Theory MacLaurin

(*  Title:      HOL/MacLaurin.thy
    Author:     Jacques D. Fleuriot, 2001 University of Edinburgh
    Author:     Lawrence C Paulson, 2004
    Author:     Lukas Bulwahn and Bernhard Häupler, 2005
*)

section ‹MacLaurin and Taylor 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 
    B::real. f h = (m<n. (j m / (fact m)) * (h^m)) + (B * ((h^n) /(fact n)))"
  by (rule exI[where x = "(f h - (m<n. (j m / (fact m)) * h^m)) * (fact n) / (h^n)"]) simp

lemma eq_diff_eq': "x = y - z  y = x + z"
  for x y z :: real
  by arith

lemma fact_diff_Suc: "n < Suc m  fact (Suc m - n) = (Suc m - n) * fact (m - n)"
  by (subst fact_reduce) auto

lemma Maclaurin_lemma2:
  fixes B
  assumes DERIV: "m t. m < n  0t  th  DERIV (diff m) t :> diff (Suc m) t"
    and INIT: "n = Suc k"
  defines "difg 
    (λm t::real. diff m t -
      ((p<n - m. diff (m + p) 0 / fact p * t ^ p) + B * (t ^ (n - m) / fact (n - m))))"
    (is "difg  (λm t. diff m t - ?difg m t)")
  shows "m t. m < n  0  t  t  h  DERIV (difg m) t :> difg (Suc m) t"
proof (rule allI impI)+
  fix m t
  assume INIT2: "m < n  0  t  t  h"
  have "DERIV (difg m) t :> diff (Suc m) t -
    ((x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / fact x) +
     real (n - m) * t ^ (n - Suc m) * B / fact (n - m))"
    by (auto simp: difg_def intro!: derivative_eq_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 "(x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / fact x) =
      (x<n - Suc m. real (Suc x) * t ^ x * diff (Suc m + x) 0 / fact (Suc x))"
    unfolding intvl by (subst sum.insert) (auto simp: sum.reindex)
  moreover
  have fact_neq_0: "x. (fact x) + real x * (fact x)  0"
    by (metis add_pos_pos fact_gt_zero less_add_same_cancel1 less_add_same_cancel2
        less_numeral_extra(3) mult_less_0_iff of_nat_less_0_iff)
  have "x. (Suc x) * t ^ x * diff (Suc m + x) 0 / fact (Suc x) = diff (Suc m + x) 0 * t^x / fact x"
    by (rule nonzero_divide_eq_eq[THEN iffD2]) auto
  moreover
  have "(n - m) * t ^ (n - Suc m) * B / fact (n - m) = B * (t ^ (n - Suc m) / fact (n - Suc m))"
    using 0 < n - m by (simp add: field_split_simps fact_reduce)
  ultimately show "DERIV (difg m) t :> difg (Suc m) t"
    unfolding difg_def  by (simp add: mult.commute)
qed

lemma Maclaurin:
  assumes h: "0 < h"
    and n: "0 < n"
    and diff_0: "diff 0 = f"
    and diff_Suc: "m t. m < n  0  t  t  h  DERIV (diff m) t :> diff (Suc m) t"
  shows
    "t::real. 0 < t  t < h 
      f h = sum (λm. (diff m 0 / fact m) * h ^ m) {..<n} + (diff n t / fact n) * h ^ n"
proof -
  from n obtain m where m: "n = Suc m"
    by (cases n) (simp add: n)
  from m have "m < n" by simp

  obtain B where f_h: "f h = (m<n. diff m 0 / fact m * h ^ m) + B * (h ^ n / fact n)"
    using Maclaurin_lemma [OF h] ..

  define g where [abs_def]: "g t =
    f t - (sum (λm. (diff m 0 / fact m) * t^m) {..<n} + B * (t^n / fact n))" for t
  have g2: "g 0 = 0" "g h = 0"
    by (simp_all add: m f_h g_def lessThan_Suc_eq_insert_0 image_iff diff_0 sum.reindex)

  define difg where [abs_def]: "difg m t =
    diff m t - (sum (λp. (diff (m + p) 0 / fact p) * (t ^ p)) {..<n-m} +
      B * ((t ^ (n - m)) / fact (n - m)))" for m t
  have difg_0: "difg 0 = g"
    by (simp add: difg_def g_def diff_0)
  have difg_Suc: "m t. m < n  0  t  t  h  DERIV (difg m) t :> difg (Suc m) t"
    using diff_Suc m unfolding difg_def [abs_def] by (rule Maclaurin_lemma2)
  have difg_eq_0: "m<n. difg m 0 = 0"
    by (auto simp: difg_def m Suc_diff_le lessThan_Suc_eq_insert_0 image_iff sum.reindex)
  have isCont_difg: "m x. m < n  0  x  x  h  isCont (difg m) x"
    by (rule DERIV_isCont [OF difg_Suc [rule_format]]) simp
  have differentiable_difg: "m x. m < n  0  x  x  h  difg m differentiable (at x)"
    using difg_Suc real_differentiable_def by auto
  have difg_Suc_eq_0:
    "m t. m < n  0  t  t  h  DERIV (difg m) t :> 0  difg (Suc m) t = 0"
    by (rule DERIV_unique [OF difg_Suc [rule_format]]) simp

  have "t. 0 < t  t < h  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 "continuous_on {0..h} (difg 0)"
        by (simp add: continuous_at_imp_continuous_on isCont_difg n)
    qed (simp add: differentiable_difg n)
  next
    case (Suc m')
    then have "t. 0 < t  t < h  DERIV (difg m') t :> 0"
      by simp
    then obtain t where t: "0 < t" "t < h" "DERIV (difg m') t :> 0"
      by fast
    have "t'. 0 < t'  t' < t  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)
      have "x. 0  x  x  t  isCont (difg (Suc m')) x"
        using t < h ‹Suc m' < n by (simp add: isCont_difg)
      then show "continuous_on {0..t} (difg (Suc m'))"
        by (simp add: continuous_at_imp_continuous_on)
    qed (use t < h ‹Suc m' < n in simp add: differentiable_difg›)
    with t < h show ?case
      by auto
  qed
  then obtain t where "0 < t" "t < h" "DERIV (difg m) t :> 0"
    by fast
  with m < n have "difg (Suc m) t = 0"
    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 = (m<n. diff m 0 / (fact m) * h ^ m) + diff n t / (fact n) * h ^ n"
      using difg (Suc m) t = 0 by (simp add: m f_h difg_def)
  qed
qed

lemma Maclaurin2:
  fixes n :: nat
    and h :: real
  assumes INIT1: "0 < h"
    and INIT2: "diff 0 = f"
    and DERIV: "m t. m < n  0  t  t  h  DERIV (diff m) t :> diff (Suc m) t"
  shows "t. 0 < t  t  h  f h = (m<n. diff m 0 / (fact m) * h ^ m) + diff n t / fact n * h ^ n"
proof (cases n)
  case 0
  with INIT1 INIT2 show ?thesis by fastforce
next
  case Suc
  then have "n > 0" by simp
  from INIT1 this INIT2 DERIV
  have "t>0. t < h  f h = (m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
    by (rule Maclaurin)
  then show ?thesis by fastforce
qed

lemma Maclaurin_minus:
  fixes n :: nat and h :: real
  assumes "h < 0" "0 < n" "diff 0 = f"
    and DERIV: "m t. m < n  h  t  t  0  DERIV (diff m) t :> diff (Suc m) t"
  shows "t. h < t  t < 0  f h = (m<n. diff m 0 / fact m * h ^ m) + diff n t / fact n * h ^ n"
proof -
  txt ‹Transform ABL'› into derivative_intros› format.›
  note DERIV' = DERIV_chain'[OF _ DERIV[rule_format], THEN DERIV_cong]
  let ?sum = "λt.
    (m<n. (- 1) ^ m * diff m (- 0) / (fact m) * (- h) ^ m) +
    (- 1) ^ n * diff n (- t) / (fact n) * (- h) ^ n"
  from assms have "t>0. t < - h  f (- (- h)) = ?sum t"
    by (intro Maclaurin) (auto intro!: derivative_eq_intros DERIV')
  then obtain t where "0 < t" "t < - h" "f (- (- h)) = ?sum t"
    by blast
  moreover have "(- 1) ^ n * diff n (- t) * (- h) ^ n / fact n = diff n (- t) * h ^ n / fact n"
    by (auto simp: power_mult_distrib[symmetric])
  moreover
    have "(m<n. (- 1) ^ m * diff m 0 * (- h) ^ m / fact m) = (m<n. diff m 0 * h ^ m / fact m)"
    by (auto intro: sum.cong simp add: power_mult_distrib[symmetric])
  ultimately have "h < - t  - t < 0 
    f h = (m<n. diff m 0 / (fact m) * h ^ m) + diff n (- t) / (fact n) * h ^ n"
    by auto
  then show ?thesis ..
qed


subsection ‹More Convenient "Bidirectional" Version.›

lemma Maclaurin_bi_le:
  fixes n :: nat and x :: real
  assumes "diff 0 = f"
    and DERIV : "m t. m < n  ¦t¦  ¦x¦  DERIV (diff m) t :> diff (Suc m) t"
  shows "t. ¦t¦  ¦x¦  f x = (m<n. diff m 0 / (fact m) * x ^ m) + diff n t / (fact n) * x ^ n"
    (is "t. _  f x = ?f x t")
proof (cases "n = 0")
  case True
  with diff 0 = f show ?thesis by force
next
  case False
  show ?thesis
  proof (cases rule: linorder_cases)
    assume "x = 0"
    with n  0 diff 0 = f DERIV have "¦0¦  ¦x¦  f x = ?f x 0"
      by auto
    then show ?thesis ..
  next
    assume "x < 0"
    with n  0 DERIV have "t>x. t < 0  diff 0 x = ?f x t"
      by (intro Maclaurin_minus) auto
    then obtain t where "x < t" "t < 0"
      "diff 0 x = (m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
      by blast
    with x < 0 diff 0 = f have "¦t¦  ¦x¦  f x = ?f x t"
      by simp
    then show ?thesis ..
  next
    assume "x > 0"
    with n  0 diff 0 = f DERIV have "t>0. t < x  diff 0 x = ?f x t"
      by (intro Maclaurin) auto
    then obtain t where "0 < t" "t < x"
      "diff 0 x = (m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
      by blast
    with x > 0 diff 0 = f have "¦t¦  ¦x¦  f x = ?f x t" by simp
    then show ?thesis ..
  qed
qed

lemma Maclaurin_all_lt:
  fixes x :: real
  assumes INIT1: "diff 0 = f"
    and INIT2: "0 < n"
    and INIT3: "x  0"
    and DERIV: "m x. DERIV (diff m) x :> diff(Suc m) x"
  shows "t. 0 < ¦t¦  ¦t¦ < ¦x¦  f x =
      (m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n"
    (is "t. _  _  f x = ?f x t")
proof (cases rule: linorder_cases)
  assume "x = 0"
  with INIT3 show ?thesis ..
next
  assume "x < 0"
  with assms have "t>x. t < 0  f x = ?f x t"
    by (intro Maclaurin_minus) auto
  then obtain t where "t > x" "t < 0" "f x = ?f x t"
    by blast
  with x < 0 have "0 < ¦t¦  ¦t¦ < ¦x¦  f x = ?f x t"
    by simp
  then show ?thesis ..
next
  assume "x > 0"
  with assms have "t>0. t < x  f x = ?f x t"
    by (intro Maclaurin) auto
  then obtain t where "t > 0" "t < x" "f x = ?f x t"
    by blast
  with x > 0 have "0 < ¦t¦  ¦t¦ < ¦x¦  f x = ?f x t"
    by simp
  then show ?thesis ..
qed

lemma Maclaurin_zero: "x = 0  n  0  (m<n. (diff m 0 / fact m) * x ^ m) = diff 0 0"
  for x :: real and n :: nat
  by simp


lemma Maclaurin_all_le:
  fixes x :: real and n :: nat
  assumes INIT: "diff 0 = f"
    and DERIV: "m x. DERIV (diff m) x :> diff (Suc m) x"
  shows "t. ¦t¦  ¦x¦  f x = (m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n"
    (is "t. _  f x = ?f x t")
proof (cases "n = 0")
  case True
  with INIT show ?thesis by force
next
  case False
  show ?thesis
  proof (cases "x = 0")
    case True
    with n  0 have "(m<n. diff m 0 / (fact m) * x ^ m) = diff 0 0"
      by (intro Maclaurin_zero) auto
    with INIT x = 0 n  0 have " ¦0¦  ¦x¦  f x = ?f x 0"
      by force
    then show ?thesis ..
  next
    case False
    with INIT n  0 DERIV have "t. 0 < ¦t¦  ¦t¦ < ¦x¦  f x = ?f x t"
      by (intro Maclaurin_all_lt) auto
    then obtain t where "0 < ¦t¦  ¦t¦ < ¦x¦  f x = ?f x t" ..
    then have "¦t¦  ¦x¦  f x = ?f x t"
      by simp
    then show ?thesis ..
  qed
qed

lemma Maclaurin_all_le_objl:
  "diff 0 = f  (m x. DERIV (diff m) x :> diff (Suc m) x) 
    (t::real. ¦t¦  ¦x¦  f x = (m<n. (diff m 0 / fact m) * x ^ m) + (diff n t / fact n) * x ^ n)"
  for x :: real and n :: nat
  by (blast intro: Maclaurin_all_le)


subsection ‹Version for Exponential Function›

lemma Maclaurin_exp_lt:
  fixes x :: real and n :: nat
  shows
    "x  0  n > 0 
      (t. 0 < ¦t¦  ¦t¦ < ¦x¦  exp x = (m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n)"
 using Maclaurin_all_lt [where diff = "λn. exp" and f = exp and x = x and n = n] by auto

lemma Maclaurin_exp_le:
  fixes x :: real and n :: nat
  shows "t. ¦t¦  ¦x¦  exp x = (m<n. (x ^ m) / fact m) + (exp t / fact n) * x ^ n"
  using Maclaurin_all_le_objl [where diff = "λn. exp" and f = exp and x = x and n = n] by auto

corollary exp_lower_Taylor_quadratic: "0  x  1 + x + x2 / 2  exp x"
  for x :: real
  using Maclaurin_exp_le [of x 3] by (auto simp: numeral_3_eq_3 power2_eq_square)

corollary ln_2_less_1: "ln 2 < (1::real)"
proof -
  have "2 < 5/(2::real)" by simp
  also have "5/2  exp (1::real)" using exp_lower_Taylor_quadratic[of 1, simplified] by simp
  finally have "exp (ln 2) < exp (1::real)" by simp
  thus "ln 2 < (1::real)" by (subst (asm) exp_less_cancel_iff) simp
qed

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"
  for m :: nat
  by 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 (auto simp: cos_add sin_add add_divide_distrib distrib_right)

lemma Maclaurin_sin_expansion2:
  "t. ¦t¦  ¦x¦ 
    sin x = (m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
proof (cases "n = 0  x = 0")
  case False
  let ?diff = "λn x. sin (x + 1/2 * real n * pi)"
  have "t. 0 < ¦t¦  ¦t¦ < ¦x¦  sin x =
      (m<n. (?diff m 0 / fact m) * x ^ m) + (?diff n t / fact n) * x ^ n"
  proof (rule Maclaurin_all_lt)
    show "m x. ((λt. sin (t + 1/2 * real m * pi)) has_real_derivative
           sin (x + 1/2 * real (Suc m) * pi)) (at x)"
      by (rule allI derivative_eq_intros | use sin_expansion_lemma in force)+
  qed (use False in auto)
  then show ?thesis
    apply (rule ex_forward, simp)
    apply (rule sum.cong[OF refl])
    apply (auto simp: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
    done
qed auto

lemma Maclaurin_sin_expansion:
  "t. sin x = (m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
  using Maclaurin_sin_expansion2 [of x n] by blast

lemma Maclaurin_sin_expansion3:
  assumes "n > 0" "x > 0"
    shows "t. 0 < t  t < x 
          sin x = (m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
proof -
  let ?diff = "λn x. sin (x + 1/2 * real n * pi)"
  have "t. 0 < t  t < x  sin x = (m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n"
  proof (rule Maclaurin)
    show "m t. m < n  0  t  t  x 
                ((λu. sin (u + 1/2 * real m * pi)) has_real_derivative
                 sin (t + 1/2 * real (Suc m) * pi)) (at t)"
      apply (simp add: sin_expansion_lemma del: of_nat_Suc)
      apply (force intro!: derivative_eq_intros)
      done
  qed (use assms in auto)
  then show ?thesis
    apply (rule ex_forward, simp)
    apply (rule sum.cong[OF refl])
    apply (auto simp: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
    done
qed

lemma Maclaurin_sin_expansion4:
  assumes "0 < x"
  shows "t. 0 < t  t  x  sin x = (m<n. sin_coeff m * x ^ m) + (sin (t + 1/2 * real n * pi) / fact n) * x ^ n"
proof -
  let ?diff = "λn x. sin (x + 1/2 * real n * pi)"
  have "t. 0 < t  t  x  sin x = (m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n"
  proof (rule Maclaurin2)
    show "m t. m < n  0  t  t  x 
                ((λu. sin (u + 1/2 * real m * pi)) has_real_derivative
                 sin (t + 1/2 * real (Suc m) * pi)) (at t)"
      apply (simp add: sin_expansion_lemma del: of_nat_Suc)
      apply (force intro!: derivative_eq_intros)
      done
  qed (use assms in auto)
  then show ?thesis
    apply (rule ex_forward, simp)
    apply (rule sum.cong[OF refl])
    apply (auto simp: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
    done
qed


subsection ‹Maclaurin Expansion for Cosine Function›

lemma sumr_cos_zero_one [simp]: "(m<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 (auto simp: cos_add sin_add distrib_right add_divide_distrib)

lemma Maclaurin_cos_expansion:
  "t::real. ¦t¦  ¦x¦ 
    cos x = (m<n. cos_coeff m * x ^ m) + (cos(t + 1/2 * real n * pi) / fact n) * x ^ n"
proof (cases "n = 0  x = 0")
  case False
  let ?diff = "λn x. cos (x + 1/2 * real n * pi)"
  have "t. 0 < ¦t¦  ¦t¦ < ¦x¦  cos x =
      (m<n. (?diff m 0 / fact m) * x ^ m) + (?diff n t / fact n) * x ^ n"
  proof (rule Maclaurin_all_lt)
    show "m x. ((λt. cos (t + 1/2 * real m * pi)) has_real_derivative
           cos (x + 1/2 * real (Suc m) * pi)) (at x)"
      apply (rule allI derivative_eq_intros | simp)+
      using cos_expansion_lemma by force
  qed (use False in auto)
  then show ?thesis
    apply (rule ex_forward, simp)
    apply (rule sum.cong[OF refl])
    apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE simp del: of_nat_Suc)
    done
qed auto

lemma Maclaurin_cos_expansion2:
  assumes "x > 0" "n > 0"
  shows "t. 0 < t  t < x 
      cos x = (m<n. cos_coeff m * x ^ m) + (cos (t + 1/2 * real n * pi) / fact n) * x ^ n"
proof -
  let ?diff = "λn x. cos (x + 1/2 * real n * pi)"
  have "t. 0 < t  t < x  cos x = (m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n"
  proof (rule Maclaurin)
    show "m t. m < n  0  t  t  x 
              ((λu. cos (u + 1 / 2 * real m * pi)) has_real_derivative 
               cos (t + 1 / 2 * real (Suc m) * pi)) (at t)"
      by (simp add: cos_expansion_lemma del: of_nat_Suc)
  qed (use assms in auto)
  then show ?thesis
    apply (rule ex_forward, simp)
    apply (rule sum.cong[OF refl])
    apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE)
    done
qed

lemma Maclaurin_minus_cos_expansion:
  assumes "n > 0" "x < 0"
  shows "t. x < t  t < 0 
         cos x = (m<n. cos_coeff m * x ^ m) + ((cos (t + 1/2 * real n * pi) / fact n) * x ^ n)"
proof -
  let ?diff = "λn x. cos (x + 1/2 * real n * pi)"
  have "t. x < t  t < 0  cos x = (m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / fact n * x ^ n"
  proof (rule Maclaurin_minus)
    show "m t. m < n  x  t  t  0 
              ((λu. cos (u + 1 / 2 * real m * pi)) has_real_derivative 
               cos (t + 1 / 2 * real (Suc m) * pi)) (at t)"
      by (simp add: cos_expansion_lemma del: of_nat_Suc)
  qed (use assms in auto)
  then show ?thesis
    apply (rule ex_forward, simp)
    apply (rule sum.cong[OF refl])
    apply (auto simp: cos_coeff_def cos_zero_iff elim: evenE)
    done
qed


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


lemma sin_bound_lemma: "x = y  ¦u¦  v  ¦(x + u) - y¦  v"
  for x y u v :: real
  by auto

lemma Maclaurin_sin_bound: "¦sin x - (m<n. sin_coeff m * x ^ m)¦  inverse (fact n) * ¦x¦ ^ n"
proof -
  have est: "x  1  0  y  x * y  1 * y" for x y :: real
    by (rule mult_right_mono) simp_all
  let ?diff = "λ(n::nat) (x::real).
    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 m) x :> ?diff (Suc m) x" for m and x
    using mod_exhaust_less_4 [of m]
    by (auto simp: mod_Suc intro!: derivative_eq_intros)
  then have DERIV_diff: "m x. DERIV (?diff m) x :> ?diff (Suc m) x"
    by blast
  from Maclaurin_all_le [OF diff_0 DERIV_diff]
  obtain t where t1: "¦t¦  ¦x¦"
    and t2: "sin x = (m<n. ?diff m 0 / (fact m) * x ^ m) + ?diff n t / (fact n) * x ^ n"
    by fast
  have diff_m_0: "?diff m 0 = (if even m then 0 else (- 1) ^ ((m - Suc 0) div 2))" for m
    using mod_exhaust_less_4 [of m]
    by (auto simp: minus_one_power_iff even_even_mod_4_iff [of m] dest: even_mod_4_div_2 odd_mod_4_div_2)
  show ?thesis
    unfolding sin_coeff_def
    apply (subst t2)
    apply (rule sin_bound_lemma)
     apply (rule sum.cong[OF refl])
     apply (subst diff_m_0, simp)
    using est
    apply (auto intro: mult_right_mono [where b=1, simplified] mult_right_mono
        simp: ac_simps divide_inverse power_abs [symmetric] abs_mult)
    done
qed


section ‹Taylor series›

text ‹
  We use MacLaurin and the translation of the expansion point c› to 0›
  to prove Taylor's theorem.
›

lemma Taylor_up:
  assumes INIT: "n > 0" "diff 0 = f"
    and DERIV: "m t. m < n  a  t  t  b  DERIV (diff m) t :> (diff (Suc m) t)"
    and INTERV: "a  c" "c < b"
  shows "t::real. c < t  t < b 
    f b = (m<n. (diff m c / fact m) * (b - c)^m) + (diff n t / fact n) * (b - c)^n"
proof -
  from INTERV have "0 < b - c" by arith
  moreover from INIT have "n > 0" "(λm x. diff m (x + c)) 0 = (λx. f (x + c))"
    by auto
  moreover
  have "m t. m < n  0  t  t  b - c  DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)"
  proof (intro strip)
    fix m t
    assume "m < n  0  t  t  b - c"
    with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)"
      by auto
    moreover from DERIV_ident and DERIV_const have "DERIV (λx. x + c) t :> 1 + 0"
      by (rule DERIV_add)
    ultimately have "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)"
      by (rule DERIV_chain2)
    then show "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)"
      by simp
  qed
  ultimately obtain x where
    "0 < x  x < b - c 
      f (b - c + c) =
        (m<n. diff m (0 + c) / fact m * (b - c) ^ m) + diff n (x + c) / fact n * (b - c) ^ n"
     by (rule Maclaurin [THEN exE])
   then have "c < x + c  x + c < b  f b =
     (m<n. diff m c / fact m * (b - c) ^ m) + diff n (x + c) / fact n * (b - c) ^ n"
    by fastforce
  then show ?thesis by fastforce
qed

lemma Taylor_down:
  fixes a :: real and n :: nat
  assumes INIT: "n > 0" "diff 0 = f"
    and DERIV: "(m t. m < n  a  t  t  b  DERIV (diff m) t :> diff (Suc m) t)"
    and INTERV: "a < c" "c  b"
  shows "t. a < t  t < c 
    f a = (m<n. (diff m c / fact m) * (a - c)^m) + (diff n t / fact n) * (a - c)^n"
proof -
  from INTERV have "a-c < 0" by arith
  moreover from INIT have "n > 0" "(λm x. diff m (x + c)) 0 = (λx. f (x + c))"
    by auto
  moreover
  have "m t. m < n  a - c  t  t  0  DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)"
  proof (rule allI impI)+
    fix m t
    assume "m < n  a - c  t  t  0"
    with DERIV and INTERV have "DERIV (diff m) (t + c) :> diff (Suc m) (t + c)"
      by auto
    moreover from DERIV_ident and DERIV_const have "DERIV (λx. x + c) t :> 1 + 0"
      by (rule DERIV_add)
    ultimately have "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c) * (1 + 0)"
      by (rule DERIV_chain2)
    then show "DERIV (λx. diff m (x + c)) t :> diff (Suc m) (t + c)"
      by simp
  qed
  ultimately obtain x where
    "a - c < x  x < 0 
      f (a - c + c) =
        (m<n. diff m (0 + c) / fact m * (a - c) ^ m) + diff n (x + c) / fact n * (a - c) ^ n"
    by (rule Maclaurin_minus [THEN exE])
  then have "a < x + c  x + c < c 
    f a = (m<n. diff m c / fact m * (a - c) ^ m) + diff n (x + c) / fact n * (a - c) ^ n"
    by fastforce
  then show ?thesis by fastforce
qed

theorem Taylor:
  fixes a :: real and n :: nat
  assumes INIT: "n > 0" "diff 0 = f"
    and DERIV: "m t. m < n  a  t  t  b  DERIV (diff m) t :> diff (Suc m) t"
    and INTERV: "a  c " "c  b" "a  x" "x  b" "x  c"
  shows "t.
    (if x < c then x < t  t < c else c < t  t < x) 
    f x = (m<n. (diff m c / fact m) * (x - c)^m) + (diff n t / fact n) * (x - c)^n"
proof (cases "x < c")
  case True
  note INIT
  moreover have "m t. m < n  x  t  t  b  DERIV (diff m) t :> diff (Suc m) t"
    using DERIV and INTERV by fastforce
  moreover note True
  moreover from INTERV have "c  b"
    by simp
  ultimately have "t>x. t < c  f x =
    (m<n. diff m c / (fact m) * (x - c) ^ m) + diff n t / (fact n) * (x - c) ^ n"
    by (rule Taylor_down)
  with True show ?thesis by simp
next
  case False
  note INIT
  moreover have "m t. m < n  a  t  t  x  DERIV (diff m) t :> diff (Suc m) t"
    using DERIV and INTERV by fastforce
  moreover from INTERV have "a  c"
    by arith
  moreover from False and INTERV have "c < x"
    by arith
  ultimately have "t>c. t < x  f x =
    (m<n. diff m c / (fact m) * (x - c) ^ m) + diff n t / (fact n) * (x - c) ^ n"
    by (rule Taylor_up)
  with False show ?thesis by simp
qed

end