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);