SET007 Axioms: SET007+705.ax


%------------------------------------------------------------------------------
% File     : SET007+705 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Pythagorean Triples
% 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    : pythtrip [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   39 (   2 unit)
%            Number of atoms       :  208 (  21 equality)
%            Maximal formula depth :   19 (   8 average)
%            Number of connectives :  197 (  28 ~  ;   1  |;  95  &)
%                                         (  12 <=>;  61 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-3 arity)
%            Number of functors    :   21 (   9 constant; 0-4 arity)
%            Number of variables   :   75 (   0 singleton;  58 !;  17 ?)
%            Maximal term depth    :    5 (   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_pythtrip,axiom,(
    ! [A] :
      ( v1_pythtrip(A)
     => ( v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v1_int_1(A) ) ) )).

fof(fc1_pythtrip,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v4_ordinal2(k5_square_1(A))
        & v1_xcmplx_0(k5_square_1(A))
        & v1_xreal_0(k5_square_1(A))
        & v1_int_1(k5_square_1(A))
        & v1_pythtrip(k5_square_1(A)) ) ) )).

fof(rc1_pythtrip,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_abian(A)
      & v1_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_int_1(A)
      & v1_pythtrip(A) ) )).

fof(rc2_pythtrip,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & ~ v1_abian(A)
      & v1_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_int_1(A)
      & v1_pythtrip(A) ) )).

fof(rc3_pythtrip,axiom,(
    ? [A] :
      ( v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_abian(A)
      & v1_xreal_0(A)
      & v1_int_1(A)
      & v1_pythtrip(A) ) )).

fof(rc4_pythtrip,axiom,(
    ? [A] :
      ( v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & ~ v1_abian(A)
      & v1_xreal_0(A)
      & v1_int_1(A)
      & v1_pythtrip(A) ) )).

fof(fc2_pythtrip,axiom,(
    ! [A,B] :
      ( ( v1_pythtrip(A)
        & v1_pythtrip(B) )
     => ( v4_ordinal2(k3_xcmplx_0(A,B))
        & v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_int_1(k3_xcmplx_0(A,B))
        & v1_pythtrip(k3_xcmplx_0(A,B)) ) ) )).

fof(fc3_pythtrip,axiom,(
    ! [A] :
      ( ( v1_abian(A)
        & v1_int_1(A) )
     => v1_abian(k5_square_1(A)) ) )).

fof(fc4_pythtrip,axiom,(
    ! [A] :
      ( ( ~ v1_abian(A)
        & v1_int_1(A) )
     => ~ v1_abian(k5_square_1(A)) ) )).

fof(fc5_pythtrip,axiom,(
    ! [A,B] :
      ( ( ~ v1_abian(A)
        & v1_pythtrip(A)
        & ~ v1_abian(B)
        & v1_pythtrip(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_abian(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v1_int_1(k2_xcmplx_0(A,B))
        & ~ v1_pythtrip(k2_xcmplx_0(A,B)) ) ) )).

fof(cc2_pythtrip,axiom,(
    ! [A] :
      ( m1_pythtrip(A)
     => v1_finset_1(A) ) )).

fof(rc5_pythtrip,axiom,(
    ? [A] :
      ( m1_pythtrip(A)
      & v1_finset_1(A)
      & ~ v2_pythtrip(A)
      & v3_pythtrip(A) ) )).

fof(d1_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_int_2(A,B)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ( r1_nat_1(C,A)
                    & r1_nat_1(C,B) )
                 => C = np__1 ) ) ) ) ) )).

fof(d2_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_int_2(A,B)
          <=> ! [C] :
                ( ( v1_int_2(C)
                  & m2_subset_1(C,k1_numbers,k5_numbers) )
               => ~ ( r1_nat_1(C,A)
                    & r1_nat_1(C,B) ) ) ) ) ) )).

fof(d3_pythtrip,axiom,(
    ! [A] :
      ( v1_pythtrip(A)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = k2_pepin(B) ) ) )).

fof(t1_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( v1_pythtrip(k2_nat_1(A,B))
              & r2_int_2(A,B) )
           => ( v1_pythtrip(A)
              & v1_pythtrip(B) ) ) ) ) )).

fof(t2_pythtrip,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v1_abian(A)
      <=> v1_abian(k2_pepin(A)) ) ) )).

fof(t3_pythtrip,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v1_abian(A)
       => k4_nat_1(k2_pepin(A),np__4) = np__0 ) ) )).

fof(t4_pythtrip,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( ~ v1_abian(A)
       => k4_nat_1(k2_pepin(A),np__4) = np__1 ) ) )).

fof(t5_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_pepin(A) = k2_pepin(B)
           => A = B ) ) ) )).

fof(t6_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_nat_1(A,B)
          <=> r1_nat_1(k2_pepin(A),k2_pepin(B)) ) ) ) )).

fof(t7_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_nat_1(A,B)
                  | C = np__0 )
              <=> r1_nat_1(k2_nat_1(C,A),k2_nat_1(C,B)) ) ) ) ) )).

fof(t8_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k6_nat_1(k2_nat_1(A,B),k2_nat_1(A,C)) = k2_nat_1(A,k6_nat_1(B,C)) ) ) ) )).

fof(t9_pythtrip,axiom,(
    ! [A] :
      ~ ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & r1_xreal_0(B,C)
                & r2_hidden(C,A) ) )
        & v1_finset_1(A) ) )).

fof(t10_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( r2_int_2(A,B)
              & v1_abian(A)
              & v1_abian(B) ) ) ) )).

fof(t11_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( k1_nat_1(k2_pepin(A),k2_pepin(B)) = k2_pepin(C)
                  & r2_int_2(A,B)
                  & ~ v1_abian(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ~ ( r1_xreal_0(D,E)
                              & A = k5_real_1(k2_pepin(E),k2_pepin(D))
                              & B = k2_nat_1(k2_nat_1(np__2,D),E)
                              & C = k1_nat_1(k2_pepin(E),k2_pepin(D)) ) ) ) ) ) ) ) )).

fof(t12_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( A = k5_real_1(k2_pepin(B),k2_pepin(C))
                          & D = k2_nat_1(k2_nat_1(np__2,C),B)
                          & E = k1_nat_1(k2_pepin(B),k2_pepin(C)) )
                       => k1_nat_1(k2_pepin(A),k2_pepin(D)) = k2_pepin(E) ) ) ) ) ) ) )).

fof(d4_pythtrip,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
     => ( m1_pythtrip(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & k1_nat_1(k2_pepin(B),k2_pepin(C)) = k2_pepin(D)
                    & A = k8_domain_1(k5_numbers,B,C,D) ) ) ) ) ) )).

fof(d5_pythtrip,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
     => ( m1_pythtrip(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & r1_xreal_0(C,D)
                    & A = k8_domain_1(k1_numbers,k4_real_1(B,k5_real_1(k2_pepin(D),k2_pepin(C))),k2_nat_1(B,k2_nat_1(k2_nat_1(np__2,C),D)),k2_nat_1(B,k1_nat_1(k2_pepin(D),k2_pepin(C)))) ) ) ) ) ) )).

fof(d6_pythtrip,axiom,(
    ! [A] :
      ( m1_pythtrip(A)
     => ( v2_pythtrip(A)
      <=> r2_hidden(np__0,A) ) ) )).

fof(t13_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( ~ r1_xreal_0(A,np__2)
          & ! [B] :
              ( m1_pythtrip(B)
             => ~ ( ~ v2_pythtrip(B)
                  & r2_hidden(A,B) ) ) ) ) )).

fof(d7_pythtrip,axiom,(
    ! [A] :
      ( m1_pythtrip(A)
     => ( v3_pythtrip(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,A)
                   => r1_nat_1(B,C) ) )
             => B = np__1 ) ) ) ) )).

fof(d8_pythtrip,axiom,(
    ! [A] :
      ( m1_pythtrip(A)
     => ( v3_pythtrip(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & r2_hidden(B,A)
                & r2_hidden(C,A)
                & r2_int_2(B,C) ) ) ) ) )).

fof(t14_pythtrip,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & ! [B] :
              ( m1_pythtrip(B)
             => ~ ( ~ v2_pythtrip(B)
                  & v3_pythtrip(B)
                  & r2_hidden(k2_nat_1(np__4,A),B) ) ) ) ) )).

fof(t15_pythtrip,axiom,
    ( ~ v2_pythtrip(k8_domain_1(k5_numbers,np__3,np__4,np__5))
    & v3_pythtrip(k8_domain_1(k5_numbers,np__3,np__4,np__5))
    & m1_pythtrip(k8_domain_1(k5_numbers,np__3,np__4,np__5)) )).

fof(dt_m1_pythtrip,axiom,(
    ! [A] :
      ( m1_pythtrip(A)
     => m1_subset_1(A,k1_zfmisc_1(k5_numbers)) ) )).

fof(existence_m1_pythtrip,axiom,(
    ? [A] : m1_pythtrip(A) )).

fof(t16_pythtrip,axiom,(
    ~ v1_finset_1(a_0_0_pythtrip) )).

fof(fraenkel_a_0_0_pythtrip,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_pythtrip)
    <=> ? [B] :
          ( m1_pythtrip(B)
          & A = B
          & ~ v2_pythtrip(B)
          & v3_pythtrip(B) ) ) )).
%------------------------------------------------------------------------------