Article CONVFUN1, MML version 4.99.1005
:: CONVFUN1:funcnot 1 => CONVFUN1:func 1
definition
let a1, a2 be non empty RLSStruct;
func Add_in_Prod_of_RLS(A1,A2) -> Function-like quasi_total Relation of [:[:the carrier of a1,the carrier of a2:],[:the carrier of a1,the carrier of a2:]:],[:the carrier of a1,the carrier of a2:] means
for b1, b2 being Element of [:the carrier of a1,the carrier of a2:]
for b3, b4 being Element of the carrier of a1
for b5, b6 being Element of the carrier of a2
st b1 = [b3,b5] & b2 = [b4,b6]
holds it .(b1,b2) = [(the addF of a1) . [b3,b4],(the addF of a2) . [b5,b6]];
end;
:: CONVFUN1:def 1
theorem
for b1, b2 being non empty RLSStruct
for b3 being Function-like quasi_total Relation of [:[:the carrier of b1,the carrier of b2:],[:the carrier of b1,the carrier of b2:]:],[:the carrier of b1,the carrier of b2:] holds
b3 = Add_in_Prod_of_RLS(b1,b2)
iff
for b4, b5 being Element of [:the carrier of b1,the carrier of b2:]
for b6, b7 being Element of the carrier of b1
for b8, b9 being Element of the carrier of b2
st b4 = [b6,b8] & b5 = [b7,b9]
holds b3 .(b4,b5) = [(the addF of b1) . [b6,b7],(the addF of b2) . [b8,b9]];
:: CONVFUN1:funcnot 2 => CONVFUN1:func 2
definition
let a1, a2 be non empty RLSStruct;
func Mult_in_Prod_of_RLS(A1,A2) -> Function-like quasi_total Relation of [:REAL,[:the carrier of a1,the carrier of a2:]:],[:the carrier of a1,the carrier of a2:] means
for b1 being Element of REAL
for b2 being Element of [:the carrier of a1,the carrier of a2:]
for b3 being Element of the carrier of a1
for b4 being Element of the carrier of a2
st b2 = [b3,b4]
holds it .(b1,b2) = [(the Mult of a1) . [b1,b3],(the Mult of a2) . [b1,b4]];
end;
:: CONVFUN1:def 2
theorem
for b1, b2 being non empty RLSStruct
for b3 being Function-like quasi_total Relation of [:REAL,[:the carrier of b1,the carrier of b2:]:],[:the carrier of b1,the carrier of b2:] holds
b3 = Mult_in_Prod_of_RLS(b1,b2)
iff
for b4 being Element of REAL
for b5 being Element of [:the carrier of b1,the carrier of b2:]
for b6 being Element of the carrier of b1
for b7 being Element of the carrier of b2
st b5 = [b6,b7]
holds b3 .(b4,b5) = [(the Mult of b1) . [b4,b6],(the Mult of b2) . [b4,b7]];
:: CONVFUN1:funcnot 3 => CONVFUN1:func 3
definition
let a1, a2 be non empty RLSStruct;
func Prod_of_RLS(A1,A2) -> RLSStruct equals
RLSStruct(#[:the carrier of a1,the carrier of a2:],[0. a1,0. a2],Add_in_Prod_of_RLS(a1,a2),Mult_in_Prod_of_RLS(a1,a2)#);
end;
:: CONVFUN1:def 3
theorem
for b1, b2 being non empty RLSStruct holds
Prod_of_RLS(b1,b2) = RLSStruct(#[:the carrier of b1,the carrier of b2:],[0. b1,0. b2],Add_in_Prod_of_RLS(b1,b2),Mult_in_Prod_of_RLS(b1,b2)#);
:: CONVFUN1:funcreg 1
registration
let a1, a2 be non empty RLSStruct;
cluster Prod_of_RLS(a1,a2) -> non empty;
end;
:: CONVFUN1:th 1
theorem
for b1, b2 being non empty RLSStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of Prod_of_RLS(b1,b2)
for b6 being Element of REAL
st b5 = [b3,b4]
holds b6 * b5 = [b6 * b3,b6 * b4];
:: CONVFUN1:th 2
theorem
for b1, b2 being non empty RLSStruct
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7, b8 being Element of the carrier of Prod_of_RLS(b1,b2)
st b7 = [b3,b5] & b8 = [b4,b6]
holds b7 + b8 = [b3 + b4,b5 + b6];
:: CONVFUN1:funcreg 2
registration
let a1, a2 be non empty Abelian RLSStruct;
cluster Prod_of_RLS(a1,a2) -> Abelian;
end;
:: CONVFUN1:funcreg 3
registration
let a1, a2 be non empty add-associative RLSStruct;
cluster Prod_of_RLS(a1,a2) -> add-associative;
end;
:: CONVFUN1:funcreg 4
registration
let a1, a2 be non empty right_zeroed RLSStruct;
cluster Prod_of_RLS(a1,a2) -> right_zeroed;
end;
:: CONVFUN1:funcreg 5
registration
let a1, a2 be non empty right_complementable RLSStruct;
cluster Prod_of_RLS(a1,a2) -> right_complementable;
end;
:: CONVFUN1:funcreg 6
registration
let a1, a2 be non empty RealLinearSpace-like RLSStruct;
cluster Prod_of_RLS(a1,a2) -> RealLinearSpace-like;
end;
:: CONVFUN1:th 3
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of NAT
for b4 being FinSequence of the carrier of b1
for b5 being FinSequence of the carrier of b2
for b6 being FinSequence of the carrier of Prod_of_RLS(b1,b2)
st len b4 = b3 &
len b5 = b3 &
len b6 = b3 &
(for b7 being Element of NAT
st b7 in Seg b3
holds b6 . b7 = [b4 . b7,b5 . b7])
holds Sum b6 = [Sum b4,Sum b5];
:: CONVFUN1:funcnot 4 => CONVFUN1:func 4
definition
func RLS_Real -> non empty RLSStruct equals
RLSStruct(#REAL,0,addreal,multreal#);
end;
:: CONVFUN1:def 4
theorem
RLS_Real = RLSStruct(#REAL,0,addreal,multreal#);
:: CONVFUN1:funcreg 7
registration
cluster RLS_Real -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like;
end;
:: CONVFUN1:funcnot 5 => CONVFUN1:func 5
definition
let a1 be FinSequence of ExtREAL;
func Sum A1 -> Element of ExtREAL means
ex b1 being Function-like quasi_total Relation of NAT,ExtREAL st
it = b1 . len a1 &
b1 . 0 = 0. &
(for b2 being Element of NAT
st b2 < len a1
holds b1 . (b2 + 1) = (b1 . b2) + (a1 . (b2 + 1)));
end;
:: CONVFUN1:def 5
theorem
for b1 being FinSequence of ExtREAL
for b2 being Element of ExtREAL holds
b2 = Sum b1
iff
ex b3 being Function-like quasi_total Relation of NAT,ExtREAL st
b2 = b3 . len b1 &
b3 . 0 = 0. &
(for b4 being Element of NAT
st b4 < len b1
holds b3 . (b4 + 1) = (b3 . b4) + (b1 . (b4 + 1)));
:: CONVFUN1:th 4
theorem
Sum <*> ExtREAL = 0.;
:: CONVFUN1:th 5
theorem
for b1 being Element of ExtREAL holds
Sum <*b1*> = b1;
:: CONVFUN1:th 6
theorem
for b1, b2 being Element of ExtREAL holds
Sum <*b1,b2*> = b1 + b2;
:: CONVFUN1:th 7
theorem
for b1, b2 being FinSequence of ExtREAL
st not -infty in rng b1 & not -infty in rng b2
holds Sum (b1 ^ b2) = (Sum b1) + Sum b2;
:: CONVFUN1:th 8
theorem
for b1, b2 being FinSequence of ExtREAL
for b3 being Function-like quasi_total bijective Relation of dom b1,dom b1
st b2 = b1 * b3 & not -infty in rng b1
holds Sum b1 = Sum b2;
:: CONVFUN1:funcnot 6 => CONVFUN1:func 6
definition
let a1 be non empty RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,ExtREAL;
func epigraph A2 -> Element of bool the carrier of Prod_of_RLS(a1,RLS_Real) equals
{[b1,b2] where b1 is Element of the carrier of a1, b2 is Element of REAL: a2 . b1 <= R_EAL b2};
end;
:: CONVFUN1:def 6
theorem
for b1 being non empty RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,ExtREAL holds
epigraph b2 = {[b3,b4] where b3 is Element of the carrier of b1, b4 is Element of REAL: b2 . b3 <= R_EAL b4};
:: CONVFUN1:attrnot 1 => CONVFUN1:attr 1
definition
let a1 be non empty RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,ExtREAL;
attr a2 is convex means
epigraph a2 is convex(Prod_of_RLS(a1,RLS_Real));
end;
:: CONVFUN1:dfs 7
definiens
let a1 be non empty RLSStruct;
let a2 be Function-like quasi_total Relation of the carrier of a1,ExtREAL;
To prove
a2 is convex
it is sufficient to prove
thus epigraph a2 is convex(Prod_of_RLS(a1,RLS_Real));
:: CONVFUN1:def 7
theorem
for b1 being non empty RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,ExtREAL holds
b2 is convex(b1)
iff
epigraph b2 is convex(Prod_of_RLS(b1,RLS_Real));
:: CONVFUN1:th 9
theorem
for b1 being non empty RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,ExtREAL
st for b3 being Element of the carrier of b1 holds
b2 . b3 <> -infty
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
holds b2 . ((b5 * b3) + ((1 - b5) * b4)) <= ((R_EAL b5) * (b2 . b3)) + ((R_EAL (1 - b5)) * (b2 . b4));
:: CONVFUN1:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,ExtREAL
st for b3 being Element of the carrier of b1 holds
b2 . b3 <> -infty
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
holds b2 . ((b5 * b3) + ((1 - b5) * b4)) <= ((R_EAL b5) * (b2 . b3)) + ((R_EAL (1 - b5)) * (b2 . b4));
:: CONVFUN1:th 11
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Function-like quasi_total Relation of the carrier of RLS_Real,ExtREAL
for b3 being Element of bool the carrier of RLS_Real
st b3 c= dom b1 &
(for b4 being Element of REAL holds
(b4 in b3 implies b2 . b4 = b1 . b4) & (b4 in b3 or b2 . b4 = +infty))
holds b2 is convex(RLS_Real)
iff
b1 is_convex_on b3 & b3 is convex(RLS_Real);
:: CONVFUN1:th 12
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)
iff
for b3 being non empty Element of NAT
for b4 being FinSequence of REAL
for b5, b6 being FinSequence of the carrier of b1
st len b4 = b3 &
len b5 = b3 &
len b6 = b3 &
Sum b4 = 1 &
(for b7 being Element of NAT
st b7 in Seg b3
holds 0 < b4 . b7 &
b6 . b7 = (b4 . b7) * (b5 /. b7) &
b5 /. b7 in b2)
holds Sum b6 in b2;
:: CONVFUN1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,ExtREAL
st for b3 being Element of the carrier of b1 holds
b2 . b3 <> -infty
holds b2 is convex(b1)
iff
for b3 being non empty Element of NAT
for b4 being FinSequence of REAL
for b5 being FinSequence of ExtREAL
for b6, b7 being FinSequence of the carrier of b1
st len b4 = b3 &
len b5 = b3 &
len b6 = b3 &
len b7 = b3 &
Sum b4 = 1 &
(for b8 being Element of NAT
st b8 in Seg b3
holds 0 < b4 . b8 &
b7 . b8 = (b4 . b8) * (b6 /. b8) &
b5 . b8 = (R_EAL (b4 . b8)) * (b2 . (b6 /. b8)))
holds b2 . Sum b7 <= Sum b5;
:: CONVFUN1:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,ExtREAL
st for b3 being Element of the carrier of b1 holds
b2 . b3 <> -infty
holds b2 is convex(b1)
iff
for b3 being non empty Element of NAT
for b4 being FinSequence of REAL
for b5 being FinSequence of ExtREAL
for b6, b7 being FinSequence of the carrier of b1
st len b4 = b3 &
len b5 = b3 &
len b6 = b3 &
len b7 = b3 &
Sum b4 = 1 &
(for b8 being Element of NAT
st b8 in Seg b3
holds 0 <= b4 . b8 &
b7 . b8 = (b4 . b8) * (b6 /. b8) &
b5 . b8 = (R_EAL (b4 . b8)) * (b2 . (b6 /. b8)))
holds b2 . Sum b7 <= Sum b5;