(*  Author      : Jacques D. Fleuriot

     Copyright   : 2001 University of Edinburgh

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

     Conversion of Mac Laurin to Isar by Lukas Bulwahn and Bernhard Häupler, 2005

 *)

 header{*MacLaurin Series*}

 theory MacLaurin

 imports Transcendental

 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)))"

 by (rule exI[where x = "(f h - (\<Sum>m=0..<n. (j m / real (fact m)) * h^m)) *

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

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

 by arith

 lemma fact_diff_Suc [rule_format]:

   "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"

   by (subst fact_reduce_nat, auto)

 lemma Maclaurin_lemma2:

   fixes B

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

     and INIT : "n = Suc k"

   defines "difg \<equiv> (\<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)))))" (is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)")

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

 proof (rule allI impI)+

   fix m t assume INIT2: "m < n & 0 \<le> t & t \<le> h"

   have "DERIV (difg m) t :> diff (Suc m) t -

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

      real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)))" unfolding difg_def

     by (auto intro!: DERIV_intros DERIV[rule_format, OF INIT2])

       moreover

   from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m"

     unfolding atLeast0LessThan[symmetric] by auto

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

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

     unfolding intvl atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.reindex)

   moreover

   have fact_neq_0: "\<And>x::nat. real (fact x) + real x * real (fact x) \<noteq> 0"

     by (metis fact_gt_zero_nat not_add_less1 real_of_nat_add real_of_nat_mult real_of_nat_zero_iff)

   have "\<And>x. real (Suc x) * t ^ x * diff (Suc m + x) 0 / real (fact (Suc x)) =

       diff (Suc m + x) 0 * t^x / real (fact x)"

     by (auto simp: field_simps real_of_nat_Suc fact_neq_0 intro!: nonzero_divide_eq_eq[THEN iffD2])

   moreover

   have "real (n - m) * t ^ (n - Suc m) * B / real (fact (n - m)) =

       B * (t ^ (n - Suc m) / real (fact (n - Suc m)))"

     using `0 < n - m` by (simp add: fact_reduce_nat)

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

     unfolding difg_def by simp

 qed

 lemma Maclaurin:

   assumes h: "0 < h"

   assumes n: "0 < n"

   assumes diff_0: "diff 0 = f"

   assumes diff_Suc:

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

   shows

     "\<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"

 proof -

   from n obtain m where m: "n = Suc m"

     by (cases n) (simp add: n)

   obtain B where f_h: "f h =

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

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

     using Maclaurin_lemma [OF h] ..

   def g \<equiv> "(\<lambda>t. f t -

     (setsum (\<lambda>m. (diff m 0 / real(fact m)) * t^m) {0..<n}

       + (B * (t^n / real(fact n)))))"

   have g2: "g 0 = 0 & g h = 0"

     apply (simp add: m f_h g_def del: setsum_op_ivl_Suc)

     apply (cut_tac n = m and k = "Suc 0" in sumr_offset2)

     apply (simp add: eq_diff_eq' diff_0 del: setsum_op_ivl_Suc)

     done

   def difg \<equiv> "(%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))))))"

   have difg_0: "difg 0 = g"

     unfolding difg_def g_def by (simp add: diff_0)

   have difg_Suc: "\<forall>(m\<Colon>nat) t\<Colon>real.

         m < n \<and> (0\<Colon>real) \<le> t \<and> t \<le> h \<longrightarrow> DERIV (difg m) t :> difg (Suc m) t"

     using diff_Suc m unfolding difg_def by (rule Maclaurin_lemma2)

   have difg_eq_0: "\<forall>m. m < n --> difg m 0 = 0"

     apply clarify

     apply (simp add: m difg_def)

     apply (frule less_iff_Suc_add [THEN iffD1], clarify)

     apply (simp del: setsum_op_ivl_Suc)

     apply (insert sumr_offset4 [of "Suc 0"])

     apply (simp del: setsum_op_ivl_Suc fact_Suc)

     done

   have isCont_difg: "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> isCont (difg m) x"

     by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp

   have differentiable_difg:

     "\<And>m x. \<lbrakk>m < n; 0 \<le> x; x \<le> h\<rbrakk> \<Longrightarrow> difg m differentiable x"

     by (rule differentiableI [OF difg_Suc [rule_format]]) simp

   have difg_Suc_eq_0: "\<And>m t. \<lbrakk>m < n; 0 \<le> t; t \<le> h; DERIV (difg m) t :> 0\<rbrakk>

         \<Longrightarrow> difg (Suc m) t = 0"

     by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp

   have "m < n" using m by simp

   have "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m) t :> 0"

   using `m < n`

   proof (induct m)

     case 0

     show ?case

     proof (rule Rolle)

       show "0 < h" by fact

       show "difg 0 0 = difg 0 h" by (simp add: difg_0 g2)

       show "\<forall>x. 0 \<le> x \<and> x \<le> h \<longrightarrow> isCont (difg (0\<Colon>nat)) x"

         by (simp add: isCont_difg n)

       show "\<forall>x. 0 < x \<and> x < h \<longrightarrow> difg (0\<Colon>nat) differentiable x"

         by (simp add: differentiable_difg n)

     qed

   next

     case (Suc m')

     hence "\<exists>t. 0 < t \<and> t < h \<and> DERIV (difg m') t :> 0" by simp

     then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0" by fast

     have "\<exists>t'. 0 < t' \<and> t' < t \<and> DERIV (difg (Suc m')) t' :> 0"

     proof (rule Rolle)

       show "0 < t" by fact

       show "difg (Suc m') 0 = difg (Suc m') t"

         using t `Suc m' < n` by (simp add: difg_Suc_eq_0 difg_eq_0)

       show "\<forall>x. 0 \<le> x \<and> x \<le> t \<longrightarrow> isCont (difg (Suc m')) x"

         using `t < h` `Suc m' < n` by (simp add: isCont_difg)

       show "\<forall>x. 0 < x \<and> x < t \<longrightarrow> difg (Suc m') differentiable x"

         using `t < h` `Suc m' < n` by (simp add: differentiable_difg)

     qed

     thus ?case

       using `t < h` by auto

   qed

   then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0" by fast

   hence "difg (Suc m) t = 0"

     using `m < n` by (simp add: difg_Suc_eq_0)

   show ?thesis

   proof (intro exI conjI)

     show "0 < t" by fact

     show "t < h" by fact

     show "f h =

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

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

       using `difg (Suc m) t = 0`

       by (simp add: m f_h difg_def del: fact_Suc)

   qed

 qed

 lemma Maclaurin_objl:

   "0 < h & n>0 & 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:

   assumes INIT1: "0 < h " and INIT2: "diff 0 = f"

   and DERIV: "\<forall>m t.

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

   shows "\<exists>t. 0 < t \<and> t \<le> h \<and> f h =

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

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

 proof (cases "n")

   case 0 with INIT1 INIT2 show ?thesis by fastsimp

 next

   case Suc

   hence "n > 0" by simp

   from INIT1 this INIT2 DERIV have "\<exists>t>0. t < h \<and>

     f h =

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

     by (rule Maclaurin)

   thus ?thesis by fastsimp

 qed

 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:

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

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

   shows "\<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"

 proof -

   txt "Transform @{text ABL'} into @{text DERIV_intros} format."

   note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong]

   from assms

   have "\<exists>t>0. t < - h \<and>

     f (- (- h)) =

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

     (- 1) ^ m * diff m (- 0) / real (fact m) * (- h) ^ m) +

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

     by (intro Maclaurin) (auto intro!: DERIV_intros DERIV')

   then guess t ..

   moreover

   have "-1 ^ n * diff n (- t) * (- h) ^ n / real (fact n) = diff n (- t) * h ^ n / real (fact n)"

     by (auto simp add: power_mult_distrib[symmetric])

   moreover

   have "(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))"

     by (auto intro: setsum_cong simp add: power_mult_distrib[symmetric])

   ultimately have " h < - t \<and>

     - t < 0 \<and>

     f h =

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

     by auto

   thus ?thesis ..

 qed

 lemma Maclaurin_minus_objl:

      "(h < 0 & n > 0 & 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]:

   "n>0 \<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:

    assumes "diff 0 = f"

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

    shows "\<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" (is "\<exists>t. _ \<and> f x = ?f x t")

 proof cases

   assume "n = 0" with `diff 0 = f` show ?thesis by force

 next

   assume "n \<noteq> 0"

   show ?thesis

   proof (cases rule: linorder_cases)

     assume "x = 0" with `n \<noteq> 0` `diff 0 = f` DERIV

     have "\<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by (force simp add: Maclaurin_bi_le_lemma)

     thus ?thesis ..

   next

     assume "x < 0"

     with `n \<noteq> 0` DERIV

     have "\<exists>t>x. t < 0 \<and> diff 0 x = ?f x t" by (intro Maclaurin_minus) auto

     then guess t ..

     with `x < 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp

     thus ?thesis ..

   next

     assume "x > 0"

     with `n \<noteq> 0` `diff 0 = f` DERIV

     have "\<exists>t>0. t < x \<and> diff 0 x = ?f x t" by (intro Maclaurin) auto

     then guess t ..

     with `x > 0` `diff 0 = f` have "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp

     thus ?thesis ..

   qed

 qed

 lemma Maclaurin_all_lt:

   assumes INIT1: "diff 0 = f" and INIT2: "0 < n" and INIT3: "x \<noteq> 0"

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

   shows "\<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" (is "\<exists>t. _ \<and> _ \<and> f x = ?f x t")

 proof (cases rule: linorder_cases)

   assume "x = 0" with INIT3 show "?thesis"..

 next

   assume "x < 0"

   with assms have "\<exists>t>x. t < 0 \<and> f x = ?f x t" by (intro Maclaurin_minus) auto

   then guess t ..

   with `x < 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp

   thus ?thesis ..

 next

   assume "x > 0"

   with assms have "\<exists>t>0. t < x \<and> f x = ?f x t " by (intro Maclaurin) auto

   then guess t ..

   with `x > 0` have "0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t" by simp

   thus ?thesis ..

 qed

 lemma Maclaurin_all_lt_objl:

      "diff 0 = f &

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

       x ~= 0 & n > 0

       --> (\<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)

       ==> n \<noteq> 0 -->

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

           diff 0 0"

 by (induct n, auto)

 lemma Maclaurin_all_le:

   assumes INIT: "diff 0 = f"

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

   shows "\<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" (is "\<exists>t. _ \<and> f x = ?f x t")

 proof cases

   assume "n = 0" with INIT show ?thesis by force

   next

   assume "n \<noteq> 0"

   show ?thesis

   proof cases

     assume "x = 0"

     with `n \<noteq> 0` have "(\<Sum>m = 0..<n. diff m 0 / real (fact m) * x ^ m) = diff 0 0"

       by (intro Maclaurin_zero) auto

     with INIT `x = 0` `n \<noteq> 0` have " \<bar>0\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x 0" by force

     thus ?thesis ..

   next

     assume "x \<noteq> 0"

     with INIT `n \<noteq> 0` DERIV have "\<exists>t. 0 < \<bar>t\<bar> \<and> \<bar>t\<bar> < \<bar>x\<bar> \<and> f x = ?f x t"

       by (intro Maclaurin_all_lt) auto

     then guess t ..

     hence "\<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = ?f x t" by simp

     thus ?thesis ..

   qed

 qed

 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; n > 0 |]

       ==> (\<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 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]:

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

 by (induct "n", auto)

 lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:

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

 by (induct "n", auto)

 lemma Suc_mult_two_diff_one [rule_format, simp]:

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

 by (induct "n", auto)

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

 lemma sin_expansion_lemma:

      "sin (x + real (Suc m) * pi / 2) =

       cos (x + real (m) * pi / 2)"

 by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)

 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"

   unfolding sin_coeff_def by simp (* TODO: move *)

 lemma Maclaurin_sin_expansion2:

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

        sin x =

        (\<Sum>m=0..<n. sin_coeff 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) add: sin_expansion_lemma)

 apply (subst (asm) setsum_0', clarify, case_tac "a", simp, simp)

 apply (cases n, 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_coeff_def sin_zero_iff odd_Suc_mult_two_ex)

 done

 lemma Maclaurin_sin_expansion:

      "\<exists>t. sin x =

        (\<Sum>m=0..<n. sin_coeff 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:

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

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

        sin x =

        (\<Sum>m=0..<n. sin_coeff 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) add: sin_expansion_lemma)

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: sin_coeff_def 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. sin_coeff 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) add: sin_expansion_lemma)

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)

 done

 subsection{*Maclaurin Expansion for Cosine Function*}

 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"

   unfolding cos_coeff_def by simp (* TODO: move *)

 lemma sumr_cos_zero_one [simp]:

   "(\<Sum>m=0..<(Suc n). cos_coeff m * 0 ^ m) = 1"

 by (induct "n", auto)

 lemma cos_expansion_lemma:

   "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"

 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)

 lemma Maclaurin_cos_expansion:

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

        cos x =

        (\<Sum>m=0..<n. cos_coeff m * 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) add: cos_expansion_lemma)

 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_coeff_def cos_zero_iff even_mult_two_ex)

 done

 lemma Maclaurin_cos_expansion2:

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

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

        cos x =

        (\<Sum>m=0..<n. cos_coeff m * 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) add: cos_expansion_lemma)

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)

 done

 lemma Maclaurin_minus_cos_expansion:

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

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

        cos x =

        (\<Sum>m=0..<n. cos_coeff m * 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) add: cos_expansion_lemma)

 apply (erule ssubst)

 apply (rule_tac x = t in exI, simp)

 apply (rule setsum_cong[OF refl])

 apply (auto simp add: cos_coeff_def 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. sin_coeff 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]

   let ?diff = "\<lambda>(n::nat) x. if n mod 4 = 0 then sin(x) else if n mod 4 = 1 then cos(x) else if n mod 4 = 2 then -sin(x) else -cos(x)"

   have diff_0: "?diff 0 = sin" by simp

   have DERIV_diff: "\<forall>m x. DERIV (?diff m) x :> ?diff (Suc m) x"

     apply (clarify)

     apply (subst (1 2 3) mod_Suc_eq_Suc_mod)

     apply (cut_tac m=m in mod_exhaust_less_4)

     apply (safe, auto intro!: DERIV_intros)

     done

   from Maclaurin_all_le [OF diff_0 DERIV_diff]

   obtain t where t1: "\<bar>t\<bar> \<le> \<bar>x\<bar>" and

     t2: "sin x = (\<Sum>m = 0..<n. ?diff m 0 / real (fact m) * x ^ m) +

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

   have diff_m_0:

     "\<And>m. ?diff m 0 = (if even m then 0

          else -1 ^ ((m - Suc 0) div 2))"

     apply (subst even_even_mod_4_iff)

     apply (cut_tac m=m in mod_exhaust_less_4)

     apply (elim disjE, simp_all)

     apply (safe dest!: mod_eqD, simp_all)

     done

   show ?thesis

     unfolding sin_coeff_def

     apply (subst t2)

     apply (rule sin_bound_lemma)

     apply (rule setsum_cong[OF refl])

     apply (subst diff_m_0, simp)

     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