Article CONVEX1, MML version 4.99.1005

:: CONVEX1:funcnot 1 => CONVEX1:func 1
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of REAL;
  func A3 * A2 -> Element of bool the carrier of a1 equals
    {a3 * b1 where b1 is Element of the carrier of a1: b1 in a2};
end;

:: CONVEX1:def 1
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of REAL holds
   b3 * b2 = {b3 * b4 where b4 is Element of the carrier of b1: b4 in b2};

:: CONVEX1:attrnot 1 => CONVEX1:attr 1
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is convex means
    for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL
          st 0 < b3 & b3 < 1 & b1 in a2 & b2 in a2
       holds (b3 * b1) + ((1 - b3) * b2) in a2;
end;

:: CONVEX1:dfs 2
definiens
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is convex
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of a1
    for b3 being Element of REAL
          st 0 < b3 & b3 < 1 & b1 in a2 & b2 in a2
       holds (b3 * b1) + ((1 - b3) * b2) in a2;

:: CONVEX1:def 2
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is convex(b1)
   iff
      for b3, b4 being Element of the carrier of b1
      for b5 being Element of REAL
            st 0 < b5 & b5 < 1 & b3 in b2 & b4 in b2
         holds (b5 * b3) + ((1 - b5) * b4) in b2;

:: CONVEX1:th 1
theorem
for b1 being non empty RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of REAL
      st b2 is convex(b1)
   holds b3 * b2 is convex(b1);

:: CONVEX1:th 2
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is convex(b1) & b3 is convex(b1)
   holds b2 + b3 is convex(b1);

:: CONVEX1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is convex(b1) & b3 is convex(b1)
   holds b2 - b3 is convex(b1);

:: CONVEX1:th 4
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is convex(b1)
   iff
      for b3 being Element of REAL
            st 0 < b3 & b3 < 1
         holds (b3 * b2) + ((1 - b3) * b2) c= b2;

:: CONVEX1:th 5
theorem
for b1 being non empty Abelian RLSStruct
for b2 being Element of bool the carrier of b1
   st b2 is convex(b1)
for b3 being Element of REAL
      st 0 < b3 & b3 < 1
   holds ((1 - b3) * b2) + (b3 * b2) c= b2;

:: CONVEX1:th 6
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
   st b2 is convex(b1) & b3 is convex(b1)
for b4 being Element of REAL holds
   (b4 * b2) + ((1 - b4) * b3) is convex(b1);

:: CONVEX1:th 7
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
for b3 being Element of the carrier of b1 holds
      b2 is convex(b1)
   iff
      b3 + b2 is convex(b1);

:: CONVEX1:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   Up (0). b1 is convex(b1);

:: CONVEX1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
   Up (Omega). b1 is convex(b1);

:: CONVEX1:th 10
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 = {}
   holds b2 is convex(b1);

:: CONVEX1:th 11
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of REAL
      st b2 is convex(b1) & b3 is convex(b1)
   holds (b4 * b2) + (b5 * b3) is convex(b1);

:: CONVEX1:th 12
theorem
for b1 being non empty RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of REAL holds
(b3 + b4) * b2 c= (b3 * b2) + (b4 * b2);

:: CONVEX1:th 13
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
for b3, b4 being Element of REAL
      st 0 <= b3 & 0 <= b4 & b2 is convex(b1)
   holds (b3 * b2) + (b4 * b2) = (b3 + b4) * b2;

:: CONVEX1:th 14
theorem
for b1 being non empty Abelian add-associative RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of bool the carrier of b1
for b5, b6, b7 being Element of REAL
      st b2 is convex(b1) & b3 is convex(b1) & b4 is convex(b1)
   holds ((b5 * b2) + (b6 * b3)) + (b7 * b4) is convex(b1);

:: CONVEX1:th 15
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1
              st b3 in b2
           holds b3 is convex(b1)
   holds meet b2 is convex(b1);

:: CONVEX1:th 16
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
      st b2 is Affine(b1)
   holds b2 is convex(b1);

:: CONVEX1:exreg 1
registration
  let a1 be non empty RLSStruct;
  cluster non empty convex Element of bool the carrier of a1;
end;

:: CONVEX1:exreg 2
registration
  let a1 be non empty RLSStruct;
  cluster empty convex Element of bool the carrier of a1;
end;

:: CONVEX1:th 17
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b4 <= b5 .|. b3}
   holds b2 is convex(b1);

:: CONVEX1:th 18
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b4 < b5 .|. b3}
   holds b2 is convex(b1);

:: CONVEX1:th 19
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b5 .|. b3 <= b4}
   holds b2 is convex(b1);

:: CONVEX1:th 20
theorem
for b1 being non empty RealUnitarySpace-like UNITSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Element of REAL
      st b2 = {b5 where b5 is Element of the carrier of b1: b5 .|. b3 < b4}
   holds b2 is convex(b1);

:: CONVEX1:attrnot 2 => CONVEX1:attr 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Linear_Combination of a1;
  attr a2 is convex means
    ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one &
        proj2 b1 = Carrier a2 &
        (ex b2 being FinSequence of REAL st
           len b2 = len b1 &
            Sum b2 = 1 &
            (for b3 being natural set
                  st b3 in dom b2
               holds b2 . b3 = a2 . (b1 . b3) & 0 <= b2 . b3));
end;

:: CONVEX1:dfs 3
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
  let a2 be Linear_Combination of a1;
To prove
     a2 is convex
it is sufficient to prove
  thus ex b1 being FinSequence of the carrier of a1 st
       b1 is one-to-one &
        proj2 b1 = Carrier a2 &
        (ex b2 being FinSequence of REAL st
           len b2 = len b1 &
            Sum b2 = 1 &
            (for b3 being natural set
                  st b3 in dom b2
               holds b2 . b3 = a2 . (b1 . b3) & 0 <= b2 . b3));

:: CONVEX1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
      b2 is convex(b1)
   iff
      ex b3 being FinSequence of the carrier of b1 st
         b3 is one-to-one &
          proj2 b3 = Carrier b2 &
          (ex b4 being FinSequence of REAL st
             len b4 = len b3 &
              Sum b4 = 1 &
              (for b5 being natural set
                    st b5 in dom b4
                 holds b4 . b5 = b2 . (b3 . b5) & 0 <= b4 . b5));

:: CONVEX1:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
      st b2 is convex(b1)
   holds Carrier b2 <> {};

:: CONVEX1:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
for b3 being Element of the carrier of b1
      st b2 is convex(b1) & b2 . b3 <= 0
   holds not b3 in Carrier b2;

:: CONVEX1:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1
      st b2 is convex(b1)
   holds b2 <> ZeroLC b1;

:: CONVEX1:th 24
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
for b3 being Linear_Combination of {b2}
      st b3 is convex(b1)
   holds b3 . b2 = 1 & Sum b3 = (b3 . b2) * b2;

:: CONVEX1:th 25
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
for b4 being Linear_Combination of {b2,b3}
      st b2 <> b3 & b4 is convex(b1)
   holds (b4 . b2) + (b4 . b3) = 1 &
    0 <= b4 . b2 &
    0 <= b4 . b3 &
    Sum b4 = ((b4 . b2) * b2) + ((b4 . b3) * b3);

:: CONVEX1:th 26
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
for b5 being Linear_Combination of {b2,b3,b4}
      st b2 <> b3 & b3 <> b4 & b4 <> b2 & b5 is convex(b1)
   holds ((b5 . b2) + (b5 . b3)) + (b5 . b4) = 1 &
    0 <= b5 . b2 &
    0 <= b5 . b3 &
    0 <= b5 . b4 &
    Sum b5 = (((b5 . b2) * b2) + ((b5 . b3) * b3)) + ((b5 . b4) * b4);

:: CONVEX1:th 27
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
for b3 being Linear_Combination of b1
      st b3 is convex(b1) & Carrier b3 = {b2}
   holds b3 . b2 = 1;

:: CONVEX1:th 28
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
for b4 being Linear_Combination of b1
      st b4 is convex(b1) & Carrier b4 = {b2,b3} & b2 <> b3
   holds (b4 . b2) + (b4 . b3) = 1 & 0 <= b4 . b2 & 0 <= b4 . b3;

:: CONVEX1:th 29
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
for b5 being Linear_Combination of b1
      st b5 is convex(b1) & Carrier b5 = {b2,b3,b4} & b2 <> b3 & b3 <> b4 & b4 <> b2
   holds ((b5 . b2) + (b5 . b3)) + (b5 . b4) = 1 &
    0 <= b5 . b2 &
    0 <= b5 . b3 &
    0 <= b5 . b4 &
    Sum b5 = (((b5 . b2) * b2) + ((b5 . b3) * b3)) + ((b5 . b4) * b4);

:: CONVEX1:funcnot 2 => CONVEX1:func 2
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  func Convex-Family A2 -> Element of bool bool the carrier of a1 means
    for b1 being Element of bool the carrier of a1 holds
          b1 in it
       iff
          b1 is convex(a1) & a2 c= b1;
end;

:: CONVEX1:def 4
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
      b3 = Convex-Family b2
   iff
      for b4 being Element of bool the carrier of b1 holds
            b4 in b3
         iff
            b4 is convex(b1) & b2 c= b4;

:: CONVEX1:funcnot 3 => CONVEX1:func 3
definition
  let a1 be non empty RLSStruct;
  let a2 be Element of bool the carrier of a1;
  func conv A2 -> convex Element of bool the carrier of a1 equals
    meet Convex-Family a2;
end;

:: CONVEX1:def 5
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1 holds
   conv b2 = meet Convex-Family b2;

:: CONVEX1:th 30
theorem
for b1 being non empty RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being convex Element of bool the carrier of b1
      st b2 c= b3
   holds conv b2 c= b3;

:: CONVEX1:th 31
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2, b3, b4 being set
      st b1 is one-to-one & proj2 b1 = {b2,b3,b4} & b2 <> b3 & b3 <> b4 & b4 <> b2 & b1 <> <*b2,b3,b4*> & b1 <> <*b2,b4,b3*> & b1 <> <*b3,b2,b4*> & b1 <> <*b3,b4,b2*> & b1 <> <*b4,b2,b3*>
   holds b1 = <*b4,b3,b2*>;

:: CONVEX1:th 32
theorem
for b1 being non empty RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
   1 * b2 = b2;

:: CONVEX1:th 33
theorem
for b1 being non empty RLSStruct
for b2 being empty Element of bool the carrier of b1
for b3 being Element of REAL holds
   b3 * b2 = {};

:: CONVEX1:th 34
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
   0 * b2 = {0. b1};

:: CONVEX1:th 35
theorem
for b1 being non empty right_zeroed addLoopStr
for b2 being Element of bool the carrier of b1 holds
   b2 + {0. b1} = b2;

:: CONVEX1:th 36
theorem
for b1 being non empty add-associative addLoopStr
for b2, b3, b4 being Element of bool the carrier of b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);

:: CONVEX1:th 37
theorem
for b1 being non empty RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of REAL holds
b3 * (b4 * b2) = (b3 * b4) * b2;

:: CONVEX1:th 38
theorem
for b1 being non empty RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of REAL holds
   b4 * (b2 + b3) = (b4 * b2) + (b4 * b3);

:: CONVEX1:th 39
theorem
for b1 being non empty RLSStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of REAL
      st b2 c= b3
   holds b4 * b2 c= b4 * b3;

:: CONVEX1:th 40
theorem
for b1 being non empty addLoopStr
for b2 being empty Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
   b2 + b3 = {};