Formalization of Taylor series by Lukas Bulwahn and
Bernhard Haeupler.

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Hyperreal/Taylor.thy	Sun Sep 25 20:14:16 2005 +0200
@@ -0,0 +1,134 @@
+(*  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: "0 < n" "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 "0<n" "((\<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_Id 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: "0 < n" "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 "0<n" "((\<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_Id 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: "0 < n" "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