Article RFUNCT_3, MML version 4.99.1005
:: RFUNCT_3:funcnot 1 => RFUNCT_3:func 1
definition
let a1, a2 be Element of NAT;
redefine func min(a1,a2) -> Element of NAT;
commutativity;
:: for a1, a2 being Element of NAT holds
:: min(a1,a2) = min(a2,a1);
idempotence;
:: for a1 being Element of NAT holds
:: min(a1,a1) = a1;
end;
:: RFUNCT_3:funcnot 2 => RFUNCT_3:func 2
definition
let a1 be real set;
func max+ A1 -> Element of REAL equals
max(a1,0);
end;
:: RFUNCT_3:def 1
theorem
for b1 being real set holds
max+ b1 = max(b1,0);
:: RFUNCT_3:funcnot 3 => RFUNCT_3:func 3
definition
let a1 be real set;
func max- A1 -> Element of REAL equals
max(- a1,0);
end;
:: RFUNCT_3:def 2
theorem
for b1 being real set holds
max- b1 = max(- b1,0);
:: RFUNCT_3:th 1
theorem
for b1 being real set holds
b1 = (max+ b1) - max- b1;
:: RFUNCT_3:th 2
theorem
for b1 being real set holds
abs b1 = (max+ b1) + max- b1;
:: RFUNCT_3:th 3
theorem
for b1 being real set holds
2 * max+ b1 = b1 + abs b1;
:: RFUNCT_3:th 4
theorem
for b1, b2 being real set
st 0 <= b1
holds max+ (b1 * b2) = b1 * max+ b2;
:: RFUNCT_3:th 5
theorem
for b1, b2 being real set holds
max+ (b1 + b2) <= (max+ b1) + max+ b2;
:: RFUNCT_3:th 8
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3, b4 being real set
st b3 <> 0
holds b2 " {b4 / b3} = (b3 (#) b2) " {b4};
:: RFUNCT_3:th 9
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
(0 (#) b2) " {0} = dom b2;
:: RFUNCT_3:th 10
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
st 0 < b3
holds (abs b2) " {b3} = b2 " {- b3,b3};
:: RFUNCT_3:th 11
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
(abs b2) " {0} = b2 " {0};
:: RFUNCT_3:th 12
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
st b3 < 0
holds (abs b2) " {b3} = {};
:: RFUNCT_3:th 13
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
for b4 being Function-like Relation of b2,REAL
for b5 being Element of REAL
st b5 <> 0
holds b3,b4 are_fiberwise_equipotent
iff
b5 (#) b3,b5 (#) b4 are_fiberwise_equipotent;
:: RFUNCT_3:th 14
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
for b4 being Function-like Relation of b2,REAL holds
b3,b4 are_fiberwise_equipotent
iff
- b3,- b4 are_fiberwise_equipotent;
:: RFUNCT_3:th 15
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
for b4 being Function-like Relation of b2,REAL
st b3,b4 are_fiberwise_equipotent
holds abs b3,abs b4 are_fiberwise_equipotent;
:: RFUNCT_3:modenot 1 => RFUNCT_3:mode 1
definition
let a1, a2 be set;
mode PartFunc-set of A1,A2 means
for b1 being Element of it holds
b1 is Function-like Relation of a1,a2;
end;
:: RFUNCT_3:dfs 3
definiens
let a1, a2, a3 be set;
To prove
a3 is PartFunc-set of a1,a2
it is sufficient to prove
thus for b1 being Element of a3 holds
b1 is Function-like Relation of a1,a2;
:: RFUNCT_3:def 3
theorem
for b1, b2, b3 being set holds
b3 is PartFunc-set of b1,b2
iff
for b4 being Element of b3 holds
b4 is Function-like Relation of b1,b2;
:: RFUNCT_3:exreg 1
registration
let a1, a2 be set;
cluster non empty PartFunc-set of a1,a2;
end;
:: RFUNCT_3:modenot 2
definition
let a1, a2 be set;
mode PFUNC_DOMAIN of a1,a2 is non empty PartFunc-set of a1,a2;
end;
:: RFUNCT_3:funcnot 4 => RFUNCT_3:func 4
definition
let a1, a2 be set;
redefine func PFuncs(a1,a2) -> PartFunc-set of a1,a2;
end;
:: RFUNCT_3:modenot 3 => RFUNCT_3:mode 2
definition
let a1, a2 be set;
let a3 be non empty PartFunc-set of a1,a2;
redefine mode Element of a3 -> Function-like Relation of a1,a2;
end;
:: RFUNCT_3:funcnot 5 => RFUNCT_3:func 5
definition
let a1, a2 be non empty set;
let a3 be Element of bool a1;
let a4 be Element of a2;
redefine func a3 --> a4 -> Element of PFuncs(a1,a2);
end;
:: RFUNCT_3:funcnot 6 => RFUNCT_3:func 6
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3, a4 be Element of PFuncs(a1,a2);
redefine func a3 + a4 -> Element of PFuncs(a1,REAL);
commutativity;
:: for a1 being non empty set
:: for a2 being real-membered set
:: for a3, a4 being Element of PFuncs(a1,a2) holds
:: a3 + a4 = a4 + a3;
end;
:: RFUNCT_3:funcnot 7 => RFUNCT_3:func 7
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3, a4 be Element of PFuncs(a1,a2);
redefine func a3 - a4 -> Element of PFuncs(a1,REAL);
end;
:: RFUNCT_3:funcnot 8 => RFUNCT_3:func 8
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3, a4 be Element of PFuncs(a1,a2);
redefine func a3 (#) a4 -> Element of PFuncs(a1,REAL);
commutativity;
:: for a1 being non empty set
:: for a2 being real-membered set
:: for a3, a4 being Element of PFuncs(a1,a2) holds
:: a3 (#) a4 = a4 (#) a3;
end;
:: RFUNCT_3:funcnot 9 => RFUNCT_3:func 9
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3, a4 be Element of PFuncs(a1,a2);
redefine func a3 / a4 -> Element of PFuncs(a1,REAL);
end;
:: RFUNCT_3:funcnot 10 => RFUNCT_3:func 10
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3 be Element of PFuncs(a1,a2);
redefine func abs a3 -> Element of PFuncs(a1,REAL);
projectivity;
:: for a1 being non empty set
:: for a2 being real-membered set
:: for a3 being Element of PFuncs(a1,a2) holds
:: abs abs a3 = abs a3;
end;
:: RFUNCT_3:funcnot 11 => RFUNCT_3:func 11
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3 be Element of PFuncs(a1,a2);
redefine func - a3 -> Element of PFuncs(a1,REAL);
involutiveness;
:: for a1 being non empty set
:: for a2 being real-membered set
:: for a3 being Element of PFuncs(a1,a2) holds
:: - - a3 = a3;
end;
:: RFUNCT_3:funcnot 12 => RFUNCT_3:func 12
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3 be Element of PFuncs(a1,a2);
redefine func a3 ^ -> Element of PFuncs(a1,REAL);
end;
:: RFUNCT_3:funcnot 13 => RFUNCT_3:func 13
definition
let a1 be non empty set;
let a2 be real-membered set;
let a3 be Element of PFuncs(a1,a2);
let a4 be real set;
redefine func a4 (#) a3 -> Element of PFuncs(a1,REAL);
end;
:: RFUNCT_3:funcnot 14 => RFUNCT_3:func 14
definition
let a1 be non empty set;
func addpfunc A1 -> Function-like quasi_total Relation of [:PFuncs(a1,REAL),PFuncs(a1,REAL):],PFuncs(a1,REAL) means
for b1, b2 being Element of PFuncs(a1,REAL) holds
it .(b1,b2) = b1 + b2;
end;
:: RFUNCT_3:def 4
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:PFuncs(b1,REAL),PFuncs(b1,REAL):],PFuncs(b1,REAL) holds
b2 = addpfunc b1
iff
for b3, b4 being Element of PFuncs(b1,REAL) holds
b2 .(b3,b4) = b3 + b4;
:: RFUNCT_3:th 16
theorem
for b1 being non empty set holds
addpfunc b1 is commutative(PFuncs(b1,REAL));
:: RFUNCT_3:th 17
theorem
for b1 being non empty set holds
addpfunc b1 is associative(PFuncs(b1,REAL));
:: RFUNCT_3:th 18
theorem
for b1 being non empty set holds
([#] b1) --> 0 is_a_unity_wrt addpfunc b1;
:: RFUNCT_3:th 19
theorem
for b1 being non empty set holds
the_unity_wrt addpfunc b1 = ([#] b1) --> 0;
:: RFUNCT_3:th 20
theorem
for b1 being non empty set holds
addpfunc b1 is having_a_unity(PFuncs(b1,REAL));
:: RFUNCT_3:funcnot 15 => RFUNCT_3:func 15
definition
let a1 be non empty set;
let a2 be FinSequence of PFuncs(a1,REAL);
func Sum A2 -> Element of PFuncs(a1,REAL) equals
(addpfunc a1) "**" a2;
end;
:: RFUNCT_3:def 5
theorem
for b1 being non empty set
for b2 being FinSequence of PFuncs(b1,REAL) holds
Sum b2 = (addpfunc b1) "**" b2;
:: RFUNCT_3:th 21
theorem
for b1 being non empty set holds
Sum <*> PFuncs(b1,REAL) = ([#] b1) --> 0;
:: RFUNCT_3:th 23
theorem
for b1 being non empty set
for b2 being FinSequence of PFuncs(b1,REAL)
for b3 being Element of PFuncs(b1,REAL) holds
Sum (b2 ^ <*b3*>) = (Sum b2) + b3;
:: RFUNCT_3:th 24
theorem
for b1 being non empty set
for b2, b3 being FinSequence of PFuncs(b1,REAL) holds
Sum (b2 ^ b3) = (Sum b2) + Sum b3;
:: RFUNCT_3:th 25
theorem
for b1 being non empty set
for b2 being FinSequence of PFuncs(b1,REAL)
for b3 being Element of PFuncs(b1,REAL) holds
Sum (<*b3*> ^ b2) = b3 + Sum b2;
:: RFUNCT_3:th 26
theorem
for b1 being non empty set
for b2, b3 being Element of PFuncs(b1,REAL) holds
Sum <*b2,b3*> = b2 + b3;
:: RFUNCT_3:th 27
theorem
for b1 being non empty set
for b2, b3, b4 being Element of PFuncs(b1,REAL) holds
Sum <*b2,b3,b4*> = (b2 + b3) + b4;
:: RFUNCT_3:th 28
theorem
for b1 being non empty set
for b2, b3 being FinSequence of PFuncs(b1,REAL)
st b2,b3 are_fiberwise_equipotent
holds Sum b2 = Sum b3;
:: RFUNCT_3:funcnot 16 => RFUNCT_3:func 16
definition
let a1 be non empty set;
let a2 be Relation-like Function-like FinSequence-like set;
func CHI(A2,A1) -> FinSequence of PFuncs(a1,REAL) means
len it = len a2 &
(for b1 being Element of NAT
st b1 in dom it
holds it . b1 = chi(a2 . b1,a1));
end;
:: RFUNCT_3:def 6
theorem
for b1 being non empty set
for b2 being Relation-like Function-like FinSequence-like set
for b3 being FinSequence of PFuncs(b1,REAL) holds
b3 = CHI(b2,b1)
iff
len b3 = len b2 &
(for b4 being Element of NAT
st b4 in dom b3
holds b3 . b4 = chi(b2 . b4,b1));
:: RFUNCT_3:funcnot 17 => RFUNCT_3:func 17
definition
let a1 be non empty set;
let a2 be FinSequence of PFuncs(a1,REAL);
let a3 be FinSequence of REAL;
func A3 (#) A2 -> FinSequence of PFuncs(a1,REAL) means
len it = min(len a3,len a2) &
(for b1 being Element of NAT
st b1 in dom it
for b2 being Function-like Relation of a1,REAL
for b3 being Element of REAL
st b3 = a3 . b1 & b2 = a2 . b1
holds it . b1 = b3 (#) b2);
end;
:: RFUNCT_3:def 7
theorem
for b1 being non empty set
for b2 being FinSequence of PFuncs(b1,REAL)
for b3 being FinSequence of REAL
for b4 being FinSequence of PFuncs(b1,REAL) holds
b4 = b3 (#) b2
iff
len b4 = min(len b3,len b2) &
(for b5 being Element of NAT
st b5 in dom b4
for b6 being Function-like Relation of b1,REAL
for b7 being Element of REAL
st b7 = b3 . b5 & b6 = b2 . b5
holds b4 . b5 = b7 (#) b6);
:: RFUNCT_3:funcnot 18 => RFUNCT_3:func 18
definition
let a1 be non empty set;
let a2 be FinSequence of PFuncs(a1,REAL);
let a3 be Element of a1;
func A2 # A3 -> FinSequence of REAL means
len it = len a2 &
(for b1 being Element of NAT
for b2 being Element of PFuncs(a1,REAL)
st b1 in dom it & a2 . b1 = b2
holds it . b1 = b2 . a3);
end;
:: RFUNCT_3:def 8
theorem
for b1 being non empty set
for b2 being FinSequence of PFuncs(b1,REAL)
for b3 being Element of b1
for b4 being FinSequence of REAL holds
b4 = b2 # b3
iff
len b4 = len b2 &
(for b5 being Element of NAT
for b6 being Element of PFuncs(b1,REAL)
st b5 in dom b4 & b2 . b5 = b6
holds b4 . b5 = b6 . b3);
:: RFUNCT_3:prednot 1 => RFUNCT_3:pred 1
definition
let a1, a2 be non empty set;
let a3 be FinSequence of PFuncs(a1,a2);
let a4 be Element of a1;
pred A4 is_common_for_dom A3 means
for b1 being Element of PFuncs(a1,a2)
for b2 being Element of NAT
st b2 in dom a3 & a3 . b2 = b1
holds a4 in dom b1;
end;
:: RFUNCT_3:dfs 9
definiens
let a1, a2 be non empty set;
let a3 be FinSequence of PFuncs(a1,a2);
let a4 be Element of a1;
To prove
a4 is_common_for_dom a3
it is sufficient to prove
thus for b1 being Element of PFuncs(a1,a2)
for b2 being Element of NAT
st b2 in dom a3 & a3 . b2 = b1
holds a4 in dom b1;
:: RFUNCT_3:def 9
theorem
for b1, b2 being non empty set
for b3 being FinSequence of PFuncs(b1,b2)
for b4 being Element of b1 holds
b4 is_common_for_dom b3
iff
for b5 being Element of PFuncs(b1,b2)
for b6 being Element of NAT
st b6 in dom b3 & b3 . b6 = b5
holds b4 in dom b5;
:: RFUNCT_3:th 29
theorem
for b1, b2 being non empty set
for b3 being FinSequence of PFuncs(b1,b2)
for b4 being Element of b1
for b5 being Element of NAT
st b4 is_common_for_dom b3 & b5 <> 0
holds b4 is_common_for_dom b3 | b5;
:: RFUNCT_3:th 30
theorem
for b1, b2 being non empty set
for b3 being FinSequence of PFuncs(b1,b2)
for b4 being Element of b1
for b5 being Element of NAT
st b4 is_common_for_dom b3
holds b4 is_common_for_dom b3 /^ b5;
:: RFUNCT_3:th 31
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being FinSequence of PFuncs(b1,REAL)
st len b3 <> 0
holds b2 is_common_for_dom b3
iff
b2 in dom Sum b3;
:: RFUNCT_3:th 32
theorem
for b1 being non empty set
for b2 being FinSequence of PFuncs(b1,REAL)
for b3 being Element of b1
for b4 being Element of NAT holds
(b2 | b4) # b3 = (b2 # b3) | b4;
:: RFUNCT_3:th 33
theorem
for b1 being non empty set
for b2 being Relation-like Function-like FinSequence-like set
for b3 being Element of b1 holds
b3 is_common_for_dom CHI(b2,b1);
:: RFUNCT_3:th 34
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being FinSequence of PFuncs(b1,REAL)
for b4 being FinSequence of REAL
st b2 is_common_for_dom b3
holds b2 is_common_for_dom b4 (#) b3;
:: RFUNCT_3:th 35
theorem
for b1 being non empty set
for b2 being Relation-like Function-like FinSequence-like set
for b3 being FinSequence of REAL
for b4 being Element of b1 holds
b4 is_common_for_dom b3 (#) CHI(b2,b1);
:: RFUNCT_3:th 36
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being FinSequence of PFuncs(b1,REAL)
st b2 is_common_for_dom b3
holds (Sum b3) . b2 = Sum (b3 # b2);
:: RFUNCT_3:funcnot 19 => RFUNCT_3:func 19
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,REAL;
func max+ A2 -> Function-like Relation of a1,REAL means
dom it = dom a2 &
(for b1 being Element of a1
st b1 in dom it
holds it . b1 = max+ (a2 . b1));
end;
:: RFUNCT_3:def 10
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
b3 = max+ b2
iff
dom b3 = dom b2 &
(for b4 being Element of b1
st b4 in dom b3
holds b3 . b4 = max+ (b2 . b4));
:: RFUNCT_3:funcnot 20 => RFUNCT_3:func 20
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,REAL;
func max- A2 -> Function-like Relation of a1,REAL means
dom it = dom a2 &
(for b1 being Element of a1
st b1 in dom it
holds it . b1 = max- (a2 . b1));
end;
:: RFUNCT_3:def 11
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
b3 = max- b2
iff
dom b3 = dom b2 &
(for b4 being Element of b1
st b4 in dom b3
holds b3 . b4 = max- (b2 . b4));
:: RFUNCT_3:th 37
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
b2 = (max+ b2) - max- b2 &
abs b2 = (max+ b2) + max- b2 &
2 (#) max+ b2 = b2 + abs b2;
:: RFUNCT_3:th 38
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
st 0 < b3
holds b2 " {b3} = (max+ b2) " {b3};
:: RFUNCT_3:th 39
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
b2 " left_closed_halfline 0 = (max+ b2) " {0};
:: RFUNCT_3:th 40
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of b1 holds
0 <= (max+ b2) . b3;
:: RFUNCT_3:th 41
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
st 0 < b3
holds b2 " {- b3} = (max- b2) " {b3};
:: RFUNCT_3:th 42
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
b2 " right_closed_halfline 0 = (max- b2) " {0};
:: RFUNCT_3:th 43
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of b1 holds
0 <= (max- b2) . b3;
:: RFUNCT_3:th 44
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
for b4 being Function-like Relation of b2,REAL
st b3,b4 are_fiberwise_equipotent
holds max+ b3,max+ b4 are_fiberwise_equipotent;
:: RFUNCT_3:th 45
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
for b4 being Function-like Relation of b2,REAL
st b3,b4 are_fiberwise_equipotent
holds max- b3,max- b4 are_fiberwise_equipotent;
:: RFUNCT_3:exreg 2
registration
let a1, a2 be set;
cluster Relation-like Function-like finite Relation of a1,a2;
end;
:: RFUNCT_3:funcreg 1
registration
let a1 be non empty set;
let a2 be Function-like finite Relation of a1,REAL;
cluster max+ a2 -> Function-like finite;
end;
:: RFUNCT_3:funcreg 2
registration
let a1 be non empty set;
let a2 be Function-like finite Relation of a1,REAL;
cluster max- a2 -> Function-like finite;
end;
:: RFUNCT_3:th 46
theorem
for b1, b2 being non empty set
for b3 being Function-like finite Relation of b1,REAL
for b4 being Function-like finite Relation of b2,REAL
st max+ b3,max+ b4 are_fiberwise_equipotent & max- b3,max- b4 are_fiberwise_equipotent
holds b3,b4 are_fiberwise_equipotent;
:: RFUNCT_3:th 47
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set holds
(max+ b2) | b3 = max+ (b2 | b3);
:: RFUNCT_3:th 48
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set holds
(max- b2) | b3 = max- (b2 | b3);
:: RFUNCT_3:th 49
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
st for b3 being Element of b1
st b3 in dom b2
holds 0 <= b2 . b3
holds max+ b2 = b2;
:: RFUNCT_3:th 50
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
st for b3 being Element of b1
st b3 in dom b2
holds b2 . b3 <= 0
holds max- b2 = - b2;
:: RFUNCT_3:th 51
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
b2 - 0 = b2;
:: RFUNCT_3:th 52
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
for b4 being set holds
(b2 | b4) - b3 = (b2 - b3) | b4;
:: RFUNCT_3:th 53
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3, b4 being Element of REAL holds
b2 " {b4 + b3} = (b2 - b3) " {b4};
:: RFUNCT_3:th 54
theorem
for b1, b2 being non empty set
for b3 being Function-like Relation of b1,REAL
for b4 being Function-like Relation of b2,REAL
for b5 being Element of REAL holds
b3,b4 are_fiberwise_equipotent
iff
b3 - b5,b4 - b5 are_fiberwise_equipotent;
:: RFUNCT_3:prednot 2 => RFUNCT_3:pred 2
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
pred A1 is_convex_on A2 means
a2 c= dom a1 &
(for b1 being Element of REAL
st 0 <= b1 & b1 <= 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) <= (b1 * (a1 . b2)) + ((1 - b1) * (a1 . b3)));
end;
:: RFUNCT_3:dfs 12
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be set;
To prove
a1 is_convex_on a2
it is sufficient to prove
thus a2 c= dom a1 &
(for b1 being Element of REAL
st 0 <= b1 & b1 <= 1
for b2, b3 being Element of REAL
st b2 in a2 &
b3 in a2 &
(b1 * b2) + ((1 - b1) * b3) in a2
holds a1 . ((b1 * b2) + ((1 - b1) * b3)) <= (b1 * (a1 . b2)) + ((1 - b1) * (a1 . b3)));
:: RFUNCT_3:def 13
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
b1 is_convex_on b2
iff
b2 c= dom b1 &
(for b3 being Element of REAL
st 0 <= b3 & b3 <= 1
for b4, b5 being Element of REAL
st b4 in b2 &
b5 in b2 &
(b3 * b4) + ((1 - b3) * b5) in b2
holds b1 . ((b3 * b4) + ((1 - b3) * b5)) <= (b3 * (b1 . b4)) + ((1 - b3) * (b1 . b5)));
:: RFUNCT_3:th 55
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
b3 is_convex_on [.b1,b2.]
iff
[.b1,b2.] c= dom b3 &
(for b4 being Element of REAL
st 0 <= b4 & b4 <= 1
for b5, b6 being Element of REAL
st b5 in [.b1,b2.] & b6 in [.b1,b2.]
holds b3 . ((b4 * b5) + ((1 - b4) * b6)) <= (b4 * (b3 . b5)) + ((1 - b4) * (b3 . b6)));
:: RFUNCT_3:th 56
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
b3 is_convex_on [.b1,b2.]
iff
[.b1,b2.] c= dom b3 &
(for b4, b5, b6 being Element of REAL
st b4 in [.b1,b2.] & b5 in [.b1,b2.] & b6 in [.b1,b2.] & b4 < b5 & b5 < b6
holds ((b3 . b4) - (b3 . b5)) / (b4 - b5) <= ((b3 . b5) - (b3 . b6)) / (b5 - b6));
:: RFUNCT_3:th 57
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being set
st b1 is_convex_on b2 & b3 c= b2
holds b1 is_convex_on b3;
:: RFUNCT_3:th 58
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
for b3 being Element of REAL holds
b1 is_convex_on b2
iff
b1 - b3 is_convex_on b2;
:: RFUNCT_3:th 59
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
for b3 being Element of REAL
st 0 < b3
holds b1 is_convex_on b2
iff
b3 (#) b1 is_convex_on b2;
:: RFUNCT_3:th 60
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
st b2 c= dom b1
holds 0 (#) b1 is_convex_on b2;
:: RFUNCT_3:th 61
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being set
st b1 is_convex_on b3 & b2 is_convex_on b3
holds b1 + b2 is_convex_on b3;
:: RFUNCT_3:th 62
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
for b3 being Element of REAL
st b1 is_convex_on b2
holds max+ (b1 - b3) is_convex_on b2;
:: RFUNCT_3:th 63
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set
st b1 is_convex_on b2
holds max+ b1 is_convex_on b2;
:: RFUNCT_3:th 64
theorem
id [#] REAL is_convex_on REAL;
:: RFUNCT_3:th 65
theorem
for b1 being Element of REAL holds
max+ ((id [#] REAL) - b1) is_convex_on REAL;
:: RFUNCT_3:funcnot 21 => RFUNCT_3:func 21
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,REAL;
let a3 be set;
assume dom (a2 | a3) is finite;
func FinS(A2,A3) -> non-increasing FinSequence of REAL means
a2 | a3,it are_fiberwise_equipotent;
end;
:: RFUNCT_3:def 14
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
st dom (b2 | b3) is finite
for b4 being non-increasing FinSequence of REAL holds
b4 = FinS(b2,b3)
iff
b2 | b3,b4 are_fiberwise_equipotent;
:: RFUNCT_3:th 66
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
st dom (b2 | b3) is finite
holds FinS(b2,dom (b2 | b3)) = FinS(b2,b3);
:: RFUNCT_3:th 67
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
st dom (b2 | b3) is finite
holds FinS(b2 | b3,b3) = FinS(b2,b3);
:: RFUNCT_3:funcnot 22 => RFUNCT_3:func 22
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,REAL;
let a3 be finite set;
redefine func a2 | a3 -> Function-like finite Relation of a1,REAL;
end;
:: RFUNCT_3:th 68
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being set
for b4 being Function-like Relation of b1,REAL
st b3 is finite & b2 in dom (b4 | b3)
holds (FinS(b4,b3 \ {b2})) ^ <*b4 . b2*>,b4 | b3 are_fiberwise_equipotent;
:: RFUNCT_3:th 69
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being set
for b4 being Function-like Relation of b1,REAL
st dom (b4 | b3) is finite & b2 in dom (b4 | b3)
holds (FinS(b4,b3 \ {b2})) ^ <*b4 . b2*>,b4 | b3 are_fiberwise_equipotent;
:: RFUNCT_3:th 70
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
for b4 being finite set
st b4 = dom (b2 | b3)
holds len FinS(b2,b3) = card b4;
:: RFUNCT_3:th 71
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
FinS(b2,{}) = <*> REAL;
:: RFUNCT_3:th 72
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of b1
st b3 in dom b2
holds FinS(b2,{b3}) = <*b2 . b3*>;
:: RFUNCT_3:th 73
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
for b4 being Element of b1
st dom (b2 | b3) is finite &
b4 in dom (b2 | b3) &
(FinS(b2,b3)) . len FinS(b2,b3) = b2 . b4
holds FinS(b2,b3) = (FinS(b2,b3 \ {b4})) ^ <*b2 . b4*>;
:: RFUNCT_3:th 74
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3, b4 being set
st dom (b2 | b3) is finite &
b4 c= b3 &
(for b5, b6 being Element of b1
st b5 in dom (b2 | b4) & b6 in dom (b2 | (b3 \ b4))
holds b2 . b6 <= b2 . b5)
holds FinS(b2,b3) = (FinS(b2,b4)) ^ FinS(b2,b3 \ b4);
:: RFUNCT_3:th 75
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
for b4 being set
for b5 being Element of b1
st dom (b2 | b4) is finite & b5 in dom (b2 | b4)
holds (FinS(b2 - b3,b4)) . len FinS(b2 - b3,b4) = (b2 - b3) . b5
iff
(FinS(b2,b4)) . len FinS(b2,b4) = b2 . b5;
:: RFUNCT_3:th 76
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of REAL
for b4 being set
for b5 being finite set
st b5 = dom (b2 | b4)
holds FinS(b2 - b3,b4) = (FinS(b2,b4)) - ((card b5) |-> b3);
:: RFUNCT_3:th 77
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
st dom (b2 | b3) is finite &
(for b4 being Element of b1
st b4 in dom (b2 | b3)
holds 0 <= b2 . b4)
holds FinS(max+ b2,b3) = FinS(b2,b3);
:: RFUNCT_3:th 78
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
for b4 being Element of REAL
for b5 being finite set
st b5 = dom (b2 | b3) & rng (b2 | b3) = {b4}
holds FinS(b2,b3) = (card b5) |-> b4;
:: RFUNCT_3:th 79
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3, b4 being set
st dom (b2 | (b3 \/ b4)) is finite & b3 misses b4
holds FinS(b2,b3 \/ b4),(FinS(b2,b3)) ^ FinS(b2,b4) are_fiberwise_equipotent;
:: RFUNCT_3:funcnot 23 => RFUNCT_3:func 23
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,REAL;
let a3 be set;
func Sum(A2,A3) -> Element of REAL equals
Sum FinS(a2,a3);
end;
:: RFUNCT_3:def 15
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set holds
Sum(b2,b3) = Sum FinS(b2,b3);
:: RFUNCT_3:th 80
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
for b4 being Element of REAL
st dom (b2 | b3) is finite
holds Sum(b4 (#) b2,b3) = b4 * Sum(b2,b3);
:: RFUNCT_3:th 81
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL
for b4 being set
for b5 being finite set
st b5 = dom (b2 | b4) & dom (b2 | b4) = dom (b3 | b4)
holds Sum(b2 + b3,b4) = (Sum(b2,b4)) + Sum(b3,b4);
:: RFUNCT_3:th 82
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL
for b4 being set
st dom (b2 | b4) is finite & dom (b2 | b4) = dom (b3 | b4)
holds Sum(b2 - b3,b4) = (Sum(b2,b4)) - Sum(b3,b4);
:: RFUNCT_3:th 83
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being set
for b4 being Element of REAL
for b5 being finite set
st b5 = dom (b2 | b3)
holds Sum(b2 - b4,b3) = (Sum(b2,b3)) - (b4 * card b5);
:: RFUNCT_3:th 84
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
Sum(b2,{}) = 0;
:: RFUNCT_3:th 85
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3 being Element of b1
st b3 in dom b2
holds Sum(b2,{b3}) = b2 . b3;
:: RFUNCT_3:th 86
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3, b4 being set
st dom (b2 | (b3 \/ b4)) is finite & b3 misses b4
holds Sum(b2,b3 \/ b4) = (Sum(b2,b3)) + Sum(b2,b4);
:: RFUNCT_3:th 87
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL
for b3, b4 being set
st dom (b2 | (b3 \/ b4)) is finite & dom (b2 | b3) misses dom (b2 | b4)
holds Sum(b2,b3 \/ b4) = (Sum(b2,b3)) + Sum(b2,b4);