Quelle  Harmonic_Numbers.thy

(*  Title:    HOL/Analysis/Harmonic_Numbers.thy
  Author: Manuel Eberl, TU München
*)

section ‹Harmonic Numbers›

theory Harmonic_Numbers
imports
  Complex_Transcendental
  Summation_Tests
begin

text ‹
  The definition of the Harmonic Numbers and the Euler-Mascheroni constant.
  Also provides a reasonably accurate approximation of 🍋‹ln 2 :: real›
  and the Euler-Mascheroni constant.
 ›

subsection ‹The Harmonic numbers›

definition🍋‹tag important› harm :: "nat ==> 'a :: real_normed_field" where
  "harm n = (∑k=1..n. inverse (of_nat k))"

lemma harm_altdef: "harm n = (∑k
  unfolding harm_def by (induction n) simp_all

lemma harm_Suc: "harm (Suc n) = harm n + inverse (of_nat (Suc n))"
  by (simp add: harm_def)

lemma harm_nonneg: "harm n ≥ (0 :: 'a :: {real_normed_field,linordered_field})"
  unfolding harm_def by (intro sum_nonneg) simp_all

lemma harm_pos: "n > 0 ==> harm n > (0 :: 'a :: {real_normed_field,linordered_field})"
  unfolding harm_def by (intro sum_pos) simp_all

lemma harm_mono: "m ≤ n ==> harm m ≤ (harm n :: 'a :: {real_normed_field,linordered_field})"
by(simp add: harm_def sum_mono2)

lemma of_real_harm: "of_real (harm n) = harm n"
  unfolding harm_def by simp

lemma abs_harm [simp]: "(abs (harm n) :: real) = harm n"
  using harm_nonneg[of n] by (rule abs_of_nonneg)

lemma norm_harm: "norm (harm n) = harm n"
  by (subst of_real_harm [symmetric]) (simp add: harm_nonneg)

lemma harm_expand:
  "harm 0 = 0"
  "harm (Suc 0) = 1"
  "harm (numeral n) = harm (pred_numeral n) + inverse (numeral n)"
proof -
  have "numeral n = Suc (pred_numeral n)" by simp
  also have "harm … = harm (pred_numeral n) + inverse (numeral n)"
    by (subst harm_Suc, subst numeral_eq_Suc[symmetric]) simp
  finally show "harm (numeral n) = harm (pred_numeral n) + inverse (numeral n)" .
qed (simp_all add: harm_def)

theorem not_convergent_harm: "¬convergent (harm :: nat ==> 'a :: real_normed_field)"
proof -
  have "convergent (λn. norm (harm n :: 'a)) ⟷
            convergent (harm :: nat ==> real)" by (simp add: norm_harm)
  also have "… ⟷ convergent (λn. ∑k=Suc 0..Suc n. inverse (of_nat k) :: real)"
    unfolding harm_def[abs_def] by (subst convergent_Suc_iff) simp_all
  also have "... ⟷ convergent (λn. ∑k≤n. inverse (of_nat (Suc k)) :: real)"
    by (subst sum.shift_bounds_cl_Suc_ivl) (simp add: atLeast0AtMost)
  also have "... ⟷ summable (λn. inverse (of_nat n) :: real)"
    by (subst summable_Suc_iff [symmetric]) (simp add: summable_iff_convergent')
  also have "¬..." by (rule not_summable_harmonic)
  finally show ?thesis by (blast dest: convergent_norm)
qed

lemma harm_pos_iff [simp]: "harm n > (0 :: 'a :: {real_normed_field,linordered_field}) ⟷ n > 0"
  by (rule iffI, cases n, simp add: harm_expand, simp, rule harm_pos)

lemma ln_diff_le_inverse:
  assumes "x ≥ (1::real)"
  shows   "ln (x + 1) - ln x < 1 / x"
proof -
  from assms have "∃z>x. z < x + 1 ∧ ln (x + 1) - ln x = (x + 1 - x) * inverse z"
    by (intro MVT2) (auto intro!: derivative_eq_intros simp: field_simps)
  then obtain z where z: "z > x" "z < x + 1" "ln (x + 1) - ln x = inverse z" by auto
  have "ln (x + 1) - ln x = inverse z" by fact
  also from z(1,2) assms have "… < 1 / x" by (simp add: field_simps)
  finally show ?thesis .
qed

lemma ln_le_harm: "ln (real n + 1) ≤ (harm n :: real)"
proof (induction n)
  fix n assume IH: "ln (real n + 1) ≤ harm n"
  have "ln (real (Suc n) + 1) = ln (real n + 1) + (ln (real n + 2) - ln (real n + 1))" by simp
  also have "(ln (real n + 2) - ln (real n + 1)) ≤ 1 / real (Suc n)"
    using ln_diff_le_inverse[of "real n + 1"] by (simp add: add_ac)
  also note IH
  also have "harm n + 1 / real (Suc n) = harm (Suc n)" by (simp add: harm_Suc field_simps)
  finally show "ln (real (Suc n) + 1) ≤ harm (Suc n)" by - simp
qed (simp_all add: harm_def)

lemma harm_at_top: "filterlim (harm :: nat ==> real) at_top sequentially"
proof (rule filterlim_at_top_mono)
  show "eventually (λn. harm n ≥ ln (real (Suc n))) at_top"
    using ln_le_harm by (intro always_eventually allI) (simp_all add: add_ac)
  show "filterlim (λn. ln (real (Suc n))) at_top sequentially"
    by (intro filterlim_compose[OF ln_at_top] filterlim_compose[OF filterlim_real_sequentially]
              filterlim_Suc)
qed


subsection ‹The Euler-Mascheroni constant›

text ‹
  The limit of the difference between the partial harmonic sum and the natural logarithm
  (approximately 0.577216). This value occurs e.g. in the definition of the Gamma function.
  ›
definition euler_mascheroni :: "'a :: real_normed_algebra_1" where
  "euler_mascheroni = of_real (lim (λn. harm n - ln (of_nat n)))"

lemma of_real_euler_mascheroni [simp]: "of_real euler_mascheroni = euler_mascheroni"
  by (simp add: euler_mascheroni_def)

lemma harm_ge_ln: "harm n ≥ ln (real n + 1)"
proof -
  have "ln (n + 1) = (∑j
    by (subst sum_lessThan_telescope) auto
  also have "… ≤ (∑j
  proof (intro sum_mono, clarify)
    fix j assume j: "j < n"
    have "∃ξ. ξ > real j + 1 ∧ ξ < real j + 2 ∧
            ln (real j + 2) - ln (real j + 1) = (real j + 2 - (real j + 1)) * (1 / ξ)"
      by (intro MVT2) (auto intro!: derivative_eq_intros)
    then obtain ξ :: real
      where ξ: "ξ ∈ {real j + 1..real j + 2}" "ln (real j + 2) - ln (real j + 1) = 1 / ξ"
      by auto
    note ξ(2)
    also have "1 / ξ ≤ 1 / (Suc j)"
      using ξ(1) by (auto simp: field_simps)
    finally show "ln (real (Suc j + 1)) - ln (real (j + 1)) ≤ 1 / (Suc j)"
      by (simp add: add_ac)
  qed
  also have "… = harm n"
    by (simp add: harm_altdef field_simps)
  finally show ?thesis by (simp add: add_ac)
qed

lemma decseq_harm_diff_ln: "decseq (λn. harm (Suc n) - ln (Suc n))"
proof (rule decseq_SucI)
  fix m :: nat
  define n where "n = Suc m"
  have "n > 0" by (simp add: n_def)
  have "convex_on {0<..} (λx :: real. -ln x)"
    by (rule convex_on_realI[where f' = "λx. -1/x"])
       (auto intro!: derivative_eq_intros simp: field_simps)
  hence "(-1 / (n + 1)) * (real n - real (n + 1)) ≤ (- ln (real n)) - (-ln (real (n + 1)))"
    using ‹n > 0› by (intro convex_on_imp_above_tangent[where A = "{0<..}"])
                     (auto intro!: derivative_eq_intros simp: interior_open)
  thus "harm (Suc n) - ln (Suc n) ≤ harm n - ln n"
    by (auto simp: harm_Suc field_simps)
qed

lemma euler_mascheroni_sequence_nonneg:
  assumes "n > 0"
  shows   "harm n - ln (real n) ≥ (0 :: real)"
proof -
  have "ln (real n) ≤ ln (real n + 1)"
    using assms by simp
  also have "… ≤ harm n"
    by (rule harm_ge_ln)
  finally show ?thesis by simp
qed

lemma euler_mascheroni_convergent: "convergent (λn. harm n - ln n)"
proof -
  have "harm (Suc n) - ln (real (Suc n)) ≥ 0" for n :: nat
    using euler_mascheroni_sequence_nonneg[of "Suc n"] by simp
  hence "convergent (λn. harm (Suc n) - ln (Suc n))"
    by (intro Bseq_monoseq_convergent decseq_bounded[of _ 0] decseq_harm_diff_ln decseq_imp_monoseq)
       auto
  thus ?thesis
    by (subst (asm) convergent_Suc_iff)
qed

lemma euler_mascheroni_sequence_decreasing:
  "m > 0 ==> m ≤ n ==> harm n - ln (of_nat n) ≤ harm m - ln (of_nat m :: real)"
  using decseqD[OF decseq_harm_diff_ln, of "m - 1" "n - 1"] by simp
  
lemma🍋‹tag important› euler_mascheroni_LIMSEQ:
  "(λn. harm n - ln (of_nat n) :: real) <---- euler_mascheroni"
  unfolding euler_mascheroni_def
  by (simp add: convergent_LIMSEQ_iff [symmetric] euler_mascheroni_convergent)

lemma euler_mascheroni_LIMSEQ_of_real:
  "(λn. of_real (harm n - ln (of_nat n))) <----
      (euler_mascheroni :: 'a :: {real_normed_algebra_1, topological_space})"
proof -
  have "(λn. of_real (harm n - ln (of_nat n))) <---- (of_real (euler_mascheroni) :: 'a)"
    by (intro tendsto_of_real euler_mascheroni_LIMSEQ)
  thus ?thesis by simp
qed

lemma euler_mascheroni_sum_real:
  "(λn. inverse (of_nat (n+1)) + ln (of_nat (n+1)) - ln (of_nat (n+2)) :: real)
       sums euler_mascheroni"
 using sums_add[OF telescope_sums[OF LIMSEQ_Suc[OF euler_mascheroni_LIMSEQ]]
                   telescope_sums'[OF LIMSEQ_inverse_real_of_nat]]
  by (simp_all add: harm_def algebra_simps)

lemma euler_mascheroni_sum:
  "(λn. inverse (of_nat (n+1)) + of_real (ln (of_nat (n+1))) - of_real (ln (of_nat (n+2))))
       sums (euler_mascheroni :: 'a :: {banach, real_normed_field})"
proof -
  have "(λn. of_real (inverse (of_nat (n+1)) + ln (of_nat (n+1)) - ln (of_nat (n+2))))
       sums (of_real euler_mascheroni :: 'a :: {banach, real_normed_field})"
    by (subst sums_of_real_iff) (rule euler_mascheroni_sum_real)
  thus ?thesis by simp
qed

theorem alternating_harmonic_series_sums: "(λk. (-1)^k / real_of_nat (Suc k)) sums ln 2"
proof -
  let ?f = "λn. harm n - ln (real_of_nat n)"
  let ?g = "λn. if even n then 0 else (2::real)"
  let ?em = "λn. harm n - ln (real_of_nat n)"
  have "eventually (λn. ?em (2*n) - ?em n + ln 2 = (∑k<2*n. (-1)^k / real_of_nat (Suc k))) at_top"
    using eventually_gt_at_top[of "0::nat"]
  proof eventually_elim
    fix n :: nat assume n: "n > 0"
    have "(∑k<2*n. (-1)^k / real_of_nat (Suc k)) =
              (∑k<2*n. ((-1)^k + ?g k) / of_nat (Suc k)) - (∑k<2*n. ?g k / of_nat (Suc k))"
      by (simp add: sum.distrib algebra_simps divide_inverse)
    also have "(∑k<2*n. ((-1)^k + ?g k) / real_of_nat (Suc k)) = harm (2*n)"
      unfolding harm_altdef by (intro sum.cong) (auto simp: field_simps)
    also have "(∑k<2*n. ?g k / real_of_nat (Suc k)) = (∑k|k<2*n ∧ odd k. ?g k / of_nat (Suc k))"
      by (intro sum.mono_neutral_right) auto
    also have "… = (∑k|k<2*n ∧ odd k. 2 / (real_of_nat (Suc k)))"
      by (intro sum.cong) auto
    also have "(∑k|k<2*n ∧ odd k. 2 / (real_of_nat (Suc k))) = harm n"
      unfolding harm_altdef
      by (intro sum.reindex_cong[of "λn. 2*n+1"]) (auto simp: inj_on_def field_simps elim!: oddE)
    also have "harm (2*n) - harm n = ?em (2*n) - ?em n + ln 2" using n
      by (simp_all add: algebra_simps ln_mult)
    finally show "?em (2*n) - ?em n + ln 2 = (∑k<2*n. (-1)^k / real_of_nat (Suc k))" ..
  qed
  moreover have "(λn. ?em (2*n) - ?em n + ln (2::real))
                     <---- euler_mascheroni - euler_mascheroni + ln 2"
    by (intro tendsto_intros euler_mascheroni_LIMSEQ filterlim_compose[OF euler_mascheroni_LIMSEQ]
              filterlim_subseq) (auto simp: strict_mono_def)
  hence "(λn. ?em (2*n) - ?em n + ln (2::real)) <---- ln 2" by simp
  ultimately have "(λn. (∑k<2*n. (-1)^k / real_of_nat (Suc k))) <---- ln 2"
    by (blast intro: Lim_transform_eventually)

  moreover have "summable (λk. (-1)^k * inverse (real_of_nat (Suc k)))"
    using LIMSEQ_inverse_real_of_nat
    by (intro summable_Leibniz(1) decseq_imp_monoseq decseq_SucI) simp_all
  hence A: "(λn. ∑k<---- (∑k. (-1)^k / real_of_nat (Suc k))"
    by (simp add: summable_sums_iff divide_inverse sums_def)
  from filterlim_compose[OF this filterlim_subseq[of "(*) (2::nat)"]]
    have "(λn. ∑k<2*n. (-1)^k / real_of_nat (Suc k)) <---- (∑k. (-1)^k / real_of_nat (Suc k))"
    by (simp add: strict_mono_def)
  ultimately have "(∑k. (- 1) ^ k / real_of_nat (Suc k)) = ln 2" by (intro LIMSEQ_unique)
  with A show ?thesis by (simp add: sums_def)
qed

lemma alternating_harmonic_series_sums':
  "(λk. inverse (real_of_nat (2*k+1)) - inverse (real_of_nat (2*k+2))) sums ln 2"
unfolding sums_def
proof (rule Lim_transform_eventually)
  show "(λn. ∑k<2*n. (-1)^k / (real_of_nat (Suc k))) <---- ln 2"
    using alternating_harmonic_series_sums unfolding sums_def
    by (rule filterlim_compose) (rule mult_nat_left_at_top, simp)
  show "eventually (λn. (∑k<2*n. (-1)^k / (real_of_nat (Suc k))) =
            (∑k<n. inverse (real_of_nat (2*k+1)) - inverse (real_of_nat (2*k+2)))) sequentially"
  proof (intro always_eventually allI)
    fix n :: nat
    show "(∑k<2*n. (-1)^k / (real_of_nat (Suc k))) =
              (∑k<n. inverse (real_of_nat (2*k+1)) - inverse (real_of_nat (2*k+2)))"
      by (induction n) (simp_all add: inverse_eq_divide)
  qed
qed


subsection🍋‹tag unimportant› ‹Bounds on the Euler-Mascheroni constant›
(* TODO: perhaps move this section away to remove unnecessary dependency on integration *)

(* TODO: Move? *)
lemma ln_inverse_approx_le:
  assumes "(x::real) > 0" "a > 0"
  shows "ln (x + a) - ln x ≤ a * (inverse x + inverse (x + a))/2" (is "_ ≤ ?A")
proof -
  define f' where "f' = (inverse (x + a) - inverse x)/a"
  let ?f = "λt. (t - x) * f' + inverse x"
  let ?F = "λt. (t - x)^2 * f' / 2 + t * inverse x"

  have deriv: "∃D. ((λx. ?F x - ln x) has_field_derivative D) (at ξ) ∧ D ≥ 0"
    if "ξ ≥ x" "ξ ≤ x + a" for ξ
  proof -
    from that assms have t: "0 ≤ (ξ - x) / a" "(ξ - x) / a ≤ 1" by simp_all
    have "inverse ξ = inverse ((1 - (ξ - x) / a) *🪙R x + ((ξ - x) / a) *🪙R (x + a))" (is "_ = ?A")
      using assms by (simp add: field_simps)
    also from assms have "convex_on {x..x+a} inverse" by (intro convex_on_inverse) auto
    from convex_onD_Icc[OF this _ t] assms
      have "?A ≤ (1 - (ξ - x) / a) * inverse x + (ξ - x) / a * inverse (x + a)" by simp
    also have "… = (ξ - x) * f' + inverse x" using assms
      by (simp add: f'_def divide_simps) (simp add: field_simps)
    finally have "?f ξ - 1 / ξ ≥ 0" by (simp add: field_simps)
    moreover have "((λx. ?F x - ln x) has_field_derivative ?f ξ - 1 / ξ) (at ξ)"
      using that assms by (auto intro!: derivative_eq_intros simp: field_simps)
    ultimately show ?thesis by blast
  qed
  have "?F x - ln x ≤ ?F (x + a) - ln (x + a)"
    by (rule DERIV_nonneg_imp_nondecreasing[of x "x + a", OF _ deriv]) (use assms in auto)
  thus ?thesis
    using assms by (simp add: f'_def divide_simps) (simp add: algebra_simps power2_eq_square)?
qed

lemma ln_inverse_approx_ge:
  assumes "(x::real) > 0" "x < y"
  shows "ln y - ln x ≥ 2 * (y - x) / (x + y)" (is "_ ≥ ?A")
proof -
  define m where "m = (x+y)/2"
  define f' where "f' = -inverse (m^2)"
  from assms have m: "m > 0" by (simp add: m_def)
  let ?F = "λt. (t - m)^2 * f' / 2 + t / m"
  let ?f = "λt. (t - m) * f' + inverse m"
  
  have deriv: "∃D. ((λx. ln x - ?F x) has_field_derivative D) (at ξ) ∧ D ≥ 0"
    if "ξ ≥ x" "ξ ≤ y" for ξ
  proof -
    from that assms have "inverse ξ - inverse m ≥ f' * (ξ - m)"
      by (intro convex_on_imp_above_tangent[of "{0<..}"] convex_on_inverse)
         (auto simp: m_def interior_open f'_def power2_eq_square intro!: derivative_eq_intros)
    hence "1 / ξ - ?f ξ ≥ 0" by (simp add: field_simps f'_def)
    moreover have "((λx. ln x - ?F x) has_field_derivative 1 / ξ - ?f ξ) (at ξ)"
      using that assms m by (auto intro!: derivative_eq_intros simp: field_simps)
    ultimately show ?thesis by blast
  qed
  have "ln x - ?F x ≤ ln y - ?F y"
    by (rule DERIV_nonneg_imp_nondecreasing[of x y, OF _ deriv]) (use assms in auto)
  hence "ln y - ln x ≥ ?F y - ?F x"
    by (simp add: algebra_simps)
  also have "?F y - ?F x = ?A"
    using assms by (simp add: f'_def m_def divide_simps) (simp add: algebra_simps power2_eq_square)
  finally show ?thesis .
qed

lemma euler_mascheroni_lower:
          "euler_mascheroni ≥ harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 2))"
    and euler_mascheroni_upper:
          "euler_mascheroni ≤ harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 1))"
proof -
  define D :: "_ ==> real"
    where "D n = inverse (of_nat (n+1)) + ln (of_nat (n+1)) - ln (of_nat (n+2))" for n
  let ?g = "λn. ln (of_nat (n+2)) - ln (of_nat (n+1)) - inverse (of_nat (n+1)) :: real"
  define inv where [abs_def]: "inv n = inverse (real_of_nat n)" for n
  fix n :: nat
  note summable = sums_summable[OF euler_mascheroni_sum_real, folded D_def]
  have sums: "(λk. (inv (Suc (k + (n+1))) - inv (Suc (Suc k + (n+1))))/2) sums ((inv (Suc (0 + (n+1))) - 0)/2)"
    unfolding inv_def
    by (intro sums_divide telescope_sums' LIMSEQ_ignore_initial_segment LIMSEQ_inverse_real_of_nat)
  have sums': "(λk. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2) sums ((inv (Suc (0 + n)) - 0)/2)"
    unfolding inv_def
    by (intro sums_divide telescope_sums' LIMSEQ_ignore_initial_segment LIMSEQ_inverse_real_of_nat)
  from euler_mascheroni_sum_real have "euler_mascheroni = (∑k. D k)"
    by (simp add: sums_iff D_def)
  also have "… = (∑k. D (k + Suc n)) + (∑k≤n. D k)"
    by (subst suminf_split_initial_segment[OF summable, of "Suc n"],
        subst lessThan_Suc_atMost) simp
  finally have sum: "(∑k≤n. D k) - euler_mascheroni = -(∑k. D (k + Suc n))" by simp

  note sum
  also have "… ≤ -(∑k. (inv (k + Suc n + 1) - inv (k + Suc n + 2)) / 2)"
  proof (intro le_imp_neg_le suminf_le allI summable_ignore_initial_segment[OF summable])
    fix k' :: nat
    define k where "k = k' + Suc n"
    hence k: "k > 0" by (simp add: k_def)
    have "real_of_nat (k+1) > 0" by (simp add: k_def)
    with ln_inverse_approx_le[OF this zero_less_one]
      have "ln (of_nat k + 2) - ln (of_nat k + 1) ≤ (inv (k+1) + inv (k+2))/2"
      by (simp add: inv_def add_ac)
    hence "(inv (k+1) - inv (k+2))/2 ≤ inv (k+1) + ln (of_nat (k+1)) - ln (of_nat (k+2))"
      by (simp add: field_simps)
    also have "… = D k" unfolding D_def inv_def ..
    finally show "D (k' + Suc n) ≥ (inv (k' + Suc n + 1) - inv (k' + Suc n + 2)) / 2"
      by (simp add: k_def)
    from sums_summable[OF sums]
      show "summable (λk. (inv (k + Suc n + 1) - inv (k + Suc n + 2))/2)" by simp
  qed
  also from sums have "… = -inv (n+2) / 2" by (simp add: sums_iff)
  finally have "euler_mascheroni ≥ (∑k≤n. D k) + 1 / (of_nat (2 * (n+2)))"
    by (simp add: inv_def field_simps)
  also have "(∑k≤n. D k) = harm (Suc n) - (∑k≤n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1)))"
    unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add: sum.distrib sum_subtractf)
  also have "(∑k≤n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1))) = ln (of_nat (n+2))"
    by (subst atLeast0AtMost [symmetric], subst sum_Suc_diff) simp_all
  finally show "euler_mascheroni ≥ harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 2))"
    by simp

  note sum
  also have "-(∑k. D (k + Suc n)) ≥ -(∑k. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2)"
  proof (intro le_imp_neg_le suminf_le allI summable_ignore_initial_segment[OF summable])
    fix k' :: nat
    define k where "k = k' + Suc n"
    hence k: "k > 0" by (simp add: k_def)
    have "real_of_nat (k+1) > 0" by (simp add: k_def)
    from ln_inverse_approx_ge[of "of_nat k + 1" "of_nat k + 2"]
      have "2 / (2 * real_of_nat k + 3) ≤ ln (of_nat (k+2)) - ln (real_of_nat (k+1))"
      by (simp add: add_ac)
    hence "D k ≤ 1 / real_of_nat (k+1) - 2 / (2 * real_of_nat k + 3)"
      by (simp add: D_def inverse_eq_divide inv_def)
    also have "… = inv ((k+1)*(2*k+3))" unfolding inv_def by (simp add: field_simps)
    also have "… ≤ inv (2*k*(k+1))" unfolding inv_def using k
      by (intro le_imp_inverse_le)
         (simp add: algebra_simps, simp del: of_nat_add)
    also have "… = (inv k - inv (k+1))/2" unfolding inv_def using k
      by (simp add: divide_simps del: of_nat_mult) (simp add: algebra_simps)
    finally show "D k ≤ (inv (Suc (k' + n)) - inv (Suc (Suc k' + n)))/2" unfolding k_def by simp
  next
    from sums_summable[OF sums']
      show "summable (λk. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2)" by simp
  qed
  also from sums' have "(∑k. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2) = inv (n+1)/2"
    by (simp add: sums_iff)
  finally have "euler_mascheroni ≤ (∑k≤n. D k) + 1 / of_nat (2 * (n+1))"
    by (simp add: inv_def field_simps)
  also have "(∑k≤n. D k) = harm (Suc n) - (∑k≤n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1)))"
    unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add: sum.distrib sum_subtractf)
  also have "(∑k≤n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1))) = ln (of_nat (n+2))"
    by (subst atLeast0AtMost [symmetric], subst sum_Suc_diff) simp_all
  finally show "euler_mascheroni ≤ harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 1))"
    by simp
qed

lemma euler_mascheroni_pos: "euler_mascheroni > (0::real)"
  using euler_mascheroni_lower[of 0] ln_2_less_1 by (simp add: harm_def)

context
begin

private lemma ln_approx_aux:
  fixes n :: nat and x :: real
  defines "y ≡ (x-1)/(x+1)"
  assumes x: "x > 0" "x ≠ 1"
  shows "inverse (2*y^(2*n+1)) * (ln x - (∑k<n. 2*y^(2*k+1) / of_nat (2*k+1))) ∈
            {0..(1 / (1 - y^2) / of_nat (2*n+1))}"
proof -
  from x have norm_y: "norm y < 1" unfolding y_def by simp
  from power_strict_mono[OF this, of 2] have norm_y': "norm y^2 < 1" by simp

  let ?f = "λk. 2 * y ^ (2*k+1) / of_nat (2*k+1)"
  note sums = ln_series_quadratic[OF x(1)]
  define c where "c = inverse (2*y^(2*n+1))"
  let ?d = "c * (ln x - (∑k<n. ?f k))"
  have "∧k. y🪙2^k / of_nat (2*(k+n)+1) ≤ y🪙2 ^ k / of_nat (2*n+1)"
    by (intro divide_left_mono mult_right_mono mult_pos_pos zero_le_power[of "y^2"]) simp_all
  moreover {
    have "(λk. ?f (k + n)) sums (ln x - (∑k<n. ?f k))"
      using sums_split_initial_segment[OF sums] by (simp add: y_def)
    hence "(λk. c * ?f (k + n)) sums ?d" by (rule sums_mult)
    also have "(λk. c * (2*y^(2*(k+n)+1) / of_nat (2*(k+n)+1))) =
                   (λk. (c * (2*y^(2*n+1))) * ((y^2)^k / of_nat (2*(k+n)+1)))"
      by (simp only: ring_distribs power_add power_mult) (simp add: mult_ac)
    also from x have "c * (2*y^(2*n+1)) = 1" by (simp add: c_def y_def)
    finally have "(λk. (y^2)^k / of_nat (2*(k+n)+1)) sums ?d" by simp
  } note sums' = this
  moreover from norm_y' have "(λk. (y^2)^k / of_nat (2*n+1)) sums (1 / (1 - y^2) / of_nat (2*n+1))"
    by (intro sums_divide geometric_sums) (simp_all add: norm_power)
  ultimately have "?d ≤ (1 / (1 - y^2) / of_nat (2*n+1))" by (rule sums_le)
  moreover have "c * (ln x - (∑k<n. 2 * y ^ (2 * k + 1) / real_of_nat (2 * k + 1))) ≥ 0"
    by (intro sums_le[OF _ sums_zero sums']) simp_all
  ultimately show ?thesis unfolding c_def by simp
qed

lemma
  fixes n :: nat and x :: real
  defines "y ≡ (x-1)/(x+1)"
  defines "approx ≡ (∑k<n. 2*y^(2*k+1) / of_nat (2*k+1))"
  defines "d ≡ y^(2*n+1) / (1 - y^2) / of_nat (2*n+1)"
  assumes x: "x > 1"
  shows ln_approx_bounds: "ln x ∈ {approx..approx + 2*d}"
  and ln_approx_abs: "abs (ln x - (approx + d)) ≤ d"
proof -
  define c where "c = 2*y^(2*n+1)"
  from x have c_pos: "c > 0" unfolding c_def y_def
    by (intro mult_pos_pos zero_less_power) simp_all
  have A: "inverse c * (ln x - (∑k<n. 2*y^(2*k+1) / of_nat (2*k+1))) ∈
              {0.. (1 / (1 - y^2) / of_nat (2*n+1))}" using assms unfolding y_def c_def
    by (intro ln_approx_aux) simp_all
  hence "inverse c * (ln x - (∑k<n. 2*y^(2*k+1)/of_nat (2*k+1))) ≤ (1 / (1-y^2) / of_nat (2*n+1))"
    by simp
  hence "(ln x - (∑k<n. 2*y^(2*k+1) / of_nat (2*k+1))) / c ≤ (1 / (1 - y^2) / of_nat (2*n+1))"
    by (auto simp add: field_split_simps)
  with c_pos have "ln x ≤ c / (1 - y^2) / of_nat (2*n+1) + approx"
    by (subst (asm) pos_divide_le_eq) (simp_all add: mult_ac approx_def)
  moreover {
    from A c_pos have "0 ≤ c * (inverse c * (ln x - (∑k<n. 2*y^(2*k+1) / of_nat (2*k+1))))"
      by (intro mult_nonneg_nonneg[of c]) simp_all
    also have "… = (c * inverse c) * (ln x - (∑k<n. 2*y^(2*k+1) / of_nat (2*k+1)))"
      by (simp add: mult_ac)
    also from c_pos have "c * inverse c = 1" by simp
    finally have "ln x ≥ approx" by (simp add: approx_def)
  }
  ultimately show "ln x ∈ {approx..approx + 2*d}" by (simp add: c_def d_def)
  thus "abs (ln x - (approx + d)) ≤ d" by auto
qed

end

lemma euler_mascheroni_bounds:
  fixes n :: nat assumes "n ≥ 1" defines "t ≡ harm n - ln (of_nat (Suc n)) :: real"
  shows "euler_mascheroni ∈ {t + inverse (of_nat (2*(n+1)))..t + inverse (of_nat (2*n))}"
  using assms euler_mascheroni_upper[of "n-1"] euler_mascheroni_lower[of "n-1"]
  unfolding t_def by (cases n) (simp_all add: harm_Suc t_def inverse_eq_divide)

lemma euler_mascheroni_bounds':
  fixes n :: nat assumes "n ≥ 1" "ln (real_of_nat (Suc n)) ∈ {l<..<u}"
  shows "euler_mascheroni ∈
           {harm n - u + inverse (of_nat (2*(n+1)))<..<harm n - l + inverse (of_nat (2*n))}"
  using euler_mascheroni_bounds[OF assms(1)] assms(2) by auto


text ‹
  Approximation of 🍋‹ln 2›. The lower bound is accurate to about 0.03; the upper
  bound is accurate to about 0.0015.
 ›
lemma ln2_ge_two_thirds: "2/3 ≤ ln (2::real)"
  and ln2_le_25_over_36: "ln (2::real) ≤ 25/36"
  using ln_approx_bounds[of 2 1, simplified, simplified eval_nat_numeral, simplified] by simp_all


text ‹
  Approximation of the Euler-Mascheroni constant. The lower bound is accurate to about 0.0015;
  the upper bound is accurate to about 0.015.
 ›
lemma euler_mascheroni_gt_19_over_33: "(euler_mascheroni :: real) > 19/33" (is ?th1)
  and euler_mascheroni_less_13_over_22: "(euler_mascheroni :: real) < 13/22" (is ?th2)
proof -
  have "ln (real (Suc 7)) = 3 * ln 2" by (simp add: ln_powr [symmetric])
  also from ln_approx_bounds[of 2 3] have "… ∈ {3*307/443<..<3*4615/6658}"
    by (simp add: eval_nat_numeral)
  finally have "ln (real (Suc 7)) ∈ …" .
  from euler_mascheroni_bounds'[OF _ this] have "?th1 ∧ ?th2" by (simp_all add: harm_expand)
  thus ?th1 ?th2 by blast+
qed

end