:: Properties of Partial Functions from a Domain to the Set of Real Numbers
:: by Jaros{\l}aw Kotowicz and Yuji Sakai
::
:: Received March 15, 1993
:: Copyright (c) 1993-2011 Association of Mizar Users

begin

definition

let n, m be Element of NAT ;
:: original: min
redefine func min (n,m) -> Element of NAT ;
coherence
min (n,m) is Element of NAT by FINSEQ_2:1;

end;

definition

let r be real number ;
func max+ r -> Real equals :: RFUNCT_3:def 1
max (r,0);
correctness
coherence
max (r,0) is Real;
by XREAL_0:def 1;
func max- r -> Real equals :: RFUNCT_3:def 2
max ((- r),0);
correctness
coherence
max ((- r),0) is Real;
by XREAL_0:def 1;

end;

:: deftheorem defines max+ RFUNCT_3:def 1 :
for r being real number holds max+ r = max (r,0);

:: deftheorem defines max- RFUNCT_3:def 2 :
for r being real number holds max- r = max ((- r),0);

theorem Th1: :: RFUNCT_3:1

for r being real number holds r = (max+ r) - (max- r)

proof end;

theorem Th2: :: RFUNCT_3:2

for r being real number holds abs r = (max+ r) + (max- r)

proof end;

theorem Th3: :: RFUNCT_3:3

for r being real number holds 2 * (max+ r) = r + (abs r)

proof end;

theorem Th4: :: RFUNCT_3:4

for r, s being real number st 0 <= r holds
max+ (r * s) = r * (max+ s)

proof end;

theorem Th5: :: RFUNCT_3:5

for r, s being real number holds max+ (r + s) <= (max+ r) + (max+ s)

proof end;

Lm1: for n being Element of NAT
for D being non empty set
for f being FinSequence of D st len f <= n holds
f | n = f

proof end;

Lm2: for f being Function
for x being set st not x in rng f holds
f " {x} = {}

proof end;

begin

theorem :: RFUNCT_3:6

canceled;

theorem :: RFUNCT_3:7

canceled;

theorem Th8: :: RFUNCT_3:8

for D being non empty set
for F being PartFunc of D,REAL
for r, s being real number st r <> 0 holds
F " {(s / r)} = (r (#) F) " {s}

proof end;

theorem Th9: :: RFUNCT_3:9

for D being non empty set
for F being PartFunc of D,REAL holds (0 (#) F) " {0} = dom F

proof end;

theorem Th10: :: RFUNCT_3:10

for D being non empty set
for F being PartFunc of D,REAL
for r being Real st 0 < r holds
(abs F) " {r} = F " {(- r),r}

proof end;

theorem Th11: :: RFUNCT_3:11

for D being non empty set
for F being PartFunc of D,REAL holds (abs F) " {0} = F " {0}

proof end;

theorem Th12: :: RFUNCT_3:12

for D being non empty set
for F being PartFunc of D,REAL
for r being Real st r < 0 holds
(abs F) " {r} = {}

proof end;

theorem Th13: :: RFUNCT_3:13

for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL
for r being Real st r <> 0 holds
( F,G are_fiberwise_equipotent iff r (#) F,r (#) G are_fiberwise_equipotent )

proof end;

theorem :: RFUNCT_3:14

for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL holds
( F,G are_fiberwise_equipotent iff - F, - G are_fiberwise_equipotent )

proof end;

theorem :: RFUNCT_3:15

for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
abs F, abs G are_fiberwise_equipotent

proof end;

definition

let X, Y be set ;
mode PartFunc-set of X,Y -> set means :Def3: :: RFUNCT_3:def 3
for x being Element of it holds x is PartFunc of X,Y;
existence
ex b₁ being set st
for x being Element of b₁ holds x is PartFunc of X,Y

proof end;

end;

:: deftheorem Def3 defines PartFunc-set RFUNCT_3:def 3 :
for X, Y being set
for b₃ being set holds
( b₃ is PartFunc-set of X,Y iff for x being Element of b₃ holds x is PartFunc of X,Y );

registration

let X, Y be set ;
cluster non empty PartFunc-set of X,Y;
existence
not for b₁ being PartFunc-set of X,Y holds b₁ is empty

proof end;

end;

definition

let X, Y be set ;
mode PFUNC_DOMAIN of X,Y is non empty PartFunc-set of X,Y;

end;

definition

let X, Y be set ;
:: original: PFuncs
redefine func PFuncs (X,Y) -> PartFunc-set of X,Y;
coherence
PFuncs (X,Y) is PartFunc-set of X,Y

proof end;

let P be non empty PartFunc-set of X,Y;
:: original: Element
redefine mode Element of P -> PartFunc of X,Y;
coherence
for b₁ being Element of P holds b₁ is PartFunc of X,Y by Def3;

end;

definition

let D, C be non empty set ;
let X be Subset of D;
let c be Element of C;
:: original: -->
redefine func X --> c -> Element of PFuncs (D,C);
coherence
X --> c is Element of PFuncs (D,C)

proof end;

end;

registration

let D be non empty set ;
let E be real-membered set ;
cluster -> real-valued Element of PFuncs (D,E);
coherence
for b₁ being Element of PFuncs (D,E) holds b₁ is real-valued ;

end;

definition

let D be non empty set ;
let E be real-membered set ;
let F1, F2 be Element of PFuncs (D,E);
:: original: +
redefine func F1 + F2 -> Element of PFuncs (D,REAL);
coherence
F1 + F2 is Element of PFuncs (D,REAL)

proof end;

:: original: -
redefine func F1 - F2 -> Element of PFuncs (D,REAL);
coherence
F1 - F2 is Element of PFuncs (D,REAL)

proof end;

:: original: (#)
redefine func F1 (#) F2 -> Element of PFuncs (D,REAL);
coherence
F1 (#) F2 is Element of PFuncs (D,REAL)

proof end;

:: original: /
redefine func F1 / F2 -> Element of PFuncs (D,REAL);
coherence
F1 / F2 is Element of PFuncs (D,REAL)

proof end;

end;

definition

let D be non empty set ;
let E be real-membered set ;
let F be Element of PFuncs (D,E);
:: original: |.
redefine func abs F -> Element of PFuncs (D,REAL);
coherence
|.F.| is Element of PFuncs (D,REAL)

proof end;

:: original: -
redefine func - F -> Element of PFuncs (D,REAL);
coherence
- F is Element of PFuncs (D,REAL)

proof end;

:: original: ^
redefine func F ^ -> Element of PFuncs (D,REAL);
coherence
F ^ is Element of PFuncs (D,REAL)

proof end;

end;

definition

let D be non empty set ;
let E be real-membered set ;
let F be Element of PFuncs (D,E);
let r be real number ;
:: original: (#)
redefine func r (#) F -> Element of PFuncs (D,REAL);
coherence
r (#) F is Element of PFuncs (D,REAL)

proof end;

end;

definition

let D be non empty set ;
func addpfunc D -> BinOp of (PFuncs (D,REAL)) means :Def4: :: RFUNCT_3:def 4
for F1, F2 being Element of PFuncs (D,REAL) holds it . (F1,F2) = F1 + F2;
existence
ex b₁ being BinOp of (PFuncs (D,REAL)) st
for F1, F2 being Element of PFuncs (D,REAL) holds b₁ . (F1,F2) = F1 + F2

proof end;

uniqueness
for b₁, b₂ being BinOp of (PFuncs (D,REAL)) st ( for F1, F2 being Element of PFuncs (D,REAL) holds b₁ . (F1,F2) = F1 + F2 ) & ( for F1, F2 being Element of PFuncs (D,REAL) holds b₂ . (F1,F2) = F1 + F2 ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines addpfunc RFUNCT_3:def 4 :
for D being non empty set
for b₂ being BinOp of (PFuncs (D,REAL)) holds
( b₂ = addpfunc D iff for F1, F2 being Element of PFuncs (D,REAL) holds b₂ . (F1,F2) = F1 + F2 );

theorem Th16: :: RFUNCT_3:16

for D being non empty set holds addpfunc D is commutative

proof end;

theorem Th17: :: RFUNCT_3:17

for D being non empty set holds addpfunc D is associative

proof end;

theorem Th18: :: RFUNCT_3:18

for D being non empty set holds ([#] D) --> 0 is_a_unity_wrt addpfunc D

proof end;

theorem Th19: :: RFUNCT_3:19

for D being non empty set holds the_unity_wrt (addpfunc D) = ([#] D) --> 0

proof end;

theorem Th20: :: RFUNCT_3:20

for D being non empty set holds addpfunc D is having_a_unity

proof end;

definition

let D be non empty set ;
let f be FinSequence of PFuncs (D,REAL);
func Sum f -> Element of PFuncs (D,REAL) equals :: RFUNCT_3:def 5
(addpfunc D) $$ f;
correctness
coherence
(addpfunc D) $$ f is Element of PFuncs (D,REAL);
;

end;

:: deftheorem defines Sum RFUNCT_3:def 5 :
for D being non empty set
for f being FinSequence of PFuncs (D,REAL) holds Sum f = (addpfunc D) $$ f;

theorem Th21: :: RFUNCT_3:21

for D being non empty set holds Sum (<*> (PFuncs (D,REAL))) = ([#] D) --> 0

proof end;

theorem :: RFUNCT_3:22

canceled;

theorem Th23: :: RFUNCT_3:23

for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for G being Element of PFuncs (D,REAL) holds Sum (f ^ <*G*>) = (Sum f) + G

proof end;

theorem Th24: :: RFUNCT_3:24

for D being non empty set
for f1, f2 being FinSequence of PFuncs (D,REAL) holds Sum (f1 ^ f2) = (Sum f1) + (Sum f2)

proof end;

theorem :: RFUNCT_3:25

for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for G being Element of PFuncs (D,REAL) holds Sum (<*G*> ^ f) = G + (Sum f)

proof end;

theorem Th26: :: RFUNCT_3:26

for D being non empty set
for G1, G2 being Element of PFuncs (D,REAL) holds Sum <*G1,G2*> = G1 + G2

proof end;

theorem :: RFUNCT_3:27

for D being non empty set
for G1, G2, G3 being Element of PFuncs (D,REAL) holds Sum <*G1,G2,G3*> = (G1 + G2) + G3

proof end;

theorem :: RFUNCT_3:28

for D being non empty set
for f, g being FinSequence of PFuncs (D,REAL) st f,g are_fiberwise_equipotent holds
Sum f = Sum g

proof end;

definition

let D be non empty set ;
let f be FinSequence;
func CHI (f,D) -> FinSequence of PFuncs (D,REAL) means :Def6: :: RFUNCT_3:def 6
( len it = len f & ( for n being Element of NAT st n in dom it holds
it . n = chi ((f . n),D) ) );
existence
ex b₁ being FinSequence of PFuncs (D,REAL) st
( len b₁ = len f & ( for n being Element of NAT st n in dom b₁ holds
b₁ . n = chi ((f . n),D) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of PFuncs (D,REAL) st len b₁ = len f & ( for n being Element of NAT st n in dom b₁ holds
b₁ . n = chi ((f . n),D) ) & len b₂ = len f & ( for n being Element of NAT st n in dom b₂ holds
b₂ . n = chi ((f . n),D) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines CHI RFUNCT_3:def 6 :
for D being non empty set
for f being FinSequence
for b₃ being FinSequence of PFuncs (D,REAL) holds
( b₃ = CHI (f,D) iff ( len b₃ = len f & ( for n being Element of NAT st n in dom b₃ holds
b₃ . n = chi ((f . n),D) ) ) );

definition

let D be non empty set ;
let f be FinSequence of PFuncs (D,REAL);
let R be FinSequence of REAL ;
func R (#) f -> FinSequence of PFuncs (D,REAL) means :Def7: :: RFUNCT_3:def 7
( len it = min ((len R),(len f)) & ( for n being Element of NAT st n in dom it holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
it . n = r (#) F ) );
existence
ex b₁ being FinSequence of PFuncs (D,REAL) st
( len b₁ = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b₁ holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b₁ . n = r (#) F ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of PFuncs (D,REAL) st len b₁ = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b₁ holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b₁ . n = r (#) F ) & len b₂ = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b₂ holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b₂ . n = r (#) F ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines (#) RFUNCT_3:def 7 :
for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for R being FinSequence of REAL
for b₄ being FinSequence of PFuncs (D,REAL) holds
( b₄ = R (#) f iff ( len b₄ = min ((len R),(len f)) & ( for n being Element of NAT st n in dom b₄ holds
for F being PartFunc of D,REAL
for r being Real st r = R . n & F = f . n holds
b₄ . n = r (#) F ) ) );

definition

let D be non empty set ;
let f be FinSequence of PFuncs (D,REAL);
let d be Element of D;
func f # d -> FinSequence of REAL means :Def8: :: RFUNCT_3:def 8
( len it = len f & ( for n being Element of NAT st n in dom it holds
it . n = (f . n) . d ) );
existence
ex b₁ being FinSequence of REAL st
( len b₁ = len f & ( for n being Element of NAT st n in dom b₁ holds
b₁ . n = (f . n) . d ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of REAL st len b₁ = len f & ( for n being Element of NAT st n in dom b₁ holds
b₁ . n = (f . n) . d ) & len b₂ = len f & ( for n being Element of NAT st n in dom b₂ holds
b₂ . n = (f . n) . d ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines # RFUNCT_3:def 8 :
for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for d being Element of D
for b₄ being FinSequence of REAL holds
( b₄ = f # d iff ( len b₄ = len f & ( for n being Element of NAT st n in dom b₄ holds
b₄ . n = (f . n) . d ) ) );

definition

let D, C be non empty set ;
let f be FinSequence of PFuncs (D,C);
let d be Element of D;
pred d is_common_for_dom f means :Def9: :: RFUNCT_3:def 9
for n being Element of NAT st n in dom f holds
d in dom (f . n);

end;

:: deftheorem Def9 defines is_common_for_dom RFUNCT_3:def 9 :
for D, C being non empty set
for f being FinSequence of PFuncs (D,C)
for d being Element of D holds
( d is_common_for_dom f iff for n being Element of NAT st n in dom f holds
d in dom (f . n) );

theorem Th29: :: RFUNCT_3:29

for D, C being non empty set
for f being FinSequence of PFuncs (D,C)
for d being Element of D
for n being Element of NAT st d is_common_for_dom f & n <> 0 holds
d is_common_for_dom f | n

proof end;

theorem :: RFUNCT_3:30

for D, C being non empty set
for f being FinSequence of PFuncs (D,C)
for d being Element of D
for n being Element of NAT st d is_common_for_dom f holds
d is_common_for_dom f /^ n

proof end;

theorem Th31: :: RFUNCT_3:31

for D being non empty set
for d being Element of D
for f being FinSequence of PFuncs (D,REAL) st len f <> 0 holds
( d is_common_for_dom f iff d in dom (Sum f) )

proof end;

theorem Th32: :: RFUNCT_3:32

for D being non empty set
for f being FinSequence of PFuncs (D,REAL)
for d being Element of D
for n being Element of NAT holds (f | n) # d = (f # d) | n

proof end;

theorem Th33: :: RFUNCT_3:33

for D being non empty set
for f being FinSequence
for d being Element of D holds d is_common_for_dom CHI (f,D)

proof end;

theorem Th34: :: RFUNCT_3:34

for D being non empty set
for d being Element of D
for f being FinSequence of PFuncs (D,REAL)
for R being FinSequence of REAL st d is_common_for_dom f holds
d is_common_for_dom R (#) f

proof end;

theorem :: RFUNCT_3:35

for D being non empty set
for f being FinSequence
for R being FinSequence of REAL
for d being Element of D holds d is_common_for_dom R (#) (CHI (f,D)) by Th33, Th34;

theorem :: RFUNCT_3:36

for D being non empty set
for d being Element of D
for f being FinSequence of PFuncs (D,REAL) st d is_common_for_dom f holds
(Sum f) . d = Sum (f # d)

proof end;

definition

let D be non empty set ;
let F be PartFunc of D,REAL;
func max+ F -> PartFunc of D,REAL means :Def10: :: RFUNCT_3:def 10
( dom it = dom F & ( for d being Element of D st d in dom it holds
it . d = max+ (F . d) ) );
existence
ex b₁ being PartFunc of D,REAL st
( dom b₁ = dom F & ( for d being Element of D st d in dom b₁ holds
b₁ . d = max+ (F . d) ) )

proof end;

uniqueness
for b₁, b₂ being PartFunc of D,REAL st dom b₁ = dom F & ( for d being Element of D st d in dom b₁ holds
b₁ . d = max+ (F . d) ) & dom b₂ = dom F & ( for d being Element of D st d in dom b₂ holds
b₂ . d = max+ (F . d) ) holds
b₁ = b₂

proof end;

func max- F -> PartFunc of D,REAL means :Def11: :: RFUNCT_3:def 11
( dom it = dom F & ( for d being Element of D st d in dom it holds
it . d = max- (F . d) ) );
existence
ex b₁ being PartFunc of D,REAL st
( dom b₁ = dom F & ( for d being Element of D st d in dom b₁ holds
b₁ . d = max- (F . d) ) )

proof end;

uniqueness
for b₁, b₂ being PartFunc of D,REAL st dom b₁ = dom F & ( for d being Element of D st d in dom b₁ holds
b₁ . d = max- (F . d) ) & dom b₂ = dom F & ( for d being Element of D st d in dom b₂ holds
b₂ . d = max- (F . d) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines max+ RFUNCT_3:def 10 :
for D being non empty set
for F, b₃ being PartFunc of D,REAL holds
( b₃ = max+ F iff ( dom b₃ = dom F & ( for d being Element of D st d in dom b₃ holds
b₃ . d = max+ (F . d) ) ) );

:: deftheorem Def11 defines max- RFUNCT_3:def 11 :
for D being non empty set
for F, b₃ being PartFunc of D,REAL holds
( b₃ = max- F iff ( dom b₃ = dom F & ( for d being Element of D st d in dom b₃ holds
b₃ . d = max- (F . d) ) ) );

theorem :: RFUNCT_3:37

for D being non empty set
for F being PartFunc of D,REAL holds
( F = (max+ F) - (max- F) & abs F = (max+ F) + (max- F) & 2 (#) (max+ F) = F + (abs F) )

proof end;

theorem Th38: :: RFUNCT_3:38

for D being non empty set
for F being PartFunc of D,REAL
for r being Real st 0 < r holds
F " {r} = (max+ F) " {r}

proof end;

theorem Th39: :: RFUNCT_3:39

for D being non empty set
for F being PartFunc of D,REAL holds F " (left_closed_halfline 0) = (max+ F) " {0}

proof end;

theorem Th40: :: RFUNCT_3:40

for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D holds 0 <= (max+ F) . d

proof end;

theorem Th41: :: RFUNCT_3:41

for D being non empty set
for F being PartFunc of D,REAL
for r being Real st 0 < r holds
F " {(- r)} = (max- F) " {r}

proof end;

theorem Th42: :: RFUNCT_3:42

for D being non empty set
for F being PartFunc of D,REAL holds F " (right_closed_halfline 0) = (max- F) " {0}

proof end;

theorem Th43: :: RFUNCT_3:43

for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D holds 0 <= (max- F) . d

proof end;

theorem :: RFUNCT_3:44

for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max+ F, max+ G are_fiberwise_equipotent

proof end;

theorem :: RFUNCT_3:45

for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL st F,G are_fiberwise_equipotent holds
max- F, max- G are_fiberwise_equipotent

proof end;

registration

let D be non empty set ;
let F be finite PartFunc of D,REAL;
cluster max+ F -> finite ;
coherence
max+ F is finite

proof end;

cluster max- F -> finite ;
coherence
max- F is finite

proof end;

end;

theorem :: RFUNCT_3:46

for D, C being non empty set
for F being finite PartFunc of D,REAL
for G being finite PartFunc of C,REAL st max+ F, max+ G are_fiberwise_equipotent & max- F, max- G are_fiberwise_equipotent holds
F,G are_fiberwise_equipotent

proof end;

theorem Th47: :: RFUNCT_3:47

for D being non empty set
for F being PartFunc of D,REAL
for X being set holds (max+ F) | X = max+ (F | X)

proof end;

theorem :: RFUNCT_3:48

for D being non empty set
for F being PartFunc of D,REAL
for X being set holds (max- F) | X = max- (F | X)

proof end;

theorem Th49: :: RFUNCT_3:49

for D being non empty set
for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds
F . d >= 0 ) holds
max+ F = F

proof end;

theorem :: RFUNCT_3:50

for D being non empty set
for F being PartFunc of D,REAL st ( for d being Element of D st d in dom F holds
F . d <= 0 ) holds
max- F = - F

proof end;

theorem Th51: :: RFUNCT_3:51

for D being non empty set
for F being PartFunc of D,REAL holds F - 0 = F

proof end;

theorem :: RFUNCT_3:52

for D being non empty set
for F being PartFunc of D,REAL
for r being Real
for X being set holds (F | X) - r = (F - r) | X

proof end;

theorem Th53: :: RFUNCT_3:53

for D being non empty set
for F being PartFunc of D,REAL
for r, s being Real holds F " {(s + r)} = (F - r) " {s}

proof end;

theorem :: RFUNCT_3:54

for D, C being non empty set
for F being PartFunc of D,REAL
for G being PartFunc of C,REAL
for r being Real holds
( F,G are_fiberwise_equipotent iff F - r,G - r are_fiberwise_equipotent )

proof end;

definition

canceled;
let F be PartFunc of REAL,REAL;
let X be set ;
pred F is_convex_on X means :Def13: :: RFUNCT_3:def 13
( X c= dom F & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) );

end;

:: deftheorem RFUNCT_3:def 12 :
canceled;

:: deftheorem Def13 defines is_convex_on RFUNCT_3:def 13 :
for F being PartFunc of REAL,REAL
for X being set holds
( F is_convex_on X iff ( X c= dom F & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in X & s in X & (p * r) + ((1 - p) * s) in X holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) );

theorem Th55: :: RFUNCT_3:55

for a, b being Real
for F being PartFunc of REAL,REAL holds
( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for p being Real st 0 <= p & p <= 1 holds
for r, s being Real st r in [.a,b.] & s in [.a,b.] holds
F . ((p * r) + ((1 - p) * s)) <= (p * (F . r)) + ((1 - p) * (F . s)) ) ) )

proof end;

theorem :: RFUNCT_3:56

for a, b being Real
for F being PartFunc of REAL,REAL holds
( F is_convex_on [.a,b.] iff ( [.a,b.] c= dom F & ( for x1, x2, x3 being Real st x1 in [.a,b.] & x2 in [.a,b.] & x3 in [.a,b.] & x1 < x2 & x2 < x3 holds
((F . x1) - (F . x2)) / (x1 - x2) <= ((F . x2) - (F . x3)) / (x2 - x3) ) ) )

proof end;

theorem :: RFUNCT_3:57

for F being PartFunc of REAL,REAL
for X, Y being set st F is_convex_on X & Y c= X holds
F is_convex_on Y

proof end;

theorem :: RFUNCT_3:58

for F being PartFunc of REAL,REAL
for X being set
for r being Real holds
( F is_convex_on X iff F - r is_convex_on X )

proof end;

theorem :: RFUNCT_3:59

for F being PartFunc of REAL,REAL
for X being set
for r being Real st 0 < r holds
( F is_convex_on X iff r (#) F is_convex_on X )

proof end;

theorem :: RFUNCT_3:60

for F being PartFunc of REAL,REAL
for X being set st X c= dom F holds
0 (#) F is_convex_on X

proof end;

theorem :: RFUNCT_3:61

for F, G being PartFunc of REAL,REAL
for X being set st F is_convex_on X & G is_convex_on X holds
F + G is_convex_on X

proof end;

theorem Th62: :: RFUNCT_3:62

for F being PartFunc of REAL,REAL
for X being set
for r being Real st F is_convex_on X holds
max+ (F - r) is_convex_on X

proof end;

theorem :: RFUNCT_3:63

for F being PartFunc of REAL,REAL
for X being set st F is_convex_on X holds
max+ F is_convex_on X

proof end;

theorem Th64: :: RFUNCT_3:64

id ([#] REAL) is_convex_on REAL

proof end;

theorem :: RFUNCT_3:65

for r being Real holds max+ ((id ([#] REAL)) - r) is_convex_on REAL by Th62, Th64;

definition

let D be non empty set ;
let F be PartFunc of D,REAL;
let X be set ;
assume A1: dom (F | X) is finite ;
func FinS (F,X) -> non-increasing FinSequence of REAL means :Def14: :: RFUNCT_3:def 14
F | X,it are_fiberwise_equipotent ;
existence
ex b₁ being non-increasing FinSequence of REAL st F | X,b₁ are_fiberwise_equipotent

proof end;

uniqueness
for b₁, b₂ being non-increasing FinSequence of REAL st F | X,b₁ are_fiberwise_equipotent & F | X,b₂ are_fiberwise_equipotent holds
b₁ = b₂ by CLASSES1:84, RFINSEQ:36;

end;

:: deftheorem Def14 defines FinS RFUNCT_3:def 14 :
for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
for b₄ being non-increasing FinSequence of REAL holds
( b₄ = FinS (F,X) iff F | X,b₄ are_fiberwise_equipotent );

theorem Th66: :: RFUNCT_3:66

for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
FinS (F,(dom (F | X))) = FinS (F,X)

proof end;

theorem Th67: :: RFUNCT_3:67

for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite holds
FinS ((F | X),X) = FinS (F,X)

proof end;

theorem Th68: :: RFUNCT_3:68

for D being non empty set
for d being Element of D
for X being set
for F being PartFunc of D,REAL st X is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent

proof end;

theorem Th69: :: RFUNCT_3:69

for D being non empty set
for d being Element of D
for X being set
for F being PartFunc of D,REAL st dom (F | X) is finite & d in dom (F | X) holds
(FinS (F,(X \ {d}))) ^ <*(F . d)*>,F | X are_fiberwise_equipotent

proof end;

theorem Th70: :: RFUNCT_3:70

for D being non empty set
for F being PartFunc of D,REAL
for X being set
for Y being finite set st Y = dom (F | X) holds
len (FinS (F,X)) = card Y

proof end;

theorem Th71: :: RFUNCT_3:71

for D being non empty set
for F being PartFunc of D,REAL holds FinS (F,{}) = <*> REAL

proof end;

theorem Th72: :: RFUNCT_3:72

for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D st d in dom F holds
FinS (F,{d}) = <*(F . d)*>

proof end;

theorem Th73: :: RFUNCT_3:73

for D being non empty set
for F being PartFunc of D,REAL
for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) & (FinS (F,X)) . (len (FinS (F,X))) = F . d holds
FinS (F,X) = (FinS (F,(X \ {d}))) ^ <*(F . d)*>

proof end;

defpred S₁[ Element of NAT ] means for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set
for Z being finite set st Z = dom (F | Y) & dom (F | X) is finite & Y c= X & $1 = card Z & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)));

Lm3: S₁[ 0 ]

proof end;

Lm4: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof end;

theorem :: RFUNCT_3:74

for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | X) is finite & Y c= X & ( for d1, d2 being Element of D st d1 in dom (F | Y) & d2 in dom (F | (X \ Y)) holds
F . d1 >= F . d2 ) holds
FinS (F,X) = (FinS (F,Y)) ^ (FinS (F,(X \ Y)))

proof end;

theorem Th75: :: RFUNCT_3:75

for D being non empty set
for F being PartFunc of D,REAL
for r being Real
for X being set
for d being Element of D st dom (F | X) is finite & d in dom (F | X) holds
( (FinS ((F - r),X)) . (len (FinS ((F - r),X))) = (F - r) . d iff (FinS (F,X)) . (len (FinS (F,X))) = F . d )

proof end;

theorem Th76: :: RFUNCT_3:76

for D being non empty set
for F being PartFunc of D,REAL
for r being Real
for X being set
for Z being finite set st Z = dom (F | X) holds
FinS ((F - r),X) = (FinS (F,X)) - ((card Z) |-> r)

proof end;

theorem :: RFUNCT_3:77

for D being non empty set
for F being PartFunc of D,REAL
for X being set st dom (F | X) is finite & ( for d being Element of D st d in dom (F | X) holds
F . d >= 0 ) holds
FinS ((max+ F),X) = FinS (F,X)

proof end;

theorem :: RFUNCT_3:78

for D being non empty set
for F being PartFunc of D,REAL
for X being set
for r being Real
for Z being finite set st Z = dom (F | X) & rng (F | X) = {r} holds
FinS (F,X) = (card Z) |-> r

proof end;

theorem Th79: :: RFUNCT_3:79

for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
FinS (F,(X \/ Y)),(FinS (F,X)) ^ (FinS (F,Y)) are_fiberwise_equipotent

proof end;

definition

let D be non empty set ;
let F be PartFunc of D,REAL;
let X be set ;
func Sum (F,X) -> Real equals :: RFUNCT_3:def 15
Sum (FinS (F,X));
correctness
coherence
Sum (FinS (F,X)) is Real;
;

end;

:: deftheorem defines Sum RFUNCT_3:def 15 :
for D being non empty set
for F being PartFunc of D,REAL
for X being set holds Sum (F,X) = Sum (FinS (F,X));

theorem Th80: :: RFUNCT_3:80

for D being non empty set
for F being PartFunc of D,REAL
for X being set
for r being Real st dom (F | X) is finite holds
Sum ((r (#) F),X) = r * (Sum (F,X))

proof end;

theorem Th81: :: RFUNCT_3:81

for D being non empty set
for F, G being PartFunc of D,REAL
for X being set
for Y being finite set st Y = dom (F | X) & dom (F | X) = dom (G | X) holds
Sum ((F + G),X) = (Sum (F,X)) + (Sum (G,X))

proof end;

theorem :: RFUNCT_3:82

for D being non empty set
for F, G being PartFunc of D,REAL
for X being set st dom (F | X) is finite & dom (F | X) = dom (G | X) holds
Sum ((F - G),X) = (Sum (F,X)) - (Sum (G,X))

proof end;

theorem :: RFUNCT_3:83

for D being non empty set
for F being PartFunc of D,REAL
for X being set
for r being Real
for Y being finite set st Y = dom (F | X) holds
Sum ((F - r),X) = (Sum (F,X)) - (r * (card Y))

proof end;

theorem :: RFUNCT_3:84

for D being non empty set
for F being PartFunc of D,REAL holds Sum (F,{}) = 0 by Th71, RVSUM_1:102;

theorem :: RFUNCT_3:85

for D being non empty set
for F being PartFunc of D,REAL
for d being Element of D st d in dom F holds
Sum (F,{d}) = F . d

proof end;

theorem :: RFUNCT_3:86

for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & X misses Y holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))

proof end;

theorem :: RFUNCT_3:87

for D being non empty set
for F being PartFunc of D,REAL
for X, Y being set st dom (F | (X \/ Y)) is finite & dom (F | X) misses dom (F | Y) holds
Sum (F,(X \/ Y)) = (Sum (F,X)) + (Sum (F,Y))

proof end;