SET007 Axioms: SET007+36.ax


%------------------------------------------------------------------------------
% File     : SET007+36 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Introduction to Arithmetics
% 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    : arytm_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (   6 unit)
%            Number of atoms       :  226 (  90 equality)
%            Maximal formula depth :   22 (   6 average)
%            Number of connectives :  206 (  23 ~  ;   4  |;  76  &)
%                                         (  12 <=>;  91 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   1 propositional; 0-2 arity)
%            Number of functors    :   18 (   6 constant; 0-5 arity)
%            Number of variables   :   95 (   0 singleton;  77 !;  18 ?)
%            Maximal term depth    :    3 (   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(t1_arytm_0,axiom,(
    r1_tarski(k2_arytm_2,k1_numbers) )).

fof(t2_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ( A != k12_arytm_3
       => r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers) ) ) )).

fof(t3_arytm_0,axiom,(
    ! [A] :
      ~ ( r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers)
        & A = k12_arytm_3 ) )).

fof(t4_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => r2_hidden(k2_arytm_1(A,B),k1_numbers) ) ) )).

fof(t5_arytm_0,axiom,(
    r1_subset_1(k2_arytm_2,k2_zfmisc_1(k1_tarski(k12_arytm_3),k2_arytm_2)) )).

fof(t6_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ( k2_arytm_1(A,B) = k12_arytm_3
           => A = B ) ) ) )).

fof(t7_arytm_0,axiom,(
    ! [A,B] : k13_arytm_3 != k4_tarski(A,B) )).

fof(t8_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] :
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] :
              ( m1_subset_1(C,k2_arytm_2)
             => ( k8_arytm_2(A,B) = k8_arytm_2(A,C)
               => ( A = k12_arytm_3
                  | B = C ) ) ) ) ) )).

fof(d1_arytm_0,axiom,(
    $true )).

fof(d2_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_arytm_2) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] :
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] :
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = E
                            & C = k7_arytm_2(D,E) ) ) ) )
                & ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] :
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] :
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = k4_tarski(np__0,E)
                            & C = k2_arytm_1(D,E) ) ) ) )
                & ( ( r2_hidden(B,k2_arytm_2)
                    & r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] :
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] :
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = k4_tarski(np__0,D)
                            & B = E
                            & C = k2_arytm_1(E,D) ) ) ) )
                & ~ ( ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_arytm_2) )
                    & ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                    & ~ ( r2_hidden(B,k2_arytm_2)
                        & r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                    & ~ ( C = k1_arytm_0(A,B)
                      <=> ? [D] :
                            ( m1_subset_1(D,k2_arytm_2)
                            & ? [E] :
                                ( m1_subset_1(E,k2_arytm_2)
                                & A = k4_tarski(np__0,D)
                                & B = k4_tarski(np__0,E)
                                & C = k4_tarski(np__0,k7_arytm_2(D,E)) ) ) ) ) ) ) ) ) )).

fof(d3_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_arytm_2) )
                 => ( C = k2_arytm_0(A,B)
                  <=> ? [D] :
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] :
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = E
                            & C = k8_arytm_2(D,E) ) ) ) )
                & ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                 => ( A = np__0
                    | ( C = k2_arytm_0(A,B)
                    <=> ? [D] :
                          ( m1_subset_1(D,k2_arytm_2)
                          & ? [E] :
                              ( m1_subset_1(E,k2_arytm_2)
                              & A = D
                              & B = k4_tarski(np__0,E)
                              & C = k4_tarski(np__0,k8_arytm_2(D,E)) ) ) ) ) )
                & ( ( r2_hidden(B,k2_arytm_2)
                    & r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                 => ( B = np__0
                    | ( C = k2_arytm_0(A,B)
                    <=> ? [D] :
                          ( m1_subset_1(D,k2_arytm_2)
                          & ? [E] :
                              ( m1_subset_1(E,k2_arytm_2)
                              & A = k4_tarski(np__0,D)
                              & B = E
                              & C = k4_tarski(np__0,k8_arytm_2(E,D)) ) ) ) ) )
                & ( ( r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                 => ( C = k2_arytm_0(A,B)
                  <=> ? [D] :
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] :
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = k4_tarski(np__0,D)
                            & B = k4_tarski(np__0,E)
                            & C = k8_arytm_2(E,D) ) ) ) )
                & ~ ( ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_arytm_2) )
                    & ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
                        & A != np__0 )
                    & ~ ( r2_hidden(B,k2_arytm_2)
                        & r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
                        & B != np__0 )
                    & ~ ( r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
                        & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                    & ~ ( C = k2_arytm_0(A,B)
                      <=> C = np__0 ) ) ) ) ) ) )).

fof(d4_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B = k3_arytm_0(A)
          <=> k1_arytm_0(A,B) = np__0 ) ) ) )).

fof(d5_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ( A != np__0
             => ( B = k4_arytm_0(A)
              <=> k2_arytm_0(A,B) = k13_arytm_3 ) )
            & ( A = np__0
             => ( B = k4_arytm_0(A)
              <=> B = np__0 ) ) ) ) ) )).

fof(t9_arytm_0,axiom,(
    $true )).

fof(t10_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ r2_hidden(k5_funct_4(k1_numbers,np__0,k13_arytm_3,A,B),k1_numbers) ) ) )).

fof(d6_arytm_0,axiom,(
    $true )).

fof(d7_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ( B = np__0
             => k5_arytm_0(A,B) = A )
            & ( B != np__0
             => k5_arytm_0(A,B) = k5_funct_4(k1_numbers,np__0,k13_arytm_3,A,B) ) ) ) ) )).

fof(t11_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & ? [C] :
              ( m1_subset_1(C,k1_numbers)
              & A = k5_arytm_0(B,C) ) ) ) )).

fof(t12_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( k5_arytm_0(A,B) = k5_arytm_0(C,D)
                   => ( A = C
                      & B = D ) ) ) ) ) ) )).

fof(t13_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B = np__0
           => k1_arytm_0(A,B) = A ) ) ) )).

fof(t14_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B = np__0
           => k2_arytm_0(A,B) = np__0 ) ) ) )).

fof(t15_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k2_arytm_0(A,k2_arytm_0(B,C)) = k2_arytm_0(k2_arytm_0(A,B),C) ) ) ) )).

fof(t16_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k2_arytm_0(A,k1_arytm_0(B,C)) = k1_arytm_0(k2_arytm_0(A,B),k2_arytm_0(A,C)) ) ) ) )).

fof(t17_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k2_arytm_0(k3_arytm_0(A),B) = k3_arytm_0(k2_arytm_0(A,B)) ) ) )).

fof(t18_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => r2_hidden(k2_arytm_0(A,A),k2_arytm_2) ) )).

fof(t19_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k1_arytm_0(k2_arytm_0(A,A),k2_arytm_0(B,B)) = np__0
           => A = np__0 ) ) ) )).

fof(t20_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( k2_arytm_0(A,B) = k13_arytm_3
                  & k2_arytm_0(A,C) = k13_arytm_3 )
               => ( A = np__0
                  | B = C ) ) ) ) ) )).

fof(t21_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B = k13_arytm_3
           => k2_arytm_0(A,B) = A ) ) ) )).

fof(t22_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B != np__0
           => k2_arytm_0(k2_arytm_0(A,B),k4_arytm_0(B)) = A ) ) ) )).

fof(t23_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( k2_arytm_0(A,B) = np__0
              & A != np__0
              & B != np__0 ) ) ) )).

fof(t24_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k4_arytm_0(k2_arytm_0(A,B)) = k2_arytm_0(k4_arytm_0(A),k4_arytm_0(B)) ) ) )).

fof(t25_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => k1_arytm_0(A,k1_arytm_0(B,C)) = k1_arytm_0(k1_arytm_0(A,B),C) ) ) ) )).

fof(t26_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r2_hidden(k5_arytm_0(A,B),k1_numbers)
           => B = np__0 ) ) ) )).

fof(t27_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k3_arytm_0(k1_arytm_0(A,B)) = k1_arytm_0(k3_arytm_0(A),k3_arytm_0(B)) ) ) )).

fof(dt_k1_arytm_0,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k1_arytm_0(A,B),k1_numbers) ) )).

fof(commutativity_k1_arytm_0,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k1_arytm_0(A,B) = k1_arytm_0(B,A) ) )).

fof(dt_k2_arytm_0,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k2_arytm_0(A,B),k1_numbers) ) )).

fof(commutativity_k2_arytm_0,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k2_arytm_0(A,B) = k2_arytm_0(B,A) ) )).

fof(dt_k3_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k3_arytm_0(A),k1_numbers) ) )).

fof(involutiveness_k3_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k3_arytm_0(k3_arytm_0(A)) = A ) )).

fof(dt_k4_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k4_arytm_0(A),k1_numbers) ) )).

fof(involutiveness_k4_arytm_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k4_arytm_0(k4_arytm_0(A)) = A ) )).

fof(dt_k5_arytm_0,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k5_arytm_0(A,B),k2_numbers) ) )).
%------------------------------------------------------------------------------