src/HOL/Hyperreal/MacLaurin.thy
author huffman
Thu May 17 00:45:27 2007 +0200 (2007-05-17)
changeset 22985 501e6dfe4e5a
parent 22983 3314057c3b57
child 23069 cdfff0241c12
permissions -rw-r--r--
cleaned up proof of Maclaurin_sin_bound
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(*  ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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header{*MacLaurin Series*}
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theory MacLaurin
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imports Transcendental
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begin
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subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}
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text{*This is a very long, messy proof even now that it's been broken down
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into lemmas.*}
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lemma Maclaurin_lemma:
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    "0 < h ==>
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     \<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +
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               (B * ((h^n) / real(fact n)))"
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apply (rule_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *
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                 real(fact n) / (h^n)"
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       in exI)
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apply (simp) 
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done
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lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
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by arith
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text{*A crude tactic to differentiate by proof.*}
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ML
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{*
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local
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val deriv_rulesI =
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  [thm "DERIV_Id", thm "DERIV_const", thm "DERIV_cos", thm "DERIV_cmult",
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  thm "DERIV_sin", thm "DERIV_exp", thm "DERIV_inverse", thm "DERIV_pow",
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  thm "DERIV_add", thm "DERIV_diff", thm "DERIV_mult", thm "DERIV_minus",
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  thm "DERIV_inverse_fun", thm "DERIV_quotient", thm "DERIV_fun_pow",
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  thm "DERIV_fun_exp", thm "DERIV_fun_sin", thm "DERIV_fun_cos",
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  thm "DERIV_Id", thm "DERIV_const", thm "DERIV_cos"];
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val DERIV_chain2 = thm "DERIV_chain2";
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in
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exception DERIV_name;
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fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f
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|   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f
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|   get_fun_name _ = raise DERIV_name;
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val deriv_tac =
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  SUBGOAL (fn (prem,i) =>
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   (resolve_tac deriv_rulesI i) ORELSE
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    ((rtac (read_instantiate [("f",get_fun_name prem)]
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                     DERIV_chain2) i) handle DERIV_name => no_tac));;
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val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));
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end
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*}
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lemma Maclaurin_lemma2:
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      "[| \<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t;
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          n = Suc k;
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        difg =
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        (\<lambda>m t. diff m t -
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               ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +
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                B * (t ^ (n - m) / real (fact (n - m)))))|] ==>
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        \<forall>m t. m < n & 0 \<le> t & t \<le> h -->
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                    DERIV (difg m) t :> difg (Suc m) t"
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apply clarify
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apply (rule DERIV_diff)
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apply (simp (no_asm_simp))
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apply (tactic DERIV_tac)
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apply (tactic DERIV_tac)
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apply (rule_tac [2] lemma_DERIV_subst)
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apply (rule_tac [2] DERIV_quotient)
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apply (rule_tac [3] DERIV_const)
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apply (rule_tac [2] DERIV_pow)
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  prefer 3 apply (simp add: fact_diff_Suc)
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 prefer 2 apply simp
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apply (frule_tac m = m in less_add_one, clarify)
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apply (simp del: setsum_op_ivl_Suc)
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apply (insert sumr_offset4 [of 1])
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apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
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apply (rule lemma_DERIV_subst)
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apply (rule DERIV_add)
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apply (rule_tac [2] DERIV_const)
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apply (rule DERIV_sumr, clarify)
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 prefer 2 apply simp
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apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)
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apply (rule DERIV_cmult)
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apply (rule lemma_DERIV_subst)
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apply (best intro: DERIV_chain2 intro!: DERIV_intros)
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apply (subst fact_Suc)
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apply (subst real_of_nat_mult)
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apply (simp add: mult_ac)
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done
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lemma Maclaurin_lemma3:
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  fixes difg :: "nat => real => real" shows
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     "[|\<forall>k t. k < Suc m \<and> 0\<le>t & t\<le>h \<longrightarrow> DERIV (difg k) t :> difg (Suc k) t;
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        \<forall>k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;
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        t < h|]
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     ==> \<exists>ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"
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apply (rule Rolle, assumption, simp)
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apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)
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apply (rule DERIV_unique)
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prefer 2 apply assumption
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apply force
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apply (subgoal_tac "\<forall>ta. 0 \<le> ta & ta \<le> t --> (difg (Suc n)) differentiable ta")
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apply (simp add: differentiable_def)
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apply (blast dest!: DERIV_isCont)
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apply (simp add: differentiable_def, clarify)
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apply (rule_tac x = "difg (Suc (Suc n)) ta" in exI)
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apply force
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apply (simp add: differentiable_def, clarify)
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apply (rule_tac x = "difg (Suc (Suc n)) x" in exI)
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apply force
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done
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lemma Maclaurin:
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   "[| 0 < h; 0 < n; diff 0 = f;
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       \<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
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    ==> \<exists>t. 0 < t &
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              t < h &
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              f h =
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              setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +
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              (diff n t / real (fact n)) * h ^ n"
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apply (case_tac "n = 0", force)
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apply (drule not0_implies_Suc)
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apply (erule exE)
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apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)
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apply (erule exE)
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apply (subgoal_tac "\<exists>g.
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     g = (%t. f t - (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n} + (B * (t^n / real(fact n)))))")
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 prefer 2 apply blast
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apply (erule exE)
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apply (subgoal_tac "g 0 = 0 & g h =0")
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 prefer 2
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 apply (simp del: setsum_op_ivl_Suc)
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 apply (cut_tac n = m and k = 1 in sumr_offset2)
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 apply (simp add: eq_diff_eq' del: setsum_op_ivl_Suc)
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apply (subgoal_tac "\<exists>difg. difg = (%m t. diff m t - (setsum (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) {0..<n-m} + (B * ((t ^ (n - m)) / real (fact (n - m))))))")
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 prefer 2 apply blast
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apply (erule exE)
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apply (subgoal_tac "difg 0 = g")
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 prefer 2 apply simp
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apply (frule Maclaurin_lemma2, assumption+)
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apply (subgoal_tac "\<forall>ma. ma < n --> (\<exists>t. 0 < t & t < h & difg (Suc ma) t = 0) ")
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 apply (drule_tac x = m and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
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 apply (erule impE)
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  apply (simp (no_asm_simp))
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 apply (erule exE)
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 apply (rule_tac x = t in exI)
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 apply (simp del: realpow_Suc fact_Suc)
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apply (subgoal_tac "\<forall>m. m < n --> difg m 0 = 0")
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 prefer 2
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 apply clarify
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 apply simp
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 apply (frule_tac m = ma in less_add_one, clarify)
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 apply (simp del: setsum_op_ivl_Suc)
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apply (insert sumr_offset4 [of 1])
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apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)
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apply (subgoal_tac "\<forall>m. m < n --> (\<exists>t. 0 < t & t < h & DERIV (difg m) t :> 0) ")
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apply (rule allI, rule impI)
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apply (drule_tac x = ma and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)
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apply (erule impE, assumption)
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apply (erule exE)
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apply (rule_tac x = t in exI)
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(* do some tidying up *)
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apply (erule_tac [!] V= "difg = (%m t. diff m t - (setsum (%p. diff (m + p) 0 / real (fact p) * t ^ p) {0..<n-m} + B * (t ^ (n - m) / real (fact (n - m)))))"
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       in thin_rl)
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apply (erule_tac [!] V="g = (%t. f t - (setsum (%m. diff m 0 / real (fact m) * t ^ m) {0..<n} + B * (t ^ n / real (fact n))))"
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       in thin_rl)
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apply (erule_tac [!] V="f h = setsum (%m. diff m 0 / real (fact m) * h ^ m) {0..<n} + B * (h ^ n / real (fact n))"
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       in thin_rl)
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(* back to business *)
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apply (simp (no_asm_simp))
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apply (rule DERIV_unique)
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prefer 2 apply blast
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apply force
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apply (rule allI, induct_tac "ma")
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apply (rule impI, rule Rolle, assumption, simp, simp)
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apply (subgoal_tac "\<forall>t. 0 \<le> t & t \<le> h --> g differentiable t")
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apply (simp add: differentiable_def)
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apply (blast dest: DERIV_isCont)
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apply (simp add: differentiable_def, clarify)
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apply (rule_tac x = "difg (Suc 0) t" in exI)
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apply force
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apply (simp add: differentiable_def, clarify)
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apply (rule_tac x = "difg (Suc 0) x" in exI)
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apply force
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apply safe
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apply force
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apply (frule Maclaurin_lemma3, assumption+, safe)
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apply (rule_tac x = ta in exI, force)
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done
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lemma Maclaurin_objl:
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     "0 < h & 0 < n & diff 0 = f &
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       (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
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    --> (\<exists>t. 0 < t &
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              t < h &
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              f h =
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              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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              diff n t / real (fact n) * h ^ n)"
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by (blast intro: Maclaurin)
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lemma Maclaurin2:
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   "[| 0 < h; diff 0 = f;
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       \<forall>m t.
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          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]
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    ==> \<exists>t. 0 < t &
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              t \<le> h &
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              f h =
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              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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              diff n t / real (fact n) * h ^ n"
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apply (case_tac "n", auto)
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apply (drule Maclaurin, auto)
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done
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lemma Maclaurin2_objl:
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     "0 < h & diff 0 = f &
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       (\<forall>m t.
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          m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)
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    --> (\<exists>t. 0 < t &
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              t \<le> h &
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              f h =
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              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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              diff n t / real (fact n) * h ^ n)"
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by (blast intro: Maclaurin2)
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lemma Maclaurin_minus:
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   "[| h < 0; 0 < n; diff 0 = f;
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       \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]
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    ==> \<exists>t. h < t &
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              t < 0 &
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              f h =
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              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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              diff n t / real (fact n) * h ^ n"
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apply (cut_tac f = "%x. f (-x)"
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        and diff = "%n x. ((- 1) ^ n) * diff n (-x)"
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        and h = "-h" and n = n in Maclaurin_objl)
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apply (simp)
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apply safe
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apply (subst minus_mult_right)
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apply (rule DERIV_cmult)
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apply (rule lemma_DERIV_subst)
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apply (rule DERIV_chain2 [where g=uminus])
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apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_Id)
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prefer 2 apply force
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apply force
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apply (rule_tac x = "-t" in exI, auto)
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apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =
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                    (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")
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apply (rule_tac [2] setsum_cong[OF refl])
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apply (auto simp add: divide_inverse power_mult_distrib [symmetric])
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done
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lemma Maclaurin_minus_objl:
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     "(h < 0 & 0 < n & diff 0 = f &
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       (\<forall>m t.
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          m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))
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    --> (\<exists>t. h < t &
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              t < 0 &
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              f h =
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              (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +
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              diff n t / real (fact n) * h ^ n)"
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by (blast intro: Maclaurin_minus)
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subsection{*More Convenient "Bidirectional" Version.*}
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(* not good for PVS sin_approx, cos_approx *)
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lemma Maclaurin_bi_le_lemma [rule_format]:
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     "0 < n \<longrightarrow>
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       diff 0 0 =
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       (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +
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       diff n 0 * 0 ^ n / real (fact n)"
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by (induct "n", auto)
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lemma Maclaurin_bi_le:
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   "[| diff 0 = f;
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       \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
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    ==> \<exists>t. abs t \<le> abs x &
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              f x =
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              (\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +
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              diff n t / real (fact n) * x ^ n"
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apply (case_tac "n = 0", force)
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   295
apply (case_tac "x = 0")
paulson@15079
   296
apply (rule_tac x = 0 in exI)
nipkow@15539
   297
apply (force simp add: Maclaurin_bi_le_lemma)
paulson@15079
   298
apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
paulson@15079
   299
txt{*Case 1, where @{term "x < 0"}*}
paulson@15079
   300
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
paulson@15079
   301
apply (simp add: abs_if)
paulson@15079
   302
apply (rule_tac x = t in exI)
paulson@15079
   303
apply (simp add: abs_if)
paulson@15079
   304
txt{*Case 2, where @{term "0 < x"}*}
paulson@15079
   305
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
paulson@15079
   306
apply (simp add: abs_if)
paulson@15079
   307
apply (rule_tac x = t in exI)
paulson@15079
   308
apply (simp add: abs_if)
paulson@15079
   309
done
paulson@15079
   310
paulson@15079
   311
lemma Maclaurin_all_lt:
paulson@15079
   312
     "[| diff 0 = f;
paulson@15079
   313
         \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
paulson@15079
   314
        x ~= 0; 0 < n
paulson@15079
   315
      |] ==> \<exists>t. 0 < abs t & abs t < abs x &
nipkow@15539
   316
               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   317
                     (diff n t / real (fact n)) * x ^ n"
paulson@15079
   318
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15079
   319
prefer 2 apply blast
paulson@15079
   320
apply (drule_tac [2] diff=diff in Maclaurin)
paulson@15079
   321
apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
paulson@15229
   322
apply (rule_tac [!] x = t in exI, auto)
paulson@15079
   323
done
paulson@15079
   324
paulson@15079
   325
lemma Maclaurin_all_lt_objl:
paulson@15079
   326
     "diff 0 = f &
paulson@15079
   327
      (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
paulson@15079
   328
      x ~= 0 & 0 < n
paulson@15079
   329
      --> (\<exists>t. 0 < abs t & abs t < abs x &
nipkow@15539
   330
               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   331
                     (diff n t / real (fact n)) * x ^ n)"
paulson@15079
   332
by (blast intro: Maclaurin_all_lt)
paulson@15079
   333
paulson@15079
   334
lemma Maclaurin_zero [rule_format]:
paulson@15079
   335
     "x = (0::real)
paulson@15079
   336
      ==> 0 < n -->
nipkow@15539
   337
          (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
paulson@15079
   338
          diff 0 0"
paulson@15079
   339
by (induct n, auto)
paulson@15079
   340
paulson@15079
   341
lemma Maclaurin_all_le: "[| diff 0 = f;
paulson@15079
   342
        \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
paulson@15079
   343
      |] ==> \<exists>t. abs t \<le> abs x &
nipkow@15539
   344
              f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   345
                    (diff n t / real (fact n)) * x ^ n"
paulson@15079
   346
apply (insert linorder_le_less_linear [of n 0])
paulson@15079
   347
apply (erule disjE, force)
paulson@15079
   348
apply (case_tac "x = 0")
paulson@15079
   349
apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
paulson@15079
   350
apply (drule gr_implies_not0 [THEN not0_implies_Suc])
paulson@15079
   351
apply (rule_tac x = 0 in exI, force)
paulson@15079
   352
apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
paulson@15079
   353
apply (rule_tac x = t in exI, auto)
paulson@15079
   354
done
paulson@15079
   355
paulson@15079
   356
lemma Maclaurin_all_le_objl: "diff 0 = f &
paulson@15079
   357
      (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
paulson@15079
   358
      --> (\<exists>t. abs t \<le> abs x &
nipkow@15539
   359
              f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   360
                    (diff n t / real (fact n)) * x ^ n)"
paulson@15079
   361
by (blast intro: Maclaurin_all_le)
paulson@15079
   362
paulson@15079
   363
paulson@15079
   364
subsection{*Version for Exponential Function*}
paulson@15079
   365
paulson@15079
   366
lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
paulson@15079
   367
      ==> (\<exists>t. 0 < abs t &
paulson@15079
   368
                abs t < abs x &
nipkow@15539
   369
                exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
paulson@15079
   370
                        (exp t / real (fact n)) * x ^ n)"
paulson@15079
   371
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
paulson@15079
   372
paulson@15079
   373
paulson@15079
   374
lemma Maclaurin_exp_le:
paulson@15079
   375
     "\<exists>t. abs t \<le> abs x &
nipkow@15539
   376
            exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
paulson@15079
   377
                       (exp t / real (fact n)) * x ^ n"
paulson@15079
   378
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
paulson@15079
   379
paulson@15079
   380
paulson@15079
   381
subsection{*Version for Sine Function*}
paulson@15079
   382
paulson@15079
   383
lemma MVT2:
paulson@15079
   384
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
huffman@21782
   385
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
paulson@15079
   386
apply (drule MVT)
paulson@15079
   387
apply (blast intro: DERIV_isCont)
paulson@15079
   388
apply (force dest: order_less_imp_le simp add: differentiable_def)
paulson@15079
   389
apply (blast dest: DERIV_unique order_less_imp_le)
paulson@15079
   390
done
paulson@15079
   391
paulson@15079
   392
lemma mod_exhaust_less_4:
paulson@15079
   393
     "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
webertj@20217
   394
by auto
paulson@15079
   395
paulson@15079
   396
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
paulson@15079
   397
     "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
paulson@15251
   398
by (induct "n", auto)
paulson@15079
   399
paulson@15079
   400
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
paulson@15079
   401
     "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
paulson@15251
   402
by (induct "n", auto)
paulson@15079
   403
paulson@15079
   404
lemma Suc_mult_two_diff_one [rule_format, simp]:
paulson@15079
   405
      "0 < n --> Suc (2 * n - 1) = 2*n"
paulson@15251
   406
by (induct "n", auto)
paulson@15079
   407
paulson@15234
   408
paulson@15234
   409
text{*It is unclear why so many variant results are needed.*}
paulson@15079
   410
paulson@15079
   411
lemma Maclaurin_sin_expansion2:
paulson@15079
   412
     "\<exists>t. abs t \<le> abs x &
paulson@15079
   413
       sin x =
nipkow@15539
   414
       (\<Sum>m=0..<n. (if even m then 0
paulson@15079
   415
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   416
                       x ^ m)
paulson@15079
   417
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   418
apply (cut_tac f = sin and n = n and x = x
paulson@15079
   419
        and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
paulson@15079
   420
apply safe
paulson@15079
   421
apply (simp (no_asm))
nipkow@15539
   422
apply (simp (no_asm))
paulson@15079
   423
apply (case_tac "n", clarify, simp, simp)
paulson@15079
   424
apply (rule ccontr, simp)
paulson@15079
   425
apply (drule_tac x = x in spec, simp)
paulson@15079
   426
apply (erule ssubst)
paulson@15079
   427
apply (rule_tac x = t in exI, simp)
nipkow@15536
   428
apply (rule setsum_cong[OF refl])
nipkow@15539
   429
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
paulson@15079
   430
done
paulson@15079
   431
paulson@15234
   432
lemma Maclaurin_sin_expansion:
paulson@15234
   433
     "\<exists>t. sin x =
nipkow@15539
   434
       (\<Sum>m=0..<n. (if even m then 0
paulson@15234
   435
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   436
                       x ^ m)
paulson@15234
   437
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15234
   438
apply (insert Maclaurin_sin_expansion2 [of x n]) 
paulson@15234
   439
apply (blast intro: elim:); 
paulson@15234
   440
done
paulson@15234
   441
paulson@15234
   442
paulson@15234
   443
paulson@15079
   444
lemma Maclaurin_sin_expansion3:
paulson@15079
   445
     "[| 0 < n; 0 < x |] ==>
paulson@15079
   446
       \<exists>t. 0 < t & t < x &
paulson@15079
   447
       sin x =
nipkow@15539
   448
       (\<Sum>m=0..<n. (if even m then 0
paulson@15079
   449
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   450
                       x ^ m)
paulson@15079
   451
      + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   452
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl)
paulson@15079
   453
apply safe
paulson@15079
   454
apply simp
nipkow@15539
   455
apply (simp (no_asm))
paulson@15079
   456
apply (erule ssubst)
paulson@15079
   457
apply (rule_tac x = t in exI, simp)
nipkow@15536
   458
apply (rule setsum_cong[OF refl])
nipkow@15539
   459
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
paulson@15079
   460
done
paulson@15079
   461
paulson@15079
   462
lemma Maclaurin_sin_expansion4:
paulson@15079
   463
     "0 < x ==>
paulson@15079
   464
       \<exists>t. 0 < t & t \<le> x &
paulson@15079
   465
       sin x =
nipkow@15539
   466
       (\<Sum>m=0..<n. (if even m then 0
paulson@15079
   467
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   468
                       x ^ m)
paulson@15079
   469
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   470
apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl)
paulson@15079
   471
apply safe
paulson@15079
   472
apply simp
nipkow@15539
   473
apply (simp (no_asm))
paulson@15079
   474
apply (erule ssubst)
paulson@15079
   475
apply (rule_tac x = t in exI, simp)
nipkow@15536
   476
apply (rule setsum_cong[OF refl])
nipkow@15539
   477
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
paulson@15079
   478
done
paulson@15079
   479
paulson@15079
   480
paulson@15079
   481
subsection{*Maclaurin Expansion for Cosine Function*}
paulson@15079
   482
paulson@15079
   483
lemma sumr_cos_zero_one [simp]:
nipkow@15539
   484
 "(\<Sum>m=0..<(Suc n).
nipkow@15539
   485
     (if even m then (- 1) ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"
paulson@15251
   486
by (induct "n", auto)
paulson@15079
   487
paulson@15079
   488
lemma Maclaurin_cos_expansion:
paulson@15079
   489
     "\<exists>t. abs t \<le> abs x &
paulson@15079
   490
       cos x =
nipkow@15539
   491
       (\<Sum>m=0..<n. (if even m
paulson@15079
   492
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   493
                       else 0) *
nipkow@15539
   494
                       x ^ m)
paulson@15079
   495
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   496
apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
paulson@15079
   497
apply safe
paulson@15079
   498
apply (simp (no_asm))
nipkow@15539
   499
apply (simp (no_asm))
paulson@15079
   500
apply (case_tac "n", simp)
nipkow@15561
   501
apply (simp del: setsum_op_ivl_Suc)
paulson@15079
   502
apply (rule ccontr, simp)
paulson@15079
   503
apply (drule_tac x = x in spec, simp)
paulson@15079
   504
apply (erule ssubst)
paulson@15079
   505
apply (rule_tac x = t in exI, simp)
nipkow@15536
   506
apply (rule setsum_cong[OF refl])
paulson@15234
   507
apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079
   508
done
paulson@15079
   509
paulson@15079
   510
lemma Maclaurin_cos_expansion2:
paulson@15079
   511
     "[| 0 < x; 0 < n |] ==>
paulson@15079
   512
       \<exists>t. 0 < t & t < x &
paulson@15079
   513
       cos x =
nipkow@15539
   514
       (\<Sum>m=0..<n. (if even m
paulson@15079
   515
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   516
                       else 0) *
nipkow@15539
   517
                       x ^ m)
paulson@15079
   518
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   519
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
paulson@15079
   520
apply safe
paulson@15079
   521
apply simp
nipkow@15539
   522
apply (simp (no_asm))
paulson@15079
   523
apply (erule ssubst)
paulson@15079
   524
apply (rule_tac x = t in exI, simp)
nipkow@15536
   525
apply (rule setsum_cong[OF refl])
paulson@15234
   526
apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079
   527
done
paulson@15079
   528
paulson@15234
   529
lemma Maclaurin_minus_cos_expansion:
paulson@15234
   530
     "[| x < 0; 0 < n |] ==>
paulson@15079
   531
       \<exists>t. x < t & t < 0 &
paulson@15079
   532
       cos x =
nipkow@15539
   533
       (\<Sum>m=0..<n. (if even m
paulson@15079
   534
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   535
                       else 0) *
nipkow@15539
   536
                       x ^ m)
paulson@15079
   537
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   538
apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl)
paulson@15079
   539
apply safe
paulson@15079
   540
apply simp
nipkow@15539
   541
apply (simp (no_asm))
paulson@15079
   542
apply (erule ssubst)
paulson@15079
   543
apply (rule_tac x = t in exI, simp)
nipkow@15536
   544
apply (rule setsum_cong[OF refl])
paulson@15234
   545
apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079
   546
done
paulson@15079
   547
paulson@15079
   548
(* ------------------------------------------------------------------------- *)
paulson@15079
   549
(* Version for ln(1 +/- x). Where is it??                                    *)
paulson@15079
   550
(* ------------------------------------------------------------------------- *)
paulson@15079
   551
paulson@15079
   552
lemma sin_bound_lemma:
paulson@15081
   553
    "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
paulson@15079
   554
by auto
paulson@15079
   555
paulson@15079
   556
lemma Maclaurin_sin_bound:
nipkow@15539
   557
  "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15081
   558
  x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
obua@14738
   559
proof -
paulson@15079
   560
  have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
obua@14738
   561
    by (rule_tac mult_right_mono,simp_all)
obua@14738
   562
  note est = this[simplified]
huffman@22985
   563
  let ?diff = "\<lambda>(n::nat) x. if n mod 4 = 0 then sin(x) else if n mod 4 = 1 then cos(x) else if n mod 4 = 2 then -sin(x) else -cos(x)"
huffman@22985
   564
  have diff_0: "?diff 0 = sin" by simp
huffman@22985
   565
  have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"
huffman@22985
   566
    apply (clarify)
huffman@22985
   567
    apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
huffman@22985
   568
    apply (cut_tac m=m in mod_exhaust_less_4)
huffman@22985
   569
    apply (safe, simp_all)
huffman@22985
   570
    apply (rule DERIV_minus, simp)
huffman@22985
   571
    apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
huffman@22985
   572
    done
huffman@22985
   573
  from Maclaurin_all_le [OF diff_0 DERIV_diff]
huffman@22985
   574
  obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and
huffman@22985
   575
    t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +
huffman@22985
   576
      ?diff n t / real (fact n) * x ^ n" by fast
huffman@22985
   577
  have diff_m_0:
huffman@22985
   578
    "\<And>m. ?diff m 0 = (if even m then 0
huffman@22985
   579
         else (- 1) ^ ((m - Suc 0) div 2))"
huffman@22985
   580
    apply (subst even_even_mod_4_iff)
huffman@22985
   581
    apply (cut_tac m=m in mod_exhaust_less_4)
huffman@22985
   582
    apply (elim disjE, simp_all)
huffman@22985
   583
    apply (safe dest!: mod_eqD, simp_all)
huffman@22985
   584
    done
obua@14738
   585
  show ?thesis
huffman@22985
   586
    apply (subst t2)
paulson@15079
   587
    apply (rule sin_bound_lemma)
nipkow@15536
   588
    apply (rule setsum_cong[OF refl])
huffman@22985
   589
    apply (subst diff_m_0, simp)
paulson@15079
   590
    apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
avigad@16775
   591
                   simp add: est mult_nonneg_nonneg mult_ac divide_inverse
paulson@16924
   592
                          power_abs [symmetric] abs_mult)
obua@14738
   593
    done
obua@14738
   594
qed
obua@14738
   595
paulson@15079
   596
end