(*  ID          : $Id$

     Author      : Jacques D. Fleuriot

     Copyright   : 2001 University of Edinburgh

     Conversion to Isar and new proofs by Lawrence C Paulson, 2004

 *)

 header{*MacLaurin Series*}

 theory MacLaurin

 imports Log

 begin

 subsection{*Maclaurin's Theorem with Lagrange Form of Remainder*}

 text{*This is a very long, messy proof even now that it's been broken down

 into lemmas.*}

 lemma Maclaurin_lemma:

     "0 < h ==>

      \<exists>B. f h = (\<Sum>m=0..<n. (j m / real (fact m)) * (h^m)) +

                (B * ((h^n) / real(fact n)))"

 apply (rule_tac x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *

                  real(fact n) / (h^n)"

        in exI)

 apply (simp) 

 done

 lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"

 by arith

 text{*A crude tactic to differentiate by proof.*}

 ML

 {*

 local

 val deriv_rulesI =

   [thm "DERIV_Id", thm "DERIV_const", thm "DERIV_cos", thm "DERIV_cmult",

   thm "DERIV_sin", thm "DERIV_exp", thm "DERIV_inverse", thm "DERIV_pow",

   thm "DERIV_add", thm "DERIV_diff", thm "DERIV_mult", thm "DERIV_minus",

   thm "DERIV_inverse_fun", thm "DERIV_quotient", thm "DERIV_fun_pow",

   thm "DERIV_fun_exp", thm "DERIV_fun_sin", thm "DERIV_fun_cos",

   thm "DERIV_Id", thm "DERIV_const", thm "DERIV_cos"];

 val DERIV_chain2 = thm "DERIV_chain2";

 in

 exception DERIV_name;

 fun get_fun_name (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _)) = f

 |   get_fun_name (_ $ (_ $ (Const ("Lim.deriv",_) $ Abs(_,_, Const (f,_) $ _) $ _ $ _))) = f

 |   get_fun_name _ = raise DERIV_name;

 val deriv_tac =

   SUBGOAL (fn (prem,i) =>

    (resolve_tac deriv_rulesI i) ORELSE

     ((rtac (read_instantiate [("f",get_fun_name prem)]

                      DERIV_chain2) i) handle DERIV_name => no_tac));;

 val DERIV_tac = ALLGOALS(fn i => REPEAT(deriv_tac i));

 end

 *}

 lemma Maclaurin_lemma2:

       "[| \<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t;

           n = Suc k;

         difg =

         (\<lambda>m t. diff m t -

                ((\<Sum>p = 0..<n - m. diff (m + p) 0 / real (fact p) * t ^ p) +

                 B * (t ^ (n - m) / real (fact (n - m)))))|] ==>

         \<forall>m t. m < n & 0 \<le> t & t \<le> h -->

                     DERIV (difg m) t :> difg (Suc m) t"

 apply clarify

 apply (rule DERIV_diff)

 apply (simp (no_asm_simp))

 apply (tactic DERIV_tac)

 apply (tactic DERIV_tac)

 apply (rule_tac [2] lemma_DERIV_subst)

 apply (rule_tac [2] DERIV_quotient)

 apply (rule_tac [3] DERIV_const)

 apply (rule_tac [2] DERIV_pow)

   prefer 3 apply (simp add: fact_diff_Suc)

  prefer 2 apply simp

 apply (frule_tac m = m in less_add_one, clarify)

 apply (simp del: setsum_op_ivl_Suc)

 apply (insert sumr_offset4 [of 1])

 apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)

 apply (rule lemma_DERIV_subst)

 apply (rule DERIV_add)

 apply (rule_tac [2] DERIV_const)

 apply (rule DERIV_sumr, clarify)

  prefer 2 apply simp

 apply (simp (no_asm) add: divide_inverse mult_assoc del: fact_Suc realpow_Suc)

 apply (rule DERIV_cmult)

 apply (rule lemma_DERIV_subst)

 apply (best intro: DERIV_chain2 intro!: DERIV_intros)

 apply (subst fact_Suc)

 apply (subst real_of_nat_mult)

 apply (simp add: mult_ac)

 done

 lemma Maclaurin_lemma3:

      "[|\<forall>k t. k < Suc m \<and> 0\<le>t & t\<le>h \<longrightarrow> DERIV (difg k) t :> difg (Suc k) t;

         \<forall>k<Suc m. difg k 0 = 0; DERIV (difg n) t :> 0;  n < m; 0 < t;

         t < h|]

      ==> \<exists>ta. 0 < ta & ta < t & DERIV (difg (Suc n)) ta :> 0"

 apply (rule Rolle, assumption, simp)

 apply (drule_tac x = n and P="%k. k<Suc m --> difg k 0 = 0" in spec)

 apply (rule DERIV_unique)

 prefer 2 apply assumption

 apply force

 apply (subgoal_tac "\<forall>ta. 0 \<le> ta & ta \<le> t --> (difg (Suc n)) differentiable ta")

 apply (simp add: differentiable_def)

 apply (blast dest!: DERIV_isCont)

 apply (simp add: differentiable_def, clarify)

 apply (rule_tac x = "difg (Suc (Suc n)) ta" in exI)

 apply force

 apply (simp add: differentiable_def, clarify)

 apply (rule_tac x = "difg (Suc (Suc n)) x" in exI)

 apply force

 done

 lemma Maclaurin:

    "[| 0 < h; 0 < n; diff 0 = f;

        \<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]

     ==> \<exists>t. 0 < t &

               t < h &

               f h =

               setsum (%m. (diff m 0 / real (fact m)) * h ^ m) {0..<n} +

               (diff n t / real (fact n)) * h ^ n"

 apply (case_tac "n = 0", force)

 apply (drule not0_implies_Suc)

 apply (erule exE)

 apply (frule_tac f=f and n=n and j="%m. diff m 0" in Maclaurin_lemma)

 apply (erule exE)

 apply (subgoal_tac "\<exists>g.

      g = (%t. f t - (setsum (%m. (diff m 0 / real(fact m)) * t^m) {0..<n} + (B * (t^n / real(fact n)))))")

  prefer 2 apply blast

 apply (erule exE)

 apply (subgoal_tac "g 0 = 0 & g h =0")

  prefer 2

  apply (simp del: setsum_op_ivl_Suc)

  apply (cut_tac n = m and k = 1 in sumr_offset2)

  apply (simp add: eq_diff_eq' del: setsum_op_ivl_Suc)

 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))))))")

  prefer 2 apply blast

 apply (erule exE)

 apply (subgoal_tac "difg 0 = g")

  prefer 2 apply simp

 apply (frule Maclaurin_lemma2, assumption+)

 apply (subgoal_tac "\<forall>ma. ma < n --> (\<exists>t. 0 < t & t < h & difg (Suc ma) t = 0) ")

  apply (drule_tac x = m and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)

  apply (erule impE)

   apply (simp (no_asm_simp))

  apply (erule exE)

  apply (rule_tac x = t in exI)

  apply (simp del: realpow_Suc fact_Suc)

 apply (subgoal_tac "\<forall>m. m < n --> difg m 0 = 0")

  prefer 2

  apply clarify

  apply simp

  apply (frule_tac m = ma in less_add_one, clarify)

  apply (simp del: setsum_op_ivl_Suc)

 apply (insert sumr_offset4 [of 1])

 apply (simp del: setsum_op_ivl_Suc fact_Suc realpow_Suc)

 apply (subgoal_tac "\<forall>m. m < n --> (\<exists>t. 0 < t & t < h & DERIV (difg m) t :> 0) ")

 apply (rule allI, rule impI)

 apply (drule_tac x = ma and P="%m. m<n --> (\<exists>t. ?QQ m t)" in spec)

 apply (erule impE, assumption)

 apply (erule exE)

 apply (rule_tac x = t in exI)

 (* do some tidying up *)

 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)))))"

        in thin_rl)

 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))))"

        in thin_rl)

 apply (erule_tac [!] V="f h = setsum (%m. diff m 0 / real (fact m) * h ^ m) {0..<n} + B * (h ^ n / real (fact n))"

        in thin_rl)

 (* back to business *)

 apply (simp (no_asm_simp))

 apply (rule DERIV_unique)

 prefer 2 apply blast

 apply force

 apply (rule allI, induct_tac "ma")

 apply (rule impI, rule Rolle, assumption, simp, simp)

 apply (subgoal_tac "\<forall>t. 0 \<le> t & t \<le> h --> g differentiable t")

 apply (simp add: differentiable_def)

 apply (blast dest: DERIV_isCont)

 apply (simp add: differentiable_def, clarify)

 apply (rule_tac x = "difg (Suc 0) t" in exI)

 apply force

 apply (simp add: differentiable_def, clarify)

 apply (rule_tac x = "difg (Suc 0) x" in exI)

 apply force

 apply safe

 apply force

 apply (frule Maclaurin_lemma3, assumption+, safe)

 apply (rule_tac x = ta in exI, force)

 done

 lemma Maclaurin_objl:

      "0 < h & 0 < n & diff 0 = f &

        (\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)

     --> (\<exists>t. 0 < t &

               t < h &

               f h =

               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +

               diff n t / real (fact n) * h ^ n)"

 by (blast intro: Maclaurin)

 lemma Maclaurin2:

    "[| 0 < h; diff 0 = f;

        \<forall>m t.

           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t |]

     ==> \<exists>t. 0 < t &

               t \<le> h &

               f h =

               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +

               diff n t / real (fact n) * h ^ n"

 apply (case_tac "n", auto)

 apply (drule Maclaurin, auto)

 done

 lemma Maclaurin2_objl:

      "0 < h & diff 0 = f &

        (\<forall>m t.

           m < n & 0 \<le> t & t \<le> h --> DERIV (diff m) t :> diff (Suc m) t)

     --> (\<exists>t. 0 < t &

               t \<le> h &

               f h =

               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +

               diff n t / real (fact n) * h ^ n)"

 by (blast intro: Maclaurin2)

 lemma Maclaurin_minus:

    "[| h < 0; 0 < n; diff 0 = f;

        \<forall>m t. m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t |]

     ==> \<exists>t. h < t &

               t < 0 &

               f h =

               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +

               diff n t / real (fact n) * h ^ n"

 apply (cut_tac f = "%x. f (-x)"

         and diff = "%n x. ((- 1) ^ n) * diff n (-x)"

         and h = "-h" and n = n in Maclaurin_objl)

 apply (simp)

 apply safe

 apply (subst minus_mult_right)

 apply (rule DERIV_cmult)

 apply (rule lemma_DERIV_subst)

 apply (rule DERIV_chain2 [where g=uminus])

 apply (rule_tac [2] DERIV_minus, rule_tac [2] DERIV_Id)

 prefer 2 apply force

 apply force

 apply (rule_tac x = "-t" in exI, auto)

 apply (subgoal_tac "(\<Sum>m = 0..<n. -1 ^ m * diff m 0 * (-h)^m / real(fact m)) =

                     (\<Sum>m = 0..<n. diff m 0 * h ^ m / real(fact m))")

 apply (rule_tac [2] setsum_cong[OF refl])

 apply (auto simp add: divide_inverse power_mult_distrib [symmetric])

 done

 lemma Maclaurin_minus_objl:

      "(h < 0 & 0 < n & diff 0 = f &

        (\<forall>m t.

           m < n & h \<le> t & t \<le> 0 --> DERIV (diff m) t :> diff (Suc m) t))

     --> (\<exists>t. h < t &

               t < 0 &

               f h =

               (\<Sum>m=0..<n. diff m 0 / real (fact m) * h ^ m) +

               diff n t / real (fact n) * h ^ n)"

 by (blast intro: Maclaurin_minus)

 subsection{*More Convenient "Bidirectional" Version.*}

 (* not good for PVS sin_approx, cos_approx *)

 lemma Maclaurin_bi_le_lemma [rule_format]:

      "0 < n \<longrightarrow>

        diff 0 0 =

        (\<Sum>m = 0..<n. diff m 0 * 0 ^ m / real (fact m)) +

        diff n 0 * 0 ^ n / real (fact n)"

 by (induct "n", auto)

 lemma Maclaurin_bi_le:

    "[| diff 0 = f;

        \<forall>m t. m < n & abs t \<le> abs x --> DERIV (diff m) t :> diff (Suc m) t |]

     ==> \<exists>t. abs t \<le> abs x &

               f x =

               (\<Sum>m=0..<n. diff m 0 / real (fact m) * x ^ m) +

               diff n t / real (fact n) * x ^ n"

 apply (case_tac "n = 0", force)

 apply (case_tac "x = 0")

 apply (rule_tac x = 0 in exI)

 apply (force simp add: Maclaurin_bi_le_lemma)

 apply (cut_tac x = x and y = 0 in linorder_less_linear, auto)

 txt{*Case 1, where @{term "x < 0"}*}

 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_minus_objl, safe)

 apply (simp add: abs_if)

 apply (rule_tac x = t in exI)

 apply (simp add: abs_if)

 txt{*Case 2, where @{term "0 < x"}*}

 apply (cut_tac f = "diff 0" and diff = diff and h = x and n = n in Maclaurin_objl, safe)

 apply (simp add: abs_if)

 apply (rule_tac x = t in exI)

 apply (simp add: abs_if)

 done

 lemma Maclaurin_all_lt:

      "[| diff 0 = f;

          \<forall>m x. DERIV (diff m) x :> diff(Suc m) x;

         x ~= 0; 0 < n

       |] ==> \<exists>t. 0 < abs t & abs t < abs x &

                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +

                      (diff n t / real (fact n)) * x ^ n"

 apply (rule_tac x = x and y = 0 in linorder_cases)

 prefer 2 apply blast

 apply (drule_tac [2] diff=diff in Maclaurin)

 apply (drule_tac diff=diff in Maclaurin_minus, simp_all, safe)

 apply (rule_tac [!] x = t in exI, auto)

 done

 lemma Maclaurin_all_lt_objl:

      "diff 0 = f &

       (\<forall>m x. DERIV (diff m) x :> diff(Suc m) x) &

       x ~= 0 & 0 < n

       --> (\<exists>t. 0 < abs t & abs t < abs x &

                f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +

                      (diff n t / real (fact n)) * x ^ n)"

 by (blast intro: Maclaurin_all_lt)

 lemma Maclaurin_zero [rule_format]:

      "x = (0::real)

       ==> 0 < n -->

           (\<Sum>m=0..<n. (diff m (0::real) / real (fact m)) * x ^ m) =

           diff 0 0"

 by (induct n, auto)

 lemma Maclaurin_all_le: "[| diff 0 = f;

         \<forall>m x. DERIV (diff m) x :> diff (Suc m) x

       |] ==> \<exists>t. abs t \<le> abs x &

               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +

                     (diff n t / real (fact n)) * x ^ n"

 apply (insert linorder_le_less_linear [of n 0])

 apply (erule disjE, force)

 apply (case_tac "x = 0")

 apply (frule_tac diff = diff and n = n in Maclaurin_zero, assumption)

 apply (drule gr_implies_not0 [THEN not0_implies_Suc])

 apply (rule_tac x = 0 in exI, force)

 apply (frule_tac diff = diff and n = n in Maclaurin_all_lt, auto)

 apply (rule_tac x = t in exI, auto)

 done

 lemma Maclaurin_all_le_objl: "diff 0 = f &

       (\<forall>m x. DERIV (diff m) x :> diff (Suc m) x)

       --> (\<exists>t. abs t \<le> abs x &

               f x = (\<Sum>m=0..<n. (diff m 0 / real (fact m)) * x ^ m) +

                     (diff n t / real (fact n)) * x ^ n)"

 by (blast intro: Maclaurin_all_le)

 subsection{*Version for Exponential Function*}

 lemma Maclaurin_exp_lt: "[| x ~= 0; 0 < n |]

       ==> (\<exists>t. 0 < abs t &

                 abs t < abs x &

                 exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +

                         (exp t / real (fact n)) * x ^ n)"

 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_lt_objl, auto)

 lemma Maclaurin_exp_le:

      "\<exists>t. abs t \<le> abs x &

             exp x = (\<Sum>m=0..<n. (x ^ m) / real (fact m)) +

                        (exp t / real (fact n)) * x ^ n"

 by (cut_tac diff = "%n. exp" and f = exp and x = x and n = n in Maclaurin_all_le_objl, auto)

 subsection{*Version for Sine Function*}

 lemma MVT2:

      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]

       ==> \<exists>z. a < z & z < b & (f b - f a = (b - a) * f'(z))"

 apply (drule MVT)

 apply (blast intro: DERIV_isCont)

 apply (force dest: order_less_imp_le simp add: differentiable_def)

 apply (blast dest: DERIV_unique order_less_imp_le)

 done

 lemma mod_exhaust_less_4:

      "m mod 4 = 0 | m mod 4 = 1 | m mod 4 = 2 | m mod 4 = (3::nat)"

 by auto

 lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:

      "0 < n --> Suc (Suc (2 * n - 2)) = 2*n"

 by (induct "n", auto)

 lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:

      "0 < n --> Suc (Suc (4*n - 2)) = 4*n"

 by (induct "n", auto)

 lemma Suc_mult_two_diff_one [rule_format, simp]:

       "0 < n --> Suc (2 * n - 1) = 2*n"

 by (induct "n", auto)

 text{*It is unclear why so many variant results are needed.*}

 lemma Maclaurin_sin_expansion2:

      "\<exists>t. abs t \<le> abs x &

        sin x =

        (\<Sum>m=0..<n. (if even m then 0

                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *

                        x ^ m)

       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"

 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)

 apply safe

 apply (simp (no_asm))

 apply (simp (no_asm))

 apply (case_tac "n", clarify, simp, simp)

 apply (rule ccontr, simp)

 apply (drule_tac x = x in spec, simp)

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)

 done

 lemma Maclaurin_sin_expansion:

      "\<exists>t. sin x =

        (\<Sum>m=0..<n. (if even m then 0

                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *

                        x ^ m)

       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"

 apply (insert Maclaurin_sin_expansion2 [of x n]) 

 apply (blast intro: elim:); 

 done

 lemma Maclaurin_sin_expansion3:

      "[| 0 < n; 0 < x |] ==>

        \<exists>t. 0 < t & t < x &

        sin x =

        (\<Sum>m=0..<n. (if even m then 0

                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *

                        x ^ m)

       + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"

 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)

 apply safe

 apply simp

 apply (simp (no_asm))

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)

 done

 lemma Maclaurin_sin_expansion4:

      "0 < x ==>

        \<exists>t. 0 < t & t \<le> x &

        sin x =

        (\<Sum>m=0..<n. (if even m then 0

                        else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *

                        x ^ m)

       + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"

 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)

 apply safe

 apply simp

 apply (simp (no_asm))

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)

 done

 subsection{*Maclaurin Expansion for Cosine Function*}

 lemma sumr_cos_zero_one [simp]:

  "(\<Sum>m=0..<(Suc n).

      (if even m then (- 1) ^ (m div 2)/(real  (fact m)) else 0) * 0 ^ m) = 1"

 by (induct "n", auto)

 lemma Maclaurin_cos_expansion:

      "\<exists>t. abs t \<le> abs x &

        cos x =

        (\<Sum>m=0..<n. (if even m

                        then (- 1) ^ (m div 2)/(real (fact m))

                        else 0) *

                        x ^ m)

       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"

 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)

 apply safe

 apply (simp (no_asm))

 apply (simp (no_asm))

 apply (case_tac "n", simp)

 apply (simp del: setsum_op_ivl_Suc)

 apply (rule ccontr, simp)

 apply (drule_tac x = x in spec, simp)

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: cos_zero_iff even_mult_two_ex)

 done

 lemma Maclaurin_cos_expansion2:

      "[| 0 < x; 0 < n |] ==>

        \<exists>t. 0 < t & t < x &

        cos x =

        (\<Sum>m=0..<n. (if even m

                        then (- 1) ^ (m div 2)/(real (fact m))

                        else 0) *

                        x ^ m)

       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"

 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)

 apply safe

 apply simp

 apply (simp (no_asm))

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: cos_zero_iff even_mult_two_ex)

 done

 lemma Maclaurin_minus_cos_expansion:

      "[| x < 0; 0 < n |] ==>

        \<exists>t. x < t & t < 0 &

        cos x =

        (\<Sum>m=0..<n. (if even m

                        then (- 1) ^ (m div 2)/(real (fact m))

                        else 0) *

                        x ^ m)

       + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"

 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)

 apply safe

 apply simp

 apply (simp (no_asm))

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: cos_zero_iff even_mult_two_ex)

 done

 (* ------------------------------------------------------------------------- *)

 (* Version for ln(1 +/- x). Where is it??                                    *)

 (* ------------------------------------------------------------------------- *)

 lemma sin_bound_lemma:

     "[|x = y; abs u \<le> (v::real) |] ==> \<bar>(x + u) - y\<bar> \<le> v"

 by auto

 lemma Maclaurin_sin_bound:

   "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else ((- 1) ^ ((m - (Suc 0)) div 2)) / real (fact m)) *

   x ^ m))  \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"

 proof -

   have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"

     by (rule_tac mult_right_mono,simp_all)

   note est = this[simplified]

   show ?thesis

     apply (cut_tac f=sin and n=n and x=x and

       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)"

       in Maclaurin_all_le_objl)

     apply safe

     apply simp

     apply (subst (1 2 3) mod_Suc_eq_Suc_mod)

     apply (cut_tac m=m in mod_exhaust_less_4, safe, simp+)

     apply (rule DERIV_minus, simp+)

     apply (rule lemma_DERIV_subst, rule DERIV_minus, rule DERIV_cos, simp)

     apply (erule ssubst)

     apply (rule sin_bound_lemma)

     apply (rule setsum_cong[OF refl])

     apply (rule_tac f = "%u. u * (x^xa)" in arg_cong)

     apply (subst even_even_mod_4_iff)

     apply (cut_tac m=xa in mod_exhaust_less_4, simp, safe)

     apply (simp_all add:even_num_iff)

     apply (drule lemma_even_mod_4_div_2[simplified])

     apply(simp add: numeral_2_eq_2 divide_inverse)

     apply (drule lemma_odd_mod_4_div_2)

     apply (simp add: numeral_2_eq_2 divide_inverse)

     apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono

                    simp add: est mult_nonneg_nonneg mult_ac divide_inverse

                           power_abs [symmetric] abs_mult)

     done

 qed

 end