SET007 Axioms: SET007+724.ax


%------------------------------------------------------------------------------
% File     : SET007+724 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Introducing Spans
% 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    : jordan13 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    2 (   0 unit)
%            Number of atoms       :   46 (   3 equality)
%            Maximal formula depth :   19 (  16 average)
%            Number of connectives :   55 (  11 ~  ;   1  |;  32  &)
%                                         (   1 <=>;  10 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-4 arity)
%            Number of functors    :   18 (   4 constant; 0-4 arity)
%            Number of variables   :    6 (   0 singleton;   6 !;   0 ?)
%            Maximal term depth    :    4 (   2 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(d1_jordan13,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_topreal2(A)
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_jordan1h(A,B)
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & v1_sprect_2(C)
                  & ~ v5_seqm_3(C)
                  & v1_topreal1(C)
                  & v2_topreal1(C)
                  & v1_finseq_6(C,u1_struct_0(k15_euclid(np__2)))
                  & v1_goboard5(C)
                  & v2_goboard5(C)
                  & m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
               => ( C = k1_jordan13(A,B)
                <=> ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B))
                    & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,B),k3_jordan1h(A,B),k3_jordan11(A,B))
                    & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__2) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(A,B),k5_binarith(k3_jordan1h(A,B),np__1),k3_jordan11(A,B))
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(np__1,D)
                            & r1_xreal_0(k1_nat_1(D,np__2),k3_finseq_1(C)) )
                         => ( ( ( r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A)
                                & r1_xboole_0(k6_gobrd13(C,k1_jordan8(A,B),D),A) )
                             => r2_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) )
                            & ( r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A)
                             => ( r1_xboole_0(k6_gobrd13(C,k1_jordan8(A,B),D),A)
                                | r3_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) ) )
                            & ( ~ r1_xboole_0(k5_gobrd13(C,k1_jordan8(A,B),D),A)
                             => r1_gobrd13(u1_struct_0(k15_euclid(np__2)),C,k1_jordan8(A,B),D) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_k1_jordan13,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_topreal2(A)
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & m1_subset_1(B,k5_numbers) )
     => ( ~ v1_xboole_0(k1_jordan13(A,B))
        & v1_sprect_2(k1_jordan13(A,B))
        & ~ v5_seqm_3(k1_jordan13(A,B))
        & v1_topreal1(k1_jordan13(A,B))
        & v2_topreal1(k1_jordan13(A,B))
        & v1_finseq_6(k1_jordan13(A,B),u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(k1_jordan13(A,B))
        & v2_goboard5(k1_jordan13(A,B))
        & m2_finseq_1(k1_jordan13(A,B),u1_struct_0(k15_euclid(np__2))) ) ) )).
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