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 = {};