SET007 Axioms: SET007+203.ax


%------------------------------------------------------------------------------
% File     : SET007+203 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Elementary Notions of the Theory of Petri Nets
% 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    : net_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   73 (  22 unit)
%            Number of atoms       :  285 (  44 equality)
%            Maximal formula depth :   16 (   5 average)
%            Number of connectives :  239 (  27 ~  ;   7  |;  80  &)
%                                         (  24 <=>; 101 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   1 propositional; 0-3 arity)
%            Number of functors    :   23 (   1 constant; 0-3 arity)
%            Number of variables   :  132 (   0 singleton; 125 !;   7 ?)
%            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_net_1,axiom,(
    ? [A] :
      ( l1_net_1(A)
      & v1_net_1(A) ) )).

fof(fc1_net_1,axiom,
    ( v1_net_1(g1_net_1(k1_xboole_0,k1_xboole_0,k1_xboole_0))
    & v2_net_1(g1_net_1(k1_xboole_0,k1_xboole_0,k1_xboole_0)) )).

fof(rc2_net_1,axiom,(
    ? [A] :
      ( l1_net_1(A)
      & v1_net_1(A)
      & v2_net_1(A) ) )).

fof(d1_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => ( v2_net_1(A)
      <=> ( r1_xboole_0(u1_net_1(A),u2_net_1(A))
          & r1_tarski(u3_net_1(A),k2_xboole_0(k2_zfmisc_1(u1_net_1(A),u2_net_1(A)),k2_zfmisc_1(u2_net_1(A),u1_net_1(A)))) ) ) ) )).

fof(d2_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => k1_net_1(A) = k2_xboole_0(u1_net_1(A),u2_net_1(A)) ) )).

fof(t1_net_1,axiom,(
    $true )).

fof(t2_net_1,axiom,(
    $true )).

fof(t3_net_1,axiom,(
    $true )).

fof(t4_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => r1_tarski(u1_net_1(A),k1_net_1(A)) ) )).

fof(t5_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => r1_tarski(u2_net_1(A),k1_net_1(A)) ) )).

fof(t6_net_1,axiom,(
    ! [A,B] :
      ( l1_net_1(B)
     => ( r2_hidden(A,k1_net_1(B))
      <=> ( r2_hidden(A,u1_net_1(B))
          | r2_hidden(A,u2_net_1(B)) ) ) ) )).

fof(t7_net_1,axiom,(
    ! [A,B] :
      ( l1_net_1(B)
     => ~ ( k1_net_1(B) != k1_xboole_0
          & m1_subset_1(A,k1_net_1(B))
          & ~ m1_subset_1(A,u1_net_1(B))
          & ~ m1_subset_1(A,u2_net_1(B)) ) ) )).

fof(t8_net_1,axiom,(
    ! [A,B] :
      ( l1_net_1(B)
     => ( m1_subset_1(A,u1_net_1(B))
       => ( u1_net_1(B) = k1_xboole_0
          | m1_subset_1(A,k1_net_1(B)) ) ) ) )).

fof(t9_net_1,axiom,(
    ! [A,B] :
      ( l1_net_1(B)
     => ( m1_subset_1(A,u2_net_1(B))
       => ( u2_net_1(B) = k1_xboole_0
          | m1_subset_1(A,k1_net_1(B)) ) ) ) )).

fof(t10_net_1,axiom,(
    $true )).

fof(t11_net_1,axiom,(
    ! [A,B] :
      ( ( v2_net_1(B)
        & l1_net_1(B) )
     => ~ ( r2_hidden(A,u1_net_1(B))
          & r2_hidden(A,u2_net_1(B)) ) ) )).

fof(t12_net_1,axiom,(
    ! [A,B,C] :
      ( ( v2_net_1(C)
        & l1_net_1(C) )
     => ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
          & r2_hidden(A,u2_net_1(C)) )
       => r2_hidden(B,u1_net_1(C)) ) ) )).

fof(t13_net_1,axiom,(
    ! [A,B,C] :
      ( ( v2_net_1(C)
        & l1_net_1(C) )
     => ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
          & r2_hidden(B,u2_net_1(C)) )
       => r2_hidden(A,u1_net_1(C)) ) ) )).

fof(t14_net_1,axiom,(
    ! [A,B,C] :
      ( ( v2_net_1(C)
        & l1_net_1(C) )
     => ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
          & r2_hidden(A,u1_net_1(C)) )
       => r2_hidden(B,u2_net_1(C)) ) ) )).

fof(t15_net_1,axiom,(
    ! [A,B,C] :
      ( ( v2_net_1(C)
        & l1_net_1(C) )
     => ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
          & r2_hidden(B,u1_net_1(C)) )
       => r2_hidden(A,u2_net_1(C)) ) ) )).

fof(d3_net_1,axiom,(
    $true )).

fof(d4_net_1,axiom,(
    $true )).

fof(d5_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B,C] :
          ( r1_net_1(A,B,C)
        <=> ( r2_hidden(k4_tarski(C,B),u3_net_1(A))
            & r2_hidden(B,u2_net_1(A)) ) ) ) )).

fof(d6_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B,C] :
          ( r2_net_1(A,B,C)
        <=> ( r2_hidden(k4_tarski(B,C),u3_net_1(A))
            & r2_hidden(B,u2_net_1(A)) ) ) ) )).

fof(d7_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ! [C] :
              ( C = k2_net_1(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ( r2_hidden(D,k1_net_1(A))
                    & r2_hidden(k4_tarski(D,B),u3_net_1(A)) ) ) ) ) ) )).

fof(d8_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ! [C] :
              ( C = k3_net_1(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ( r2_hidden(D,k1_net_1(A))
                    & r2_hidden(k4_tarski(B,D),u3_net_1(A)) ) ) ) ) ) )).

fof(t16_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => r1_tarski(k2_net_1(A,B),k1_net_1(A)) ) ) )).

fof(t17_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => r1_tarski(k2_net_1(A,B),k1_net_1(A)) ) ) )).

fof(t18_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => r1_tarski(k3_net_1(A,B),k1_net_1(A)) ) ) )).

fof(t19_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => r1_tarski(k3_net_1(A,B),k1_net_1(A)) ) ) )).

fof(t20_net_1,axiom,(
    ! [A,B] :
      ( ( v2_net_1(B)
        & l1_net_1(B) )
     => ! [C] :
          ( m1_subset_1(C,k1_net_1(B))
         => ( r2_hidden(C,u2_net_1(B))
           => ( r2_hidden(A,k2_net_1(B,C))
            <=> r1_net_1(B,C,A) ) ) ) ) )).

fof(t21_net_1,axiom,(
    ! [A,B] :
      ( ( v2_net_1(B)
        & l1_net_1(B) )
     => ! [C] :
          ( m1_subset_1(C,k1_net_1(B))
         => ( r2_hidden(C,u2_net_1(B))
           => ( r2_hidden(A,k3_net_1(B,C))
            <=> r2_net_1(B,C,A) ) ) ) ) )).

fof(d9_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ( k1_net_1(A) != k1_xboole_0
           => ! [C] :
                ( C = k4_net_1(A,B)
              <=> ( ( r2_hidden(B,u1_net_1(A))
                   => C = k1_tarski(B) )
                  & ( r2_hidden(B,u2_net_1(A))
                   => C = k2_net_1(A,B) ) ) ) ) ) ) )).

fof(t22_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ~ ( k1_net_1(A) != k1_xboole_0
              & k4_net_1(A,B) != k1_tarski(B)
              & k4_net_1(A,B) != k2_net_1(A,B) ) ) ) )).

fof(t23_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ( k1_net_1(A) != k1_xboole_0
           => r1_tarski(k4_net_1(A,B),k1_net_1(A)) ) ) ) )).

fof(t24_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ( k1_net_1(A) != k1_xboole_0
           => r1_tarski(k4_net_1(A,B),k1_net_1(A)) ) ) ) )).

fof(d10_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ( k1_net_1(A) != k1_xboole_0
           => ! [C] :
                ( C = k5_net_1(A,B)
              <=> ( ( r2_hidden(B,u1_net_1(A))
                   => C = k1_tarski(B) )
                  & ( r2_hidden(B,u2_net_1(A))
                   => C = k3_net_1(A,B) ) ) ) ) ) ) )).

fof(t25_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ~ ( k1_net_1(A) != k1_xboole_0
              & k5_net_1(A,B) != k1_tarski(B)
              & k5_net_1(A,B) != k3_net_1(A,B) ) ) ) )).

fof(t26_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ( k1_net_1(A) != k1_xboole_0
           => r1_tarski(k5_net_1(A,B),k1_net_1(A)) ) ) ) )).

fof(t27_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ( k1_net_1(A) != k1_xboole_0
           => r1_tarski(k5_net_1(A,B),k1_net_1(A)) ) ) ) )).

fof(d11_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => k6_net_1(A,B) = k2_xboole_0(k4_net_1(A,B),k5_net_1(A,B)) ) ) )).

fof(d12_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => ! [B] :
          ( m1_subset_1(B,u2_net_1(A))
         => ! [C] :
              ( C = k7_net_1(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ( r2_hidden(D,u1_net_1(A))
                    & r2_hidden(k4_tarski(D,B),u3_net_1(A)) ) ) ) ) ) )).

fof(d13_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => ! [B] :
          ( m1_subset_1(B,u2_net_1(A))
         => ! [C] :
              ( C = k8_net_1(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ( r2_hidden(D,u1_net_1(A))
                    & r2_hidden(k4_tarski(B,D),u3_net_1(A)) ) ) ) ) ) )).

fof(d14_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B,C] :
          ( C = k9_net_1(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> ( r1_tarski(D,k1_net_1(A))
                & ? [E] :
                    ( m1_subset_1(E,k1_net_1(A))
                    & r2_hidden(E,B)
                    & D = k4_net_1(A,E) ) ) ) ) ) )).

fof(d15_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B,C] :
          ( C = k10_net_1(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> ( r1_tarski(D,k1_net_1(A))
                & ? [E] :
                    ( m1_subset_1(E,k1_net_1(A))
                    & r2_hidden(E,B)
                    & D = k5_net_1(A,E) ) ) ) ) ) )).

fof(t28_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ! [C] :
              ( ( r1_tarski(C,k1_net_1(A))
                & r2_hidden(B,C) )
             => ( k1_net_1(A) = k1_xboole_0
                | r2_hidden(k4_net_1(A,B),k9_net_1(A,C)) ) ) ) ) )).

fof(t29_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_net_1(A))
         => ! [C] :
              ( ( r1_tarski(C,k1_net_1(A))
                & r2_hidden(B,C) )
             => ( k1_net_1(A) = k1_xboole_0
                | r2_hidden(k5_net_1(A,B),k10_net_1(A,C)) ) ) ) ) )).

fof(d16_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] : k11_net_1(A,B) = k3_tarski(k9_net_1(A,B)) ) )).

fof(d17_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] : k12_net_1(A,B) = k3_tarski(k10_net_1(A,B)) ) )).

fof(t30_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B,C] :
          ( r1_tarski(C,k1_net_1(A))
         => ( k1_net_1(A) = k1_xboole_0
            | ( r2_hidden(B,k11_net_1(A,C))
            <=> ? [D] :
                  ( m1_subset_1(D,k1_net_1(A))
                  & r2_hidden(D,C)
                  & r2_hidden(B,k4_net_1(A,D)) ) ) ) ) ) )).

fof(t31_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B,C] :
          ( r1_tarski(C,k1_net_1(A))
         => ( k1_net_1(A) = k1_xboole_0
            | ( r2_hidden(B,k12_net_1(A,C))
            <=> ? [D] :
                  ( m1_subset_1(D,k1_net_1(A))
                  & r2_hidden(D,C)
                  & r2_hidden(B,k5_net_1(A,D)) ) ) ) ) ) )).

fof(t32_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_net_1(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_net_1(A))
             => ( k1_net_1(A) != k1_xboole_0
               => ( r2_hidden(C,k11_net_1(A,B))
                <=> ~ ( ~ ( r2_hidden(C,B)
                          & r2_hidden(C,u1_net_1(A)) )
                      & ! [D] :
                          ( m1_subset_1(D,k1_net_1(A))
                         => ~ ( r2_hidden(D,B)
                              & r2_hidden(D,u2_net_1(A))
                              & r1_net_1(A,D,C) ) ) ) ) ) ) ) ) )).

fof(t33_net_1,axiom,(
    ! [A] :
      ( ( v2_net_1(A)
        & l1_net_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_net_1(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_net_1(A))
             => ( k1_net_1(A) != k1_xboole_0
               => ( r2_hidden(C,k12_net_1(A,B))
                <=> ~ ( ~ ( r2_hidden(C,B)
                          & r2_hidden(C,u1_net_1(A)) )
                      & ! [D] :
                          ( m1_subset_1(D,k1_net_1(A))
                         => ~ ( r2_hidden(D,B)
                              & r2_hidden(D,u2_net_1(A))
                              & r2_net_1(A,D,C) ) ) ) ) ) ) ) ) )).

fof(dt_l1_net_1,axiom,(
    $true )).

fof(existence_l1_net_1,axiom,(
    ? [A] : l1_net_1(A) )).

fof(abstractness_v1_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => ( v1_net_1(A)
       => A = g1_net_1(u1_net_1(A),u2_net_1(A),u3_net_1(A)) ) ) )).

fof(dt_k1_net_1,axiom,(
    $true )).

fof(dt_k2_net_1,axiom,(
    $true )).

fof(dt_k3_net_1,axiom,(
    $true )).

fof(dt_k4_net_1,axiom,(
    $true )).

fof(dt_k5_net_1,axiom,(
    $true )).

fof(dt_k6_net_1,axiom,(
    $true )).

fof(dt_k7_net_1,axiom,(
    $true )).

fof(dt_k8_net_1,axiom,(
    $true )).

fof(dt_k9_net_1,axiom,(
    $true )).

fof(dt_k10_net_1,axiom,(
    $true )).

fof(dt_k11_net_1,axiom,(
    $true )).

fof(dt_k12_net_1,axiom,(
    $true )).

fof(dt_u1_net_1,axiom,(
    $true )).

fof(dt_u2_net_1,axiom,(
    $true )).

fof(dt_u3_net_1,axiom,(
    ! [A] :
      ( l1_net_1(A)
     => v1_relat_1(u3_net_1(A)) ) )).

fof(dt_g1_net_1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ( v1_net_1(g1_net_1(A,B,C))
        & l1_net_1(g1_net_1(A,B,C)) ) ) )).

fof(free_g1_net_1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ! [D,E,F] :
          ( g1_net_1(A,B,C) = g1_net_1(D,E,F)
         => ( A = D
            & B = E
            & C = F ) ) ) )).
%------------------------------------------------------------------------------