SET007 Axioms: SET007+781.ax


%------------------------------------------------------------------------------
% File     : SET007+781 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Convex Hull, Set of Convex Combinations and Convex Cone
% 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    : convex3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   18 (   2 unit)
%            Number of atoms       :  186 (  11 equality)
%            Maximal formula depth :   21 (  10 average)
%            Number of connectives :  201 (  33 ~  ;   2  |; 110  &)
%                                         (   9 <=>;  47 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-3 arity)
%            Number of functors    :   17 (   3 constant; 0-4 arity)
%            Number of variables   :   53 (   0 singleton;  46 !;   7 ?)
%            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(rc1_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v1_convex3(B,A) ) ) )).

fof(rc2_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v1_xboole_0(B)
          & v1_membered(B)
          & v2_membered(B)
          & v3_membered(B)
          & v4_membered(B)
          & v5_membered(B)
          & v1_convex3(B,A) ) ) )).

fof(rc3_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_convex3(B,A) ) ) )).

fof(d1_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( B = k1_convex3(A)
        <=> ! [C] :
              ( r2_hidden(C,B)
            <=> ( v2_convex1(C,A)
                & m1_rlvect_2(C,A) ) ) ) ) )).

fof(d2_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( C = k2_convex3(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ( v2_convex1(D,A)
                    & m2_rlvect_2(D,A,B) ) ) ) ) ) )).

fof(t1_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ? [C] :
              ( v2_convex1(C,A)
              & m1_rlvect_2(C,A)
              & k13_rlvect_2(A,C) = B
              & ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                 => ( r2_hidden(B,D)
                   => ( v2_convex1(C,A)
                      & m2_rlvect_2(C,A,D) ) ) ) ) ) ) )).

fof(t2_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & ! [D] :
                      ( ( v2_convex1(D,A)
                        & m1_rlvect_2(D,A) )
                     => ? [E] :
                          ( ~ v1_xboole_0(E)
                          & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                          & r1_tarski(k8_rlvect_2(A,B,C),E)
                          & ~ ( v2_convex1(D,A)
                              & m2_rlvect_2(D,A,E) ) ) ) ) ) ) ) )).

fof(t3_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ~ ( B != C
                      & B != D
                      & C != D
                      & ! [E] :
                          ( ( v2_convex1(E,A)
                            & m1_rlvect_2(E,A) )
                         => ? [F] :
                              ( ~ v1_xboole_0(F)
                              & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                              & r1_tarski(k9_rlvect_2(A,B,C,D),F)
                              & ~ ( v2_convex1(E,A)
                                  & m2_rlvect_2(E,A,F) ) ) ) ) ) ) ) ) )).

fof(d3_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_convex3(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,B)
                     => ( r1_xreal_0(C,np__0)
                        | r2_hidden(k3_rlvect_1(A,D,C),B) ) ) ) ) ) ) ) )).

fof(t6_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( B = k1_xboole_0
           => v1_convex3(B,A) ) ) ) )).

fof(t7_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( v1_convex3(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v7_rlvect_1(A)
           => ( v1_convex1(B,A)
            <=> ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ( ( r2_hidden(C,B)
                          & r2_hidden(D,B) )
                       => r2_hidden(k2_rlvect_1(A,C,D),B) ) ) ) ) ) ) ) )).

fof(t8_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v1_convex1(B,A)
              & v1_convex3(B,A) )
          <=> ! [C] :
                ( m2_rlvect_2(C,A,B)
               => ( ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(D,k10_rlvect_2(A,C))
                          & r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,C,D),np__0) ) )
                 => ( k10_rlvect_2(A,C) = k1_xboole_0
                    | r2_hidden(k13_rlvect_2(A,C),B) ) ) ) ) ) ) )).

fof(t9_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( v1_convex3(B,A)
                  & v1_convex3(C,A) )
               => v1_convex3(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) )).

fof(dt_k1_convex3,axiom,(
    $true )).

fof(dt_k2_convex3,axiom,(
    $true )).

fof(t4_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v1_convex1(B,A)
          <=> r1_tarski(a_2_0_convex3(A,B),B) ) ) ) )).

fof(t5_convex3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k3_convex1(A,B) = a_2_0_convex3(A,B) ) ) )).

fof(fraenkel_a_2_0_convex3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_rlvect_1(B)
        & l2_rlvect_1(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_0_convex3(B,C))
      <=> ? [D] :
            ( v2_convex1(D,B)
            & m2_rlvect_2(D,B,C)
            & A = k13_rlvect_2(B,D)
            & r2_hidden(D,k1_convex3(B)) ) ) ) )).
%------------------------------------------------------------------------------