:: TAYLOR_1 semantic presentation
:: deftheorem Def1 defines #Z TAYLOR_1:def 1 :
theorem Th1: :: TAYLOR_1:1
theorem Th2: :: TAYLOR_1:2
theorem Th3: :: TAYLOR_1:3
Lemma4:
for b1 being natural number
for b2 being real number holds (exp_R b2) #R b1 = exp_R (b1 * b2)
theorem Th4: :: TAYLOR_1:4
Lemma6:
for b1 being Integer
for b2 being real number holds (exp_R b2) #R b1 = exp_R (b1 * b2)
theorem Th5: :: TAYLOR_1:5
theorem Th6: :: TAYLOR_1:6
Lemma9:
for b1 being real number
for b2 being Rational holds (exp_R b1) #R b2 = exp_R (b2 * b1)
theorem Th7: :: TAYLOR_1:7
theorem Th8: :: TAYLOR_1:8
theorem Th9: :: TAYLOR_1:9
theorem Th10: :: TAYLOR_1:10
theorem Th11: :: TAYLOR_1:11
then Lemma13:
( number_e > 1 & number_e > 0 )
by XXREAL_0:2;
theorem Th12: :: TAYLOR_1:12
theorem Th13: :: TAYLOR_1:13
theorem Th14: :: TAYLOR_1:14
theorem Th15: :: TAYLOR_1:15
theorem Th16: :: TAYLOR_1:16
theorem Th17: :: TAYLOR_1:17
:: deftheorem Def2 defines log_ TAYLOR_1:def 2 :
theorem Th18: :: TAYLOR_1:18
theorem Th19: :: TAYLOR_1:19
theorem Th20: :: TAYLOR_1:20
:: deftheorem Def3 defines #R TAYLOR_1:def 3 :
theorem Th21: :: TAYLOR_1:21
theorem Th22: :: TAYLOR_1:22
:: deftheorem Def4 defines diff TAYLOR_1:def 4 :
:: deftheorem Def5 defines is_differentiable_on TAYLOR_1:def 5 :
theorem Th23: :: TAYLOR_1:23
definition
let c
1 be
PartFunc of
REAL ,
REAL ;
let c
2 be
Subset of
REAL ;
let c
3, c
4 be
real number ;
func Taylor c
1,c
2,c
3,c
4 -> Real_Sequence means :
Def6:
:: TAYLOR_1:def 6
for b
1 being
natural number holds a
5 . b
1 = ((((diff a1,a2) . b1) . a3) * ((a4 - a3) |^ b1)) / (b1 ! );
existence
ex b1 being Real_Sequence st
for b2 being natural number holds b1 . b2 = ((((diff c1,c2) . b2) . c3) * ((c4 - c3) |^ b2)) / (b2 ! )
uniqueness
for b1, b2 being Real_Sequence holds
( ( for b3 being natural number holds b1 . b3 = ((((diff c1,c2) . b3) . c3) * ((c4 - c3) |^ b3)) / (b3 ! ) ) & ( for b3 being natural number holds b2 . b3 = ((((diff c1,c2) . b3) . c3) * ((c4 - c3) |^ b3)) / (b3 ! ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines Taylor TAYLOR_1:def 6 :
Lemma28:
for b1 being Real holds
ex b2 being PartFunc of REAL , REAL st
( dom b2 = REAL & ( for b3 being Real holds
( b2 . b3 = b1 - b3 & ( for b4 being Real holds
( b2 is_differentiable_in b4 & diff b2,b4 = - 1 ) ) ) ) )
Lemma29:
for b1 being Nat
for b2, b3 being Real holds
ex b4 being PartFunc of REAL , REAL st
( dom b4 = REAL & ( for b5 being Real holds
( b4 . b5 = (b2 * ((b3 - b5) |^ (b1 + 1))) / ((b1 + 1) ! ) & ( for b6 being Real holds
( b4 is_differentiable_in b6 & diff b4,b6 = - ((b2 * ((b3 - b6) |^ b1)) / (b1 ! )) ) ) ) ) )
Lemma30:
for b1 being Nat
for b2 being PartFunc of REAL , REAL
for b3 being Subset of REAL
for b4 being Real holds
ex b5 being PartFunc of REAL , REAL st
( dom b5 = b3 & ( for b6 being Real holds
( b6 in b3 implies b5 . b6 = (b2 . b4) - ((Partial_Sums (Taylor b2,b3,b6,b4)) . b1) ) ) )
theorem Th24: :: TAYLOR_1:24
Lemma32:
for b1 being PartFunc of REAL , REAL
for b2 being Subset of REAL
for b3 being Nat holds
( b1 is_differentiable_on b3,b2 implies for b4, b5 being Real holds
( b4 < b5 & [.b4,b5.] c= b2 & (diff b1,b2) . b3 is_continuous_on [.b4,b5.] & b1 is_differentiable_on b3 + 1,].b4,b5.[ implies for b6 being PartFunc of REAL , REAL holds
( dom b6 = b2 & ( for b7 being Real holds
( b7 in b2 implies b6 . b7 = (b1 . b5) - ((Partial_Sums (Taylor b1,b2,b7,b5)) . b3) ) ) implies ( b6 . b5 = 0 & b6 is_continuous_on [.b4,b5.] & b6 is_differentiable_on ].b4,b5.[ & ( for b7 being Real holds
( b7 in ].b4,b5.[ implies diff b6,b7 = - (((((diff b1,].b4,b5.[) . (b3 + 1)) . b7) * ((b5 - b7) |^ b3)) / (b3 ! )) ) ) ) ) ) )
Lemma33:
for b1 being Nat
for b2 being PartFunc of REAL , REAL
for b3 being Subset of REAL holds
( b2 is_differentiable_on b1,b3 implies for b4, b5 being Real holds
not ( b4 < b5 & [.b4,b5.] c= b3 & (diff b2,b3) . b1 is_continuous_on [.b4,b5.] & b2 is_differentiable_on b1 + 1,].b4,b5.[ & ( for b6 being PartFunc of REAL , REAL holds
not ( dom b6 = b3 & ( for b7 being Real holds
( b7 in b3 implies b6 . b7 = (b2 . b5) - ((Partial_Sums (Taylor b2,b3,b7,b5)) . b1) ) ) & b6 . b5 = 0 & b6 is_continuous_on [.b4,b5.] & b6 is_differentiable_on ].b4,b5.[ & ( for b7 being Real holds
( b7 in ].b4,b5.[ implies diff b6,b7 = - (((((diff b2,].b4,b5.[) . (b1 + 1)) . b7) * ((b5 - b7) |^ b1)) / (b1 ! )) ) ) ) ) ) )
theorem Th25: :: TAYLOR_1:25
for b
1 being
Natfor b
2 being
PartFunc of
REAL ,
REAL for b
3 being
Subset of
REAL holds
( b
2 is_differentiable_on b
1,b
3 implies for b
4, b
5 being
Real holds
( b
4 < b
5 &
[.b4,b5.] c= b
3 &
(diff b2,b3) . b
1 is_continuous_on [.b4,b5.] & b
2 is_differentiable_on b
1 + 1,
].b4,b5.[ implies for b
6 being
Realfor b
7 being
PartFunc of
REAL ,
REAL holds
(
dom b
7 = REAL & ( for b
8 being
Real holds b
7 . b
8 = ((b2 . b5) - ((Partial_Sums (Taylor b2,b3,b8,b5)) . b1)) - ((b6 * ((b5 - b8) |^ (b1 + 1))) / ((b1 + 1) ! )) ) &
((b2 . b5) - ((Partial_Sums (Taylor b2,b3,b4,b5)) . b1)) - ((b6 * ((b5 - b4) |^ (b1 + 1))) / ((b1 + 1) ! )) = 0 implies ( b
7 is_differentiable_on ].b4,b5.[ & b
7 . b
4 = 0 & b
7 . b
5 = 0 & b
7 is_continuous_on [.b4,b5.] & ( for b
8 being
Real holds
( b
8 in ].b4,b5.[ implies
diff b
7,b
8 = (- (((((diff b2,].b4,b5.[) . (b1 + 1)) . b8) * ((b5 - b8) |^ b1)) / (b1 ! ))) + ((b6 * ((b5 - b8) |^ b1)) / (b1 ! )) ) ) ) ) ) )
theorem Th26: :: TAYLOR_1:26
theorem Th27: :: TAYLOR_1:27
for b
1 being
Natfor b
2 being
PartFunc of
REAL ,
REAL for b
3 being
Subset of
REAL holds
( b
2 is_differentiable_on b
1,b
3 implies for b
4, b
5 being
Real holds
not ( b
4 < b
5 &
[.b4,b5.] c= b
3 &
(diff b2,b3) . b
1 is_continuous_on [.b4,b5.] & b
2 is_differentiable_on b
1 + 1,
].b4,b5.[ & ( for b
6 being
Real holds
not ( b
6 in ].b4,b5.[ & b
2 . b
5 = ((Partial_Sums (Taylor b2,b3,b4,b5)) . b1) + (((((diff b2,].b4,b5.[) . (b1 + 1)) . b6) * ((b5 - b4) |^ (b1 + 1))) / ((b1 + 1) ! )) ) ) ) )
Lemma37:
for b1 being PartFunc of REAL , REAL
for b2 being Subset of REAL
for b3 being Nat holds
( b1 is_differentiable_on b3,b2 implies for b4, b5 being Real holds
( b4 < b5 & [.b4,b5.] c= b2 & (diff b1,b2) . b3 is_continuous_on [.b4,b5.] & b1 is_differentiable_on b3 + 1,].b4,b5.[ implies for b6 being PartFunc of REAL , REAL holds
( dom b6 = b2 & ( for b7 being Real holds
( b7 in b2 implies b6 . b7 = (b1 . b4) - ((Partial_Sums (Taylor b1,b2,b7,b4)) . b3) ) ) implies ( b6 . b4 = 0 & b6 is_continuous_on [.b4,b5.] & b6 is_differentiable_on ].b4,b5.[ & ( for b7 being Real holds
( b7 in ].b4,b5.[ implies diff b6,b7 = - (((((diff b1,].b4,b5.[) . (b3 + 1)) . b7) * ((b4 - b7) |^ b3)) / (b3 ! )) ) ) ) ) ) )
Lemma38:
for b1 being Nat
for b2 being PartFunc of REAL , REAL
for b3 being Subset of REAL holds
( b2 is_differentiable_on b1,b3 implies for b4, b5 being Real holds
not ( b4 < b5 & [.b4,b5.] c= b3 & (diff b2,b3) . b1 is_continuous_on [.b4,b5.] & b2 is_differentiable_on b1 + 1,].b4,b5.[ & ( for b6 being PartFunc of REAL , REAL holds
not ( dom b6 = b3 & ( for b7 being Real holds
( b7 in b3 implies b6 . b7 = (b2 . b4) - ((Partial_Sums (Taylor b2,b3,b7,b4)) . b1) ) ) & b6 . b4 = 0 & b6 is_continuous_on [.b4,b5.] & b6 is_differentiable_on ].b4,b5.[ & ( for b7 being Real holds
( b7 in ].b4,b5.[ implies diff b6,b7 = - (((((diff b2,].b4,b5.[) . (b1 + 1)) . b7) * ((b4 - b7) |^ b1)) / (b1 ! )) ) ) ) ) ) )
theorem Th28: :: TAYLOR_1:28
for b
1 being
Natfor b
2 being
PartFunc of
REAL ,
REAL for b
3 being
Subset of
REAL holds
( b
2 is_differentiable_on b
1,b
3 implies for b
4, b
5 being
Real holds
( b
4 < b
5 &
[.b4,b5.] c= b
3 &
(diff b2,b3) . b
1 is_continuous_on [.b4,b5.] & b
2 is_differentiable_on b
1 + 1,
].b4,b5.[ implies for b
6 being
Realfor b
7 being
PartFunc of
REAL ,
REAL holds
(
dom b
7 = REAL & ( for b
8 being
Real holds b
7 . b
8 = ((b2 . b4) - ((Partial_Sums (Taylor b2,b3,b8,b4)) . b1)) - ((b6 * ((b4 - b8) |^ (b1 + 1))) / ((b1 + 1) ! )) ) &
((b2 . b4) - ((Partial_Sums (Taylor b2,b3,b5,b4)) . b1)) - ((b6 * ((b4 - b5) |^ (b1 + 1))) / ((b1 + 1) ! )) = 0 implies ( b
7 is_differentiable_on ].b4,b5.[ & b
7 . b
5 = 0 & b
7 . b
4 = 0 & b
7 is_continuous_on [.b4,b5.] & ( for b
8 being
Real holds
( b
8 in ].b4,b5.[ implies
diff b
7,b
8 = (- (((((diff b2,].b4,b5.[) . (b1 + 1)) . b8) * ((b4 - b8) |^ b1)) / (b1 ! ))) + ((b6 * ((b4 - b8) |^ b1)) / (b1 ! )) ) ) ) ) ) )
theorem Th29: :: TAYLOR_1:29
for b
1 being
Natfor b
2 being
PartFunc of
REAL ,
REAL for b
3 being
Subset of
REAL holds
( b
2 is_differentiable_on b
1,b
3 implies for b
4, b
5 being
Real holds
not ( b
4 < b
5 &
[.b4,b5.] c= b
3 &
(diff b2,b3) . b
1 is_continuous_on [.b4,b5.] & b
2 is_differentiable_on b
1 + 1,
].b4,b5.[ & ( for b
6 being
Real holds
not ( b
6 in ].b4,b5.[ & b
2 . b
4 = ((Partial_Sums (Taylor b2,b3,b5,b4)) . b1) + (((((diff b2,].b4,b5.[) . (b1 + 1)) . b6) * ((b4 - b5) |^ (b1 + 1))) / ((b1 + 1) ! )) ) ) ) )
theorem Th30: :: TAYLOR_1:30
theorem Th31: :: TAYLOR_1:31
theorem Th32: :: TAYLOR_1:32
theorem Th33: :: TAYLOR_1:33
for b
1 being
Natfor b
2 being
PartFunc of
REAL ,
REAL for b
3, b
4 being
Real holds
( 0
< b
4 & b
2 is_differentiable_on b
1 + 1,
].(b3 - b4),(b3 + b4).[ implies for b
5 being
Real holds
not ( b
5 in ].(b3 - b4),(b3 + b4).[ & ( for b
6 being
Real holds
not ( 0
< b
6 & b
6 < 1 & b
2 . b
5 = ((Partial_Sums (Taylor b2,].(b3 - b4),(b3 + b4).[,b3,b5)) . b1) + (((((diff b2,].(b3 - b4),(b3 + b4).[) . (b1 + 1)) . (b3 + (b6 * (b5 - b3)))) * ((b5 - b3) |^ (b1 + 1))) / ((b1 + 1) ! )) ) ) ) )