src/HOL/Hyperreal/MacLaurin.thy
author avigad
Wed Jul 13 19:49:07 2005 +0200 (2005-07-13)
changeset 16819 00d8f9300d13
parent 16775 c1b87ef4a1c3
child 16924 04246269386e
permissions -rw-r--r--
Additions to the Real (and Hyperreal) libraries:
RealDef.thy: lemmas relating nats, ints, and reals
reversed direction of real_of_int_mult, etc. (now they agree with nat versions)
SEQ.thy and Series.thy: various additions
RComplete.thy: lemmas involving floor and ceiling
introduced natfloor and natceiling
Log.thy: various additions
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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 Log
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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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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_rulesI = [DERIV_Id,DERIV_const,DERIV_cos,DERIV_cmult,
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                    DERIV_sin, DERIV_exp, DERIV_inverse,DERIV_pow,
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                    DERIV_add, DERIV_diff, DERIV_mult, DERIV_minus,
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                    DERIV_inverse_fun,DERIV_quotient,DERIV_fun_pow,
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                    DERIV_fun_exp,DERIV_fun_sin,DERIV_fun_cos,
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                    DERIV_Id,DERIV_const,DERIV_cos];
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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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*}
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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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     "[|\<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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apply (case_tac "x = 0")
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apply (rule_tac x = 0 in exI)
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apply (force simp add: Maclaurin_bi_le_lemma)
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apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
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txt{*Case 1, where @{term "x < 0"}*}
paulson@15079
   291
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
paulson@15079
   292
apply (simp add: abs_if)
paulson@15079
   293
apply (rule_tac x = t in exI)
paulson@15079
   294
apply (simp add: abs_if)
paulson@15079
   295
txt{*Case 2, where @{term "0 < x"}*}
paulson@15079
   296
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
paulson@15079
   297
apply (simp add: abs_if)
paulson@15079
   298
apply (rule_tac x = t in exI)
paulson@15079
   299
apply (simp add: abs_if)
paulson@15079
   300
done
paulson@15079
   301
paulson@15079
   302
lemma Maclaurin_all_lt:
paulson@15079
   303
     "[| diff 0 = f;
paulson@15079
   304
         \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
paulson@15079
   305
        x ~= 0; 0 < n
paulson@15079
   306
      |] ==> \<exists>t. 0 < abs t & abs t < abs x &
nipkow@15539
   307
               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   308
                     (diff n t / real (fact n)) * x ^ n"
paulson@15079
   309
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15079
   310
prefer 2 apply blast
paulson@15079
   311
apply (drule_tac [2] diff=diff in Maclaurin)
paulson@15079
   312
apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
paulson@15229
   313
apply (rule_tac [!] x = t in exI, auto)
paulson@15079
   314
done
paulson@15079
   315
paulson@15079
   316
lemma Maclaurin_all_lt_objl:
paulson@15079
   317
     "diff 0 = f &
paulson@15079
   318
      (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
paulson@15079
   319
      x ~= 0 & 0 < n
paulson@15079
   320
      --> (\<exists>t. 0 < abs t & abs t < abs x &
nipkow@15539
   321
               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   322
                     (diff n t / real (fact n)) * x ^ n)"
paulson@15079
   323
by (blast intro: Maclaurin_all_lt)
paulson@15079
   324
paulson@15079
   325
lemma Maclaurin_zero [rule_format]:
paulson@15079
   326
     "x = (0::real)
paulson@15079
   327
      ==> 0 < n -->
nipkow@15539
   328
          (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =
paulson@15079
   329
          diff 0 0"
paulson@15079
   330
by (induct n, auto)
paulson@15079
   331
paulson@15079
   332
lemma Maclaurin_all_le: "[| diff 0 = f;
paulson@15079
   333
        \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
paulson@15079
   334
      |] ==> \<exists>t. abs t \<le> abs x &
nipkow@15539
   335
              f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   336
                    (diff n t / real (fact n)) * x ^ n"
paulson@15079
   337
apply (insert linorder_le_less_linear [of n 0])
paulson@15079
   338
apply (erule disjE, force)
paulson@15079
   339
apply (case_tac "x = 0")
paulson@15079
   340
apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
paulson@15079
   341
apply (drule gr_implies_not0 [THEN not0_implies_Suc])
paulson@15079
   342
apply (rule_tac x = 0 in exI, force)
paulson@15079
   343
apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
paulson@15079
   344
apply (rule_tac x = t in exI, auto)
paulson@15079
   345
done
paulson@15079
   346
paulson@15079
   347
lemma Maclaurin_all_le_objl: "diff 0 = f &
paulson@15079
   348
      (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
paulson@15079
   349
      --> (\<exists>t. abs t \<le> abs x &
nipkow@15539
   350
              f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   351
                    (diff n t / real (fact n)) * x ^ n)"
paulson@15079
   352
by (blast intro: Maclaurin_all_le)
paulson@15079
   353
paulson@15079
   354
paulson@15079
   355
subsection{*Version for Exponential Function*}
paulson@15079
   356
paulson@15079
   357
lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
paulson@15079
   358
      ==> (\<exists>t. 0 < abs t &
paulson@15079
   359
                abs t < abs x &
nipkow@15539
   360
                exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
paulson@15079
   361
                        (exp t / real (fact n)) * x ^ n)"
paulson@15079
   362
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)
paulson@15079
   363
paulson@15079
   364
paulson@15079
   365
lemma Maclaurin_exp_le:
paulson@15079
   366
     "\<exists>t. abs t \<le> abs x &
nipkow@15539
   367
            exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +
paulson@15079
   368
                       (exp t / real (fact n)) * x ^ n"
paulson@15079
   369
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
paulson@15079
   370
paulson@15079
   371
paulson@15079
   372
subsection{*Version for Sine Function*}
paulson@15079
   373
paulson@15079
   374
lemma MVT2:
paulson@15079
   375
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
paulson@15079
   376
      ==> \<exists>z. a < z & z < b & (f b - f a = (b - a) * f'(z))"
paulson@15079
   377
apply (drule MVT)
paulson@15079
   378
apply (blast intro: DERIV_isCont)
paulson@15079
   379
apply (force dest: order_less_imp_le simp add: differentiable_def)
paulson@15079
   380
apply (blast dest: DERIV_unique order_less_imp_le)
paulson@15079
   381
done
paulson@15079
   382
paulson@15079
   383
lemma mod_exhaust_less_4:
paulson@15079
   384
     "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
paulson@15079
   385
by (case_tac "m mod 4", auto, arith)
paulson@15079
   386
paulson@15079
   387
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
paulson@15079
   388
     "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
paulson@15251
   389
by (induct "n", auto)
paulson@15079
   390
paulson@15079
   391
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
paulson@15079
   392
     "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
paulson@15251
   393
by (induct "n", auto)
paulson@15079
   394
paulson@15079
   395
lemma Suc_mult_two_diff_one [rule_format, simp]:
paulson@15079
   396
      "0 < n --> Suc (2 * n - 1) = 2*n"
paulson@15251
   397
by (induct "n", auto)
paulson@15079
   398
paulson@15234
   399
paulson@15234
   400
text{*It is unclear why so many variant results are needed.*}
paulson@15079
   401
paulson@15079
   402
lemma Maclaurin_sin_expansion2:
paulson@15079
   403
     "\<exists>t. abs t \<le> abs x &
paulson@15079
   404
       sin x =
nipkow@15539
   405
       (\<Sum>m=0..<n. (if even m then 0
paulson@15079
   406
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   407
                       x ^ m)
paulson@15079
   408
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   409
apply (cut_tac f = sin and n = n and x = x
paulson@15079
   410
        and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
paulson@15079
   411
apply safe
paulson@15079
   412
apply (simp (no_asm))
nipkow@15539
   413
apply (simp (no_asm))
paulson@15079
   414
apply (case_tac "n", clarify, simp, simp)
paulson@15079
   415
apply (rule ccontr, simp)
paulson@15079
   416
apply (drule_tac x = x in spec, simp)
paulson@15079
   417
apply (erule ssubst)
paulson@15079
   418
apply (rule_tac x = t in exI, simp)
nipkow@15536
   419
apply (rule setsum_cong[OF refl])
nipkow@15539
   420
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
paulson@15079
   421
done
paulson@15079
   422
paulson@15234
   423
lemma Maclaurin_sin_expansion:
paulson@15234
   424
     "\<exists>t. sin x =
nipkow@15539
   425
       (\<Sum>m=0..<n. (if even m then 0
paulson@15234
   426
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   427
                       x ^ m)
paulson@15234
   428
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15234
   429
apply (insert Maclaurin_sin_expansion2 [of x n]) 
paulson@15234
   430
apply (blast intro: elim:); 
paulson@15234
   431
done
paulson@15234
   432
paulson@15234
   433
paulson@15234
   434
paulson@15079
   435
lemma Maclaurin_sin_expansion3:
paulson@15079
   436
     "[| 0 < n; 0 < x |] ==>
paulson@15079
   437
       \<exists>t. 0 < t & t < x &
paulson@15079
   438
       sin x =
nipkow@15539
   439
       (\<Sum>m=0..<n. (if even m then 0
paulson@15079
   440
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   441
                       x ^ m)
paulson@15079
   442
      + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   443
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
   444
apply safe
paulson@15079
   445
apply simp
nipkow@15539
   446
apply (simp (no_asm))
paulson@15079
   447
apply (erule ssubst)
paulson@15079
   448
apply (rule_tac x = t in exI, simp)
nipkow@15536
   449
apply (rule setsum_cong[OF refl])
nipkow@15539
   450
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
paulson@15079
   451
done
paulson@15079
   452
paulson@15079
   453
lemma Maclaurin_sin_expansion4:
paulson@15079
   454
     "0 < x ==>
paulson@15079
   455
       \<exists>t. 0 < t & t \<le> x &
paulson@15079
   456
       sin x =
nipkow@15539
   457
       (\<Sum>m=0..<n. (if even m then 0
paulson@15079
   458
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
nipkow@15539
   459
                       x ^ m)
paulson@15079
   460
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   461
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
   462
apply safe
paulson@15079
   463
apply simp
nipkow@15539
   464
apply (simp (no_asm))
paulson@15079
   465
apply (erule ssubst)
paulson@15079
   466
apply (rule_tac x = t in exI, simp)
nipkow@15536
   467
apply (rule setsum_cong[OF refl])
nipkow@15539
   468
apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
paulson@15079
   469
done
paulson@15079
   470
paulson@15079
   471
paulson@15079
   472
subsection{*Maclaurin Expansion for Cosine Function*}
paulson@15079
   473
paulson@15079
   474
lemma sumr_cos_zero_one [simp]:
nipkow@15539
   475
 "(\<Sum>m=0..<(Suc n).
nipkow@15539
   476
     (if even m then (- 1) ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"
paulson@15251
   477
by (induct "n", auto)
paulson@15079
   478
paulson@15079
   479
lemma Maclaurin_cos_expansion:
paulson@15079
   480
     "\<exists>t. abs t \<le> abs x &
paulson@15079
   481
       cos x =
nipkow@15539
   482
       (\<Sum>m=0..<n. (if even m
paulson@15079
   483
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   484
                       else 0) *
nipkow@15539
   485
                       x ^ m)
paulson@15079
   486
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   487
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
   488
apply safe
paulson@15079
   489
apply (simp (no_asm))
nipkow@15539
   490
apply (simp (no_asm))
paulson@15079
   491
apply (case_tac "n", simp)
nipkow@15561
   492
apply (simp del: setsum_op_ivl_Suc)
paulson@15079
   493
apply (rule ccontr, simp)
paulson@15079
   494
apply (drule_tac x = x in spec, simp)
paulson@15079
   495
apply (erule ssubst)
paulson@15079
   496
apply (rule_tac x = t in exI, simp)
nipkow@15536
   497
apply (rule setsum_cong[OF refl])
paulson@15234
   498
apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079
   499
done
paulson@15079
   500
paulson@15079
   501
lemma Maclaurin_cos_expansion2:
paulson@15079
   502
     "[| 0 < x; 0 < n |] ==>
paulson@15079
   503
       \<exists>t. 0 < t & t < x &
paulson@15079
   504
       cos x =
nipkow@15539
   505
       (\<Sum>m=0..<n. (if even m
paulson@15079
   506
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   507
                       else 0) *
nipkow@15539
   508
                       x ^ m)
paulson@15079
   509
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   510
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
   511
apply safe
paulson@15079
   512
apply simp
nipkow@15539
   513
apply (simp (no_asm))
paulson@15079
   514
apply (erule ssubst)
paulson@15079
   515
apply (rule_tac x = t in exI, simp)
nipkow@15536
   516
apply (rule setsum_cong[OF refl])
paulson@15234
   517
apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079
   518
done
paulson@15079
   519
paulson@15234
   520
lemma Maclaurin_minus_cos_expansion:
paulson@15234
   521
     "[| x < 0; 0 < n |] ==>
paulson@15079
   522
       \<exists>t. x < t & t < 0 &
paulson@15079
   523
       cos x =
nipkow@15539
   524
       (\<Sum>m=0..<n. (if even m
paulson@15079
   525
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   526
                       else 0) *
nipkow@15539
   527
                       x ^ m)
paulson@15079
   528
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   529
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
   530
apply safe
paulson@15079
   531
apply simp
nipkow@15539
   532
apply (simp (no_asm))
paulson@15079
   533
apply (erule ssubst)
paulson@15079
   534
apply (rule_tac x = t in exI, simp)
nipkow@15536
   535
apply (rule setsum_cong[OF refl])
paulson@15234
   536
apply (auto simp add: cos_zero_iff even_mult_two_ex)
paulson@15079
   537
done
paulson@15079
   538
paulson@15079
   539
(* ------------------------------------------------------------------------- *)
paulson@15079
   540
(* Version for ln(1 +/- x). Where is it??                                    *)
paulson@15079
   541
(* ------------------------------------------------------------------------- *)
paulson@15079
   542
paulson@15079
   543
lemma sin_bound_lemma:
paulson@15081
   544
    "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
paulson@15079
   545
by auto
paulson@15079
   546
paulson@15079
   547
lemma Maclaurin_sin_bound:
nipkow@15539
   548
  "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15081
   549
  x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
obua@14738
   550
proof -
paulson@15079
   551
  have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
obua@14738
   552
    by (rule_tac mult_right_mono,simp_all)
obua@14738
   553
  note est = this[simplified]
obua@14738
   554
  show ?thesis
paulson@15079
   555
    apply (cut_tac f=sin and n=n and x=x and
obua@14738
   556
      diff = "%n 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)"
obua@14738
   557
      in Maclaurin_all_le_objl)
paulson@15079
   558
    apply safe
paulson@15079
   559
    apply simp
paulson@15944
   560
    apply (subst (1 2 3) mod_Suc_eq_Suc_mod)
paulson@15079
   561
    apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)
obua@14738
   562
    apply (rule DERIV_minus, simp+)
obua@14738
   563
    apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
paulson@15079
   564
    apply (erule ssubst)
paulson@15079
   565
    apply (rule sin_bound_lemma)
nipkow@15536
   566
    apply (rule setsum_cong[OF refl])
nipkow@15536
   567
    apply (rule_tac f = "%u. u * (x^xa)" in arg_cong)
obua@14738
   568
    apply (subst even_even_mod_4_iff)
nipkow@15536
   569
    apply (cut_tac m=xa in mod_exhaust_less_4, simp, safe)
obua@14738
   570
    apply (simp_all add:even_num_iff)
obua@14738
   571
    apply (drule lemma_even_mod_4_div_2[simplified])
paulson@15079
   572
    apply(simp add: numeral_2_eq_2 divide_inverse)
paulson@15079
   573
    apply (drule lemma_odd_mod_4_div_2)
paulson@15079
   574
    apply (simp add: numeral_2_eq_2 divide_inverse)
paulson@15079
   575
    apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
avigad@16775
   576
                   simp add: est mult_nonneg_nonneg mult_ac divide_inverse
paulson@15079
   577
                          power_abs [symmetric])
obua@14738
   578
    done
obua@14738
   579
qed
obua@14738
   580
paulson@15079
   581
end