(*  Title       : MacLaurin.thy

    Author      : Jacques D. Fleuriot

    Copyright   : 2001 University of Edinburgh

    Description : MacLaurin series

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

*)

theory MacLaurin

imports Log

begin

lemma sumr_offset: "sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"

by (induct_tac "n", auto)

lemma sumr_offset2: "\<forall>f. sumr 0 n (%m. f (m+k)) = sumr 0 (n+k) f - sumr 0 k f"

by (induct_tac "n", auto)

lemma sumr_offset3: "sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"

by (simp  add: sumr_offset)

lemma sumr_offset4: "\<forall>n f. sumr 0 (n+k) f = sumr 0 n (%m. f (m+k)) + sumr 0 k f"

by (simp add: sumr_offset)

lemma sumr_from_1_from_0: "0 < n ==>

      sumr (Suc 0) (Suc n) (%n. (if even(n) then 0 else

             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n) =

      sumr 0 (Suc n) (%n. (if even(n) then 0 else

             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n)"

by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)

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 = sumr 0 n (%m. (j m / real (fact m)) * (h^m)) +

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

apply (rule_tac x = "(f h - sumr 0 n (%m. (j m / real (fact m)) * h^m)) *

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

       in exI)

apply (simp add: times_divide_eq) 

done

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

by arith

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

ML

{*

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_rulesI = [DERIV_Id,DERIV_const,DERIV_cos,DERIV_cmult,

                    DERIV_sin, DERIV_exp, DERIV_inverse,DERIV_pow,

                    DERIV_add, DERIV_diff, DERIV_mult, DERIV_minus,

                    DERIV_inverse_fun,DERIV_quotient,DERIV_fun_pow,

                    DERIV_fun_exp,DERIV_fun_sin,DERIV_fun_cos,

                    DERIV_Id,DERIV_const,DERIV_cos];

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));

*}

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: sumr_Suc)

apply (insert sumr_offset4 [of 1])

apply (simp del: sumr_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: inverse_mult_distrib 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 =

              sumr 0 n (%m. (diff m 0 / real (fact m)) * h ^ m) +

              (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 - (sumr 0 n (%m. (diff m 0 / real(fact m)) * t^m) + (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: sumr_Suc)

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

 apply (simp add: eq_diff_eq' del: sumr_Suc)

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))))))")

 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 add: times_divide_eq 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: sumr_Suc)

apply (insert sumr_offset4 [of 1])

apply (simp del: sumr_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 - (sumr 0 (n - m) (%p. diff (m + p) 0 / real (fact p) * t ^ p) + B * (t ^ (n - m) / real (fact (n - m)))))"

       in thin_rl)

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))))"

       in thin_rl)

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

              sumr 0 n (%m. 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 =

              sumr 0 n (%m. 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 =

              sumr 0 n (%m. 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 =

              sumr 0 n (%m. 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 add: times_divide_eq) 

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] sumr_fun_eq)

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 =

              sumr 0 n (%m. 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_tac "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 =

              sumr 0 n (%m. 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 times_divide_eq)

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 = sumr 0 n (%m. (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 = sumr 0 n (%m. (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 -->

          sumr 0 n (%m. (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 = sumr 0 n (%m. (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 = sumr 0 n (%m. (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 = sumr 0 n (%m. (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 = sumr 0 n (%m. (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 (case_tac "m mod 4", auto, arith)

lemma Suc_Suc_mult_two_diff_two [rule_format, simp]:

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

by (induct_tac "n", auto)

lemma lemma_Suc_Suc_4n_diff_2 [rule_format, simp]:

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

by (induct_tac "n", auto)

lemma Suc_mult_two_diff_one [rule_format, simp]:

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

by (induct_tac "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 =

       (sumr 0 n (%m. (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) add: times_divide_eq)

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 sumr_fun_eq)

apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)

done

lemma Maclaurin_sin_expansion:

     "\<exists>t. sin x =

       (sumr 0 n (%m. (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 =

       (sumr 0 n (%m. (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) add: times_divide_eq)

apply (erule ssubst)

apply (rule_tac x = t in exI, simp)

apply (rule sumr_fun_eq)

apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)

done

lemma Maclaurin_sin_expansion4:

     "0 < x ==>

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

       sin x =

       (sumr 0 n (%m. (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) add: times_divide_eq)

apply (erule ssubst)

apply (rule_tac x = t in exI, simp)

apply (rule sumr_fun_eq)

apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex times_divide_eq)

done

subsection{*Maclaurin Expansion for Cosine Function*}

lemma sumr_cos_zero_one [simp]:

     "sumr 0 (Suc n)

         (%m. (if even m

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

               else 0) *

              0 ^ m) = 1"

by (induct_tac "n", auto)

lemma Maclaurin_cos_expansion:

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

       cos x =

       (sumr 0 n (%m. (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) add: times_divide_eq)

apply (case_tac "n", simp)

apply (simp del: sumr_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 sumr_fun_eq)

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 =

       (sumr 0 n (%m. (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) add: times_divide_eq)

apply (erule ssubst)

apply (rule_tac x = t in exI, simp)

apply (rule sumr_fun_eq)

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 =

       (sumr 0 n (%m. (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) add: times_divide_eq)

apply (erule ssubst)

apply (rule_tac x = t in exI, simp)

apply (rule sumr_fun_eq)

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 - sumr 0 n (%m. (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 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 sumr_fun_eq, safe)

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

    apply (subst even_even_mod_4_iff)

    apply (cut_tac m=r 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_pos_le mult_ac divide_inverse

                          power_abs [symmetric])

    done

qed

end