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src/HOL/Taylor.thy

author | webertj |

Fri, 19 Oct 2012 15:12:52 +0200 | |

changeset 49962 | a8cc904a6820 |

parent 44890 | 22f665a2e91c |

child 56193 | c726ecfb22b6 |

permissions | -rw-r--r-- |

Renamed {left,right}_distrib to distrib_{right,left}.

(* Title: HOL/Taylor.thy 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 fastforce thus ?thesis by fastforce 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 fastforce thus ?thesis by fastforce 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 fastforce 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 fastforce 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