--- a/src/HOL/Hyperreal/Taylor.thy Wed Dec 03 09:53:58 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,134 +0,0 @@
-(* Title: HOL/Hyperreal/Taylor.thy
- ID: $Id$
- Author: Lukas Bulwahn, Bernhard Haeupler, Technische Universitaet Muenchen
-*)
-
-header {* Taylor series *}
-
-theory Taylor
-imports MacLaurin
-begin
-
-text {*
-We use MacLaurin and the translation of the expansion point @{text c} to @{text 0}
-to prove Taylor's theorem.
-*}
-
-lemma taylor_up:
- assumes INIT: "n>0" "diff 0 = f"
- and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
- and INTERV: "a \<le> c" "c < b"
- shows "\<exists> t. c < t & t < b &
- f b = setsum (%m. (diff m c / real (fact m)) * (b - c)^m) {0..<n} +
- (diff n t / real (fact n)) * (b - c)^n"
-proof -
- from INTERV have "0 < b-c" by arith
- moreover
- from INIT have "n>0" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. f (x + c))" by auto
- moreover
- have "ALL m t. m < n & 0 <= t & t <= b - c --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)"
- proof (intro strip)
- fix m t
- assume "m < n & 0 <= t & t <= b - c"
- with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto
- moreover
- from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add)
- ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)"
- by (rule DERIV_chain2)
- thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp
- qed
- ultimately
- have EX:"EX t>0. t < b - c &
- f (b - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) +
- diff n (t + c) / real (fact n) * (b - c) ^ n"
- by (rule Maclaurin)
- show ?thesis
- proof -
- from EX obtain x where
- X: "0 < x & x < b - c &
- f (b - c + c) = (\<Sum>m = 0..<n. diff m (0 + c) / real (fact m) * (b - c) ^ m) +
- diff n (x + c) / real (fact n) * (b - c) ^ n" ..
- let ?H = "x + c"
- from X have "c<?H & ?H<b \<and> f b = (\<Sum>m = 0..<n. diff m c / real (fact m) * (b - c) ^ m) +
- diff n ?H / real (fact n) * (b - c) ^ n"
- by fastsimp
- thus ?thesis by fastsimp
- qed
-qed
-
-lemma taylor_down:
- assumes INIT: "n>0" "diff 0 = f"
- and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
- and INTERV: "a < c" "c \<le> b"
- shows "\<exists> t. a < t & t < c &
- f a = setsum (% m. (diff m c / real (fact m)) * (a - c)^m) {0..<n} +
- (diff n t / real (fact n)) * (a - c)^n"
-proof -
- from INTERV have "a-c < 0" by arith
- moreover
- from INIT have "n>0" "((\<lambda>m x. diff m (x + c)) 0) = (\<lambda>x. f (x + c))" by auto
- moreover
- have "ALL m t. m < n & a-c <= t & t <= 0 --> DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)"
- proof (rule allI impI)+
- fix m t
- assume "m < n & a-c <= t & t <= 0"
- with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)" by auto
- moreover
- from DERIV_ident and DERIV_const have "DERIV (%x. x + c) t :> 1+0" by (rule DERIV_add)
- ultimately have "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c) * (1+0)" by (rule DERIV_chain2)
- thus "DERIV (%x. diff m (x + c)) t :> diff (Suc m) (t + c)" by simp
- qed
- ultimately
- have EX: "EX t>a - c. t < 0 &
- f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) +
- diff n (t + c) / real (fact n) * (a - c) ^ n"
- by (rule Maclaurin_minus)
- show ?thesis
- proof -
- from EX obtain x where X: "a - c < x & x < 0 &
- f (a - c + c) = (SUM m = 0..<n. diff m (0 + c) / real (fact m) * (a - c) ^ m) +
- diff n (x + c) / real (fact n) * (a - c) ^ n" ..
- let ?H = "x + c"
- from X have "a<?H & ?H<c \<and> f a = (\<Sum>m = 0..<n. diff m c / real (fact m) * (a - c) ^ m) +
- diff n ?H / real (fact n) * (a - c) ^ n"
- by fastsimp
- thus ?thesis by fastsimp
- qed
-qed
-
-lemma taylor:
- assumes INIT: "n>0" "diff 0 = f"
- and DERIV: "(\<forall> m t. m < n & a \<le> t & t \<le> b \<longrightarrow> DERIV (diff m) t :> (diff (Suc m) t))"
- and INTERV: "a \<le> c " "c \<le> b" "a \<le> x" "x \<le> b" "x \<noteq> c"
- shows "\<exists> t. (if x<c then (x < t & t < c) else (c < t & t < x)) &
- f x = setsum (% m. (diff m c / real (fact m)) * (x - c)^m) {0..<n} +
- (diff n t / real (fact n)) * (x - c)^n"
-proof (cases "x<c")
- case True
- note INIT
- moreover from DERIV and INTERV
- have "\<forall>m t. m < n \<and> x \<le> t \<and> t \<le> b \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
- by fastsimp
- moreover note True
- moreover from INTERV have "c \<le> b" by simp
- ultimately have EX: "\<exists>t>x. t < c \<and> f x =
- (\<Sum>m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) +
- diff n t / real (fact n) * (x - c) ^ n"
- by (rule taylor_down)
- with True show ?thesis by simp
-next
- case False
- note INIT
- moreover from DERIV and INTERV
- have "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> x \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
- by fastsimp
- moreover from INTERV have "a \<le> c" by arith
- moreover from False and INTERV have "c < x" by arith
- ultimately have EX: "\<exists>t>c. t < x \<and> f x =
- (\<Sum>m = 0..<n. diff m c / real (fact m) * (x - c) ^ m) +
- diff n t / real (fact n) * (x - c) ^ n"
- by (rule taylor_up)
- with False show ?thesis by simp
-qed
-
-end