src/HOL/Taylor.thy
 author haftmann Mon Apr 27 10:11:44 2009 +0200 (2009-04-27) changeset 31001 7e6ffd8f51a9 parent 28952 15a4b2cf8c34 child 44890 22f665a2e91c permissions -rw-r--r--
cleaned up theory power further
```     1 (*  Title:      HOL/Taylor.thy
```
```     2     Author:     Lukas Bulwahn, Bernhard Haeupler, Technische Universitaet Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Taylor series *}
```
```     6
```
```     7 theory Taylor
```
```     8 imports MacLaurin
```
```     9 begin
```
```    10
```
```    11 text {*
```
```    12 We use MacLaurin and the translation of the expansion point @{text c} to @{text 0}
```
```    13 to prove Taylor's theorem.
```
```    14 *}
```
```    15
```
```    16 lemma taylor_up:
```
```    17   assumes INIT: "n>0" "diff 0 = f"
```
```    18   and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
```
```    19   and INTERV: "a \<le> c" "c < b"
```
```    20   shows "\<exists> t. c < t & t < b &
```
```    21     f b = setsum (%m. (diff m c / real (fact m)) * (b - c)^m) {0..<n} +
```
```    22       (diff n t / real (fact n)) * (b - c)^n"
```
```    23 proof -
```
```    24   from INTERV have "0 < b-c" by arith
```
```    25   moreover
```
```    26   from INIT have "n>0" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. f (x + c))" by auto
```
```    27   moreover
```
```    28   have "ALL m t. m < n & 0 <= t & t <= b - c --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)"
```
```    29   proof (intro strip)
```
```    30     fix m t
```
```    31     assume "m < n & 0 <= t & t <= b - c"
```
```    32     with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto
```
```    33     moreover
```
```    34     from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add)
```
```    35     ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)"
```
```    36       by (rule DERIV_chain2)
```
```    37     thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp
```
```    38   qed
```
```    39   ultimately
```
```    40   have EX:"EX t>0. t < b - c &
```
```    41     f (b - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) +
```
```    42       diff n (t + c) / real (fact n) * (b - c) ^ n"
```
```    43     by (rule Maclaurin)
```
```    44   show ?thesis
```
```    45   proof -
```
```    46     from EX obtain x where
```
```    47       X: "0 < x & x < b - c &
```
```    48         f (b - c + c) = (\<Sum>m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) +
```
```    49           diff n (x + c) / real (fact n) * (b - c) ^ n" ..
```
```    50     let ?H = "x + c"
```
```    51     from X have "c<?H & ?H<b \<and> f b = (\<Sum>m = 0..<n. diff m c / real (fact m) * (b - c) ^ m) +
```
```    52       diff n ?H / real (fact n) * (b - c) ^ n"
```
```    53       by fastsimp
```
```    54     thus ?thesis by fastsimp
```
```    55   qed
```
```    56 qed
```
```    57
```
```    58 lemma taylor_down:
```
```    59   assumes INIT: "n>0" "diff 0 = f"
```
```    60   and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
```
```    61   and INTERV: "a < c" "c \<le> b"
```
```    62   shows "\<exists> t. a < t & t < c &
```
```    63     f a = setsum (% m. (diff m c / real (fact m)) * (a - c)^m) {0..<n} +
```
```    64       (diff n t / real (fact n)) * (a - c)^n"
```
```    65 proof -
```
```    66   from INTERV have "a-c < 0" by arith
```
```    67   moreover
```
```    68   from INIT have "n>0" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. f (x + c))" by auto
```
```    69   moreover
```
```    70   have "ALL m t. m < n & a-c <= t & t <= 0 --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)"
```
```    71   proof (rule allI impI)+
```
```    72     fix m t
```
```    73     assume "m < n & a-c <= t & t <= 0"
```
```    74     with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto
```
```    75     moreover
```
```    76     from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add)
```
```    77     ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)" by (rule DERIV_chain2)
```
```    78     thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp
```
```    79   qed
```
```    80   ultimately
```
```    81   have EX: "EX t>a - c. t < 0 &
```
```    82     f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) +
```
```    83       diff n (t + c) / real (fact n) * (a - c) ^ n"
```
```    84     by (rule Maclaurin_minus)
```
```    85   show ?thesis
```
```    86   proof -
```
```    87     from EX obtain x where X: "a - c < x & x < 0 &
```
```    88       f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) +
```
```    89         diff n (x + c) / real (fact n) * (a - c) ^ n" ..
```
```    90     let ?H = "x + c"
```
```    91     from X have "a<?H & ?H<c \<and> f a = (\<Sum>m = 0..<n. diff m c / real (fact m) * (a - c) ^ m) +
```
```    92       diff n ?H / real (fact n) * (a - c) ^ n"
```
```    93       by fastsimp
```
```    94     thus ?thesis by fastsimp
```
```    95   qed
```
```    96 qed
```
```    97
```
```    98 lemma taylor:
```
```    99   assumes INIT: "n>0" "diff 0 = f"
```
```   100   and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
```
```   101   and INTERV: "a \<le> c " "c \<le> b" "a \<le> x" "x \<le> b" "x \<noteq> c"
```
```   102   shows "\<exists> t. (if x<c then (x < t & t < c) else (c < t & t < x)) &
```
```   103     f x = setsum (% m. (diff m c / real (fact m)) * (x - c)^m) {0..<n} +
```
```   104       (diff n t / real (fact n)) * (x - c)^n"
```
```   105 proof (cases "x<c")
```
```   106   case True
```
```   107   note INIT
```
```   108   moreover from DERIV and INTERV
```
```   109   have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
```
```   110     by fastsimp
```
```   111   moreover note True
```
```   112   moreover from INTERV have "c \<le> b" by simp
```
```   113   ultimately have EX: "\<exists>t>x. t < c \<and> f x =
```
```   114     (\<Sum>m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) +
```
```   115       diff n t / real (fact n) * (x - c) ^ n"
```
```   116     by (rule taylor_down)
```
```   117   with True show ?thesis by simp
```
```   118 next
```
```   119   case False
```
```   120   note INIT
```
```   121   moreover from DERIV and INTERV
```
```   122   have "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
```
```   123     by fastsimp
```
```   124   moreover from INTERV have "a \<le> c" by arith
```
```   125   moreover from False and INTERV have "c < x" by arith
```
```   126   ultimately have EX: "\<exists>t>c. t < x \<and> f x =
```
```   127     (\<Sum>m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) +
```
```   128       diff n t / real (fact n) * (x - c) ^ n"
```
```   129     by (rule taylor_up)
```
```   130   with False show ?thesis by simp
```
```   131 qed
```
```   132
```
```   133 end
```