src/HOL/Hyperreal/MacLaurin.thy
author paulson
Wed Jul 28 16:25:40 2004 +0200 (2004-07-28)
changeset 15081 32402f5624d1
parent 15079 2ef899e4526d
child 15131 c69542757a4d
permissions -rw-r--r--
abs notation
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(*  Title       : MacLaurin.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 2001 University of Edinburgh
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    Description : MacLaurin series
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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theory MacLaurin = Log:
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lemma sumr_offset: "sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
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by (induct_tac "n", auto)
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lemma sumr_offset2: "\<forall>f. sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"
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by (induct_tac "n", auto)
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lemma sumr_offset3: "sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
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by (simp  add: sumr_offset)
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lemma sumr_offset4: "\<forall>n f. sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"
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by (simp add: sumr_offset)
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lemma sumr_from_1_from_0: "0 < n ==>
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      sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else
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             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) =
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      sumr 0 (Suc n) (%n. (if even(n) then 0 else
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             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)"
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by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
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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 = sumr 0 n (%m. (j m / real (fact m)) * (h^m)) +
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               (B * ((h^n) / real(fact n)))"
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by (rule_tac x = "(f h - sumr 0 n (%m. (j m / real (fact m)) * h^m)) *
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                 real(fact n) / (h^n)"
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       in exI, auto)
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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: sumr_Suc)
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apply (insert sumr_offset4 [of 1])
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apply (simp del: sumr_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: inverse_mult_distrib 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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              sumr 0 n (%m. (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 = 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 - (sumr 0 n (%m. (diff m 0 / real(fact m)) * t^m) + (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: sumr_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: sumr_Suc)
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apply (subgoal_tac "\<exists>difg. difg = (%m t. diff m t - (sumr 0 (n - m) (%p. (diff (m + p) 0 / real (fact p)) * (t ^ p)) + (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: sumr_Suc)
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apply (insert sumr_offset4 [of 1])
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apply (simp del: sumr_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 - (sumr 0 (n - m) (%p. diff (m + p) 0 / real (fact p) * t ^ p) + 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 - (sumr 0 n (%m. diff m 0 / real (fact m) * t ^ m) + B * (t ^ n / real (fact n))))"
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       in thin_rl)
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apply (erule_tac [!] V="f h = sumr 0 n (%m. diff m 0 / real (fact m) * h ^ m) + 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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              sumr 0 n (%m. 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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              sumr 0 n (%m. 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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              sumr 0 n (%m. 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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              sumr 0 n (%m. 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] sumr_fun_eq)
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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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              sumr 0 n (%m. 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)"
paulson@15079
   292
by (induct_tac "n", auto)
obua@14738
   293
paulson@15079
   294
lemma Maclaurin_bi_le:
paulson@15079
   295
   "[| diff 0 = f;
paulson@15079
   296
       \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]
paulson@15079
   297
    ==> \<exists>t. abs t \<le> abs x &
paulson@15079
   298
              f x =
paulson@15079
   299
              sumr 0 n (%m. diff m 0 / real (fact m) * x ^ m) +
paulson@15079
   300
              diff n t / real (fact n) * x ^ n"
paulson@15079
   301
apply (case_tac "n = 0", force)
paulson@15079
   302
apply (case_tac "x = 0")
paulson@15079
   303
apply (rule_tac x = 0 in exI)
paulson@15079
   304
apply (force simp add: Maclaurin_bi_le_lemma)
paulson@15079
   305
apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)
paulson@15079
   306
txt{*Case 1, where @{term "x < 0"}*}
paulson@15079
   307
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)
paulson@15079
   308
apply (simp add: abs_if)
paulson@15079
   309
apply (rule_tac x = t in exI)
paulson@15079
   310
apply (simp add: abs_if)
paulson@15079
   311
txt{*Case 2, where @{term "0 < x"}*}
paulson@15079
   312
apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)
paulson@15079
   313
apply (simp add: abs_if)
paulson@15079
   314
apply (rule_tac x = t in exI)
paulson@15079
   315
apply (simp add: abs_if)
paulson@15079
   316
done
paulson@15079
   317
paulson@15079
   318
lemma Maclaurin_all_lt:
paulson@15079
   319
     "[| diff 0 = f;
paulson@15079
   320
         \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;
paulson@15079
   321
        x ~= 0; 0 < n
paulson@15079
   322
      |] ==> \<exists>t. 0 < abs t & abs t < abs x &
paulson@15079
   323
               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   324
                     (diff n t / real (fact n)) * x ^ n"
paulson@15079
   325
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15079
   326
prefer 2 apply blast
paulson@15079
   327
apply (drule_tac [2] diff=diff in Maclaurin)
paulson@15079
   328
apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)
paulson@15079
   329
apply (rule_tac [!] x = t in exI, auto, arith+)
paulson@15079
   330
done
paulson@15079
   331
paulson@15079
   332
lemma Maclaurin_all_lt_objl:
paulson@15079
   333
     "diff 0 = f &
paulson@15079
   334
      (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &
paulson@15079
   335
      x ~= 0 & 0 < n
paulson@15079
   336
      --> (\<exists>t. 0 < abs t & abs t < abs x &
paulson@15079
   337
               f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   338
                     (diff n t / real (fact n)) * x ^ n)"
paulson@15079
   339
by (blast intro: Maclaurin_all_lt)
paulson@15079
   340
paulson@15079
   341
lemma Maclaurin_zero [rule_format]:
paulson@15079
   342
     "x = (0::real)
paulson@15079
   343
      ==> 0 < n -->
paulson@15079
   344
          sumr 0 n (%m. (diff m (0::real) / real (fact m)) * x ^ m) =
paulson@15079
   345
          diff 0 0"
paulson@15079
   346
by (induct n, auto)
paulson@15079
   347
paulson@15079
   348
lemma Maclaurin_all_le: "[| diff 0 = f;
paulson@15079
   349
        \<forall>m x. DERIV (diff m) x :> diff (Suc m) x
paulson@15079
   350
      |] ==> \<exists>t. abs t \<le> abs x &
paulson@15079
   351
              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   352
                    (diff n t / real (fact n)) * x ^ n"
paulson@15079
   353
apply (insert linorder_le_less_linear [of n 0])
paulson@15079
   354
apply (erule disjE, force)
paulson@15079
   355
apply (case_tac "x = 0")
paulson@15079
   356
apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)
paulson@15079
   357
apply (drule gr_implies_not0 [THEN not0_implies_Suc])
paulson@15079
   358
apply (rule_tac x = 0 in exI, force)
paulson@15079
   359
apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)
paulson@15079
   360
apply (rule_tac x = t in exI, auto)
paulson@15079
   361
done
paulson@15079
   362
paulson@15079
   363
lemma Maclaurin_all_le_objl: "diff 0 = f &
paulson@15079
   364
      (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)
paulson@15079
   365
      --> (\<exists>t. abs t \<le> abs x &
paulson@15079
   366
              f x = sumr 0 n (%m. (diff m 0 / real (fact m)) * x ^ m) +
paulson@15079
   367
                    (diff n t / real (fact n)) * x ^ n)"
paulson@15079
   368
by (blast intro: Maclaurin_all_le)
paulson@15079
   369
paulson@15079
   370
paulson@15079
   371
subsection{*Version for Exponential Function*}
paulson@15079
   372
paulson@15079
   373
lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]
paulson@15079
   374
      ==> (\<exists>t. 0 < abs t &
paulson@15079
   375
                abs t < abs x &
paulson@15079
   376
                exp x = sumr 0 n (%m. (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_lt_objl, auto)
paulson@15079
   379
paulson@15079
   380
paulson@15079
   381
lemma Maclaurin_exp_le:
paulson@15079
   382
     "\<exists>t. abs t \<le> abs x &
paulson@15079
   383
            exp x = sumr 0 n (%m. (x ^ m) / real (fact m)) +
paulson@15079
   384
                       (exp t / real (fact n)) * x ^ n"
paulson@15079
   385
by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)
paulson@15079
   386
paulson@15079
   387
paulson@15079
   388
subsection{*Version for Sine Function*}
paulson@15079
   389
paulson@15079
   390
lemma MVT2:
paulson@15079
   391
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
paulson@15079
   392
      ==> \<exists>z. a < z & z < b & (f b - f a = (b - a) * f'(z))"
paulson@15079
   393
apply (drule MVT)
paulson@15079
   394
apply (blast intro: DERIV_isCont)
paulson@15079
   395
apply (force dest: order_less_imp_le simp add: differentiable_def)
paulson@15079
   396
apply (blast dest: DERIV_unique order_less_imp_le)
paulson@15079
   397
done
paulson@15079
   398
paulson@15079
   399
lemma mod_exhaust_less_4:
paulson@15079
   400
     "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"
paulson@15079
   401
by (case_tac "m mod 4", auto, arith)
paulson@15079
   402
paulson@15079
   403
lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:
paulson@15079
   404
     "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"
paulson@15079
   405
by (induct_tac "n", auto)
paulson@15079
   406
paulson@15079
   407
lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:
paulson@15079
   408
     "0 < n --> Suc (Suc (4*n - 2)) = 4*n"
paulson@15079
   409
by (induct_tac "n", auto)
paulson@15079
   410
paulson@15079
   411
lemma Suc_mult_two_diff_one [rule_format, simp]:
paulson@15079
   412
      "0 < n --> Suc (2 * n - 1) = 2*n"
paulson@15079
   413
by (induct_tac "n", auto)
paulson@15079
   414
paulson@15079
   415
lemma Maclaurin_sin_expansion:
paulson@15079
   416
     "\<exists>t. sin x =
paulson@15079
   417
       (sumr 0 n (%m. (if even m then 0
paulson@15079
   418
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15079
   419
                       x ^ m))
paulson@15079
   420
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   421
apply (cut_tac f = sin and n = n and x = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
paulson@15079
   422
apply safe
paulson@15079
   423
apply (simp (no_asm))
paulson@15079
   424
apply (simp (no_asm))
paulson@15079
   425
apply (case_tac "n", clarify, simp)
paulson@15079
   426
apply (drule_tac x = 0 in spec, simp, simp)
paulson@15079
   427
apply (rule ccontr, simp)
paulson@15079
   428
apply (drule_tac x = x in spec, simp)
paulson@15079
   429
apply (erule ssubst)
paulson@15079
   430
apply (rule_tac x = t in exI, simp)
paulson@15079
   431
apply (rule sumr_fun_eq)
paulson@15079
   432
apply (auto simp add: odd_Suc_mult_two_ex)
paulson@15079
   433
apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
paulson@15079
   434
(*Could sin_zero_iff help?*)
paulson@15079
   435
done
paulson@15079
   436
paulson@15079
   437
lemma Maclaurin_sin_expansion2:
paulson@15079
   438
     "\<exists>t. abs t \<le> abs x &
paulson@15079
   439
       sin x =
paulson@15079
   440
       (sumr 0 n (%m. (if even m then 0
paulson@15079
   441
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15079
   442
                       x ^ m))
paulson@15079
   443
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   444
apply (cut_tac f = sin and n = n and x = x
paulson@15079
   445
        and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
paulson@15079
   446
apply safe
paulson@15079
   447
apply (simp (no_asm))
paulson@15079
   448
apply (simp (no_asm))
paulson@15079
   449
apply (case_tac "n", clarify, simp, simp)
paulson@15079
   450
apply (rule ccontr, simp)
paulson@15079
   451
apply (drule_tac x = x in spec, simp)
paulson@15079
   452
apply (erule ssubst)
paulson@15079
   453
apply (rule_tac x = t in exI, simp)
paulson@15079
   454
apply (rule sumr_fun_eq)
paulson@15079
   455
apply (auto simp add: odd_Suc_mult_two_ex)
paulson@15079
   456
apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
paulson@15079
   457
done
paulson@15079
   458
paulson@15079
   459
lemma Maclaurin_sin_expansion3:
paulson@15079
   460
     "[| 0 < n; 0 < x |] ==>
paulson@15079
   461
       \<exists>t. 0 < t & t < x &
paulson@15079
   462
       sin x =
paulson@15079
   463
       (sumr 0 n (%m. (if even m then 0
paulson@15079
   464
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15079
   465
                       x ^ m))
paulson@15079
   466
      + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   467
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
   468
apply safe
paulson@15079
   469
apply simp
paulson@15079
   470
apply (simp (no_asm))
paulson@15079
   471
apply (erule ssubst)
paulson@15079
   472
apply (rule_tac x = t in exI, simp)
paulson@15079
   473
apply (rule sumr_fun_eq)
paulson@15079
   474
apply (auto simp add: odd_Suc_mult_two_ex)
paulson@15079
   475
apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
paulson@15079
   476
done
paulson@15079
   477
paulson@15079
   478
lemma Maclaurin_sin_expansion4:
paulson@15079
   479
     "0 < x ==>
paulson@15079
   480
       \<exists>t. 0 < t & t \<le> x &
paulson@15079
   481
       sin x =
paulson@15079
   482
       (sumr 0 n (%m. (if even m then 0
paulson@15079
   483
                       else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15079
   484
                       x ^ m))
paulson@15079
   485
      + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   486
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
   487
apply safe
paulson@15079
   488
apply simp
paulson@15079
   489
apply (simp (no_asm))
paulson@15079
   490
apply (erule ssubst)
paulson@15079
   491
apply (rule_tac x = t in exI, simp)
paulson@15079
   492
apply (rule sumr_fun_eq)
paulson@15079
   493
apply (auto simp add: odd_Suc_mult_two_ex)
paulson@15079
   494
apply (auto simp add: even_mult_two_ex simp del: fact_Suc realpow_Suc)
paulson@15079
   495
done
paulson@15079
   496
paulson@15079
   497
paulson@15079
   498
subsection{*Maclaurin Expansion for Cosine Function*}
paulson@15079
   499
paulson@15079
   500
lemma sumr_cos_zero_one [simp]:
paulson@15079
   501
     "sumr 0 (Suc n)
paulson@15079
   502
         (%m. (if even m
paulson@15079
   503
               then (- 1) ^ (m div 2)/(real  (fact m))
paulson@15079
   504
               else 0) *
paulson@15079
   505
              0 ^ m) = 1"
paulson@15079
   506
by (induct_tac "n", auto)
paulson@15079
   507
paulson@15079
   508
lemma Maclaurin_cos_expansion:
paulson@15079
   509
     "\<exists>t. abs t \<le> abs x &
paulson@15079
   510
       cos x =
paulson@15079
   511
       (sumr 0 n (%m. (if even m
paulson@15079
   512
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   513
                       else 0) *
paulson@15079
   514
                       x ^ m))
paulson@15079
   515
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   516
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
   517
apply safe
paulson@15079
   518
apply (simp (no_asm))
paulson@15079
   519
apply (simp (no_asm))
paulson@15079
   520
apply (case_tac "n", simp)
paulson@15079
   521
apply (simp del: sumr_Suc)
paulson@15079
   522
apply (rule ccontr, simp)
paulson@15079
   523
apply (drule_tac x = x in spec, simp)
paulson@15079
   524
apply (erule ssubst)
paulson@15079
   525
apply (rule_tac x = t in exI, simp)
paulson@15079
   526
apply (rule sumr_fun_eq)
paulson@15079
   527
apply (auto simp add: odd_Suc_mult_two_ex)
paulson@15079
   528
apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
paulson@15079
   529
apply (simp add: mult_commute [of _ pi])
paulson@15079
   530
done
paulson@15079
   531
paulson@15079
   532
lemma Maclaurin_cos_expansion2:
paulson@15079
   533
     "[| 0 < x; 0 < n |] ==>
paulson@15079
   534
       \<exists>t. 0 < t & t < x &
paulson@15079
   535
       cos x =
paulson@15079
   536
       (sumr 0 n (%m. (if even m
paulson@15079
   537
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   538
                       else 0) *
paulson@15079
   539
                       x ^ m))
paulson@15079
   540
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   541
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
   542
apply safe
paulson@15079
   543
apply simp
paulson@15079
   544
apply (simp (no_asm))
paulson@15079
   545
apply (erule ssubst)
paulson@15079
   546
apply (rule_tac x = t in exI, simp)
paulson@15079
   547
apply (rule sumr_fun_eq)
paulson@15079
   548
apply (auto simp add: odd_Suc_mult_two_ex)
paulson@15079
   549
apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
paulson@15079
   550
apply (simp add: mult_commute [of _ pi])
paulson@15079
   551
done
paulson@15079
   552
paulson@15079
   553
lemma Maclaurin_minus_cos_expansion: "[| x < 0; 0 < n |] ==>
paulson@15079
   554
       \<exists>t. x < t & t < 0 &
paulson@15079
   555
       cos x =
paulson@15079
   556
       (sumr 0 n (%m. (if even m
paulson@15079
   557
                       then (- 1) ^ (m div 2)/(real (fact m))
paulson@15079
   558
                       else 0) *
paulson@15079
   559
                       x ^ m))
paulson@15079
   560
      + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
paulson@15079
   561
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
   562
apply safe
paulson@15079
   563
apply simp
paulson@15079
   564
apply (simp (no_asm))
paulson@15079
   565
apply (erule ssubst)
paulson@15079
   566
apply (rule_tac x = t in exI, simp)
paulson@15079
   567
apply (rule sumr_fun_eq)
paulson@15079
   568
apply (auto simp add: odd_Suc_mult_two_ex)
paulson@15079
   569
apply (auto simp add: even_mult_two_ex left_distrib cos_add simp del: fact_Suc realpow_Suc)
paulson@15079
   570
apply (simp add: mult_commute [of _ pi])
paulson@15079
   571
done
paulson@15079
   572
paulson@15079
   573
(* ------------------------------------------------------------------------- *)
paulson@15079
   574
(* Version for ln(1 +/- x). Where is it??                                    *)
paulson@15079
   575
(* ------------------------------------------------------------------------- *)
paulson@15079
   576
paulson@15079
   577
lemma sin_bound_lemma:
paulson@15081
   578
    "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"
paulson@15079
   579
by auto
paulson@15079
   580
paulson@15079
   581
lemma Maclaurin_sin_bound:
paulson@15079
   582
  "abs(sin x - sumr 0 n (%m. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *
paulson@15081
   583
  x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
obua@14738
   584
proof -
paulson@15079
   585
  have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
obua@14738
   586
    by (rule_tac mult_right_mono,simp_all)
obua@14738
   587
  note est = this[simplified]
obua@14738
   588
  show ?thesis
paulson@15079
   589
    apply (cut_tac f=sin and n=n and x=x and
obua@14738
   590
      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
   591
      in Maclaurin_all_le_objl)
paulson@15079
   592
    apply safe
paulson@15079
   593
    apply simp
obua@14738
   594
    apply (subst mod_Suc_eq_Suc_mod)
paulson@15079
   595
    apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)
obua@14738
   596
    apply (rule DERIV_minus, simp+)
obua@14738
   597
    apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)
paulson@15079
   598
    apply (erule ssubst)
paulson@15079
   599
    apply (rule sin_bound_lemma)
paulson@15079
   600
    apply (rule sumr_fun_eq, safe)
paulson@15079
   601
    apply (rule_tac f = "%u. u * (x^r)" in arg_cong)
obua@14738
   602
    apply (subst even_even_mod_4_iff)
paulson@15079
   603
    apply (cut_tac m=r in mod_exhaust_less_4, simp, safe)
obua@14738
   604
    apply (simp_all add:even_num_iff)
obua@14738
   605
    apply (drule lemma_even_mod_4_div_2[simplified])
paulson@15079
   606
    apply(simp add: numeral_2_eq_2 divide_inverse)
paulson@15079
   607
    apply (drule lemma_odd_mod_4_div_2)
paulson@15079
   608
    apply (simp add: numeral_2_eq_2 divide_inverse)
paulson@15079
   609
    apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
paulson@15079
   610
                   simp add: est mult_pos_le mult_ac divide_inverse
paulson@15079
   611
                          power_abs [symmetric])
obua@14738
   612
    done
obua@14738
   613
qed
obua@14738
   614
paulson@15079
   615
end