:: INTEGRA1 semantic presentation
:: deftheorem Def1 defines closed-interval INTEGRA1:def 1 :
theorem Th1: :: INTEGRA1:1
theorem Th2: :: INTEGRA1:2
theorem Th3: :: INTEGRA1:3
theorem Th4: :: INTEGRA1:4
theorem Th5: :: INTEGRA1:5
theorem Th6: :: INTEGRA1:6
:: deftheorem Def2 defines DivisionPoint INTEGRA1:def 2 :
:: deftheorem Def3 defines divs INTEGRA1:def 3 :
:: deftheorem Def4 defines Division INTEGRA1:def 4 :
theorem Th7: :: INTEGRA1:7
canceled;
theorem Th8: :: INTEGRA1:8
theorem Th9: :: INTEGRA1:9
:: deftheorem Def5 defines divset INTEGRA1:def 5 :
theorem Th10: :: INTEGRA1:10
:: deftheorem Def6 defines vol INTEGRA1:def 6 :
theorem Th11: :: INTEGRA1:11
definition
let c
1 be
closed-interval Subset of
REAL ;
let c
2 be
PartFunc of c
1,
REAL ;
let c
3 be non
empty Division of c
1;
let c
4 be
Element of c
3;
func upper_volume c
2,c
4 -> FinSequence of
REAL means :
Def7:
:: INTEGRA1:def 7
(
len a
5 = len a
4 & ( for b
1 being
Nat holds
( b
1 in Seg (len a4) implies a
5 . b
1 = (sup (rng (a2 | (divset a4,b1)))) * (vol (divset a4,b1)) ) ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = len c4 & ( for b2 being Nat holds
( b2 in Seg (len c4) implies b1 . b2 = (sup (rng (c2 | (divset c4,b2)))) * (vol (divset c4,b2)) ) ) )
uniqueness
for b1, b2 being FinSequence of REAL holds
( len b1 = len c4 & ( for b3 being Nat holds
( b3 in Seg (len c4) implies b1 . b3 = (sup (rng (c2 | (divset c4,b3)))) * (vol (divset c4,b3)) ) ) & len b2 = len c4 & ( for b3 being Nat holds
( b3 in Seg (len c4) implies b2 . b3 = (sup (rng (c2 | (divset c4,b3)))) * (vol (divset c4,b3)) ) ) implies b1 = b2 )
func lower_volume c
2,c
4 -> FinSequence of
REAL means :
Def8:
:: INTEGRA1:def 8
(
len a
5 = len a
4 & ( for b
1 being
Nat holds
( b
1 in Seg (len a4) implies a
5 . b
1 = (inf (rng (a2 | (divset a4,b1)))) * (vol (divset a4,b1)) ) ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = len c4 & ( for b2 being Nat holds
( b2 in Seg (len c4) implies b1 . b2 = (inf (rng (c2 | (divset c4,b2)))) * (vol (divset c4,b2)) ) ) )
uniqueness
for b1, b2 being FinSequence of REAL holds
( len b1 = len c4 & ( for b3 being Nat holds
( b3 in Seg (len c4) implies b1 . b3 = (inf (rng (c2 | (divset c4,b3)))) * (vol (divset c4,b3)) ) ) & len b2 = len c4 & ( for b3 being Nat holds
( b3 in Seg (len c4) implies b2 . b3 = (inf (rng (c2 | (divset c4,b3)))) * (vol (divset c4,b3)) ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines upper_volume INTEGRA1:def 7 :
:: deftheorem Def8 defines lower_volume INTEGRA1:def 8 :
:: deftheorem Def9 defines upper_sum INTEGRA1:def 9 :
:: deftheorem Def10 defines lower_sum INTEGRA1:def 10 :
definition
let c
1 be
closed-interval Subset of
REAL ;
let c
2 be
PartFunc of c
1,
REAL ;
func upper_sum_set c
2 -> PartFunc of
divs a
1,
REAL means :
Def11:
:: INTEGRA1:def 11
(
dom a
3 = divs a
1 & ( for b
1 being
Element of
divs a
1 holds
( b
1 in dom a
3 implies a
3 . b
1 = upper_sum a
2,b
1 ) ) );
existence
ex b1 being PartFunc of divs c1, REAL st
( dom b1 = divs c1 & ( for b2 being Element of divs c1 holds
( b2 in dom b1 implies b1 . b2 = upper_sum c2,b2 ) ) )
uniqueness
for b1, b2 being PartFunc of divs c1, REAL holds
( dom b1 = divs c1 & ( for b3 being Element of divs c1 holds
( b3 in dom b1 implies b1 . b3 = upper_sum c2,b3 ) ) & dom b2 = divs c1 & ( for b3 being Element of divs c1 holds
( b3 in dom b2 implies b2 . b3 = upper_sum c2,b3 ) ) implies b1 = b2 )
func lower_sum_set c
2 -> PartFunc of
divs a
1,
REAL means :
Def12:
:: INTEGRA1:def 12
(
dom a
3 = divs a
1 & ( for b
1 being
Element of
divs a
1 holds
( b
1 in dom a
3 implies a
3 . b
1 = lower_sum a
2,b
1 ) ) );
existence
ex b1 being PartFunc of divs c1, REAL st
( dom b1 = divs c1 & ( for b2 being Element of divs c1 holds
( b2 in dom b1 implies b1 . b2 = lower_sum c2,b2 ) ) )
uniqueness
for b1, b2 being PartFunc of divs c1, REAL holds
( dom b1 = divs c1 & ( for b3 being Element of divs c1 holds
( b3 in dom b1 implies b1 . b3 = lower_sum c2,b3 ) ) & dom b2 = divs c1 & ( for b3 being Element of divs c1 holds
( b3 in dom b2 implies b2 . b3 = lower_sum c2,b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines upper_sum_set INTEGRA1:def 11 :
:: deftheorem Def12 defines lower_sum_set INTEGRA1:def 12 :
:: deftheorem Def13 defines is_upper_integrable_on INTEGRA1:def 13 :
:: deftheorem Def14 defines is_lower_integrable_on INTEGRA1:def 14 :
:: deftheorem Def15 defines upper_integral INTEGRA1:def 15 :
:: deftheorem Def16 defines lower_integral INTEGRA1:def 16 :
:: deftheorem Def17 defines is_integrable_on INTEGRA1:def 17 :
:: deftheorem Def18 defines integral INTEGRA1:def 18 :
theorem Th12: :: INTEGRA1:12
theorem Th13: :: INTEGRA1:13
theorem Th14: :: INTEGRA1:14
theorem Th15: :: INTEGRA1:15
theorem Th16: :: INTEGRA1:16
theorem Th17: :: INTEGRA1:17
theorem Th18: :: INTEGRA1:18
theorem Th19: :: INTEGRA1:19
theorem Th20: :: INTEGRA1:20
theorem Th21: :: INTEGRA1:21
theorem Th22: :: INTEGRA1:22
theorem Th23: :: INTEGRA1:23
theorem Th24: :: INTEGRA1:24
theorem Th25: :: INTEGRA1:25
theorem Th26: :: INTEGRA1:26
theorem Th27: :: INTEGRA1:27
theorem Th28: :: INTEGRA1:28
theorem Th29: :: INTEGRA1:29
theorem Th30: :: INTEGRA1:30
:: deftheorem Def19 defines delta INTEGRA1:def 19 :
:: deftheorem Def20 defines <= INTEGRA1:def 20 :
theorem Th31: :: INTEGRA1:31
theorem Th32: :: INTEGRA1:32
theorem Th33: :: INTEGRA1:33
theorem Th34: :: INTEGRA1:34
theorem Th35: :: INTEGRA1:35
:: deftheorem Def21 defines indx INTEGRA1:def 21 :
theorem Th36: :: INTEGRA1:36
theorem Th37: :: INTEGRA1:37
theorem Th38: :: INTEGRA1:38
for b
1 being
closed-interval Subset of
REAL for b
2 being non
empty Division of b
1for b
3 being
Element of b
2for b
4, b
5 being
Nat holds
not ( b
4 in dom b
3 & b
5 in dom b
3 & b
4 <= b
5 & ( for b
6 being
closed-interval Subset of
REAL holds
not (
inf b
6 = (mid b3,b4,b5) . 1 &
sup b
6 = (mid b3,b4,b5) . (len (mid b3,b4,b5)) &
len (mid b3,b4,b5) = (b5 - b4) + 1 &
mid b
3,b
4,b
5 is
DivisionPoint of b
6 ) ) )
theorem Th39: :: INTEGRA1:39
:: deftheorem Def22 defines PartSums INTEGRA1:def 22 :
theorem Th40: :: INTEGRA1:40
theorem Th41: :: INTEGRA1:41
theorem Th42: :: INTEGRA1:42
theorem Th43: :: INTEGRA1:43
theorem Th44: :: INTEGRA1:44
theorem Th45: :: INTEGRA1:45
theorem Th46: :: INTEGRA1:46
theorem Th47: :: INTEGRA1:47
theorem Th48: :: INTEGRA1:48
theorem Th49: :: INTEGRA1:49
theorem Th50: :: INTEGRA1:50
theorem Th51: :: INTEGRA1:51
theorem Th52: :: INTEGRA1:52
theorem Th53: :: INTEGRA1:53
theorem Th54: :: INTEGRA1:54
theorem Th55: :: INTEGRA1:55
theorem Th56: :: INTEGRA1:56
theorem Th57: :: INTEGRA1:57
theorem Th58: :: INTEGRA1:58
theorem Th59: :: INTEGRA1:59