Article CONVEX3, MML version 4.99.1005

:: CONVEX3:funcnot 1 => CONVEX3:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  func ConvexComb A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is convex Linear_Combination of a1;
end;

:: CONVEX3:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being set holds
      b2 = ConvexComb b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 is convex Linear_Combination of b1;

:: CONVEX3:funcnot 2 => CONVEX3:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be non empty Element of bool the carrier of a1;
  func ConvexComb A2 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 is convex Linear_Combination of a2;
end;

:: CONVEX3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being set holds
      b3 = ConvexComb b2
   iff
      for b4 being set holds
            b4 in b3
         iff
            b4 is convex Linear_Combination of b2;

:: CONVEX3:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
   ex b3 being convex Linear_Combination of b1 st
      Sum b3 = b2 &
       (for b4 being non empty Element of bool the carrier of b1
             st b2 in b4
          holds b3 is convex Linear_Combination of b4);

:: CONVEX3:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
      st b2 <> b3
   holds ex b4 being convex Linear_Combination of b1 st
      for b5 being non empty Element of bool the carrier of b1
            st {b2,b3} c= b5
         holds b4 is convex Linear_Combination of b5;

:: CONVEX3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
      st b2 <> b3 & b2 <> b4 & b3 <> b4
   holds ex b5 being convex Linear_Combination of b1 st
      for b6 being non empty Element of bool the carrier of b1
            st {b2,b3,b4} c= b6
         holds b5 is convex Linear_Combination of b6;

:: CONVEX3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
      b2 is convex(b1)
   iff
      {Sum b3 where b3 is convex Linear_Combination of b2: b3 in ConvexComb b1} c= b2;

:: CONVEX3:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being non empty Element of bool the carrier of b1 holds
   conv b2 = {Sum b3 where b3 is convex Linear_Combination of b2: b3 in ConvexComb b1};

:: CONVEX3:attrnot 1 => CONVEX3:attr 1
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is cone means
    for b1 being Element of REAL
    for b2 being Element of the carrier of a1
          st 0 < b1 & b2 in a2
       holds b1 * b2 in a2;
end;

:: CONVEX3:dfs 3
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is cone
it is sufficient to prove
  thus for b1 being Element of REAL
    for b2 being Element of the carrier of a1
          st 0 < b1 & b2 in a2
       holds b1 * b2 in a2;

:: CONVEX3:def 3
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is cone(b1)
   iff
      for b3 being Element of REAL
      for b4 being Element of the carrier of b1
            st 0 < b3 & b4 in b2
         holds b3 * b4 in b2;

:: CONVEX3:th 6
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 = {}
   holds b2 is cone(b1);

:: CONVEX3:exreg 1
registration
  let a1 be non empty RLSStruct;
  cluster cone Element of bool the carrier of a1;
end;

:: CONVEX3:exreg 2
registration
  let a1 be non empty RLSStruct;
  cluster empty cone Element of bool the carrier of a1;
end;

:: CONVEX3:exreg 3
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  cluster non empty cone Element of bool the carrier of a1;
end;

:: CONVEX3:th 7
theorem
for b1 being non empty RLSStruct
for b2 being cone Element of bool the carrier of b1
      st b1 is RealLinearSpace-like
   holds    b2 is convex(b1)
   iff
      for b3, b4 being Element of the carrier of b1
            st b3 in b2 & b4 in b2
         holds b3 + b4 in b2;

:: CONVEX3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is convex(b1) & b2 is cone(b1)
   iff
      for b3 being Linear_Combination of b2
            st Carrier b3 <> {} &
               (for b4 being Element of the carrier of b1
                     st b4 in Carrier b3
                  holds 0 < b3 . b4)
         holds Sum b3 in b2;

:: CONVEX3:th 9
theorem
for b1 being non empty RLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is cone(b1) & b3 is cone(b1)
   holds b2 /\ b3 is cone(b1);