SET007 Axioms: SET007+772.ax


%------------------------------------------------------------------------------
% File     : SET007+772 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Properties for Convex Combinations
% 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    : convex2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   1 unit)
%            Number of atoms       :  277 (  22 equality)
%            Maximal formula depth :   19 (  12 average)
%            Number of connectives :  304 (  52 ~  ;   0  |; 166  &)
%                                         (   7 <=>;  79 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-3 arity)
%            Number of functors    :   30 (   5 constant; 0-3 arity)
%            Number of variables   :   99 (   0 singleton;  91 !;   8 ?)
%            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(rc1_convex2,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_rlvect_2(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(A),k1_numbers)
          & v1_seq_1(B)
          & v2_convex1(B,A) ) ) )).

fof(rc2_convex2,axiom,(
    ! [A,B] :
      ( ( ~ 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)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] :
          ( m2_rlvect_2(C,A,B)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,u1_struct_0(A),k1_numbers)
          & v1_seq_1(C)
          & v2_convex1(C,A) ) ) )).

fof(t1_convex2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( ( v1_convex1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_convex1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => v1_convex1(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) )).

fof(t6_convex2,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)))
         => ( ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ! [D] :
                    ( m2_rlvect_2(D,A,C)
                   => ( ( v2_convex1(D,A)
                        & r1_tarski(C,B) )
                     => r2_hidden(k13_rlvect_2(A,D),B) ) ) )
          <=> v1_convex1(B,A) ) ) ) )).

fof(d1_convex2,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)))
         => ! [C] :
              ( C = k1_convex2(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> m2_rlvect_2(D,A,B) ) ) ) ) )).

fof(t7_convex2,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)))
         => k2_convex1(A,B) != k1_xboole_0 ) ) )).

fof(t8_convex2,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)))
         => r1_tarski(B,k3_convex1(A,B)) ) ) )).

fof(t9_convex2,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] :
          ( ( v2_convex1(B,A)
            & m1_rlvect_2(B,A) )
         => ! [C] :
              ( ( v2_convex1(C,A)
                & m1_rlvect_2(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( ~ r1_xreal_0(D,np__0)
                      & ~ r1_xreal_0(np__1,D)
                      & ~ ( v2_convex1(k14_rlvect_2(A,k15_rlvect_2(A,D,B),k15_rlvect_2(A,k5_real_1(np__1,D),C)),A)
                          & m1_rlvect_2(k14_rlvect_2(A,k15_rlvect_2(A,D,B),k15_rlvect_2(A,k5_real_1(np__1,D),C)),A) ) ) ) ) ) ) )).

fof(t10_convex2,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] :
              ( ( v2_convex1(C,A)
                & m2_rlvect_2(C,A,B) )
             => ! [D] :
                  ( ( v2_convex1(D,A)
                    & m2_rlvect_2(D,A,B) )
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ~ ( ~ r1_xreal_0(E,np__0)
                          & ~ r1_xreal_0(np__1,E)
                          & ~ ( v2_convex1(k14_rlvect_2(A,k15_rlvect_2(A,E,C),k15_rlvect_2(A,k5_real_1(np__1,E),D)),A)
                              & m2_rlvect_2(k14_rlvect_2(A,k15_rlvect_2(A,E,C),k15_rlvect_2(A,k5_real_1(np__1,E),D)),A,B) ) ) ) ) ) ) ) )).

fof(t11_convex2,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_rlvect_2(B,A)
          & v2_convex1(B,A) ) ) )).

fof(t12_convex2,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] :
              ( m2_rlvect_2(C,A,B)
              & v2_convex1(C,A) ) ) ) )).

fof(t13_convex2,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_rlsub_1(B,A)
         => ! [C] :
              ( m1_rlsub_1(C,A)
             => k3_rusub_4(A,k1_rlsub_2(A,B,C)) = k7_rusub_4(A,k3_rusub_4(A,B),k3_rusub_4(A,C)) ) ) ) )).

fof(t14_convex2,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_rlsub_1(B,A)
         => ! [C] :
              ( m1_rlsub_1(C,A)
             => k3_rusub_4(A,k2_rlsub_2(A,B,C)) = k5_subset_1(u1_struct_0(A),k3_rusub_4(A,B),k3_rusub_4(A,C)) ) ) ) )).

fof(t15_convex2,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] :
          ( ( v2_convex1(B,A)
            & m1_rlvect_2(B,A) )
         => ! [C] :
              ( ( v2_convex1(C,A)
                & m1_rlvect_2(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ~ r1_xreal_0(k4_real_1(D,E),np__0)
                       => k10_rlvect_2(A,k14_rlvect_2(A,k15_rlvect_2(A,D,B),k15_rlvect_2(A,E,C))) = k2_rlvect_2(A,k10_rlvect_2(A,k15_rlvect_2(A,D,B)),k10_rlvect_2(A,k15_rlvect_2(A,E,C))) ) ) ) ) ) ) )).

fof(t16_convex2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_rfinseq(A,B)
           => ! [C,D] :
                ~ ( r2_hidden(C,k1_relat_1(A))
                  & r2_hidden(D,k1_relat_1(A))
                  & C != D
                  & ! [E,F] :
                      ~ ( r2_hidden(E,k1_relat_1(B))
                        & r2_hidden(F,k1_relat_1(B))
                        & E != F
                        & k1_funct_1(A,C) = k1_funct_1(B,E)
                        & k1_funct_1(A,D) = k1_funct_1(B,F) ) ) ) ) ) )).

fof(t17_convex2,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_rlvect_2(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ~ ( r1_tarski(C,k10_rlvect_2(A,B))
                  & ! [D] :
                      ( m1_rlvect_2(D,A)
                     => k10_rlvect_2(A,D) != C ) ) ) ) ) )).

fof(dt_k1_convex2,axiom,(
    $true )).

fof(t2_convex2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_finseq_1(D,k1_numbers)
                 => ( B = a_3_0_convex2(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(t3_convex2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_finseq_1(D,k1_numbers)
                 => ( B = a_3_1_convex2(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(t4_convex2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_finseq_1(D,k1_numbers)
                 => ( B = a_3_2_convex2(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(t5_convex2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_finseq_1(D,k1_numbers)
                 => ( B = a_3_3_convex2(A,C,D)
                   => v1_convex1(B,A) ) ) ) ) ) )).

fof(fraenkel_a_3_0_convex2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m2_finseq_1(C,u1_struct_0(B))
        & m2_finseq_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_0_convex2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & ! [F] :
                ( m2_subset_1(F,k1_numbers,k5_numbers)
               => ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
                    & ! [G] :
                        ( m1_subset_1(G,u1_struct_0(B))
                       => ~ ( G = k1_funct_1(C,F)
                            & r1_xreal_0(k1_bhsp_1(B,E,G),k1_funct_1(D,F)) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_1_convex2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m2_finseq_1(C,u1_struct_0(B))
        & m2_finseq_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_1_convex2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & ! [F] :
                ( m2_subset_1(F,k1_numbers,k5_numbers)
               => ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
                    & ! [G] :
                        ( m1_subset_1(G,u1_struct_0(B))
                       => ~ ( G = k1_funct_1(C,F)
                            & ~ r1_xreal_0(k1_funct_1(D,F),k1_bhsp_1(B,E,G)) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_2_convex2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m2_finseq_1(C,u1_struct_0(B))
        & m2_finseq_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_2_convex2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & ! [F] :
                ( m2_subset_1(F,k1_numbers,k5_numbers)
               => ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
                    & ! [G] :
                        ( m1_subset_1(G,u1_struct_0(B))
                       => ~ ( G = k1_funct_1(C,F)
                            & r1_xreal_0(k1_funct_1(D,F),k1_bhsp_1(B,E,G)) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_3_convex2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_bhsp_1(B)
        & l1_bhsp_1(B)
        & m2_finseq_1(C,u1_struct_0(B))
        & m2_finseq_1(D,k1_numbers) )
     => ( r2_hidden(A,a_3_3_convex2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & ! [F] :
                ( m2_subset_1(F,k1_numbers,k5_numbers)
               => ~ ( r2_hidden(F,k5_subset_1(k5_numbers,k4_finseq_1(C),k4_finseq_1(D)))
                    & ! [G] :
                        ( m1_subset_1(G,u1_struct_0(B))
                       => ~ ( G = k1_funct_1(C,F)
                            & ~ r1_xreal_0(k1_bhsp_1(B,E,G),k1_funct_1(D,F)) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------