_xreal_0(np__1,D)
                        & r1_xreal_0(D,k3_finseq_1(B))
                        & ! [E] :
                            ( m1_subset_1(E,u1_graph_1(A))
                           => ! [F] :
                                ( m1_subset_1(F,u1_graph_1(A))
                               => ~ ( E = k1_funct_1(C,D)
                                    & F = k1_funct_1(C,k1_nat_1(D,np__1))
                                    & r2_graph_1(A,E,F,k1_funct_1(B,D)) ) ) ) ) )
                & k1_funct_1(C,np__1) = k1_funct_1(C,k3_finseq_1(C)) ) ) ) ) )).

fof(d16_graph_1,axiom,(
    $true )).

fof(d17_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( m3_graph_1(B,A)
          <=> ( r1_tarski(u1_graph_1(B),u1_graph_1(A))
              & r1_tarski(u2_graph_1(B),u2_graph_1(A))
              & ! [C] :
                  ( r2_hidden(C,u2_graph_1(B))
                 => ( k1_funct_1(u3_graph_1(B),C) = k1_funct_1(u3_graph_1(A),C)
                    & k1_funct_1(u4_graph_1(B),C) = k1_funct_1(u4_graph_1(A),C)
                    & r2_hidden(k1_funct_1(u3_graph_1(A),C),u1_graph_1(B))
                    & r2_hidden(k1_funct_1(u4_graph_1(A),C),u1_graph_1(B)) ) ) ) ) ) ) )).

fof(d18_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k2_graph_1(A)
          <=> ? [C] :
                ( v1_finset_1(C)
                & C = u1_graph_1(A)
                & B = k4_card_1(C) ) ) ) ) )).

fof(d19_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k3_graph_1(A)
          <=> ? [C] :
                ( v1_finset_1(C)
                & C = u2_graph_1(A)
                & B = k4_card_1(C) ) ) ) ) )).

fof(d20_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k4_graph_1(A,B)
              <=> ? [D] :
                    ( v1_finset_1(D)
                    & ! [E] :
                        ( r2_hidden(E,D)
                      <=> ( r2_hidden(E,u2_graph_1(A))
                          & k1_funct_1(u4_graph_1(A),E) = B ) )
                    & C = k4_card_1(D) ) ) ) ) ) )).

fof(d21_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k5_graph_1(A,B)
              <=> ? [D] :
                    ( v1_finset_1(D)
                    & ! [E] :
                        ( r2_hidden(E,D)
                      <=> ( r2_hidden(E,u2_graph_1(A))
                          & k1_funct_1(u3_graph_1(A),E) = B ) )
                    & C = k4_card_1(D) ) ) ) ) ) )).

fof(d22_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => k6_graph_1(A,B) = k1_nat_1(k4_graph_1(A,B),k5_graph_1(A,B)) ) ) )).

fof(d23_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( r4_graph_1(A,B)
          <=> m3_graph_1(A,B) ) ) ) )).

fof(d24_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( B = k7_graph_1(A)
        <=> ! [C] :
              ( r2_hidden(C,B)
            <=> ( v1_graph_1(C)
                & m3_graph_1(C,A) ) ) ) ) )).

fof(t1_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( k1_relat_1(u3_graph_1(A)) = u2_graph_1(A)
        & k1_relat_1(u4_graph_1(A)) = u2_graph_1(A)
        & r1_tarski(k2_relat_1(u3_graph_1(A)),u1_graph_1(A))
        & r1_tarski(k2_relat_1(u4_graph_1(A)),u1_graph_1(A)) ) ) )).

fof(t2_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => r2_hidden(B,u1_graph_1(A)) ) ) )).

fof(t3_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( r2_hidden(B,u2_graph_1(A))
         => ( r2_hidden(k1_funct_1(u3_graph_1(A),B),u1_graph_1(A))
            & r2_hidden(k1_funct_1(u4_graph_1(A),B),u1_graph_1(A)) ) ) ) )).

fof(t4_graph_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( m2_graph_1(C,B)
             => m2_graph_1(k2_partfun1(k5_numbers,u2_graph_1(B),C,k2_finseq_1(A)),B) ) ) ) )).

fof(t5_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( r4_graph_1(A,B)
           => ( r1_tarski(u3_graph_1(A),u3_graph_1(B))
              & r1_tarski(u4_graph_1(A),u4_graph_1(B)) ) ) ) ) )).

fof(t6_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
              & r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
           => ( u3_graph_1(k1_graph_1(A,B)) = k2_xboole_0(u3_graph_1(A),u3_graph_1(B))
              & u4_graph_1(k1_graph_1(A,B)) = k2_xboole_0(u4_graph_1(A),u4_graph_1(B)) ) ) ) ) )).

fof(t7_graph_1,axiom,(
    ! [A] :
      ( ( v1_graph_1(A)
        & v2_graph_1(A)
        & l1_graph_1(A) )
     => A = k1_graph_1(A,A) ) )).

fof(t8_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
              & r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
           => k1_graph_1(A,B) = k1_graph_1(B,A) ) ) ) )).

fof(t9_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
                  & r1_partfun1(u4_graph_1(A),u4_graph_1(B))
                  & r1_partfun1(u3_graph_1(A),u3_graph_1(C))
                  & r1_partfun1(u4_graph_1(A),u4_graph_1(C))
                  & r1_partfun1(u3_graph_1(B),u3_graph_1(C))
                  & r1_partfun1(u4_graph_1(B),u4_graph_1(C)) )
               => k1_graph_1(k1_graph_1(A,B),C) = k1_graph_1(A,k1_graph_1(B,C)) ) ) ) ) )).

fof(t10_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( r1_graph_1(A,B,C)
               => r1_graph_1(A,C,B) ) ) ) ) )).

fof(t11_graph_1,axiom,(
    ! [A] :
      ( ( v1_graph_1(A)
        & v2_graph_1(A)
        & l1_graph_1(A) )
     => r1_graph_1(A,A,A) ) )).

fof(t12_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ? [C] :
                ( v2_graph_1(C)
                & l1_graph_1(C)
                & r4_graph_1(A,C)
                & r4_graph_1(B,C) )
           => k1_graph_1(A,B) = k1_graph_1(B,A) ) ) ) )).

fof(t13_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( ? [D] :
                    ( v2_graph_1(D)
                    & l1_graph_1(D)
                    & r4_graph_1(A,D)
                    & r4_graph_1(B,D)
                    & r4_graph_1(C,D) )
               => k1_graph_1(k1_graph_1(A,B),C) = k1_graph_1(A,k1_graph_1(B,C)) ) ) ) ) )).

fof(t14_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v2_funct_1(k1_xboole_0)
        & v8_graph_1(k1_xboole_0,A)
        & v10_graph_1(k1_xboole_0,A)
        & m2_graph_1(k1_xboole_0,A) ) ) )).

fof(t15_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_graph_1(B)
            & m3_graph_1(B,A) )
         => ! [C] :
              ( ( v1_graph_1(C)
                & m3_graph_1(C,A) )
             => ( ( u1_graph_1(B) = u1_graph_1(C)
                  & u2_graph_1(B) = u2_graph_1(C) )
               => B = C ) ) ) ) )).

fof(t16_graph_1,axiom,(
    ! [A] :
      ( ( v1_graph_1(A)
        & v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_graph_1(B)
            & v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ( r4_graph_1(A,B)
              & r4_graph_1(B,A) )
           => A = B ) ) ) )).

fof(t17_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( ( r4_graph_1(A,B)
                  & r4_graph_1(B,C) )
               => r4_graph_1(A,C) ) ) ) ) )).

fof(t18_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( r1_graph_1(A,B,C)
               => ( r4_graph_1(B,A)
                  & r4_graph_1(C,A) ) ) ) ) ) )).

fof(t19_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
              & r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
           => ( r4_graph_1(A,k1_graph_1(A,B))
              & r4_graph_1(B,k1_graph_1(A,B)) ) ) ) ) )).

fof(t20_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ? [C] :
                ( v2_graph_1(C)
                & l1_graph_1(C)
                & r4_graph_1(A,C)
                & r4_graph_1(B,C) )
           => ( r4_graph_1(A,k1_graph_1(A,B))
              & r4_graph_1(B,k1_graph_1(A,B)) ) ) ) ) )).

fof(t21_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ! [D] :
                  ( ( v2_graph_1(D)
                    & l1_graph_1(D) )
                 => ( ( r4_graph_1(A,B)
                      & r4_graph_1(C,B)
                      & r1_graph_1(D,A,C) )
                   => r4_graph_1(D,B) ) ) ) ) ) )).

fof(t22_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( ( r4_graph_1(A,B)
                  & r4_graph_1(C,B) )
               => r4_graph_1(k1_graph_1(A,C),B) ) ) ) ) )).

fof(t23_graph_1,axiom,(
    ! [A] :
      ( ( v1_graph_1(A)
        & v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_graph_1(B)
            & v2_graph_1(B)
            & l1_graph_1(B) )
         => ( r4_graph_1(A,B)
           => ( k1_graph_1(A,B) = B
              & k1_graph_1(B,A) = B ) ) ) ) )).

fof(t24_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
              & r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
           => ( ( k1_graph_1(A,B) != B
                & k1_graph_1(B,A) != B )
              | r4_graph_1(A,B) ) ) ) ) )).

fof(t25_graph_1,axiom,(
    $true )).

fof(t26_graph_1,axiom,(
    $true )).

fof(t27_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & v3_graph_1(B)
            & l1_graph_1(B) )
         => ( r4_graph_1(A,B)
           => v3_graph_1(A) ) ) ) )).

fof(t28_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & v4_graph_1(B)
            & l1_graph_1(B) )
         => ( r4_graph_1(A,B)
           => v4_graph_1(A) ) ) ) )).

fof(t29_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & v5_graph_1(B)
            & l1_graph_1(B) )
         => ( r4_graph_1(A,B)
           => v5_graph_1(A) ) ) ) )).

fof(t30_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_graph_1(B)
            & v2_graph_1(B)
            & l1_graph_1(B) )
         => ( r2_hidden(B,k7_graph_1(A))
          <=> r4_graph_1(B,A) ) ) ) )).

fof(t31_graph_1,axiom,(
    ! [A] :
      ( ( v1_graph_1(A)
        & v2_graph_1(A)
        & l1_graph_1(A) )
     => r2_hidden(A,k7_graph_1(A)) ) )).

fof(t32_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_graph_1(B)
            & v2_graph_1(B)
            & l1_graph_1(B) )
         => ( r4_graph_1(B,A)
          <=> r1_tarski(k7_graph_1(B),k7_graph_1(A)) ) ) ) )).

fof(t33_graph_1,axiom,(
    $true )).

fof(t34_graph_1,axiom,(
    ! [A] :
      ( ( v1_graph_1(A)
        & v2_graph_1(A)
        & l1_graph_1(A) )
     => r1_tarski(k1_tarski(A),k7_graph_1(A)) ) )).

fof(t35_graph_1,axiom,(
    ! [A] :
      ( ( v1_graph_1(A)
        & v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_graph_1(B)
            & v2_graph_1(B)
            & l1_graph_1(B) )
         => ~ ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
              & r1_partfun1(u4_graph_1(A),u4_graph_1(B))
              & r1_tarski(k7_graph_1(k1_graph_1(A,B)),k2_xboole_0(k7_graph_1(A),k7_graph_1(B)))
              & ~ r4_graph_1(A,B)
              & ~ r4_graph_1(B,A) ) ) ) )).

fof(t36_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ( ( r1_partfun1(u3_graph_1(A),u3_graph_1(B))
              & r1_partfun1(u4_graph_1(A),u4_graph_1(B)) )
           => r1_tarski(k2_xboole_0(k7_graph_1(A),k7_graph_1(B)),k7_graph_1(k1_graph_1(A,B))) ) ) ) )).

fof(t37_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( ( r2_hidden(A,k7_graph_1(B))
                  & r2_hidden(C,k7_graph_1(B)) )
               => r2_hidden(k1_graph_1(A,C),k7_graph_1(B)) ) ) ) ) )).

fof(s1_graph_1,axiom,(
    ? [A] :
    ! [B] :
      ( r2_hidden(B,A)
    <=> ( v1_graph_1(B)
        & m3_graph_1(B,f1_s1_graph_1)
        & p1_s1_graph_1(B) ) ) )).

fof(dt_m1_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_graph_1(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) ) ) ) )).

fof(existence_m1_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] : m1_graph_1(B,A) ) )).

fof(dt_m2_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
         => m2_finseq_1(B,u2_graph_1(A)) ) ) )).

fof(existence_m2_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] : m2_graph_1(B,A) ) )).

fof(redefinition_m2_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
        <=> m1_graph_1(B,A) ) ) )).

fof(dt_m3_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m3_graph_1(B,A)
         => ( v2_graph_1(B)
            & l1_graph_1(B) ) ) ) )).

fof(existence_m3_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] : m3_graph_1(B,A) ) )).

fof(dt_l1_graph_1,axiom,(
    $true )).

fof(existence_l1_graph_1,axiom,(
    ? [A] : l1_graph_1(A) )).

fof(abstractness_v1_graph_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( v1_graph_1(A)
       => A = g1_graph_1(u1_graph_1(A),u2_graph_1(A),u3_graph_1(A),u4_graph_1(A)) ) ) )).

fof(redefinition_v9_graph_1,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A)
        & m1_graph_1(B,A) )
     => ( v9_graph_1(B,A)
      <=> v2_funct_1(B) ) ) )).

fof(reflexivity_r4_graph_1,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A)
        & v2_graph_1(B)
        & l1_graph_1(B) )
     => r4_graph_1(A,A) ) )).

fof(dt_k1_graph_1,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A)
        & v2_graph_1(B)
        & l1_graph_1(B) )
     => ( v1_graph_1(k1_graph_1(A,B))
        & v2_graph_1(k1_graph_1(A,B))
        & l1_graph_1(k1_graph_1(A,B)) ) ) )).

fof(dt_k2_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A) )
     => m2_subset_1(k2_graph_1(A),k1_numbers,k5_numbers) ) )).

fof(dt_k3_graph_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A) )
     => m2_subset_1(k3_graph_1(A),k1_numbers,k5_numbers) ) )).

fof(dt_k4_graph_1,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A)
        & m1_subset_1(B,u1_graph_1(A)) )
     => m2_subset_1(k4_graph_1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k5_graph_1,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A)
        & m1_subset_1(B,u1_graph_1(A)) )
     => m2_subset_1(k5_graph_1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k6_graph_1,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & v7_graph_1(A)
        & l1_graph_1(A)
        & m1_subset_1(B,u1_graph_1(A)) )
     => m2_subset_1(k6_graph_1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k7_graph_1,axiom,(
    $true )).

fof(dt_u1_graph_1,axiom,(
    $true )).

fof(dt_u2_graph_1,axiom,(
    $true )).

fof(dt_u3_graph_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( v1_funct_1(u3_graph_1(A))
        & v1_funct_2(u3_graph_1(A),u2_graph_1(A),u1_graph_1(A))
        & m2_relset_1(u3_graph_1(A),u2_graph_1(A),u1_graph_1(A)) ) ) )).

fof(dt_u4_graph_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( v1_funct_1(u4_graph_1(A))
        & v1_funct_2(u4_graph_1(A),u2_graph_1(A),u1_graph_1(A))
        & m2_relset_1(u4_graph_1(A),u2_graph_1(A),u1_graph_1(A)) ) ) )).

fof(dt_g1_graph_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,B,A)
        & m1_relset_1(C,B,A)
        & v1_funct_1(D)
        & v1_funct_2(D,B,A)
        & m1_relset_1(D,B,A) )
     => ( v1_graph_1(g1_graph_1(A,B,C,D))
        & l1_graph_1(g1_graph_1(A,B,C,D)) ) ) )).

fof(free_g1_graph_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,B,A)
        & m1_relset_1(C,B,A)
        & v1_funct_1(D)
        & v1_funct_2(D,B,A)
        & m1_relset_1(D,B,A) )
     => ! [E,F,G,H] :
          ( g1_graph_1(A,B,C,D) = g1_graph_1(E,F,G,H)
         => ( A = E
            & B = F
            & C = G
            & D = H ) ) ) )).
%------------------------------------------------------------------------------