INTEGRA1: The Definition of Riemann Definite Integral and some Related Lemmas

:: The Definition of Riemann Definite Integral and some Related Lemmas
:: by Noboru Endou and Artur Korni{\l}owicz
::
:: Received March 13, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

registration

cluster closed_interval -> compact for Element of bool REAL;

proof end;

end;

theorem :: INTEGRA1:1

canceled;

theorem :: INTEGRA1:2

canceled;

::$CT 2

theorem Th1: :: INTEGRA1:3

for A being non empty closed_interval Subset of REAL holds
( A is bounded_below & A is bounded_above )

proof end;

registration

cluster non empty closed_interval -> real-bounded for Element of bool REAL;

coherence by Th1;

end;

theorem Th2: :: INTEGRA1:4

for A being non empty closed_interval Subset of REAL holds A = [.(lower_bound A),(upper_bound A).]

proof end;

theorem Th3: :: INTEGRA1:5

for A being non empty closed_interval Subset of REAL
for a1, a2, b1, b2 being Real st A = [.a1,b1.] & A = [.a2,b2.] holds
( a1 = a2 & b1 = b2 )

proof end;

definition

let A be non empty compact Subset of REAL;

mode Division of A -> non empty increasing FinSequence of REAL means :Def1: :: INTEGRA1:def 2
( rng it c= A & it . (len it) = upper_bound A );

proof end;

end;

:: deftheorem INTEGRA1:def 1 :

:: deftheorem Def1 defines Division INTEGRA1:def 2 :

definition

let A be non empty compact Subset of REAL;

func divs A -> set means :Def2: :: INTEGRA1:def 3
for x1 being set holds
( x1 in it iff x1 is Division of A );

proof end;

proof end;

end;

:: deftheorem Def2 defines divs INTEGRA1:def 3 :

registration

let A be non empty compact Subset of REAL;

cluster divs A -> non empty ;

proof end;

end;

registration

let A be non empty compact Subset of REAL;

cluster -> Relation-like Function-like for Element of divs A;

coherence by Def2;

end;

registration

let A be non empty compact Subset of REAL;

cluster -> FinSequence-like real-valued for Element of divs A;

coherence by Def2;

end;

theorem Th4: :: INTEGRA1:6

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
D . i in A

proof end;

theorem Th5: :: INTEGRA1:7

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & i <> 1 holds
( i - 1 in dom D & D . (i - 1) in A & i - 1 in NAT )

proof end;

definition

let A be non empty closed_interval Subset of REAL;
let D be Division of A;
let i be Nat;
assume A1: i in dom D ;

func divset (D,i) -> non empty closed_interval Subset of REAL means :Def3: :: INTEGRA1:def 4
( lower_bound it = lower_bound A & upper_bound it = D . i ) if i = 1
otherwise ( lower_bound it = D . (i - 1) & upper_bound it = D . i );

proof end;

proof end;

end;

:: deftheorem Def3 defines divset INTEGRA1:def 4 :

theorem Th6: :: INTEGRA1:8

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
divset (D,i) c= A

proof end;

definition

let A be Subset of REAL;

func vol A -> Real equals :: INTEGRA1:def 5
(upper_bound A) - (lower_bound A);

end;

:: deftheorem defines vol INTEGRA1:def 5 :

theorem :: INTEGRA1:9

for A being non empty real-bounded Subset of REAL holds 0 <= vol A by SEQ_4:11, XREAL_1:48;

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
let D be Division of A;

func upper_volume (f,D) -> FinSequence of REAL means :Def5: :: INTEGRA1:def 6
( len it = len D & ( for i being Nat st i in dom D holds
it . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) );

proof end;

proof end;

func lower_volume (f,D) -> FinSequence of REAL means :Def6: :: INTEGRA1:def 7
( len it = len D & ( for i being Nat st i in dom D holds
it . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) );

proof end;

proof end;

end;

:: deftheorem Def5 defines upper_volume INTEGRA1:def 6 :

:: deftheorem Def6 defines lower_volume INTEGRA1:def 7 :

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
let D be Division of A;

func upper_sum (f,D) -> Real equals :: INTEGRA1:def 8
Sum (upper_volume (f,D));

func lower_sum (f,D) -> Real equals :: INTEGRA1:def 9
Sum (lower_volume (f,D));

end;

:: deftheorem defines upper_sum INTEGRA1:def 8 :

:: deftheorem defines lower_sum INTEGRA1:def 9 :

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
set S = divs A;

func upper_sum_set f -> Function of (divs A),REAL means :Def9: :: INTEGRA1:def 10
for D being Division of A holds it . D = upper_sum (f,D);

proof end;

proof end;

func lower_sum_set f -> Function of (divs A),REAL means :Def10: :: INTEGRA1:def 11
for D being Division of A holds it . D = lower_sum (f,D);

proof end;

proof end;

end;

:: deftheorem Def9 defines upper_sum_set INTEGRA1:def 10 :

:: deftheorem Def10 defines lower_sum_set INTEGRA1:def 11 :

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;

attr f is upper_integrable means :: INTEGRA1:def 12
rng (upper_sum_set f) is bounded_below ;

attr f is lower_integrable means :: INTEGRA1:def 13
rng (lower_sum_set f) is bounded_above ;

end;

:: deftheorem defines upper_integrable INTEGRA1:def 12 :

:: deftheorem defines lower_integrable INTEGRA1:def 13 :

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;

func upper_integral f -> Real equals :: INTEGRA1:def 14
lower_bound (rng (upper_sum_set f));

end;

:: deftheorem defines upper_integral INTEGRA1:def 14 :

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;

func lower_integral f -> Real equals :: INTEGRA1:def 15
upper_bound (rng (lower_sum_set f));

end;

:: deftheorem defines lower_integral INTEGRA1:def 15 :

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;

attr f is integrable means :: INTEGRA1:def 16
( f is upper_integrable & f is lower_integrable & upper_integral f = lower_integral f );

end;

:: deftheorem defines integrable INTEGRA1:def 16 :

definition

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;

func integral f -> Real equals :: INTEGRA1:def 17
upper_integral f;

end;

:: deftheorem defines integral INTEGRA1:def 17 :

theorem Th8: :: INTEGRA1:10

for X being non empty set
for f, g being PartFunc of X,REAL holds rng (f + g) c= (rng f) ++ (rng g)

proof end;

theorem Th9: :: INTEGRA1:11

for f being real-valued Function st f is bounded_below holds
rng f is bounded_below

proof end;

theorem :: INTEGRA1:12

for f being real-valued Function st rng f is bounded_below holds
f is bounded_below

proof end;

theorem Th11: :: INTEGRA1:13

for f being real-valued Function st f is bounded_above holds
rng f is bounded_above

proof end;

theorem :: INTEGRA1:14

for f being real-valued Function st rng f is bounded_above holds
f is bounded_above

proof end;

theorem :: INTEGRA1:15

for f being real-valued Function st f is bounded holds
rng f is real-bounded by Th11, Th9;

theorem Th14: :: INTEGRA1:16

for A being non empty set holds (chi (A,A)) | A is V8()

proof end;

theorem Th15: :: INTEGRA1:17

for X being non empty set
for A being non empty Subset of X holds rng (chi (A,A)) = {1}

proof end;

theorem Th16: :: INTEGRA1:18

for X being non empty set
for A being non empty Subset of X
for B being set st B meets dom (chi (A,A)) holds
rng ((chi (A,A)) | B) = {1}

proof end;

theorem Th17: :: INTEGRA1:19

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
vol (divset (D,i)) = (lower_volume ((chi (A,A)),D)) . i

proof end;

theorem Th18: :: INTEGRA1:20

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
vol (divset (D,i)) = (upper_volume ((chi (A,A)),D)) . i

proof end;

theorem :: INTEGRA1:21

for F, G, H being FinSequence of REAL st len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) + (G /. k) ) holds
Sum H = (Sum F) + (Sum G)

proof end;

theorem Th20: :: INTEGRA1:22

for F, G, H being FinSequence of REAL st len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)

proof end;

theorem Th21: :: INTEGRA1:23

for A being non empty closed_interval Subset of REAL
for D being Division of A holds Sum (lower_volume ((chi (A,A)),D)) = vol A

proof end;

theorem Th22: :: INTEGRA1:24

for A being non empty closed_interval Subset of REAL
for D being Division of A holds Sum (upper_volume ((chi (A,A)),D)) = vol A

proof end;

registration

let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
let D be Division of A;

cluster upper_volume (f,D) -> non empty ;

proof end;

cluster lower_volume (f,D) -> non empty ;

proof end;

end;

theorem Th23: :: INTEGRA1:25

for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_below holds
(lower_bound (rng f)) * (vol A) <= lower_sum (f,D)

proof end;

theorem :: INTEGRA1:26

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))

proof end;

theorem Th25: :: INTEGRA1:27

for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_above holds
upper_sum (f,D) <= (upper_bound (rng f)) * (vol A)

proof end;

theorem Th26: :: INTEGRA1:28

for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D) <= upper_sum (f,D)

proof end;

definition

let D1, D2 be FinSequence;

pred D1 <= D2 means :: INTEGRA1:def 18
( len D1 <= len D2 & rng D1 c= rng D2 );

end;

:: deftheorem defines <= INTEGRA1:def 18 :

notation

let D1, D2 be FinSequence;
synonym D2 >= D1 for D1 <= D2;

end;

theorem :: INTEGRA1:29

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st len D1 = 1 holds
D1 <= D2

proof end;

theorem Th28: :: INTEGRA1:30

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded_above & len D1 = 1 holds
upper_sum (f,D1) >= upper_sum (f,D2)

proof end;

theorem Th29: :: INTEGRA1:31

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded_below & len D1 = 1 holds
lower_sum (f,D1) <= lower_sum (f,D2)

proof end;

theorem :: INTEGRA1:32

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
ex A1, A2 being non empty closed_interval Subset of REAL st
( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 )

proof end;

theorem Th31: :: INTEGRA1:33

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st i in dom D1 & D1 <= D2 holds
ex j being Element of NAT st
( j in dom D2 & D1 . i = D2 . j )

proof end;

definition

let A be non empty closed_interval Subset of REAL;
let D1, D2 be Division of A;
let i be Nat;
assume A1: D1 <= D2 ;

func indx (D2,D1,i) -> Element of NAT means :Def18: :: INTEGRA1:def 19
( it in dom D2 & D1 . i = D2 . it ) if i in dom D1
otherwise it = 0 ;

existence by A1, Th31;
uniqueness

proof end;

end;

:: deftheorem Def18 defines indx INTEGRA1:def 19 :

theorem Th32: :: INTEGRA1:34

for p being increasing FinSequence of REAL
for n being Element of NAT st n <= len p holds
p /^ n is increasing FinSequence of REAL

proof end;

theorem Th33: :: INTEGRA1:35

for p being increasing FinSequence of REAL
for i, j being Element of NAT st j in dom p & i <= j holds
mid (p,i,j) is increasing FinSequence of REAL

proof end;

Lm1: for i, j being Element of NAT
for f being FinSequence st i in dom f & j in dom f & i <= j holds
len (mid (f,i,j)) = (j - i) + 1

proof end;

theorem Th34: :: INTEGRA1:36

for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j holds
ex B being non empty closed_interval Subset of REAL st
( lower_bound B = (mid (D,i,j)) . 1 & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B )

proof end;

theorem Th35: :: INTEGRA1:37

for i, j being Element of NAT
for A, B being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j & D . i >= lower_bound B & D . j = upper_bound B holds
mid (D,i,j) is Division of B

proof end;

definition

let p be FinSequence of REAL ;

func PartSums p -> FinSequence of REAL means :Def19: :: INTEGRA1:def 20
( len it = len p & ( for i being Nat st i in dom p holds
it . i = Sum (p | i) ) );

proof end;

proof end;

end;

:: deftheorem Def19 defines PartSums INTEGRA1:def 20 :

theorem Th36: :: INTEGRA1:38

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
for i being non zero Element of NAT st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i)))

proof end;

theorem Th37: :: INTEGRA1:39

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
for i being non zero Element of NAT st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i)))

proof end;

theorem Th38: :: INTEGRA1:40

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_above holds
(PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i))

proof end;

theorem Th39: :: INTEGRA1:41

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i))

proof end;

theorem Th40: :: INTEGRA1:42

for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL holds (PartSums (upper_volume (f,D))) . (len D) = upper_sum (f,D)

proof end;

theorem Th41: :: INTEGRA1:43

for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL holds (PartSums (lower_volume (f,D))) . (len D) = lower_sum (f,D)

proof end;

theorem Th42: :: INTEGRA1:44

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st D1 <= D2 holds
indx (D2,D1,(len D1)) = len D2

proof end;

theorem Th43: :: INTEGRA1:45

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
upper_sum (f,D2) <= upper_sum (f,D1)

proof end;

theorem Th44: :: INTEGRA1:46

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
lower_sum (f,D2) >= lower_sum (f,D1)

proof end;

theorem Th45: :: INTEGRA1:47

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A ex D being Division of A st
( D1 <= D & D2 <= D )

proof end;

theorem Th46: :: INTEGRA1:48

for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D1) <= upper_sum (f,D2)

proof end;

theorem Th47: :: INTEGRA1:49

for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL st f | A is bounded holds
upper_integral f >= lower_integral f

proof end;

theorem Th48: :: INTEGRA1:50

for X, Y being Subset of REAL holds (-- X) ++ (-- Y) = -- (X ++ Y)

proof end;

theorem Th49: :: INTEGRA1:51

for X, Y being Subset of REAL st X is bounded_above & Y is bounded_above holds
X ++ Y is bounded_above

proof end;

theorem Th50: :: INTEGRA1:52

for X, Y being non empty Subset of REAL st X is bounded_above & Y is bounded_above holds
upper_bound (X ++ Y) = (upper_bound X) + (upper_bound Y)

proof end;

theorem Th51: :: INTEGRA1:53

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_above & g | A is bounded_above holds
(upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i)

proof end;

theorem Th52: :: INTEGRA1:54

for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i

proof end;

theorem Th53: :: INTEGRA1:55

for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st f | A is bounded_above & g | A is bounded_above holds
upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D))

proof end;

theorem Th54: :: INTEGRA1:56

for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st f | A is bounded_below & g | A is bounded_below holds
(lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D)

proof end;

theorem :: INTEGRA1:57

for A being non empty closed_interval Subset of REAL
for f, g being Function of A,REAL st f | A is bounded & g | A is bounded & f is integrable & g is integrable holds
( f + g is integrable & integral (f + g) = (integral f) + (integral g) )

proof end;

theorem :: INTEGRA1:58

for i, j being Element of NAT
for f being FinSequence st i in dom f & j in dom f & i <= j holds
len (mid (f,i,j)) = (j - i) + 1 by Lm1;