:: Riemann Integral of Functions from $\mathbbbR$ into Real Normed Space
:: by Keiichi Miyajima , Takahiro Kato and Yasunari Shidama
::
:: Received May 20, 2010
:: Copyright (c) 2010-2021 Association of Mizar Users

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be Function of A, the carrier of X;
let D be Division of A;

mode middle_volume of f,D -> FinSequence of X means :Def1: :: INTEGR18:def 1
( len it = len D & ( for i being Nat st i in dom D holds
ex c being Point of X st
( c in rng (f | (divset (D,i))) & it . i = (vol (divset (D,i))) * c ) ) );

correctness
existence
ex b₁ being FinSequence of X st
( len b₁ = len D & ( for i being Nat st i in dom D holds
ex c being Point of X st
( c in rng (f | (divset (D,i))) & b₁ . i = (vol (divset (D,i))) * c ) ) );

proof end;

end;

:: deftheorem Def1 defines middle_volume INTEGR18:def 1 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being Function of A, the carrier of X
for D being Division of A
for b₅ being FinSequence of X holds
( b₅ is middle_volume of f,D iff ( len b₅ = len D & ( for i being Nat st i in dom D holds
ex c being Point of X st
( c in rng (f | (divset (D,i))) & b₅ . i = (vol (divset (D,i))) * c ) ) ) );

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be Function of A, the carrier of X;
let D be Division of A;
let F be middle_volume of f,D;

func middle_sum (f,F) -> Point of X equals :: INTEGR18:def 2
Sum F;

coherence
Sum F is Point of X ;

end;

:: deftheorem defines middle_sum INTEGR18:def 2 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being Function of A, the carrier of X
for D being Division of A
for F being middle_volume of f,D holds middle_sum (f,F) = Sum F;

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be Function of A, the carrier of X;
let T be DivSequence of A;

mode middle_volume_Sequence of f,T -> sequence of ( the carrier of X *) means :Def3: :: INTEGR18:def 3
for k being Element of NAT holds it . k is middle_volume of f,T . k;

correctness
existence
ex b₁ being sequence of ( the carrier of X *) st
for k being Element of NAT holds b₁ . k is middle_volume of f,T . k;

proof end;

end;

:: deftheorem Def3 defines middle_volume_Sequence INTEGR18:def 3 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being Function of A, the carrier of X
for T being DivSequence of A
for b₅ being sequence of ( the carrier of X *) holds
( b₅ is middle_volume_Sequence of f,T iff for k being Element of NAT holds b₅ . k is middle_volume of f,T . k );

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be Function of A, the carrier of X;
let T be DivSequence of A;
let S be middle_volume_Sequence of f,T;
let k be Nat;
:: original: .
redefine func S . k -> middle_volume of f,T . k;
coherence
S . k is middle_volume of f,T . k

proof end;

end;

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be Function of A, the carrier of X;
let T be DivSequence of A;
let S be middle_volume_Sequence of f,T;

func middle_sum (f,S) -> sequence of X means :Def4: :: INTEGR18:def 4
for i being Nat holds it . i = middle_sum (f,(S . i));

existence
ex b₁ being sequence of X st
for i being Nat holds b₁ . i = middle_sum (f,(S . i))

proof end;

uniqueness
for b₁, b₂ being sequence of X st ( for i being Nat holds b₁ . i = middle_sum (f,(S . i)) ) & ( for i being Nat holds b₂ . i = middle_sum (f,(S . i)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines middle_sum INTEGR18:def 4 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being Function of A, the carrier of X
for T being DivSequence of A
for S being middle_volume_Sequence of f,T
for b₆ being sequence of X holds
( b₆ = middle_sum (f,S) iff for i being Nat holds b₆ . i = middle_sum (f,(S . i)) );

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be Function of A, the carrier of X;

attr f is integrable means :: INTEGR18:def 5
ex I being Point of X st
for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = I );

end;

:: deftheorem defines integrable INTEGR18:def 5 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being Function of A, the carrier of X holds
( f is integrable iff ex I being Point of X st
for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = I ) );

theorem Th1: :: INTEGR18:1

for X being RealNormSpace
for R1, R2, R3 being FinSequence of X st len R1 = len R2 & R3 = R1 + R2 holds
Sum R3 = (Sum R1) + (Sum R2)

proof end;

theorem :: INTEGR18:2

for X being RealNormSpace
for R1, R2, R3 being FinSequence of X st len R1 = len R2 & R3 = R1 - R2 holds
Sum R3 = (Sum R1) - (Sum R2)

proof end;

theorem Th3: :: INTEGR18:3

for X being RealNormSpace
for R1, R2 being FinSequence of X
for a being Real st R2 = a (#) R1 holds
Sum R2 = a * (Sum R1)

proof end;

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be Function of A, the carrier of X;
assume A1: f is integrable ;

func integral f -> Point of X means :Def6: :: INTEGR18:def 6
for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = it );

existence
ex b₁ being Point of X st
for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = b₁ ) by A1;
uniqueness
for b₁, b₂ being Point of X st ( for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = b₁ ) ) & ( for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = b₂ ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines integral INTEGR18:def 6 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being Function of A, the carrier of X st f is integrable holds
for b₄ being Point of X holds
( b₄ = integral f iff for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = b₄ ) );

theorem Th4: :: INTEGR18:4

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for r being Real
for f, h being Function of A, the carrier of X st h = r (#) f & f is integrable holds
( h is integrable & integral h = r * (integral f) )

proof end;

reconsider jj = 1 as Real ;

theorem Th5: :: INTEGR18:5

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f, h being Function of A, the carrier of X st h = - f & f is integrable holds
( h is integrable & integral h = - (integral f) )

proof end;

theorem Th6: :: INTEGR18:6

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f, g, h being Function of A, the carrier of X st h = f + g & f is integrable & g is integrable holds
( h is integrable & integral h = (integral f) + (integral g) )

proof end;

theorem :: INTEGR18:7

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f, g, h being Function of A, the carrier of X st h = f - g & f is integrable & g is integrable holds
( h is integrable & integral h = (integral f) - (integral g) )

proof end;

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of REAL, the carrier of X;

pred f is_integrable_on A means :: INTEGR18:def 7
ex g being Function of A, the carrier of X st
( g = f | A & g is integrable );

end;

:: deftheorem defines is_integrable_on INTEGR18:def 7 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL, the carrier of X holds
( f is_integrable_on A iff ex g being Function of A, the carrier of X st
( g = f | A & g is integrable ) );

definition

let X be RealNormSpace;
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of REAL, the carrier of X;
assume A1: A c= dom f ;

func integral (f,A) -> Element of X means :Def8: :: INTEGR18:def 8
ex g being Function of A, the carrier of X st
( g = f | A & it = integral g );

correctness
existence
ex b₁ being Element of X ex g being Function of A, the carrier of X st
( g = f | A & b₁ = integral g );
uniqueness
for b₁, b₂ being Element of X st ex g being Function of A, the carrier of X st
( g = f | A & b₁ = integral g ) & ex g being Function of A, the carrier of X st
( g = f | A & b₂ = integral g ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def8 defines integral INTEGR18:def 8 :
for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL, the carrier of X st A c= dom f holds
for b₄ being Element of X holds
( b₄ = integral (f,A) iff ex g being Function of A, the carrier of X st
( g = f | A & b₄ = integral g ) );

theorem :: INTEGR18:8

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL, the carrier of X
for g being Function of A, the carrier of X st f | A = g holds
( f is_integrable_on A iff g is integrable ) ;

theorem :: INTEGR18:9

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL, the carrier of X
for g being Function of A, the carrier of X st A c= dom f & f | A = g holds
integral (f,A) = integral g by Def8;

theorem Th10: :: INTEGR18:10

for X, Y being non empty set
for V being RealNormSpace
for g, f being PartFunc of X, the carrier of V
for g1, f1 being PartFunc of Y, the carrier of V st g = g1 & f = f1 holds
g1 + f1 = g + f

proof end;

theorem :: INTEGR18:11

for X, Y being non empty set
for V being RealNormSpace
for g, f being PartFunc of X, the carrier of V
for g1, f1 being PartFunc of Y, the carrier of V st g = g1 & f = f1 holds
g1 - f1 = g - f

proof end;

theorem Th12: :: INTEGR18:12

for r being Real
for X, Y being non empty set
for V being RealNormSpace
for g being PartFunc of X, the carrier of V
for g1 being PartFunc of Y, the carrier of V st g = g1 holds
r (#) g1 = r (#) g

proof end;

theorem Th13: :: INTEGR18:13

for X being RealNormSpace
for r being Real
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL, the carrier of X st A c= dom f & f is_integrable_on A holds
( r (#) f is_integrable_on A & integral ((r (#) f),A) = r * (integral (f,A)) )

proof end;

theorem Th14: :: INTEGR18:14

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f1, f2 being PartFunc of REAL, the carrier of X st f1 is_integrable_on A & f2 is_integrable_on A & A c= dom f1 & A c= dom f2 holds
( f1 + f2 is_integrable_on A & integral ((f1 + f2),A) = (integral (f1,A)) + (integral (f2,A)) )

proof end;

theorem :: INTEGR18:15

for X being RealNormSpace
for A being non empty closed_interval Subset of REAL
for f1, f2 being PartFunc of REAL, the carrier of X st f1 is_integrable_on A & f2 is_integrable_on A & A c= dom f1 & A c= dom f2 holds
( f1 - f2 is_integrable_on A & integral ((f1 - f2),A) = (integral (f1,A)) - (integral (f2,A)) )

proof end;

definition

let X be RealNormSpace;
let f be PartFunc of REAL, the carrier of X;
let a, b be Real;

func integral (f,a,b) -> Element of X equals :Def9: :: INTEGR18:def 9
integral (f,['a,b']) if a <= b
otherwise - (integral (f,['b,a']));

correctness
coherence
( ( a <= b implies integral (f,['a,b']) is Element of X ) & ( not a <= b implies - (integral (f,['b,a'])) is Element of X ) );
consistency
for b₁ being Element of X holds verum;
;

end;

:: deftheorem Def9 defines integral INTEGR18:def 9 :
for X being RealNormSpace
for f being PartFunc of REAL, the carrier of X
for a, b being Real holds
( ( a <= b implies integral (f,a,b) = integral (f,['a,b']) ) & ( not a <= b implies integral (f,a,b) = - (integral (f,['b,a'])) ) );

theorem Th16: :: INTEGR18:16

for X being RealNormSpace
for f being PartFunc of REAL, the carrier of X
for A being non empty closed_interval Subset of REAL
for a, b being Real st A = [.a,b.] holds
integral (f,A) = integral (f,a,b)

proof end;

theorem Th17: :: INTEGR18:17

for X being RealNormSpace
for f being PartFunc of REAL, the carrier of X
for A being non empty closed_interval Subset of REAL st vol A = 0 & A c= dom f holds
( f is_integrable_on A & integral (f,A) = 0. X )

proof end;

theorem :: INTEGR18:18

for X being RealNormSpace
for f being PartFunc of REAL, the carrier of X
for A being non empty closed_interval Subset of REAL
for a, b being Real st A = [.b,a.] & A c= dom f holds
- (integral (f,A)) = integral (f,a,b)

proof end;