SET007 Axioms: SET007+714.ax


%------------------------------------------------------------------------------
% File     : SET007+714 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Half Open Intervals in Real Numbers
% 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    : rcomp_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   33 (   1 unit)
%            Number of atoms       :  173 (  22 equality)
%            Maximal formula depth :   15 (   9 average)
%            Number of connectives :  161 (  21 ~  ;   0  |;  29  &)
%                                         (   5 <=>; 106 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   1 propositional; 0-2 arity)
%            Number of functors    :   16 (   2 constant; 0-3 arity)
%            Number of variables   :   97 (   0 singleton;  95 !;   2 ?)
%            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(t1_rcomp_2,axiom,(
    $true )).

fof(t2_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( ~ r1_xreal_0(B,A)
                  & ~ r1_xreal_0(B,C) )
              <=> ~ r1_xreal_0(B,k2_square_1(A,C)) ) ) ) ) )).

fof(t3_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r2_hidden(A,k1_rcomp_2(B,C))
              <=> ( r1_xreal_0(B,A)
                  & ~ r1_xreal_0(C,A) ) ) ) ) ) )).

fof(t4_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r2_hidden(A,k2_rcomp_2(B,C))
              <=> ( ~ r1_xreal_0(A,B)
                  & r1_xreal_0(A,C) ) ) ) ) ) )).

fof(t5_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => k1_rcomp_2(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k1_tarski(A)) ) ) ) )).

fof(t6_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => k2_rcomp_2(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k1_tarski(B)) ) ) ) )).

fof(t7_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_rcomp_2(A,A) = k1_xboole_0 ) )).

fof(t8_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k2_rcomp_2(A,A) = k1_xboole_0 ) )).

fof(t9_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k1_rcomp_2(B,A) = k1_xboole_0 ) ) ) )).

fof(t10_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k2_rcomp_2(B,A) = k1_xboole_0 ) ) ) )).

fof(t11_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(B,C)) = k1_rcomp_2(A,C) ) ) ) ) )).

fof(t12_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(B,C)) = k2_rcomp_2(A,C) ) ) ) ) )).

fof(t13_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(A,C)
                      & r1_xreal_0(B,D)
                      & r1_xreal_0(C,D) )
                   => k1_rcomp_1(A,D) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_1(B,C)),k2_rcomp_2(C,D)) ) ) ) ) ) )).

fof(t14_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ~ ( ~ r1_xreal_0(B,A)
                      & ~ r1_xreal_0(C,A)
                      & ~ r1_xreal_0(D,B)
                      & ~ r1_xreal_0(D,C)
                      & k2_rcomp_1(A,D) != k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_1(B,C)),k1_rcomp_2(C,D)) ) ) ) ) ) )).

fof(t15_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => k5_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(C,D)) = k1_rcomp_2(k2_square_1(A,C),k1_square_1(B,D)) ) ) ) ) )).

fof(t16_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => k5_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(C,D)) = k2_rcomp_2(k2_square_1(A,C),k1_square_1(B,D)) ) ) ) ) )).

fof(t17_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_tarski(k2_rcomp_1(A,B),k1_rcomp_2(A,B))
            & r1_tarski(k2_rcomp_1(A,B),k2_rcomp_2(A,B))
            & r1_tarski(k1_rcomp_2(A,B),k1_rcomp_1(A,B))
            & r1_tarski(k2_rcomp_2(A,B),k1_rcomp_1(A,B)) ) ) ) )).

fof(t18_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r2_hidden(A,k1_rcomp_2(B,C))
                      & r2_hidden(D,k1_rcomp_2(B,C)) )
                   => r1_tarski(k1_rcomp_1(A,D),k1_rcomp_2(B,C)) ) ) ) ) ) )).

fof(t19_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r2_hidden(A,k2_rcomp_2(B,C))
                      & r2_hidden(D,k2_rcomp_2(B,C)) )
                   => r1_tarski(k1_rcomp_1(A,D),k2_rcomp_2(B,C)) ) ) ) ) ) )).

fof(t20_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k2_rcomp_2(B,C)) = k1_rcomp_1(A,C) ) ) ) ) )).

fof(t21_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_1(B,C)) = k1_rcomp_1(A,C) ) ) ) ) )).

fof(t22_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
                   => r1_xreal_0(C,B) ) ) ) ) ) )).

fof(t23_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xboole_0(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
                   => r1_xreal_0(C,B) ) ) ) ) ) )).

fof(t24_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
                   => k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(C,D)) = k1_rcomp_2(k1_square_1(A,C),k2_square_1(B,D)) ) ) ) ) ) )).

fof(t25_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xboole_0(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
                   => k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(C,D)) = k2_rcomp_2(k1_square_1(A,C),k2_square_1(B,D)) ) ) ) ) ) )).

fof(t26_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
                   => k6_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_2(C,D)) = k4_subset_1(k1_numbers,k1_rcomp_2(A,C),k1_rcomp_2(D,B)) ) ) ) ) ) )).

fof(t27_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xboole_0(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
                   => k6_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(C,D)) = k4_subset_1(k1_numbers,k2_rcomp_2(A,C),k2_rcomp_2(D,B)) ) ) ) ) ) )).

fof(dt_k1_rcomp_2,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) )).

fof(dt_k2_rcomp_2,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => m1_subset_1(k2_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) )).

fof(d1_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k1_rcomp_2(A,B) = a_2_0_rcomp_2(A,B) ) ) )).

fof(d2_rcomp_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k2_rcomp_2(A,B) = a_2_1_rcomp_2(A,B) ) ) )).

fof(fraenkel_a_2_0_rcomp_2,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v1_xreal_0(C) )
     => ( r2_hidden(A,a_2_0_rcomp_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = D
            & r1_xreal_0(B,D)
            & ~ r1_xreal_0(C,D) ) ) ) )).

fof(fraenkel_a_2_1_rcomp_2,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(B)
        & v1_xreal_0(C) )
     => ( r2_hidden(A,a_2_1_rcomp_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_numbers)
            & A = D
            & ~ r1_xreal_0(D,B)
            & r1_xreal_0(D,C) ) ) ) )).
%------------------------------------------------------------------------------