:: Integrability and the Integral of Partial Functions from $\Bbb R$ into $\Bbb R$
:: by Noboru Endou , Yasunari Shidama and Masahiko Yamazaki
::
:: Copyright (c) 2006-2021 Association of Mizar Users

theorem Th1: :: INTEGRA6:1
for a, b, c, d being Real st a <= b & c <= d & a + c = b + d holds
( a = b & c = d )
proof end;

theorem Th2: :: INTEGRA6:2
for a, b, x being Real st a <= b holds
].(x - a),(x + a).[ c= ].(x - b),(x + b).[
proof end;

theorem Th3: :: INTEGRA6:3
for R being Relation
for A, B, C being set st A c= B & A c= C holds
(R | B) | A = (R | C) | A
proof end;

theorem Th4: :: INTEGRA6:4
for A, B, C being set st A c= B & A c= C holds
(chi (B,B)) | A = (chi (C,C)) | A
proof end;

theorem Th5: :: INTEGRA6:5
for a, b being Real st a <= b holds
vol ['a,b'] = b - a
proof end;

theorem Th6: :: INTEGRA6:6
for a, b being Real holds vol ['(min (a,b)),(max (a,b))'] = |.(b - a).|
proof end;

Lm1: for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL st A c= dom f holds
f || A is Function of A,REAL

proof end;

theorem Th7: :: INTEGRA6:7
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL st A c= dom f & f is_integrable_on A & f | A is bounded holds
( abs f is_integrable_on A & |.(integral (f,A)).| <= integral ((abs f),A) )
proof end;

theorem Th8: :: INTEGRA6:8
for a, b being Real
for f being PartFunc of REAL,REAL st a <= b & ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded holds
|.(integral (f,a,b)).| <= integral ((abs f),a,b)
proof end;

theorem Th9: :: INTEGRA6:9
for r being Real
for A being non empty closed_interval Subset of REAL
for f being PartFunc of REAL,REAL st A c= dom f & f is_integrable_on A & f | A is bounded holds
( r (#) f is_integrable_on A & integral ((r (#) f),A) = r * (integral (f,A)) )
proof end;

theorem Th10: :: INTEGRA6:10
for a, b, c being Real
for f being PartFunc of REAL,REAL st a <= b & ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded holds
integral ((c (#) f),a,b) = c * (integral (f,a,b))
proof end;

theorem Th11: :: INTEGRA6:11
for A being non empty closed_interval Subset of REAL
for f, g being PartFunc of REAL,REAL st A c= dom f & A c= dom g & f is_integrable_on A & f | A is bounded & g is_integrable_on A & g | A is bounded holds
( f + g is_integrable_on A & f - g is_integrable_on A & integral ((f + g),A) = (integral (f,A)) + (integral (g,A)) & integral ((f - g),A) = (integral (f,A)) - (integral (g,A)) )
proof end;

theorem Th12: :: INTEGRA6:12
for a, b being Real
for f, g being PartFunc of REAL,REAL st a <= b & ['a,b'] c= dom f & ['a,b'] c= dom g & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded holds
( integral ((f + g),a,b) = (integral (f,a,b)) + (integral (g,a,b)) & integral ((f - g),a,b) = (integral (f,a,b)) - (integral (g,a,b)) )
proof end;

theorem :: INTEGRA6:13
for A being non empty closed_interval Subset of REAL
for f, g being PartFunc of REAL,REAL st f | A is bounded & g | A is bounded holds
(f (#) g) | A is bounded
proof end;

theorem :: INTEGRA6:14
for A being non empty closed_interval Subset of REAL
for f, g being PartFunc of REAL,REAL st A c= dom f & A c= dom g & f is_integrable_on A & f | A is bounded & g is_integrable_on A & g | A is bounded holds
f (#) g is_integrable_on A
proof end;

theorem Th15: :: INTEGRA6:15
for A being non empty closed_interval Subset of REAL
for n being Nat st n > 0 & vol A > 0 holds
ex D being Division of A st
( len D = n & ( for i being Nat st i in dom D holds
D . i = () + (((vol A) / n) * i) ) )
proof end;

theorem Th16: :: INTEGRA6:16
for A being non empty closed_interval Subset of REAL st vol A > 0 holds
ex T being DivSequence of A st
( delta T is convergent & lim () = 0 & ( for n being Element of NAT ex Tn being Division of A st
( Tn divide_into_equal n + 1 & T . n = Tn ) ) )
proof end;

Lm2: for a, b, c being Real
for f being PartFunc of REAL,REAL st a <= b & ['a,b'] c= dom f & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & c in ['a,b'] & b <> c holds
( integral (f,a,b) = (integral (f,a,c)) + (integral (f,c,b)) & f is_integrable_on ['a,c'] & f is_integrable_on ['c,b'] )

proof end;

theorem Th17: :: INTEGRA6:17
for a, b, c being Real
for f being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] holds
( f is_integrable_on ['a,c'] & f is_integrable_on ['c,b'] & integral (f,a,b) = (integral (f,a,c)) + (integral (f,c,b)) )
proof end;

Lm3: for a, b, c being Real st a <= c & c <= b holds
( c in ['a,b'] & ['a,c'] c= ['a,b'] & ['c,b'] c= ['a,b'] )

proof end;

theorem Th18: :: INTEGRA6:18
for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f holds
( f is_integrable_on ['c,d'] & f | ['c,d'] is bounded & ['c,d'] c= dom f )
proof end;

theorem :: INTEGRA6:19
for a, b, c, d being Real
for f, g being PartFunc of REAL,REAL st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g holds
( f + g is_integrable_on ['c,d'] & (f + g) | ['c,d'] is bounded )
proof end;

Lm4: for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d))

proof end;

theorem Th20: :: INTEGRA6:20
for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d))
proof end;

Lm5: for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) )

proof end;

theorem Th21: :: INTEGRA6:21
for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( ['(min (c,d)),(max (c,d))'] c= dom (abs f) & abs f is_integrable_on ['(min (c,d)),(max (c,d))'] & (abs f) | ['(min (c,d)),(max (c,d))'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),(min (c,d)),(max (c,d))) )
proof end;

Lm6: for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) )

proof end;

Lm7: for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
|.(integral (f,d,c)).| <= integral ((abs f),c,d)

proof end;

theorem :: INTEGRA6:22
for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) & |.(integral (f,d,c)).| <= integral ((abs f),c,d) ) by ;

Lm8: for a, b, c, d, e being Real
for f being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] & ( for x being Real st x in ['(min (c,d)),(max (c,d))'] holds
|.(f . x).| <= e ) holds
|.(integral (f,c,d)).| <= e * |.(d - c).|

proof end;

Lm9: for a, b, c, d, e being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] & ( for x being Real st x in ['c,d'] holds
|.(f . x).| <= e ) holds
|.(integral (f,c,d)).| <= e * (d - c)

proof end;

Lm10: for a, b, c, d, e being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] & ( for x being Real st x in ['c,d'] holds
|.(f . x).| <= e ) holds
|.(integral (f,d,c)).| <= e * (d - c)

proof end;

theorem :: INTEGRA6:23
for a, b, c, d, e being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] & ( for x being Real st x in ['c,d'] holds
|.(f . x).| <= e ) holds
( |.(integral (f,c,d)).| <= e * (d - c) & |.(integral (f,d,c)).| <= e * (d - c) ) by ;

Lm11: for a, b, c, d being Real
for f, g being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] holds
( f + g is_integrable_on ['c,d'] & (f + g) | ['c,d'] is bounded & integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )

proof end;

theorem Th24: :: INTEGRA6:24
for a, b, c, d being Real
for f, g being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] holds
( integral ((f + g),c,d) = (integral (f,c,d)) + (integral (g,c,d)) & integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d)) )
proof end;

Lm12: for a, b, c, d, e being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral ((e (#) f),c,d) = e * (integral (f,c,d))

proof end;

theorem :: INTEGRA6:25
for a, b, c, d, e being Real
for f being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral ((e (#) f),c,d) = e * (integral (f,c,d))
proof end;

theorem Th26: :: INTEGRA6:26
for a, b, e being Real
for f being PartFunc of REAL,REAL st a <= b & ['a,b'] c= dom f & ( for x being Real st x in ['a,b'] holds
f . x = e ) holds
( f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & integral (f,a,b) = e * (b - a) )
proof end;

theorem Th27: :: INTEGRA6:27
for a, b, c, d, e being Real
for f being PartFunc of REAL,REAL st a <= b & ( for x being Real st x in ['a,b'] holds
f . x = e ) & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral (f,c,d) = e * (d - c)
proof end;

theorem Th28: :: INTEGRA6:28
for a, b being Real
for f being PartFunc of REAL,REAL
for x0 being Real
for F being PartFunc of REAL,REAL st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) & x0 in ].a,b.[ & f is_continuous_in x0 holds
( F is_differentiable_in x0 & diff (F,x0) = f . x0 )
proof end;

Lm13: for a, b being Real
for f being PartFunc of REAL,REAL ex F being PartFunc of REAL,REAL st
( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) )

proof end;

theorem :: INTEGRA6:29
for a, b being Real
for f being PartFunc of REAL,REAL
for x0 being Real st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 holds
ex F being PartFunc of REAL,REAL st
( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds
F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 )
proof end;