src/HOL/Decision_Procs/Approximation_Bounds.thy

moved some theorems into HOL main corpus

(* 
  Author:     Johannes Hoelzl, TU Muenchen
  Coercions removed by Dmitriy Traytel

  This file contains only general material about computing lower/upper bounds
  on real functions. Approximation.thy contains the actual approximation algorithm
  and the approximation oracle. This is in order to make a clear separation between 
  "morally immaculate" material about upper/lower bounds and the trusted oracle/reflection.
*)

theory Approximation_Bounds
imports
  Complex_Main
  "HOL-Library.Float"
  Dense_Linear_Order
begin

declare powr_neg_one [simp]
declare powr_neg_numeral [simp]

section "Horner Scheme"

subsection \<open>Define auxiliary helper \<open>horner\<close> function\<close>

primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
"horner F G 0 i k x       = 0" |
"horner F G (Suc n) i k x = 1 / k - x * horner F G n (F i) (G i k) x"

lemma horner_schema':
  fixes x :: real and a :: "nat \<Rightarrow> real"
  shows "a 0 - x * (\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)"
proof -
  have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)"
    by auto
  show ?thesis
    unfolding sum_distrib_left shift_pow uminus_add_conv_diff [symmetric] sum_negf[symmetric]
    sum.atLeast_Suc_lessThan[OF zero_less_Suc]
    sum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n  *a n * x^n"] by auto
qed

lemma horner_schema:
  fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
  assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
  shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. (- 1) ^ j * (1 / (f (j' + j))) * x ^ j)"
proof (induct n arbitrary: j')
  case 0
  then show ?case by auto
next
  case (Suc n)
  show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
    using horner_schema'[of "\<lambda> j. 1 / (f (j' + j))"] by auto
qed

lemma horner_bounds':
  fixes lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
  assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
    and lb_0: "\<And> i k x. lb 0 i k x = 0"
    and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
        (lapprox_rat prec 1 k)
        (- float_round_up prec (x * (ub n (F i) (G i k) x)))"
    and ub_0: "\<And> i k x. ub 0 i k x = 0"
    and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
        (rapprox_rat prec 1 k)
        (- float_round_down prec (x * (lb n (F i) (G i k) x)))"
  shows "(lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') x \<and>
         horner F G n ((F ^^ j') s) (f j') x \<le> (ub n ((F ^^ j') s) (f j') x)"
  (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
proof (induct n arbitrary: j')
  case 0
  thus ?case unfolding lb_0 ub_0 horner.simps by auto
next
  case (Suc n)
  thus ?case using lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec]
    Suc[where j'="Suc j'"] \<open>0 \<le> real_of_float x\<close>
    by (auto intro!: add_mono mult_left_mono float_round_down_le float_round_up_le
      order_trans[OF add_mono[OF _ float_plus_down_le]]
      order_trans[OF _ add_mono[OF _ float_plus_up_le]]
      simp add: lb_Suc ub_Suc field_simps f_Suc)
qed

subsection "Theorems for floating point functions implementing the horner scheme"

text \<open>

Here \<^term_type>\<open>f :: nat \<Rightarrow> nat\<close> is the sequence defining the Taylor series, the coefficients are
all alternating and reciprocs. We use \<^term>\<open>G\<close> and \<^term>\<open>F\<close> to describe the computation of \<^term>\<open>f\<close>.

\<close>

lemma horner_bounds:
  fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
  assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
    and lb_0: "\<And> i k x. lb 0 i k x = 0"
    and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
        (lapprox_rat prec 1 k)
        (- float_round_up prec (x * (ub n (F i) (G i k) x)))"
    and ub_0: "\<And> i k x. ub 0 i k x = 0"
    and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
        (rapprox_rat prec 1 k)
        (- float_round_down prec (x * (lb n (F i) (G i k) x)))"
  shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j))"
      (is "?lb")
    and "(\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j)) \<le> (ub n ((F ^^ j') s) (f j') x)"
      (is "?ub")
proof -
  have "?lb  \<and> ?ub"
    using horner_bounds'[where lb=lb, OF \<open>0 \<le> real_of_float x\<close> f_Suc lb_0 lb_Suc ub_0 ub_Suc]
    unfolding horner_schema[where f=f, OF f_Suc] by simp
  thus "?lb" and "?ub" by auto
qed

lemma horner_bounds_nonpos:
  fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
  assumes "real_of_float x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
    and lb_0: "\<And> i k x. lb 0 i k x = 0"
    and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
        (lapprox_rat prec 1 k)
        (float_round_down prec (x * (ub n (F i) (G i k) x)))"
    and ub_0: "\<And> i k x. ub 0 i k x = 0"
    and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
        (rapprox_rat prec 1 k)
        (float_round_up prec (x * (lb n (F i) (G i k) x)))"
  shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j)" (is "?lb")
    and "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
proof -
  have diff_mult_minus: "x - y * z = x + - y * z" for x y z :: float by simp
  have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) =
    (\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real_of_float (- x) ^ j)"
    by (auto simp add: field_simps power_mult_distrib[symmetric])
  have "0 \<le> real_of_float (-x)" using assms by auto
  from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
    and lb="\<lambda> n i k x. lb n i k (-x)" and ub="\<lambda> n i k x. ub n i k (-x)",
    unfolded lb_Suc ub_Suc diff_mult_minus,
    OF this f_Suc lb_0 _ ub_0 _]
  show "?lb" and "?ub" unfolding minus_minus sum_eq
    by (auto simp: minus_float_round_up_eq minus_float_round_down_eq)
qed


subsection \<open>Selectors for next even or odd number\<close>

text \<open>
The horner scheme computes alternating series. To get the upper and lower bounds we need to
guarantee to access a even or odd member. To do this we use \<^term>\<open>get_odd\<close> and \<^term>\<open>get_even\<close>.
\<close>

definition get_odd :: "nat \<Rightarrow> nat" where
  "get_odd n = (if odd n then n else (Suc n))"

definition get_even :: "nat \<Rightarrow> nat" where
  "get_even n = (if even n then n else (Suc n))"

lemma get_odd[simp]: "odd (get_odd n)"
  unfolding get_odd_def by (cases "odd n") auto

lemma get_even[simp]: "even (get_even n)"
  unfolding get_even_def by (cases "even n") auto

lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
  by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"])

lemma get_even_double: "\<exists>i. get_even n = 2 * i"
  using get_even by (blast elim: evenE)

lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1"
  using get_odd by (blast elim: oddE)


section "Power function"

definition float_power_bnds :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
"float_power_bnds prec n l u =
  (if 0 < l then (power_down_fl prec l n, power_up_fl prec u n)
  else if odd n then
    (- power_up_fl prec \<bar>l\<bar> n,
      if u < 0 then - power_down_fl prec \<bar>u\<bar> n else power_up_fl prec u n)
  else if u < 0 then (power_down_fl prec \<bar>u\<bar> n, power_up_fl prec \<bar>l\<bar> n)
  else (0, power_up_fl prec (max \<bar>l\<bar> \<bar>u\<bar>) n))"

lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n"
  by (subst le_minus_iff) (auto intro: power_down_le power_mono_odd)

lemma float_power_bnds:
  "(l1, u1) = float_power_bnds prec n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}"
  by (auto
    simp: float_power_bnds_def max_def real_power_up_fl real_power_down_fl minus_le_iff
    split: if_split_asm
    intro!: power_up_le power_down_le le_minus_power_downI
    intro: power_mono_odd power_mono power_mono_even zero_le_even_power)

lemma bnds_power:
  "\<forall>(x::real) l u. (l1, u1) = float_power_bnds prec n l u \<and> x \<in> {l .. u} \<longrightarrow>
    l1 \<le> x ^ n \<and> x ^ n \<le> u1"
  using float_power_bnds by auto

section \<open>Approximation utility functions\<close>

definition bnds_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<times> float" where
  "bnds_mult prec a1 a2 b1 b2 =
      (float_plus_down prec (nprt a1 * pprt b2)
          (float_plus_down prec (nprt a2 * nprt b2)
            (float_plus_down prec (pprt a1 * pprt b1) (pprt a2 * nprt b1))),
        float_plus_up prec (pprt a2 * pprt b2)
            (float_plus_up prec (pprt a1 * nprt b2)
              (float_plus_up prec (nprt a2 * pprt b1) (nprt a1 * nprt b1))))"

lemma bnds_mult:
  fixes prec :: nat and a1 aa2 b1 b2 :: float
  assumes "(l, u) = bnds_mult prec a1 a2 b1 b2"
  assumes "a \<in> {real_of_float a1..real_of_float a2}"
  assumes "b \<in> {real_of_float b1..real_of_float b2}"
  shows   "a * b \<in> {real_of_float l..real_of_float u}"
proof -
  from assms have "real_of_float l \<le> a * b" 
    by (intro order.trans[OF _ mult_ge_prts[of a1 a a2 b1 b b2]])
       (auto simp: bnds_mult_def intro!: float_plus_down_le)
  moreover from assms have "real_of_float u \<ge> a * b" 
    by (intro order.trans[OF mult_le_prts[of a1 a a2 b1 b b2]])
       (auto simp: bnds_mult_def intro!: float_plus_up_le)
  ultimately show ?thesis by simp
qed

definition map_bnds :: "(nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow> (nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow>
                           nat \<Rightarrow> (float \<times> float) \<Rightarrow> (float \<times> float)" where
  "map_bnds lb ub prec = (\<lambda>(l,u). (lb prec l, ub prec u))"

lemma map_bnds:
  assumes "(lf, uf) = map_bnds lb ub prec (l, u)"
  assumes "mono f"
  assumes "x \<in> {real_of_float l..real_of_float u}"
  assumes "real_of_float (lb prec l) \<le> f (real_of_float l)"
  assumes "real_of_float (ub prec u) \<ge> f (real_of_float u)"
  shows   "f x \<in> {real_of_float lf..real_of_float uf}"
proof -
  from assms have "real_of_float lf = real_of_float (lb prec l)"
    by (simp add: map_bnds_def)
  also have "real_of_float (lb prec l) \<le> f (real_of_float l)"  by fact
  also from assms have "\<dots> \<le> f x"
    by (intro monoD[OF \<open>mono f\<close>]) auto
  finally have lf: "real_of_float lf \<le> f x" .

  from assms have "f x \<le> f (real_of_float u)"
    by (intro monoD[OF \<open>mono f\<close>]) auto
  also have "\<dots> \<le> real_of_float (ub prec u)" by fact
  also from assms have "\<dots> = real_of_float uf"
    by (simp add: map_bnds_def)
  finally have uf: "f x \<le> real_of_float uf" .

  from lf uf show ?thesis by simp
qed


section "Square root"

text \<open>
The square root computation is implemented as newton iteration. As first first step we use the
nearest power of two greater than the square root.
\<close>

fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
"sqrt_iteration prec 0 x = Float 1 ((bitlen \<bar>mantissa x\<bar> + exponent x) div 2 + 1)" |
"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
                                  in Float 1 (- 1) * float_plus_up prec y (float_divr prec x y))"

lemma compute_sqrt_iteration_base[code]:
  shows "sqrt_iteration prec n (Float m e) =
    (if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1)
    else (let y = sqrt_iteration prec (n - 1) (Float m e) in
      Float 1 (- 1) * float_plus_up prec y (float_divr prec (Float m e) y)))"
  using bitlen_Float by (cases n) simp_all

function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
"ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
              else if x < 0 then - lb_sqrt prec (- x)
                            else 0)" |
"lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
              else if x < 0 then - ub_sqrt prec (- x)
                            else 0)"
by pat_completeness auto
termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)

declare lb_sqrt.simps[simp del]
declare ub_sqrt.simps[simp del]

lemma sqrt_ub_pos_pos_1:
  assumes "sqrt x < b" and "0 < b" and "0 < x"
  shows "sqrt x < (b + x / b)/2"
proof -
  from assms have "0 < (b - sqrt x)\<^sup>2 " by simp
  also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra
  also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp
  finally have "0 < b\<^sup>2 - 2 * b * sqrt x + x" .
  hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
    by (simp add: field_simps power2_eq_square)
  thus ?thesis by (simp add: field_simps)
qed

lemma sqrt_iteration_bound:
  assumes "0 < real_of_float x"
  shows "sqrt x < sqrt_iteration prec n x"
proof (induct n)
  case 0
  show ?case
  proof (cases x)
    case (Float m e)
    hence "0 < m"
      using assms
      by (auto simp: sign_simps)
    hence "0 < sqrt m" by auto

    have int_nat_bl: "(nat (bitlen m)) = bitlen m"
      using bitlen_nonneg by auto

    have "x = (m / 2^nat (bitlen m)) * 2 powr (e + (nat (bitlen m)))"
      unfolding Float by (auto simp: powr_realpow[symmetric] field_simps powr_add)
    also have "\<dots> < 1 * 2 powr (e + nat (bitlen m))"
    proof (rule mult_strict_right_mono, auto)
      show "m < 2^nat (bitlen m)"
        using bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
        unfolding of_int_less_iff[of m, symmetric] by auto
    qed
    finally have "sqrt x < sqrt (2 powr (e + bitlen m))"
      unfolding int_nat_bl by auto
    also have "\<dots> \<le> 2 powr ((e + bitlen m) div 2 + 1)"
    proof -
      let ?E = "e + bitlen m"
      have E_mod_pow: "2 powr (?E mod 2) < 4"
      proof (cases "?E mod 2 = 1")
        case True
        thus ?thesis by auto
      next
        case False
        have "0 \<le> ?E mod 2" by auto
        have "?E mod 2 < 2" by auto
        from this[THEN zless_imp_add1_zle]
        have "?E mod 2 \<le> 0" using False by auto
        from xt1(5)[OF \<open>0 \<le> ?E mod 2\<close> this]
        show ?thesis by auto
      qed
      hence "sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)"
        by (intro real_sqrt_less_mono) auto
      hence E_mod_pow: "sqrt (2 powr (?E mod 2)) < 2" by auto

      have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)"
        by auto
      have "sqrt (2 powr ?E) = sqrt (2 powr (?E div 2) * 2 powr (?E div 2) * 2 powr (?E mod 2))"
        unfolding E_eq unfolding powr_add[symmetric] by (metis of_int_add)
      also have "\<dots> = 2 powr (?E div 2) * sqrt (2 powr (?E mod 2))"
        unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto
      also have "\<dots> < 2 powr (?E div 2) * 2 powr 1"
        by (rule mult_strict_left_mono) (auto intro: E_mod_pow)
      also have "\<dots> = 2 powr (?E div 2 + 1)"
        unfolding add.commute[of _ 1] powr_add[symmetric] by simp
      finally show ?thesis by auto
    qed
    finally show ?thesis using \<open>0 < m\<close>
      unfolding Float
      by (subst compute_sqrt_iteration_base) (simp add: ac_simps)
  qed
next
  case (Suc n)
  let ?b = "sqrt_iteration prec n x"
  have "0 < sqrt x"
    using \<open>0 < real_of_float x\<close> by auto
  also have "\<dots> < real_of_float ?b"
    using Suc .
  finally have "sqrt x < (?b + x / ?b)/2"
    using sqrt_ub_pos_pos_1[OF Suc _ \<open>0 < real_of_float x\<close>] by auto
  also have "\<dots> \<le> (?b + (float_divr prec x ?b))/2"
    by (rule divide_right_mono, auto simp add: float_divr)
  also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))"
    by simp
  also have "\<dots> \<le> (Float 1 (- 1)) * (float_plus_up prec ?b (float_divr prec x ?b))"
    by (auto simp add: algebra_simps float_plus_up_le)
  finally show ?case
    unfolding sqrt_iteration.simps Let_def distrib_left .
qed

lemma sqrt_iteration_lower_bound:
  assumes "0 < real_of_float x"
  shows "0 < real_of_float (sqrt_iteration prec n x)" (is "0 < ?sqrt")
proof -
  have "0 < sqrt x" using assms by auto
  also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
  finally show ?thesis .
qed

lemma lb_sqrt_lower_bound:
  assumes "0 \<le> real_of_float x"
  shows "0 \<le> real_of_float (lb_sqrt prec x)"
proof (cases "0 < x")
  case True
  hence "0 < real_of_float x" and "0 \<le> x"
    using \<open>0 \<le> real_of_float x\<close> by auto
  hence "0 < sqrt_iteration prec prec x"
    using sqrt_iteration_lower_bound by auto
  hence "0 \<le> real_of_float (float_divl prec x (sqrt_iteration prec prec x))"
    using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] unfolding less_eq_float_def by auto
  thus ?thesis
    unfolding lb_sqrt.simps using True by auto
next
  case False
  with \<open>0 \<le> real_of_float x\<close> have "real_of_float x = 0" by auto
  thus ?thesis
    unfolding lb_sqrt.simps by auto
qed

lemma bnds_sqrt': "sqrt x \<in> {(lb_sqrt prec x) .. (ub_sqrt prec x)}"
proof -
  have lb: "lb_sqrt prec x \<le> sqrt x" if "0 < x" for x :: float
  proof -
    from that have "0 < real_of_float x" and "0 \<le> real_of_float x" by auto
    hence sqrt_gt0: "0 < sqrt x" by auto
    hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x"
      using sqrt_iteration_bound by auto
    have "(float_divl prec x (sqrt_iteration prec prec x)) \<le>
          x / (sqrt_iteration prec prec x)" by (rule float_divl)
    also have "\<dots> < x / sqrt x"
      by (rule divide_strict_left_mono[OF sqrt_ub \<open>0 < real_of_float x\<close>
               mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
    also have "\<dots> = sqrt x"
      unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric]
                sqrt_divide_self_eq[OF \<open>0 \<le> real_of_float x\<close>, symmetric] by auto
    finally show ?thesis
      unfolding lb_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto
  qed
  have ub: "sqrt x \<le> ub_sqrt prec x" if "0 < x" for x :: float
  proof -
    from that have "0 < real_of_float x" by auto
    hence "0 < sqrt x" by auto
    hence "sqrt x < sqrt_iteration prec prec x"
      using sqrt_iteration_bound by auto
    then show ?thesis
      unfolding ub_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto
  qed
  show ?thesis
    using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x]
    by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus)
qed

lemma bnds_sqrt: "\<forall>(x::real) lx ux.
  (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> sqrt x \<and> sqrt x \<le> u"
proof ((rule allI) +, rule impI, erule conjE, rule conjI)
  fix x :: real
  fix lx ux
  assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
    and x: "x \<in> {lx .. ux}"
  hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto

  have "sqrt lx \<le> sqrt x" using x by auto
  from order_trans[OF _ this]
  show "l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto

  have "sqrt x \<le> sqrt ux" using x by auto
  from order_trans[OF this]
  show "sqrt x \<le> u" unfolding u using bnds_sqrt'[of ux prec] by auto
qed


section "Arcus tangens and \<pi>"

subsection "Compute arcus tangens series"

text \<open>
As first step we implement the computation of the arcus tangens series. This is only valid in the range
\<^term>\<open>{-1 :: real .. 1}\<close>. This is used to compute \<pi> and then the entire arcus tangens.
\<close>

fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
  "ub_arctan_horner prec 0 k x = 0"
| "ub_arctan_horner prec (Suc n) k x = float_plus_up prec
      (rapprox_rat prec 1 k) (- float_round_down prec (x * (lb_arctan_horner prec n (k + 2) x)))"
| "lb_arctan_horner prec 0 k x = 0"
| "lb_arctan_horner prec (Suc n) k x = float_plus_down prec
      (lapprox_rat prec 1 k) (- float_round_up prec (x * (ub_arctan_horner prec n (k + 2) x)))"

lemma arctan_0_1_bounds':
  assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
    and "even n"
  shows "arctan (sqrt y) \<in>
      {(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}"
proof -
  let ?c = "\<lambda>i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * sqrt y ^ (i * 2 + 1))"
  let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i"

  have "0 \<le> sqrt y" using assms by auto
  have "sqrt y \<le> 1" using assms by auto
  from \<open>even n\<close> obtain m where "2 * m = n" by (blast elim: evenE)

  have "arctan (sqrt y) \<in> { ?S n .. ?S (Suc n) }"
  proof (cases "sqrt y = 0")
    case True
    then show ?thesis by simp
  next
    case False
    hence "0 < sqrt y" using \<open>0 \<le> sqrt y\<close> by auto
    hence prem: "0 < 1 / (0 * 2 + (1::nat)) * sqrt y ^ (0 * 2 + 1)" by auto

    have "\<bar> sqrt y \<bar> \<le> 1"  using \<open>0 \<le> sqrt y\<close> \<open>sqrt y \<le> 1\<close> by auto
    from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this]
      monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded \<open>2 * m = n\<close>]
    show ?thesis unfolding arctan_series[OF \<open>\<bar> sqrt y \<bar> \<le> 1\<close>] Suc_eq_plus1 atLeast0LessThan .
  qed
  note arctan_bounds = this[unfolded atLeastAtMost_iff]

  have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto

  note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
    and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
    and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
    OF \<open>0 \<le> real_of_float y\<close> F lb_arctan_horner.simps ub_arctan_horner.simps]

  have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> arctan (sqrt y)"
  proof -
    have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> ?S n"
      using bounds(1) \<open>0 \<le> sqrt y\<close>
      apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
      apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
      apply (auto intro!: mult_left_mono)
      done
    also have "\<dots> \<le> arctan (sqrt y)" using arctan_bounds ..
    finally show ?thesis .
  qed
  moreover
  have "arctan (sqrt y) \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
  proof -
    have "arctan (sqrt y) \<le> ?S (Suc n)" using arctan_bounds ..
    also have "\<dots> \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
      using bounds(2)[of "Suc n"] \<open>0 \<le> sqrt y\<close>
      apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
      apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
      apply (auto intro!: mult_left_mono)
      done
    finally show ?thesis .
  qed
  ultimately show ?thesis by auto
qed

lemma arctan_0_1_bounds:
  assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
  shows "arctan (sqrt y) \<in>
    {(sqrt y * lb_arctan_horner prec (get_even n) 1 y) ..
      (sqrt y * ub_arctan_horner prec (get_odd n) 1 y)}"
  using
    arctan_0_1_bounds'[OF assms, of n prec]
    arctan_0_1_bounds'[OF assms, of "n + 1" prec]
    arctan_0_1_bounds'[OF assms, of "n - 1" prec]
  by (auto simp: get_even_def get_odd_def odd_pos
    simp del: ub_arctan_horner.simps lb_arctan_horner.simps)

lemma arctan_lower_bound:
  assumes "0 \<le> x"
  shows "x / (1 + x\<^sup>2) \<le> arctan x" (is "?l x \<le> _")
proof -
  have "?l x - arctan x \<le> ?l 0 - arctan 0"
    using assms
    by (intro DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. ?l x - arctan x"])
      (auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps)
  thus ?thesis by simp
qed

lemma arctan_divide_mono: "0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> arctan y / y \<le> arctan x / x"
  by (rule DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. arctan x / x"])
    (auto intro!: derivative_eq_intros divide_nonpos_nonneg
      simp: inverse_eq_divide arctan_lower_bound)

lemma arctan_mult_mono: "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> x * arctan y \<le> y * arctan x"
  using arctan_divide_mono[of x y] by (cases "x = 0") (simp_all add: field_simps)

lemma arctan_mult_le:
  assumes "0 \<le> x" "x \<le> y" "y * z \<le> arctan y"
  shows "x * z \<le> arctan x"
proof (cases "x = 0")
  case True
  then show ?thesis by simp
next
  case False
  with assms have "z \<le> arctan y / y" by (simp add: field_simps)
  also have "\<dots> \<le> arctan x / x" using assms \<open>x \<noteq> 0\<close> by (auto intro!: arctan_divide_mono)
  finally show ?thesis using assms \<open>x \<noteq> 0\<close> by (simp add: field_simps)
qed

lemma arctan_le_mult:
  assumes "0 < x" "x \<le> y" "arctan x \<le> x * z"
  shows "arctan y \<le> y * z"
proof -
  from assms have "arctan y / y \<le> arctan x / x" by (auto intro!: arctan_divide_mono)
  also have "\<dots> \<le> z" using assms by (auto simp: field_simps)
  finally show ?thesis using assms by (simp add: field_simps)
qed

lemma arctan_0_1_bounds_le:
  assumes "0 \<le> x" "x \<le> 1" "0 < real_of_float xl" "real_of_float xl \<le> x * x" "x * x \<le> real_of_float xu" "real_of_float xu \<le> 1"
  shows "arctan x \<in>
      {x * lb_arctan_horner p1 (get_even n) 1 xu .. x * ub_arctan_horner p2 (get_odd n) 1 xl}"
proof -
  from assms have "real_of_float xl \<le> 1" "sqrt (real_of_float xl) \<le> x" "x \<le> sqrt (real_of_float xu)" "0 \<le> real_of_float xu"
    "0 \<le> real_of_float xl" "0 < sqrt (real_of_float xl)"
    by (auto intro!: real_le_rsqrt real_le_lsqrt simp: power2_eq_square)
  from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xu\<close>  \<open>real_of_float xu \<le> 1\<close>]
  have "sqrt (real_of_float xu) * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan (sqrt (real_of_float xu))"
    by simp
  from arctan_mult_le[OF \<open>0 \<le> x\<close> \<open>x \<le> sqrt _\<close>  this]
  have "x * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan x" .
  moreover
  from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xl\<close>  \<open>real_of_float xl \<le> 1\<close>]
  have "arctan (sqrt (real_of_float xl)) \<le> sqrt (real_of_float xl) * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)"
    by simp
  from arctan_le_mult[OF \<open>0 < sqrt xl\<close> \<open>sqrt xl \<le> x\<close> this]
  have "arctan x \<le> x * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)" .
  ultimately show ?thesis by simp
qed

lemma arctan_0_1_bounds_round:
  assumes "0 \<le> real_of_float x" "real_of_float x \<le> 1"
  shows "arctan x \<in>
      {real_of_float x * lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)) ..
        real_of_float x * ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x))}"
  using assms
  apply (cases "x > 0")
   apply (intro arctan_0_1_bounds_le)
   apply (auto simp: float_round_down.rep_eq float_round_up.rep_eq
    intro!: truncate_up_le1 mult_le_one truncate_down_le truncate_up_le truncate_down_pos
      mult_pos_pos)
  done


subsection "Compute \<pi>"

definition ub_pi :: "nat \<Rightarrow> float" where
  "ub_pi prec =
    (let
      A = rapprox_rat prec 1 5 ;
      B = lapprox_rat prec 1 239
    in ((Float 1 2) * float_plus_up prec
      ((Float 1 2) * float_round_up prec (A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1
        (float_round_down (Suc prec) (A * A)))))
      (- float_round_down prec (B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1
        (float_round_up (Suc prec) (B * B)))))))"

definition lb_pi :: "nat \<Rightarrow> float" where
  "lb_pi prec =
    (let
      A = lapprox_rat prec 1 5 ;
      B = rapprox_rat prec 1 239
    in ((Float 1 2) * float_plus_down prec
      ((Float 1 2) * float_round_down prec (A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1
        (float_round_up (Suc prec) (A * A)))))
      (- float_round_up prec (B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1
        (float_round_down (Suc prec) (B * B)))))))"

lemma pi_boundaries: "pi \<in> {(lb_pi n) .. (ub_pi n)}"
proof -
  have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))"
    unfolding machin[symmetric] by auto

  {
    fix prec n :: nat
    fix k :: int
    assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
    let ?k = "rapprox_rat prec 1 k"
    let ?kl = "float_round_down (Suc prec) (?k * ?k)"
    have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto

    have "0 \<le> real_of_float ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: \<open>0 \<le> k\<close>)
    have "real_of_float ?k \<le> 1"
      by (auto simp add: \<open>0 < k\<close> \<open>1 \<le> k\<close> less_imp_le
        intro!: mult_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1)
    have "1 / k \<le> ?k" using rapprox_rat[where x=1 and y=k] by auto
    hence "arctan (1 / k) \<le> arctan ?k" by (rule arctan_monotone')
    also have "\<dots> \<le> (?k * ub_arctan_horner prec (get_odd n) 1 ?kl)"
      using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
      by auto
    finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 ?kl" .
  } note ub_arctan = this

  {
    fix prec n :: nat
    fix k :: int
    assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
    let ?k = "lapprox_rat prec 1 k"
    let ?ku = "float_round_up (Suc prec) (?k * ?k)"
    have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto
    have "1 / k \<le> 1" using \<open>1 < k\<close> by auto
    have "0 \<le> real_of_float ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one \<open>0 \<le> k\<close>]
      by (auto simp add: \<open>1 div k = 0\<close>)
    have "0 \<le> real_of_float (?k * ?k)" by simp
    have "real_of_float ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: \<open>1 / k \<le> 1\<close>)
    hence "real_of_float (?k * ?k) \<le> 1" using \<open>0 \<le> real_of_float ?k\<close> by (auto intro!: mult_le_one)

    have "?k \<le> 1 / k" using lapprox_rat[where x=1 and y=k] by auto

    have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan ?k"
      using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
      by auto
    also have "\<dots> \<le> arctan (1 / k)" using \<open>?k \<le> 1 / k\<close> by (rule arctan_monotone')
    finally have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan (1 / k)" .
  } note lb_arctan = this

  have "pi \<le> ub_pi n "
    unfolding ub_pi_def machin_pi Let_def times_float.rep_eq Float_num
    using lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
    by (intro mult_left_mono float_plus_up_le float_plus_down_le)
      (auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
  moreover have "lb_pi n \<le> pi"
    unfolding lb_pi_def machin_pi Let_def times_float.rep_eq Float_num
    using lb_arctan[of 5] ub_arctan[of 239]
    by (intro mult_left_mono float_plus_up_le float_plus_down_le)
      (auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
  ultimately show ?thesis by auto
qed


subsection "Compute arcus tangens in the entire domain"

function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
  "lb_arctan prec x =
    (let
      ub_horner = \<lambda> x. float_round_up prec
        (x *
          ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)));
      lb_horner = \<lambda> x. float_round_down prec
        (x *
          lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))
    in
      if x < 0 then - ub_arctan prec (-x)
      else if x \<le> Float 1 (- 1) then lb_horner x
      else if x \<le> Float 1 1 then
        Float 1 1 *
        lb_horner
          (float_divl prec x
            (float_plus_up prec 1
              (ub_sqrt prec (float_plus_up prec 1 (float_round_up prec (x * x))))))
      else let inv = float_divr prec 1 x in
        if inv > 1 then 0
        else float_plus_down prec (lb_pi prec * Float 1 (- 1)) ( - ub_horner inv))"

| "ub_arctan prec x =
    (let
      lb_horner = \<lambda> x. float_round_down prec
        (x *
          lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))) ;
      ub_horner = \<lambda> x. float_round_up prec
        (x *
          ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))
    in if x < 0 then - lb_arctan prec (-x)
    else if x \<le> Float 1 (- 1) then ub_horner x
    else if x \<le> Float 1 1 then
      let y = float_divr prec x
        (float_plus_down
          (Suc prec) 1 (lb_sqrt prec (float_plus_down prec 1 (float_round_down prec (x * x)))))
      in if y > 1 then ub_pi prec * Float 1 (- 1) else Float 1 1 * ub_horner y
    else float_plus_up prec (ub_pi prec * Float 1 (- 1)) ( - lb_horner (float_divl prec 1 x)))"
by pat_completeness auto
termination
by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)

declare ub_arctan_horner.simps[simp del]
declare lb_arctan_horner.simps[simp del]

lemma lb_arctan_bound':
  assumes "0 \<le> real_of_float x"
  shows "lb_arctan prec x \<le> arctan x"
proof -
  have "\<not> x < 0" and "0 \<le> x"
    using \<open>0 \<le> real_of_float x\<close> by (auto intro!: truncate_up_le )

  let "?ub_horner x" =
      "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))"
    and "?lb_horner x" =
      "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))"

  show ?thesis
  proof (cases "x \<le> Float 1 (- 1)")
    case True
    hence "real_of_float x \<le> 1" by simp
    from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
    show ?thesis
      unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True] using \<open>0 \<le> x\<close>
      by (auto intro!: float_round_down_le)
  next
    case False
    hence "0 < real_of_float x" by auto
    let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
    let ?sxx = "float_plus_up prec 1 (float_round_up prec (x * x))"
    let ?fR = "float_plus_up prec 1 (ub_sqrt prec ?sxx)"
    let ?DIV = "float_divl prec x ?fR"

    have divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)

    have "sqrt (1 + x*x) \<le> sqrt ?sxx"
      by (auto simp: float_plus_up.rep_eq plus_up_def float_round_up.rep_eq intro!: truncate_up_le)
    also have "\<dots> \<le> ub_sqrt prec ?sxx"
      using bnds_sqrt'[of ?sxx prec] by auto
    finally
    have "sqrt (1 + x*x) \<le> ub_sqrt prec ?sxx" .
    hence "?R \<le> ?fR" by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
    hence "0 < ?fR" and "0 < real_of_float ?fR" using \<open>0 < ?R\<close> by auto

    have monotone: "?DIV \<le> x / ?R"
    proof -
      have "?DIV \<le> real_of_float x / ?fR" by (rule float_divl)
      also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF \<open>?R \<le> ?fR\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 \<open>?R \<le> real_of_float ?fR\<close>] divisor_gt0]])
      finally show ?thesis .
    qed

    show ?thesis
    proof (cases "x \<le> Float 1 1")
      case True
      have "x \<le> sqrt (1 + x * x)"
        using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
      also note \<open>\<dots> \<le> (ub_sqrt prec ?sxx)\<close>
      finally have "real_of_float x \<le> ?fR"
        by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
      moreover have "?DIV \<le> real_of_float x / ?fR"
        by (rule float_divl)
      ultimately have "real_of_float ?DIV \<le> 1"
        unfolding divide_le_eq_1_pos[OF \<open>0 < real_of_float ?fR\<close>, symmetric] by auto

      have "0 \<le> real_of_float ?DIV"
        using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] \<open>0 < ?fR\<close>
        unfolding less_eq_float_def by auto

      from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float (?DIV)\<close> \<open>real_of_float (?DIV) \<le> 1\<close>]
      have "Float 1 1 * ?lb_horner ?DIV \<le> 2 * arctan ?DIV"
        by simp
      also have "\<dots> \<le> 2 * arctan (x / ?R)"
        using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono arctan_monotone')
      also have "2 * arctan (x / ?R) = arctan x"
        using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
      finally show ?thesis
        unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
          if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF True]
        by (auto simp: float_round_down.rep_eq
          intro!: order_trans[OF mult_left_mono[OF truncate_down]])
    next
      case False
      hence "2 < real_of_float x" by auto
      hence "1 \<le> real_of_float x" by auto

      let "?invx" = "float_divr prec 1 x"
      have "0 \<le> arctan x" using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>]
        using arctan_tan[of 0, unfolded tan_zero] by auto

      show ?thesis
      proof (cases "1 < ?invx")
        case True
        show ?thesis
          unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
            if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False] if_P[OF True]
          using \<open>0 \<le> arctan x\<close> by auto
      next
        case False
        hence "real_of_float ?invx \<le> 1" by auto
        have "0 \<le> real_of_float ?invx"
          by (rule order_trans[OF _ float_divr]) (auto simp add: \<open>0 \<le> real_of_float x\<close>)

        have "1 / x \<noteq> 0" and "0 < 1 / x"
          using \<open>0 < real_of_float x\<close> by auto

        have "arctan (1 / x) \<le> arctan ?invx"
          unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr)
        also have "\<dots> \<le> ?ub_horner ?invx"
          using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
          by (auto intro!: float_round_up_le)
        also note float_round_up
        finally have "pi / 2 - float_round_up prec (?ub_horner ?invx) \<le> arctan x"
          using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>]
          unfolding sgn_pos[OF \<open>0 < 1 / real_of_float x\<close>] le_diff_eq by auto
        moreover
        have "lb_pi prec * Float 1 (- 1) \<le> pi / 2"
          unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by simp
        ultimately
        show ?thesis
          unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
            if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x \<le> Float 1 1\<close>] if_not_P[OF False]
          by (auto intro!: float_plus_down_le)
      qed
    qed
  qed
qed

lemma ub_arctan_bound':
  assumes "0 \<le> real_of_float x"
  shows "arctan x \<le> ub_arctan prec x"
proof -
  have "\<not> x < 0" and "0 \<le> x"
    using \<open>0 \<le> real_of_float x\<close> by auto

  let "?ub_horner x" =
    "float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))"
  let "?lb_horner x" =
    "float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))"

  show ?thesis
  proof (cases "x \<le> Float 1 (- 1)")
    case True
    hence "real_of_float x \<le> 1" by auto
    show ?thesis
      unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True]
      using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
      by (auto intro!: float_round_up_le)
  next
    case False
    hence "0 < real_of_float x" by auto
    let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
    let ?sxx = "float_plus_down prec 1 (float_round_down prec (x * x))"
    let ?fR = "float_plus_down (Suc prec) 1 (lb_sqrt prec ?sxx)"
    let ?DIV = "float_divr prec x ?fR"

    have sqr_ge0: "0 \<le> 1 + real_of_float x * real_of_float x"
      using sum_power2_ge_zero[of 1 "real_of_float x", unfolded numeral_2_eq_2] by auto
    hence "0 \<le> real_of_float (1 + x*x)" by auto

    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)

    have "lb_sqrt prec ?sxx \<le> sqrt ?sxx"
      using bnds_sqrt'[of ?sxx] by auto
    also have "\<dots> \<le> sqrt (1 + x*x)"
      by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq truncate_down_le)
    finally have "lb_sqrt prec ?sxx \<le> sqrt (1 + x*x)" .
    hence "?fR \<le> ?R"
      by (auto simp: float_plus_down.rep_eq plus_down_def truncate_down_le)
    have "0 < real_of_float ?fR"
      by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq
        intro!: truncate_down_ge1 lb_sqrt_lower_bound order_less_le_trans[OF zero_less_one]
        truncate_down_nonneg add_nonneg_nonneg)
    have monotone: "x / ?R \<le> (float_divr prec x ?fR)"
    proof -
      from divide_left_mono[OF \<open>?fR \<le> ?R\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF divisor_gt0 \<open>0 < real_of_float ?fR\<close>]]
      have "x / ?R \<le> x / ?fR" .
      also have "\<dots> \<le> ?DIV" by (rule float_divr)
      finally show ?thesis .
    qed

    show ?thesis
    proof (cases "x \<le> Float 1 1")
      case True
      show ?thesis
      proof (cases "?DIV > 1")
        case True
        have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)"
          unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by auto
        from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
        show ?thesis
          unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
            if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_P[OF True] .
      next
        case False
        hence "real_of_float ?DIV \<le> 1" by auto

        have "0 \<le> x / ?R"
          using \<open>0 \<le> real_of_float x\<close> \<open>0 < ?R\<close> unfolding zero_le_divide_iff by auto
        hence "0 \<le> real_of_float ?DIV"
          using monotone by (rule order_trans)

        have "arctan x = 2 * arctan (x / ?R)"
          using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
        also have "\<dots> \<le> 2 * arctan (?DIV)"
          using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
        also have "\<dots> \<le> (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num
          using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?DIV\<close> \<open>real_of_float ?DIV \<le> 1\<close>]
          by (auto intro!: float_round_up_le)
        finally show ?thesis
          unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
            if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_not_P[OF False] .
      qed
    next
      case False
      hence "2 < real_of_float x" by auto
      hence "1 \<le> real_of_float x" by auto
      hence "0 < real_of_float x" by auto
      hence "0 < x" by auto

      let "?invx" = "float_divl prec 1 x"
      have "0 \<le> arctan x"
        using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>] and arctan_tan[of 0, unfolded tan_zero] by auto

      have "real_of_float ?invx \<le> 1"
        unfolding less_float_def
        by (rule order_trans[OF float_divl])
          (auto simp add: \<open>1 \<le> real_of_float x\<close> divide_le_eq_1_pos[OF \<open>0 < real_of_float x\<close>])
      have "0 \<le> real_of_float ?invx"
        using \<open>0 < x\<close> by (intro float_divl_lower_bound) auto

      have "1 / x \<noteq> 0" and "0 < 1 / x"
        using \<open>0 < real_of_float x\<close> by auto

      have "(?lb_horner ?invx) \<le> arctan (?invx)"
        using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
        by (auto intro!: float_round_down_le)
      also have "\<dots> \<le> arctan (1 / x)"
        unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone') (rule float_divl)
      finally have "arctan x \<le> pi / 2 - (?lb_horner ?invx)"
        using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>]
        unfolding sgn_pos[OF \<open>0 < 1 / x\<close>] le_diff_eq by auto
      moreover
      have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)"
        unfolding Float_num times_divide_eq_right mult_1_right
        using pi_boundaries by auto
      ultimately
      show ?thesis
        unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
          if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False]
        by (auto intro!: float_round_up_le float_plus_up_le)
    qed
  qed
qed

lemma arctan_boundaries: "arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}"
proof (cases "0 \<le> x")
  case True
  hence "0 \<le> real_of_float x" by auto
  show ?thesis
    using ub_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>]
    unfolding atLeastAtMost_iff by auto
next
  case False
  let ?mx = "-x"
  from False have "x < 0" and "0 \<le> real_of_float ?mx"
    by auto
  hence bounds: "lb_arctan prec ?mx \<le> arctan ?mx \<and> arctan ?mx \<le> ub_arctan prec ?mx"
    using ub_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] by auto
  show ?thesis
    unfolding minus_float.rep_eq arctan_minus lb_arctan.simps[where x=x]
      ub_arctan.simps[where x=x] Let_def if_P[OF \<open>x < 0\<close>]
    unfolding atLeastAtMost_iff using bounds[unfolded minus_float.rep_eq arctan_minus]
    by (simp add: arctan_minus)
qed

lemma bnds_arctan: "\<forall> (x::real) lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> arctan x \<and> arctan x \<le> u"
proof (rule allI, rule allI, rule allI, rule impI)
  fix x :: real
  fix lx ux
  assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux}"
  hence l: "lb_arctan prec lx = l "
    and u: "ub_arctan prec ux = u"
    and x: "x \<in> {lx .. ux}"
    by auto
  show "l \<le> arctan x \<and> arctan x \<le> u"
  proof
    show "l \<le> arctan x"
    proof -
      from arctan_boundaries[of lx prec, unfolded l]
      have "l \<le> arctan lx" by (auto simp del: lb_arctan.simps)
      also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
      finally show ?thesis .
    qed
    show "arctan x \<le> u"
    proof -
      have "arctan x \<le> arctan ux" using x by (auto intro: arctan_monotone')
      also have "\<dots> \<le> u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
      finally show ?thesis .
    qed
  qed
qed


section "Sinus and Cosinus"

subsection "Compute the cosinus and sinus series"

fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
  "ub_sin_cos_aux prec 0 i k x = 0"
| "ub_sin_cos_aux prec (Suc n) i k x = float_plus_up prec
    (rapprox_rat prec 1 k) (-
      float_round_down prec (x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"
| "lb_sin_cos_aux prec 0 i k x = 0"
| "lb_sin_cos_aux prec (Suc n) i k x = float_plus_down prec
    (lapprox_rat prec 1 k) (-
      float_round_up prec (x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"

lemma cos_aux:
  shows "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x ^(2 * i))" (is "?lb")
  and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
proof -
  have "0 \<le> real_of_float (x * x)" by auto
  let "?f n" = "fact (2 * n) :: nat"
  have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)" for n
  proof -
    have "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
    then show ?thesis by auto
  qed
  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
    OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
  show ?lb and ?ub
    by (auto simp add: power_mult power2_eq_square[of "real_of_float x"])
qed

lemma lb_sin_cos_aux_zero_le_one: "lb_sin_cos_aux prec n i j 0 \<le> 1"
  by (cases j n rule: nat.exhaust[case_product nat.exhaust])
    (auto intro!: float_plus_down_le order_trans[OF lapprox_rat])

lemma one_le_ub_sin_cos_aux: "odd n \<Longrightarrow> 1 \<le> ub_sin_cos_aux prec n i (Suc 0) 0"
  by (cases n) (auto intro!: float_plus_up_le order_trans[OF _ rapprox_rat])

lemma cos_boundaries:
  assumes "0 \<le> real_of_float x" and "x \<le> pi / 2"
  shows "cos x \<in> {(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
proof (cases "real_of_float x = 0")
  case False
  hence "real_of_float x \<noteq> 0" by auto
  hence "0 < x" and "0 < real_of_float x"
    using \<open>0 \<le> real_of_float x\<close> by auto
  have "0 < x * x"
    using \<open>0 < x\<close> by simp

  have morph_to_if_power: "(\<Sum> i=0..<n. (-1::real) ^ i * (1/(fact (2 * i))) * x ^ (2 * i)) =
    (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * x ^ i)"
    (is "?sum = ?ifsum") for x n
  proof -
    have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
    also have "\<dots> =
      (\<Sum> j = 0 ..< n. (- 1) ^ ((2 * j) div 2) / ((fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / ((fact i)) * x ^ i else 0)"
      unfolding sum_split_even_odd atLeast0LessThan ..
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / ((fact i)) else 0) * x ^ i)"
      by (rule sum.cong) auto
    finally show ?thesis .
  qed

  { fix n :: nat assume "0 < n"
    hence "0 < 2 * n" by auto
    obtain t where "0 < t" and "t < real_of_float x" and
      cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * (real_of_float x) ^ i)
      + (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real_of_float x)^(2*n)"
      (is "_ = ?SUM + ?rest / ?fact * ?pow")
      using Maclaurin_cos_expansion2[OF \<open>0 < real_of_float x\<close> \<open>0 < 2 * n\<close>]
      unfolding cos_coeff_def atLeast0LessThan by auto

    have "cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
    also have "\<dots> = cos (t + n * pi)" by (simp add: cos_add)
    also have "\<dots> = ?rest" by auto
    finally have "cos t * (- 1) ^ n = ?rest" .
    moreover
    have "t \<le> pi / 2" using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
    hence "0 \<le> cos t" using \<open>0 < t\<close> and cos_ge_zero by auto
    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto

    have "0 < ?fact" by auto
    have "0 < ?pow" using \<open>0 < real_of_float x\<close> by auto

    {
      assume "even n"
      have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
        unfolding morph_to_if_power[symmetric] using cos_aux by auto
      also have "\<dots> \<le> cos x"
      proof -
        from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close>
        have "0 \<le> (?rest / ?fact) * ?pow" by simp
        thus ?thesis unfolding cos_eq by auto
      qed
      finally have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos x" .
    } note lb = this

    {
      assume "odd n"
      have "cos x \<le> ?SUM"
      proof -
        from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>]
        have "0 \<le> (- ?rest) / ?fact * ?pow"
          by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
        thus ?thesis unfolding cos_eq by auto
      qed
      also have "\<dots> \<le> (ub_sin_cos_aux prec n 1 1 (x * x))"
        unfolding morph_to_if_power[symmetric] using cos_aux by auto
      finally have "cos x \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" .
    } note ub = this and lb
  } note ub = this(1) and lb = this(2)

  have "cos x \<le> (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))"
    using ub[OF odd_pos[OF get_odd] get_odd] .
  moreover have "(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos x"
  proof (cases "0 < get_even n")
    case True
    show ?thesis using lb[OF True get_even] .
  next
    case False
    hence "get_even n = 0" by auto
    have "- (pi / 2) \<le> x"
      by (rule order_trans[OF _ \<open>0 < real_of_float x\<close>[THEN less_imp_le]]) auto
    with \<open>x \<le> pi / 2\<close> show ?thesis
      unfolding \<open>get_even n = 0\<close> lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq
      using cos_ge_zero by auto
  qed
  ultimately show ?thesis by auto
next
  case True
  hence "x = 0"
    by (simp add: real_of_float_eq)
  thus ?thesis
    using lb_sin_cos_aux_zero_le_one one_le_ub_sin_cos_aux
    by simp
qed

lemma sin_aux:
  assumes "0 \<le> real_of_float x"
  shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
      (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb")
    and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1)) \<le>
      (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
proof -
  have "0 \<le> real_of_float (x * x)" by auto
  let "?f n" = "fact (2 * n + 1) :: nat"
  have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)" for n
  proof -
    have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
    show ?thesis
      unfolding F by auto
  qed
  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
    OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
  show "?lb" and "?ub" using \<open>0 \<le> real_of_float x\<close>
    apply (simp_all only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
    apply (simp_all only: mult.commute[where 'a=real] of_nat_fact)
    apply (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"])
    done
qed

lemma sin_boundaries:
  assumes "0 \<le> real_of_float x"
    and "x \<le> pi / 2"
  shows "sin x \<in> {(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
proof (cases "real_of_float x = 0")
  case False
  hence "real_of_float x \<noteq> 0" by auto
  hence "0 < x" and "0 < real_of_float x"
    using \<open>0 \<le> real_of_float x\<close> by auto
  have "0 < x * x"
    using \<open>0 < x\<close> by simp

  have sum_morph: "(\<Sum>j = 0 ..< n. (- 1) ^ (((2 * j + 1) - Suc 0) div 2) / ((fact (2 * j + 1))) * x ^(2 * j + 1)) =
    (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * x ^ i)"
    (is "?SUM = _") for x :: real and n
  proof -
    have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)"
      by auto
    have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM"
      by auto
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i)) * x ^ i)"
      unfolding sum_split_even_odd atLeast0LessThan ..
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i))) * x ^ i)"
      by (rule sum.cong) auto
    finally show ?thesis .
  qed

  { fix n :: nat assume "0 < n"
    hence "0 < 2 * n + 1" by auto
    obtain t where "0 < t" and "t < real_of_float x" and
      sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)
      + (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real_of_float x)^(2*n + 1)"
      (is "_ = ?SUM + ?rest / ?fact * ?pow")
      using Maclaurin_sin_expansion3[OF \<open>0 < 2 * n + 1\<close> \<open>0 < real_of_float x\<close>]
      unfolding sin_coeff_def atLeast0LessThan by auto

    have "?rest = cos t * (- 1) ^ n"
      unfolding sin_add cos_add of_nat_add distrib_right distrib_left by auto
    moreover
    have "t \<le> pi / 2"
      using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
    hence "0 \<le> cos t"
      using \<open>0 < t\<close> and cos_ge_zero by auto
    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest"
      by auto

    have "0 < ?fact"
      by (simp del: fact_Suc)
    have "0 < ?pow"
      using \<open>0 < real_of_float x\<close> by (rule zero_less_power)

    {
      assume "even n"
      have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
            (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
        using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto
      also have "\<dots> \<le> ?SUM" by auto
      also have "\<dots> \<le> sin x"
      proof -
        from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close>
        have "0 \<le> (?rest / ?fact) * ?pow" by simp
        thus ?thesis unfolding sin_eq by auto
      qed
      finally have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin x" .
    } note lb = this

    {
      assume "odd n"
      have "sin x \<le> ?SUM"
      proof -
        from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>]
        have "0 \<le> (- ?rest) / ?fact * ?pow"
          by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
        thus ?thesis unfolding sin_eq by auto
      qed
      also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
         by auto
      also have "\<dots> \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))"
        using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto
      finally have "sin x \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
    } note ub = this and lb
  } note ub = this(1) and lb = this(2)

  have "sin x \<le> (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))"
    using ub[OF odd_pos[OF get_odd] get_odd] .
  moreover have "(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin x"
  proof (cases "0 < get_even n")
    case True
    show ?thesis
      using lb[OF True get_even] .
  next
    case False
    hence "get_even n = 0" by auto
    with \<open>x \<le> pi / 2\<close> \<open>0 \<le> real_of_float x\<close>
    show ?thesis
      unfolding \<open>get_even n = 0\<close> ub_sin_cos_aux.simps minus_float.rep_eq
      using sin_ge_zero by auto
  qed
  ultimately show ?thesis by auto
next
  case True
  show ?thesis
  proof (cases "n = 0")
    case True
    thus ?thesis
      unfolding \<open>n = 0\<close> get_even_def get_odd_def
      using \<open>real_of_float x = 0\<close> lapprox_rat[where x="-1" and y=1] by auto
  next
    case False
    with not0_implies_Suc obtain m where "n = Suc m" by blast
    thus ?thesis
      unfolding \<open>n = Suc m\<close> get_even_def get_odd_def
      using \<open>real_of_float x = 0\<close> rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1]
      by (cases "even (Suc m)") auto
  qed
qed


subsection "Compute the cosinus in the entire domain"

definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
"lb_cos prec x = (let
    horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
    half = \<lambda> x. if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)
  in if x < Float 1 (- 1) then horner x
else if x < 1          then half (horner (x * Float 1 (- 1)))
                       else half (half (horner (x * Float 1 (- 2)))))"

definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
"ub_cos prec x = (let
    horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
    half = \<lambda> x. float_plus_up prec (Float 1 1 * x * x) (- 1)
  in if x < Float 1 (- 1) then horner x
else if x < 1          then half (horner (x * Float 1 (- 1)))
                       else half (half (horner (x * Float 1 (- 2)))))"

lemma lb_cos:
  assumes "0 \<le> real_of_float x" and "x \<le> pi"
  shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }")
proof -
  have x_half[symmetric]: "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" for x :: real
  proof -
    have "cos x = cos (x / 2 + x / 2)"
      by auto
    also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1"
      unfolding cos_add by auto
    also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1"
      by algebra
    finally show ?thesis .
  qed

  have "\<not> x < 0" using \<open>0 \<le> real_of_float x\<close> by auto
  let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
  let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
  let "?ub_half x" = "float_plus_up prec (Float 1 1 * x * x) (- 1)"
  let "?lb_half x" = "if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)"

  show ?thesis
  proof (cases "x < Float 1 (- 1)")
    case True
    hence "x \<le> pi / 2"
      using pi_ge_two by auto
    show ?thesis
      unfolding lb_cos_def[where x=x] ub_cos_def[where x=x]
        if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF \<open>x < Float 1 (- 1)\<close>] Let_def
      using cos_boundaries[OF \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi / 2\<close>] .
  next
    case False
    { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
      assume "y \<le> cos ?x2" and "-pi \<le> x" and "x \<le> pi"
      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2"
        using pi_ge_two unfolding Float_num by auto
      hence "0 \<le> cos ?x2"
        by (rule cos_ge_zero)

      have "(?lb_half y) \<le> cos x"
      proof (cases "y < 0")
        case True
        show ?thesis
          using cos_ge_minus_one unfolding if_P[OF True] by auto
      next
        case False
        hence "0 \<le> real_of_float y" by auto
        from mult_mono[OF \<open>y \<le> cos ?x2\<close> \<open>y \<le> cos ?x2\<close> \<open>0 \<le> cos ?x2\<close> this]
        have "real_of_float y * real_of_float y \<le> cos ?x2 * cos ?x2" .
        hence "2 * real_of_float y * real_of_float y \<le> 2 * cos ?x2 * cos ?x2"
          by auto
        hence "2 * real_of_float y * real_of_float y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1"
          unfolding Float_num by auto
        thus ?thesis
          unfolding if_not_P[OF False] x_half Float_num
          by (auto intro!: float_plus_down_le)
      qed
    } note lb_half = this

    { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
      assume ub: "cos ?x2 \<le> y" and "- pi \<le> x" and "x \<le> pi"
      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2"
        using pi_ge_two unfolding Float_num by auto
      hence "0 \<le> cos ?x2" by (rule cos_ge_zero)

      have "cos x \<le> (?ub_half y)"
      proof -
        have "0 \<le> real_of_float y"
          using \<open>0 \<le> cos ?x2\<close> ub by (rule order_trans)
        from mult_mono[OF ub ub this \<open>0 \<le> cos ?x2\<close>]
        have "cos ?x2 * cos ?x2 \<le> real_of_float y * real_of_float y" .
        hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real_of_float y * real_of_float y"
          by auto
        hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real_of_float y * real_of_float y - 1"
          unfolding Float_num by auto
        thus ?thesis
          unfolding x_half Float_num
          by (auto intro!: float_plus_up_le)
      qed
    } note ub_half = this

    let ?x2 = "x * Float 1 (- 1)"
    let ?x4 = "x * Float 1 (- 1) * Float 1 (- 1)"

    have "-pi \<le> x"
      using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] \<open>0 \<le> real_of_float x\<close>
      by (rule order_trans)

    show ?thesis
    proof (cases "x < 1")
      case True
      hence "real_of_float x \<le> 1" by auto
      have "0 \<le> real_of_float ?x2" and "?x2 \<le> pi / 2"
        using pi_ge_two \<open>0 \<le> real_of_float x\<close> using assms by auto
      from cos_boundaries[OF this]
      have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)"
        by auto

      have "(?lb x) \<le> ?cos x"
      proof -
        from lb_half[OF lb \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>]
        show ?thesis
          unfolding lb_cos_def[where x=x] Let_def
          using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto
      qed
      moreover have "?cos x \<le> (?ub x)"
      proof -
        from ub_half[OF ub \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>]
        show ?thesis
          unfolding ub_cos_def[where x=x] Let_def
          using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto
      qed
      ultimately show ?thesis by auto
    next
      case False
      have "0 \<le> real_of_float ?x4" and "?x4 \<le> pi / 2"
        using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> unfolding Float_num by auto
      from cos_boundaries[OF this]
      have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)"
        by auto

      have eq_4: "?x2 * Float 1 (- 1) = x * Float 1 (- 2)"
        by (auto simp: real_of_float_eq)

      have "(?lb x) \<le> ?cos x"
      proof -
        have "-pi \<le> ?x2" and "?x2 \<le> pi"
          using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> by auto
        from lb_half[OF lb_half[OF lb this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
        show ?thesis
          unfolding lb_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
            if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def .
      qed
      moreover have "?cos x \<le> (?ub x)"
      proof -
        have "-pi \<le> ?x2" and "?x2 \<le> pi"
          using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open> x \<le> pi\<close> by auto
        from ub_half[OF ub_half[OF ub this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
        show ?thesis
          unfolding ub_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
            if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def .
      qed
      ultimately show ?thesis by auto
    qed
  qed
qed

lemma lb_cos_minus:
  assumes "-pi \<le> x"
    and "real_of_float x \<le> 0"
  shows "cos (real_of_float(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
proof -
  have "0 \<le> real_of_float (-x)" and "(-x) \<le> pi"
    using \<open>-pi \<le> x\<close> \<open>real_of_float x \<le> 0\<close> by auto
  from lb_cos[OF this] show ?thesis .
qed

definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
"bnds_cos prec lx ux = (let
    lpi = float_round_down prec (lb_pi prec) ;
    upi = float_round_up prec (ub_pi prec) ;
    k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
    lx = float_plus_down prec lx (- k * 2 * (if k < 0 then lpi else upi)) ;
    ux = float_plus_up prec ux (- k * 2 * (if k < 0 then upi else lpi))
  in   if - lpi \<le> lx \<and> ux \<le> 0    then (lb_cos prec (-lx), ub_cos prec (-ux))
  else if 0 \<le> lx \<and> ux \<le> lpi      then (lb_cos prec ux, ub_cos prec lx)
  else if - lpi \<le> lx \<and> ux \<le> lpi  then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
  else if 0 \<le> lx \<and> ux \<le> 2 * lpi  then (Float (- 1) 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi))))
  else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float (- 1) 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux)))
                                 else (Float (- 1) 0, Float 1 0))"

lemma floor_int: obtains k :: int where "real_of_int k = (floor_fl f)"
  by (simp add: floor_fl_def)

lemma cos_periodic_nat[simp]:
  fixes n :: nat
  shows "cos (x + n * (2 * pi)) = cos x"
proof (induct n arbitrary: x)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi"
    unfolding Suc_eq_plus1 of_nat_add of_int_1 distrib_right by auto
  show ?case
    unfolding split_pi_off using Suc by auto
qed

lemma cos_periodic_int[simp]:
  fixes i :: int
  shows "cos (x + i * (2 * pi)) = cos x"
proof (cases "0 \<le> i")
  case True
  hence i_nat: "real_of_int i = nat i" by auto
  show ?thesis
    unfolding i_nat by auto
next
  case False
    hence i_nat: "i = - real (nat (-i))" by auto
  have "cos x = cos (x + i * (2 * pi) - i * (2 * pi))"
    by auto
  also have "\<dots> = cos (x + i * (2 * pi))"
    unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
  finally show ?thesis by auto
qed

lemma bnds_cos: "\<forall>(x::real) lx ux. (l, u) =
  bnds_cos prec lx ux \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> cos x \<and> cos x \<le> u"
proof (rule allI | rule impI | erule conjE)+
  fix x :: real
  fix lx ux
  assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {lx .. ux}"

  let ?lpi = "float_round_down prec (lb_pi prec)"
  let ?upi = "float_round_up prec (ub_pi prec)"
  let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
  let ?lx2 = "(- ?k * 2 * (if ?k < 0 then ?lpi else ?upi))"
  let ?ux2 = "(- ?k * 2 * (if ?k < 0 then ?upi else ?lpi))"
  let ?lx = "float_plus_down prec lx ?lx2"
  let ?ux = "float_plus_up prec ux ?ux2"

  obtain k :: int where k: "k = real_of_float ?k"
    by (rule floor_int)

  have upi: "pi \<le> ?upi" and lpi: "?lpi \<le> pi"
    using float_round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
      float_round_down[of prec "lb_pi prec"]
    by auto
  hence "lx + ?lx2 \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ux + ?ux2"
    using x
    by (cases "k = 0")
      (auto intro!: add_mono
        simp add: k [symmetric] uminus_add_conv_diff [symmetric]
        simp del: uminus_add_conv_diff)
  hence "?lx \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux"
    by (auto intro!: float_plus_down_le float_plus_up_le)
  note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
  hence lx_less_ux: "?lx \<le> real_of_float ?ux" by (rule order_trans)

  { assume "- ?lpi \<le> ?lx" and x_le_0: "x - k * (2 * pi) \<le> 0"
    with lpi[THEN le_imp_neg_le] lx
    have pi_lx: "- pi \<le> ?lx" and lx_0: "real_of_float ?lx \<le> 0"
      by simp_all

    have "(lb_cos prec (- ?lx)) \<le> cos (real_of_float (- ?lx))"
      using lb_cos_minus[OF pi_lx lx_0] by simp
    also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
      using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
      by (simp only: uminus_float.rep_eq of_int_minus
        cos_minus mult_minus_left) simp
    finally have "(lb_cos prec (- ?lx)) \<le> cos x"
      unfolding cos_periodic_int . }
  note negative_lx = this

  { assume "0 \<le> ?lx" and pi_x: "x - k * (2 * pi) \<le> pi"
    with lx
    have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real_of_float ?lx"
      by auto

    have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx"
      using cos_monotone_0_pi_le[OF lx_0 lx pi_x]
      by (simp only: of_int_minus
        cos_minus mult_minus_left) simp
    also have "\<dots> \<le> (ub_cos prec ?lx)"
      using lb_cos[OF lx_0 pi_lx] by simp
    finally have "cos x \<le> (ub_cos prec ?lx)"
      unfolding cos_periodic_int . }
  note positive_lx = this

  { assume pi_x: "- pi \<le> x - k * (2 * pi)" and "?ux \<le> 0"
    with ux
    have pi_ux: "- pi \<le> ?ux" and ux_0: "real_of_float ?ux \<le> 0"
      by simp_all

    have "cos (x + (-k) * (2 * pi)) \<le> cos (real_of_float (- ?ux))"
      using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
      by (simp only: uminus_float.rep_eq of_int_minus
          cos_minus mult_minus_left) simp
    also have "\<dots> \<le> (ub_cos prec (- ?ux))"
      using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
    finally have "cos x \<le> (ub_cos prec (- ?ux))"
      unfolding cos_periodic_int . }
  note negative_ux = this

  { assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - k * (2 * pi)"
    with lpi ux
    have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real_of_float ?ux"
      by simp_all

    have "(lb_cos prec ?ux) \<le> cos ?ux"
      using lb_cos[OF ux_0 pi_ux] by simp
    also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
      using cos_monotone_0_pi_le[OF x_ge_0 ux pi_ux]
      by (simp only: of_int_minus
        cos_minus mult_minus_left) simp
    finally have "(lb_cos prec ?ux) \<le> cos x"
      unfolding cos_periodic_int . }
  note positive_ux = this

  show "l \<le> cos x \<and> cos x \<le> u"
  proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
    case True
    with bnds have l: "l = lb_cos prec (-?lx)" and u: "u = ub_cos prec (-?ux)"
      by (auto simp add: bnds_cos_def Let_def)
    from True lpi[THEN le_imp_neg_le] lx ux
    have "- pi \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> 0"
      by auto
    with True negative_ux negative_lx show ?thesis
      unfolding l u by simp
  next
    case 1: False
    show ?thesis
    proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
      case True with bnds 1
      have l: "l = lb_cos prec ?ux"
        and u: "u = ub_cos prec ?lx"
        by (auto simp add: bnds_cos_def Let_def)
      from True lpi lx ux
      have "0 \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> pi"
        by auto
      with True positive_ux positive_lx show ?thesis
        unfolding l u by simp
    next
      case 2: False
      show ?thesis
      proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
        case Cond: True
        with bnds 1 2 have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
          and u: "u = Float 1 0"
          by (auto simp add: bnds_cos_def Let_def)
        show ?thesis
          unfolding u l using negative_lx positive_ux Cond
          by (cases "x - k * (2 * pi) < 0") (auto simp add: real_of_float_min)
      next
        case 3: False
        show ?thesis
        proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
          case Cond: True
          with bnds 1 2 3
          have l: "l = Float (- 1) 0"
            and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
            by (auto simp add: bnds_cos_def Let_def)

          have "cos x \<le> real_of_float u"
          proof (cases "x - k * (2 * pi) < pi")
            case True
            hence "x - k * (2 * pi) \<le> pi" by simp
            from positive_lx[OF Cond[THEN conjunct1] this] show ?thesis
              unfolding u by (simp add: real_of_float_max)
          next
            case False
            hence "pi \<le> x - k * (2 * pi)" by simp
            hence pi_x: "- pi \<le> x - k * (2 * pi) - 2 * pi" by simp

            have "?ux \<le> 2 * pi"
              using Cond lpi by auto
            hence "x - k * (2 * pi) - 2 * pi \<le> 0"
              using ux by simp

            have ux_0: "real_of_float (?ux - 2 * ?lpi) \<le> 0"
              using Cond by auto

            from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
            hence "- ?lpi \<le> ?ux - 2 * ?lpi" by auto
            hence pi_ux: "- pi \<le> (?ux - 2 * ?lpi)"
              using lpi[THEN le_imp_neg_le] by auto

            have x_le_ux: "x - k * (2 * pi) - 2 * pi \<le> (?ux - 2 * ?lpi)"
              using ux lpi by auto
            have "cos x = cos (x + (-k) * (2 * pi) + (-1::int) * (2 * pi))"
              unfolding cos_periodic_int ..
            also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))"
              using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
              by (simp only: minus_float.rep_eq of_int_minus of_int_1
                mult_minus_left mult_1_left) simp
            also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))"
              unfolding uminus_float.rep_eq cos_minus ..
            also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))"
              using lb_cos_minus[OF pi_ux ux_0] by simp
            finally show ?thesis unfolding u by (simp add: real_of_float_max)
          qed
          thus ?thesis unfolding l by auto
        next
          case 4: False
          show ?thesis
          proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
            case Cond: True
            with bnds 1 2 3 4 have l: "l = Float (- 1) 0"
              and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
              by (auto simp add: bnds_cos_def Let_def)

            have "cos x \<le> u"
            proof (cases "-pi < x - k * (2 * pi)")
              case True
              hence "-pi \<le> x - k * (2 * pi)" by simp
              from negative_ux[OF this Cond[THEN conjunct2]] show ?thesis
                unfolding u by (simp add: real_of_float_max)
            next
              case False
              hence "x - k * (2 * pi) \<le> -pi" by simp
              hence pi_x: "x - k * (2 * pi) + 2 * pi \<le> pi" by simp

              have "-2 * pi \<le> ?lx" using Cond lpi by auto

              hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp

              have lx_0: "0 \<le> real_of_float (?lx + 2 * ?lpi)"
                using Cond lpi by auto

              from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
              hence "?lx + 2 * ?lpi \<le> ?lpi" by auto
              hence pi_lx: "(?lx + 2 * ?lpi) \<le> pi"
                using lpi[THEN le_imp_neg_le] by auto

              have lx_le_x: "(?lx + 2 * ?lpi) \<le> x - k * (2 * pi) + 2 * pi"
                using lx lpi by auto

              have "cos x = cos (x + (-k) * (2 * pi) + (1 :: int) * (2 * pi))"
                unfolding cos_periodic_int ..
              also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))"
                using cos_monotone_0_pi_le[OF lx_0 lx_le_x pi_x]
                by (simp only: minus_float.rep_eq of_int_minus of_int_1
                  mult_minus_left mult_1_left) simp
              also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))"
                using lb_cos[OF lx_0 pi_lx] by simp
              finally show ?thesis unfolding u by (simp add: real_of_float_max)
            qed
            thus ?thesis unfolding l by auto
          next
            case False
            with bnds 1 2 3 4 show ?thesis
              by (auto simp add: bnds_cos_def Let_def)
          qed
        qed
      qed
    qed
  qed
qed


section "Exponential function"

subsection "Compute the series of the exponential function"

fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
  and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
where
"ub_exp_horner prec 0 i k x       = 0" |
"ub_exp_horner prec (Suc n) i k x = float_plus_up prec
    (rapprox_rat prec 1 (int k)) (float_round_up prec (x * lb_exp_horner prec n (i + 1) (k * i) x))" |
"lb_exp_horner prec 0 i k x       = 0" |
"lb_exp_horner prec (Suc n) i k x = float_plus_down prec
    (lapprox_rat prec 1 (int k)) (float_round_down prec (x * ub_exp_horner prec n (i + 1) (k * i) x))"

lemma bnds_exp_horner:
  assumes "real_of_float x \<le> 0"
  shows "exp x \<in> {lb_exp_horner prec (get_even n) 1 1 x .. ub_exp_horner prec (get_odd n) 1 1 x}"
proof -
  have f_eq: "fact (Suc n) = fact n * ((\<lambda>i::nat. i + 1) ^^ n) 1" for n
  proof -
    have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m"
      by (induct n) auto
    show ?thesis
      unfolding F by auto
  qed

  note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1,
    OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]

  have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x"
  proof -
    have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / (fact j) * real_of_float x ^ j)"
      using bounds(1) by auto
    also have "\<dots> \<le> exp x"
    proof -
      obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real_of_float x ^ m / (fact m)) + exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
        using Maclaurin_exp_le unfolding atLeast0LessThan by blast
      moreover have "0 \<le> exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
        by (auto simp: zero_le_even_power)
      ultimately show ?thesis using get_odd exp_gt_zero by auto
    qed
    finally show ?thesis .
  qed
  moreover
  have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x"
  proof -
    have x_less_zero: "real_of_float x ^ get_odd n \<le> 0"
    proof (cases "real_of_float x = 0")
      case True
      have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
      thus ?thesis unfolding True power_0_left by auto
    next
      case False hence "real_of_float x < 0" using \<open>real_of_float x \<le> 0\<close> by auto
      show ?thesis by (rule less_imp_le, auto simp add: \<open>real_of_float x < 0\<close>)
    qed
    obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>"
      and "exp x = (\<Sum>m = 0..<get_odd n. (real_of_float x) ^ m / (fact m)) + exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n)"
      using Maclaurin_exp_le unfolding atLeast0LessThan by blast
    moreover have "exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n) \<le> 0"
      by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero)
    ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / (fact j) * real_of_float x ^ j)"
      using get_odd exp_gt_zero by auto
    also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x"
      using bounds(2) by auto
    finally show ?thesis .
  qed
  ultimately show ?thesis by auto
qed

lemma ub_exp_horner_nonneg: "real_of_float x \<le> 0 \<Longrightarrow>
  0 \<le> real_of_float (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)"
  using bnds_exp_horner[of x prec n]
  by (intro order_trans[OF exp_ge_zero]) auto


subsection "Compute the exponential function on the entire domain"

function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
"lb_exp prec x =
  (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
  else
    let
      horner = (\<lambda> x. let  y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in
        if y \<le> 0 then Float 1 (- 2) else y)
    in
      if x < - 1 then
        power_down_fl prec (horner (float_divl prec x (- floor_fl x))) (nat (- int_floor_fl x))
      else horner x)" |
"ub_exp prec x =
  (if 0 < x then float_divr prec 1 (lb_exp prec (-x))
  else if x < - 1 then
    power_up_fl prec
      (ub_exp_horner prec (get_odd (prec + 2)) 1 1
        (float_divr prec x (- floor_fl x))) (nat (- int_floor_fl x))
  else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
  by pat_completeness auto
termination
  by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if 0 < x then 1 else 0))") auto

lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
proof -
  have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
  have "1 / 4 = (Float 1 (- 2))"
    unfolding Float_num by auto
  also have "\<dots> \<le> lb_exp_horner 3 (get_even 3) 1 1 (- 1)"
    by (subst less_eq_float.rep_eq [symmetric]) code_simp
  also have "\<dots> \<le> exp (- 1 :: float)"
    using bnds_exp_horner[where x="- 1"] by auto
  finally show ?thesis
    by simp
qed

lemma lb_exp_pos:
  assumes "\<not> 0 < x"
  shows "0 < lb_exp prec x"
proof -
  let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
  let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 (- 2) else y"
  have pos_horner: "0 < ?horner x" for x
    unfolding Let_def by (cases "?lb_horner x \<le> 0") auto
  moreover have "0 < real_of_float ((?horner x) ^ num)" for x :: float and num :: nat
  proof -
    have "0 < real_of_float (?horner x) ^ num" using \<open>0 < ?horner x\<close> by simp
    also have "\<dots> = (?horner x) ^ num" by auto
    finally show ?thesis .
  qed
  ultimately show ?thesis
    unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] Let_def
    by (cases "floor_fl x", cases "x < - 1")
      (auto simp: real_power_up_fl real_power_down_fl intro!: power_up_less power_down_pos)
qed

lemma exp_boundaries':
  assumes "x \<le> 0"
  shows "exp x \<in> { (lb_exp prec x) .. (ub_exp prec x)}"
proof -
  let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
  let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"

  have "real_of_float x \<le> 0" and "\<not> x > 0"
    using \<open>x \<le> 0\<close> by auto
  show ?thesis
  proof (cases "x < - 1")
    case False
    hence "- 1 \<le> real_of_float x" by auto
    show ?thesis
    proof (cases "?lb_exp_horner x \<le> 0")
      case True
      from \<open>\<not> x < - 1\<close>
      have "- 1 \<le> real_of_float x" by auto
      hence "exp (- 1) \<le> exp x"
        unfolding exp_le_cancel_iff .
      from order_trans[OF exp_m1_ge_quarter this] have "Float 1 (- 2) \<le> exp x"
        unfolding Float_num .
      with True show ?thesis
        using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by auto
    next
      case False
      thus ?thesis
        using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by (auto simp add: Let_def)
    qed
  next
    case True
    let ?num = "nat (- int_floor_fl x)"

    have "real_of_int (int_floor_fl x) < - 1"
      using int_floor_fl[of x] \<open>x < - 1\<close> by simp
    hence "real_of_int (int_floor_fl x) < 0" by simp
    hence "int_floor_fl x < 0" by auto
    hence "1 \<le> - int_floor_fl x" by auto
    hence "0 < nat (- int_floor_fl x)" by auto
    hence "0 < ?num"  by auto
    hence "real ?num \<noteq> 0" by auto
    have num_eq: "real ?num = - int_floor_fl x"
      using \<open>0 < nat (- int_floor_fl x)\<close> by auto
    have "0 < - int_floor_fl x"
      using \<open>0 < ?num\<close>[unfolded of_nat_less_iff[symmetric]] by simp
    hence "real_of_int (int_floor_fl x) < 0"
      unfolding less_float_def by auto
    have fl_eq: "real_of_int (- int_floor_fl x) = real_of_float (- floor_fl x)"
      by (simp add: floor_fl_def int_floor_fl_def)
    from \<open>0 < - int_floor_fl x\<close> have "0 \<le> real_of_float (- floor_fl x)"
      by (simp add: floor_fl_def int_floor_fl_def)
    from \<open>real_of_int (int_floor_fl x) < 0\<close> have "real_of_float (floor_fl x) < 0"
      by (simp add: floor_fl_def int_floor_fl_def)
    have "exp x \<le> ub_exp prec x"
    proof -
      have div_less_zero: "real_of_float (float_divr prec x (- floor_fl x)) \<le> 0"
        using float_divr_nonpos_pos_upper_bound[OF \<open>real_of_float x \<le> 0\<close> \<open>0 \<le> real_of_float (- floor_fl x)\<close>]
        unfolding less_eq_float_def zero_float.rep_eq .

      have "exp x = exp (?num * (x / ?num))"
        using \<open>real ?num \<noteq> 0\<close> by auto
      also have "\<dots> = exp (x / ?num) ^ ?num"
        unfolding exp_of_nat_mult ..
      also have "\<dots> \<le> exp (float_divr prec x (- floor_fl x)) ^ ?num"
        unfolding num_eq fl_eq
        by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
      also have "\<dots> \<le> (?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num"
        unfolding real_of_float_power
        by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
      also have "\<dots> \<le> real_of_float (power_up_fl prec (?ub_exp_horner (float_divr prec x (- floor_fl x))) ?num)"
        by (auto simp add: real_power_up_fl intro!: power_up ub_exp_horner_nonneg div_less_zero)
      finally show ?thesis
        unfolding ub_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] floor_fl_def Let_def .
    qed
    moreover
    have "lb_exp prec x \<le> exp x"
    proof -
      let ?divl = "float_divl prec x (- floor_fl x)"
      let ?horner = "?lb_exp_horner ?divl"

      show ?thesis
      proof (cases "?horner \<le> 0")
        case False
        hence "0 \<le> real_of_float ?horner" by auto

        have div_less_zero: "real_of_float (float_divl prec x (- floor_fl x)) \<le> 0"
          using \<open>real_of_float (floor_fl x) < 0\<close> \<open>real_of_float x \<le> 0\<close>
          by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)

        have "(?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num \<le>
          exp (float_divl prec x (- floor_fl x)) ^ ?num"
          using \<open>0 \<le> real_of_float ?horner\<close>[unfolded floor_fl_def[symmetric]]
            bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1]
          by (auto intro!: power_mono)
        also have "\<dots> \<le> exp (x / ?num) ^ ?num"
          unfolding num_eq fl_eq
          using float_divl by (auto intro!: power_mono simp del: uminus_float.rep_eq)
        also have "\<dots> = exp (?num * (x / ?num))"
          unfolding exp_of_nat_mult ..
        also have "\<dots> = exp x"
          using \<open>real ?num \<noteq> 0\<close> by auto
        finally show ?thesis
          using False
          unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>]
            int_floor_fl_def Let_def if_not_P[OF False]
          by (auto simp: real_power_down_fl intro!: power_down_le)
      next
        case True
        have "power_down_fl prec (Float 1 (- 2))  ?num \<le> (Float 1 (- 2)) ^ ?num"
          by (metis Float_le_zero_iff less_imp_le linorder_not_less
            not_numeral_le_zero numeral_One power_down_fl)
        then have "power_down_fl prec (Float 1 (- 2))  ?num \<le> real_of_float (Float 1 (- 2)) ^ ?num"
          by simp
        also
        have "real_of_float (floor_fl x) \<noteq> 0" and "real_of_float (floor_fl x) \<le> 0"
          using \<open>real_of_float (floor_fl x) < 0\<close> by auto
        from divide_right_mono_neg[OF floor_fl[of x] \<open>real_of_float (floor_fl x) \<le> 0\<close>, unfolded divide_self[OF \<open>real_of_float (floor_fl x) \<noteq> 0\<close>]]
        have "- 1 \<le> x / (- floor_fl x)"
          unfolding minus_float.rep_eq by auto
        from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
        have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))"
          unfolding Float_num .
        hence "real_of_float (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
          by (metis Float_num(5) power_mono zero_le_divide_1_iff zero_le_numeral)
        also have "\<dots> = exp x"
          unfolding num_eq fl_eq exp_of_nat_mult[symmetric]
          using \<open>real_of_float (floor_fl x) \<noteq> 0\<close> by auto
        finally show ?thesis
          unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>]
            int_floor_fl_def Let_def if_P[OF True] real_of_float_power .
      qed
    qed
    ultimately show ?thesis by auto
  qed
qed

lemma exp_boundaries: "exp x \<in> { lb_exp prec x .. ub_exp prec x }"
proof -
  show ?thesis
  proof (cases "0 < x")
    case False
    hence "x \<le> 0" by auto
    from exp_boundaries'[OF this] show ?thesis .
  next
    case True
    hence "-x \<le> 0" by auto

    have "lb_exp prec x \<le> exp x"
    proof -
      from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
      have ub_exp: "exp (- real_of_float x) \<le> ub_exp prec (-x)"
        unfolding atLeastAtMost_iff minus_float.rep_eq by auto

      have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)"
        using float_divl[where x=1] by auto
      also have "\<dots> \<le> exp x"
        using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp]
          exp_gt_zero, symmetric]]
        unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide
        by auto
      finally show ?thesis
        unfolding lb_exp.simps if_P[OF True] .
    qed
    moreover
    have "exp x \<le> ub_exp prec x"
    proof -
      have "\<not> 0 < -x" using \<open>0 < x\<close> by auto

      from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
      have lb_exp: "lb_exp prec (-x) \<le> exp (- real_of_float x)"
        unfolding atLeastAtMost_iff minus_float.rep_eq by auto

      have "exp x \<le> (1 :: float) / lb_exp prec (-x)"
        using lb_exp lb_exp_pos[OF \<open>\<not> 0 < -x\<close>, of prec]
        by (simp del: lb_exp.simps add: exp_minus field_simps)
      also have "\<dots> \<le> float_divr prec 1 (lb_exp prec (-x))"
        using float_divr .
      finally show ?thesis
        unfolding ub_exp.simps if_P[OF True] .
    qed
    ultimately show ?thesis
      by auto
  qed
qed

lemma bnds_exp: "\<forall>(x::real) lx ux. (l, u) =
  (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> exp x \<and> exp x \<le> u"
proof (rule allI, rule allI, rule allI, rule impI)
  fix x :: real and lx ux
  assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux}"
  hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {lx .. ux}"
    by auto
  show "l \<le> exp x \<and> exp x \<le> u"
  proof
    show "l \<le> exp x"
    proof -
      from exp_boundaries[of lx prec, unfolded l]
      have "l \<le> exp lx" by (auto simp del: lb_exp.simps)
      also have "\<dots> \<le> exp x" using x by auto
      finally show ?thesis .
    qed
    show "exp x \<le> u"
    proof -
      have "exp x \<le> exp ux" using x by auto
      also have "\<dots> \<le> u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
      finally show ?thesis .
    qed
  qed
qed


section "Logarithm"

subsection "Compute the logarithm series"

fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
"ub_ln_horner prec 0 i x       = 0" |
"ub_ln_horner prec (Suc n) i x = float_plus_up prec
    (rapprox_rat prec 1 (int i)) (- float_round_down prec (x * lb_ln_horner prec n (Suc i) x))" |
"lb_ln_horner prec 0 i x       = 0" |
"lb_ln_horner prec (Suc n) i x = float_plus_down prec
    (lapprox_rat prec 1 (int i)) (- float_round_up prec (x * ub_ln_horner prec n (Suc i) x))"

lemma ln_bounds:
  assumes "0 \<le> x"
    and "x < 1"
  shows "(\<Sum>i=0..<2*n. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
  and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
proof -
  let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"

  have ln_eq: "(\<Sum> i. (- 1) ^ i * ?a i) = ln (x + 1)"
    using ln_series[of "x + 1"] \<open>0 \<le> x\<close> \<open>x < 1\<close> by auto

  have "norm x < 1" using assms by auto
  have "?a \<longlonglongrightarrow> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
    using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF \<open>norm x < 1\<close>]]] by auto
  have "0 \<le> ?a n" for n
    by (rule mult_nonneg_nonneg) (auto simp: \<open>0 \<le> x\<close>)
  have "?a (Suc n) \<le> ?a n" for n
    unfolding inverse_eq_divide[symmetric]
  proof (rule mult_mono)
    show "0 \<le> x ^ Suc (Suc n)"
      by (auto simp add: \<open>0 \<le> x\<close>)
    have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1"
      unfolding power_Suc2 mult.assoc[symmetric]
      by (rule mult_left_mono, fact less_imp_le[OF \<open>x < 1\<close>]) (auto simp: \<open>0 \<le> x\<close>)
    thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
  qed auto
  from summable_Leibniz'(2,4)[OF \<open>?a \<longlonglongrightarrow> 0\<close> \<open>\<And>n. 0 \<le> ?a n\<close>, OF \<open>\<And>n. ?a (Suc n) \<le> ?a n\<close>, unfolded ln_eq]
  show ?lb and ?ub
    unfolding atLeast0LessThan by auto
qed

lemma ln_float_bounds:
  assumes "0 \<le> real_of_float x"
    and "real_of_float x < 1"
  shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln")
    and "ln (x + 1) \<le> x * ub_ln_horner prec (get_odd n) 1 x" (is "?ln \<le> ?ub")
proof -
  obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
  obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..

  let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real_of_float x)^(Suc n)"

  have "?lb \<le> sum ?s {0 ..< 2 * ev}"
    unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric]
    unfolding mult.commute[of "real_of_float x"] ev 
    using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" 
                    and lb="\<lambda>n i k x. lb_ln_horner prec n k x" 
                    and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
      OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
    unfolding real_of_float_power
    by (rule mult_right_mono)
  also have "\<dots> \<le> ?ln"
    using ln_bounds(1)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
  finally show "?lb \<le> ?ln" .

  have "?ln \<le> sum ?s {0 ..< 2 * od + 1}"
    using ln_bounds(2)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
  also have "\<dots> \<le> ?ub"
    unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric]
    unfolding mult.commute[of "real_of_float x"] od
    using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
      OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
    unfolding real_of_float_power
    by (rule mult_right_mono)
  finally show "?ln \<le> ?ub" .
qed

lemma ln_add:
  fixes x :: real
  assumes "0 < x" and "0 < y"
  shows "ln (x + y) = ln x + ln (1 + y / x)"
proof -
  have "x \<noteq> 0" using assms by auto
  have "x + y = x * (1 + y / x)"
    unfolding distrib_left times_divide_eq_right nonzero_mult_div_cancel_left[OF \<open>x \<noteq> 0\<close>]
    by auto
  moreover
  have "0 < y / x" using assms by auto
  hence "0 < 1 + y / x" by auto
  ultimately show ?thesis
    using ln_mult assms by auto
qed


subsection "Compute the logarithm of 2"

definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
                                        in float_plus_up prec
                                          ((Float 1 (- 1) * ub_ln_horner prec (get_odd prec) 1 (Float 1 (- 1))))
                                           (float_round_up prec (third * ub_ln_horner prec (get_odd prec) 1 third)))"
definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
                                        in float_plus_down prec
                                          ((Float 1 (- 1) * lb_ln_horner prec (get_even prec) 1 (Float 1 (- 1))))
                                           (float_round_down prec (third * lb_ln_horner prec (get_even prec) 1 third)))"

lemma ub_ln2: "ln 2 \<le> ub_ln2 prec" (is "?ub_ln2")
  and lb_ln2: "lb_ln2 prec \<le> ln 2" (is "?lb_ln2")
proof -
  let ?uthird = "rapprox_rat (max prec 1) 1 3"
  let ?lthird = "lapprox_rat prec 1 3"

  have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1::real)"
    using ln_add[of "3 / 2" "1 / 2"] by auto
  have lb3: "?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
  hence lb3_ub: "real_of_float ?lthird < 1" by auto
  have lb3_lb: "0 \<le> real_of_float ?lthird" using lapprox_rat_nonneg[of 1 3] by auto
  have ub3: "1 / 3 \<le> ?uthird" using rapprox_rat[of 1 3] by auto
  hence ub3_lb: "0 \<le> real_of_float ?uthird" by auto

  have lb2: "0 \<le> real_of_float (Float 1 (- 1))" and ub2: "real_of_float (Float 1 (- 1)) < 1"
    unfolding Float_num by auto

  have "0 \<le> (1::int)" and "0 < (3::int)" by auto
  have ub3_ub: "real_of_float ?uthird < 1"
    by (simp add: Float.compute_rapprox_rat Float.compute_lapprox_rat rapprox_posrat_less1)

  have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
  have uthird_gt0: "0 < real_of_float ?uthird + 1" using ub3_lb by auto
  have lthird_gt0: "0 < real_of_float ?lthird + 1" using lb3_lb by auto

  show ?ub_ln2
    unfolding ub_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
  proof (rule float_plus_up_le, rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
    have "ln (1 / 3 + 1) \<le> ln (real_of_float ?uthird + 1)"
      unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
    also have "\<dots> \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird"
      using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
    also note float_round_up
    finally show "ln (1 / 3 + 1) \<le> float_round_up prec (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
  qed
  show ?lb_ln2
    unfolding lb_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
  proof (rule float_plus_down_le, rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
    have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real_of_float ?lthird + 1)"
      using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
    note float_round_down_le[OF this]
    also have "\<dots> \<le> ln (1 / 3 + 1)"
      unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0]
      using lb3 by auto
    finally show "float_round_down prec (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le>
      ln (1 / 3 + 1)" .
  qed
qed


subsection "Compute the logarithm in the entire domain"

function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
"ub_ln prec x = (if x \<le> 0          then None
            else if x < 1          then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
            else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
                 if x \<le> Float 3 (- 1) then Some (horner (x - 1))
            else if x < Float 1 1  then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
                                   else let l = bitlen (mantissa x) - 1 in
                                        Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))" |
"lb_ln prec x = (if x \<le> 0          then None
            else if x < 1          then Some (- the (ub_ln prec (float_divr prec 1 x)))
            else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
                 if x \<le> Float 3 (- 1) then Some (horner (x - 1))
            else if x < Float 1 1  then Some (float_round_down prec (horner (Float 1 (- 1)) +
                                              horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
                                   else let l = bitlen (mantissa x) - 1 in
                                        Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))"
  by pat_completeness auto

termination
proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto)
  fix prec and x :: float
  assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1"
  hence "0 < real_of_float x" "1 \<le> max prec (Suc 0)" "real_of_float x < 1"
    by auto
  from float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x < 1\<close>[THEN less_imp_le] \<open>1 \<le> max prec (Suc 0)\<close>]
  show False
    using \<open>real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1\<close> by auto
next
  fix prec x
  assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divr prec 1 x) < 1"
  hence "0 < x" by auto
  from float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close>, of prec] \<open>real_of_float x < 1\<close> show False
    using \<open>real_of_float (float_divr prec 1 x) < 1\<close> by auto
qed

lemmas float_pos_eq_mantissa_pos = mantissa_pos_iff[symmetric]

lemma Float_pos_eq_mantissa_pos: "Float m e > 0 \<longleftrightarrow> m > 0"
  using powr_gt_zero[of 2 "e"]
  by (auto simp add: zero_less_mult_iff zero_float_def simp del: powr_gt_zero dest: less_zeroE)

lemma Float_representation_aux:
  fixes m e
  defines [THEN meta_eq_to_obj_eq]: "x \<equiv> Float m e"
  assumes "x > 0"
  shows "Float (exponent x + (bitlen (mantissa x) - 1)) 0 = Float (e + (bitlen m - 1)) 0" (is ?th1)
    and "Float (mantissa x) (- (bitlen (mantissa x) - 1)) = Float m ( - (bitlen m - 1))"  (is ?th2)
proof -
  from assms have mantissa_pos: "m > 0" "mantissa x > 0"
    using Float_pos_eq_mantissa_pos[of m e] float_pos_eq_mantissa_pos[of x] by simp_all
  thus ?th1
    using bitlen_Float[of m e] assms
    by (auto simp add: zero_less_mult_iff intro!: arg_cong2[where f=Float])
  have "x \<noteq> 0"
    unfolding zero_float_def[symmetric] using \<open>0 < x\<close> by auto
  from denormalize_shift[OF x_def this] obtain i where
    i: "m = mantissa x * 2 ^ i" "e = exponent x - int i" .

  have "2 powr (1 - (real_of_int (bitlen (mantissa x)) + real_of_int i)) =
    2 powr (1 - (real_of_int (bitlen (mantissa x)))) * inverse (2 powr (real i))"
    by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps)
  hence "real_of_int (mantissa x) * 2 powr (1 - real_of_int (bitlen (mantissa x))) =
    (real_of_int (mantissa x) * 2 ^ i) * 2 powr (1 - real_of_int (bitlen (mantissa x * 2 ^ i)))"
    using \<open>mantissa x > 0\<close> by (simp add: powr_realpow)
  then show ?th2
    unfolding i
    by (auto simp: real_of_float_eq)
qed

lemma compute_ln[code]:
  fixes m e
  defines "x \<equiv> Float m e"
  shows "ub_ln prec x = (if x \<le> 0          then None
              else if x < 1          then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
            else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
                 if x \<le> Float 3 (- 1) then Some (horner (x - 1))
            else if x < Float 1 1  then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
                                   else let l = bitlen m - 1 in
                                        Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
    (is ?th1)
  and "lb_ln prec x = (if x \<le> 0          then None
            else if x < 1          then Some (- the (ub_ln prec (float_divr prec 1 x)))
            else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
                 if x \<le> Float 3 (- 1) then Some (horner (x - 1))
            else if x < Float 1 1  then Some (float_round_down prec (horner (Float 1 (- 1)) +
                                              horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
                                   else let l = bitlen m - 1 in
                                        Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
    (is ?th2)
proof -
  from assms Float_pos_eq_mantissa_pos have "x > 0 \<Longrightarrow> m > 0"
    by simp
  thus ?th1 ?th2
    using Float_representation_aux[of m e]
    unfolding x_def[symmetric]
    by (auto dest: not_le_imp_less)
qed

lemma ln_shifted_float:
  assumes "0 < m"
  shows "ln (Float m e) = ln 2 * (e + (bitlen m - 1)) + ln (Float m (- (bitlen m - 1)))"
proof -
  let ?B = "2^nat (bitlen m - 1)"
  define bl where "bl = bitlen m - 1"
  have "0 < real_of_int m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0"
    using assms by auto
  hence "0 \<le> bl" by (simp add: bitlen_alt_def bl_def)
  show ?thesis
  proof (cases "0 \<le> e")
    case True
    thus ?thesis
      unfolding bl_def[symmetric] using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
      apply (simp add: ln_mult)
      apply (cases "e=0")
        apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr)
        apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr field_simps)
      done
  next
    case False
    hence "0 < -e" by auto
    have lne: "ln (2 powr real_of_int e) = ln (inverse (2 powr - e))"
      by (simp add: powr_minus)
    hence pow_gt0: "(0::real) < 2^nat (-e)"
      by auto
    hence inv_gt0: "(0::real) < inverse (2^nat (-e))"
      by auto
    show ?thesis
      using False unfolding bl_def[symmetric]
      using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
      by (auto simp add: lne ln_mult ln_powr ln_div field_simps)
  qed
qed

lemma ub_ln_lb_ln_bounds':
  assumes "1 \<le> x"
  shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
    (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
proof (cases "x < Float 1 1")
  case True
  hence "real_of_float (x - 1) < 1" and "real_of_float x < 2" by auto
  have "\<not> x \<le> 0" and "\<not> x < 1" using \<open>1 \<le> x\<close> by auto
  hence "0 \<le> real_of_float (x - 1)" using \<open>1 \<le> x\<close> by auto

  have [simp]: "(Float 3 (- 1)) = 3 / 2" by simp

  show ?thesis
  proof (cases "x \<le> Float 3 (- 1)")
    case True
    show ?thesis
      unfolding lb_ln.simps
      unfolding ub_ln.simps Let_def
      using ln_float_bounds[OF \<open>0 \<le> real_of_float (x - 1)\<close> \<open>real_of_float (x - 1) < 1\<close>, of prec]
        \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True
      by (auto intro!: float_round_down_le float_round_up_le)
  next
    case False
    hence *: "3 / 2 < x" by auto

    with ln_add[of "3 / 2" "x - 3 / 2"]
    have add: "ln x = ln (3 / 2) + ln (real_of_float x * 2 / 3)"
      by (auto simp add: algebra_simps diff_divide_distrib)

    let "?ub_horner x" = "float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x)"
    let "?lb_horner x" = "float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x)"

    { have up: "real_of_float (rapprox_rat prec 2 3) \<le> 1"
        by (rule rapprox_rat_le1) simp_all
      have low: "2 / 3 \<le> rapprox_rat prec 2 3"
        by (rule order_trans[OF _ rapprox_rat]) simp
      from mult_less_le_imp_less[OF * low] *
      have pos: "0 < real_of_float (x * rapprox_rat prec 2 3 - 1)" by auto

      have "ln (real_of_float x * 2/3)
        \<le> ln (real_of_float (x * rapprox_rat prec 2 3 - 1) + 1)"
      proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
        show "real_of_float x * 2 / 3 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1) + 1"
          using * low by auto
        show "0 < real_of_float x * 2 / 3" using * by simp
        show "0 < real_of_float (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
      qed
      also have "\<dots> \<le> ?ub_horner (x * rapprox_rat prec 2 3 - 1)"
      proof (rule float_round_up_le, rule ln_float_bounds(2))
        from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] low *
        show "real_of_float (x * rapprox_rat prec 2 3 - 1) < 1" by auto
        show "0 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1)" using pos by auto
      qed
     finally have "ln x \<le> ?ub_horner (Float 1 (-1))
          + ?ub_horner ((x * rapprox_rat prec 2 3 - 1))"
        using ln_float_bounds(2)[of "Float 1 (- 1)" prec prec] add
        by (auto intro!: add_mono float_round_up_le)
      note float_round_up_le[OF this, of prec]
    }
    moreover
    { let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"

      have up: "lapprox_rat prec 2 3 \<le> 2/3"
        by (rule order_trans[OF lapprox_rat], simp)

      have low: "0 \<le> real_of_float (lapprox_rat prec 2 3)"
        using lapprox_rat_nonneg[of 2 3 prec] by simp

      have "?lb_horner ?max
        \<le> ln (real_of_float ?max + 1)"
      proof (rule float_round_down_le, rule ln_float_bounds(1))
        from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] * low
        show "real_of_float ?max < 1" by (cases "real_of_float (lapprox_rat prec 2 3) = 0",
          auto simp add: real_of_float_max)
        show "0 \<le> real_of_float ?max" by (auto simp add: real_of_float_max)
      qed
      also have "\<dots> \<le> ln (real_of_float x * 2/3)"
      proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
        show "0 < real_of_float ?max + 1" by (auto simp add: real_of_float_max)
        show "0 < real_of_float x * 2/3" using * by auto
        show "real_of_float ?max + 1 \<le> real_of_float x * 2/3" using * up
          by (cases "0 < real_of_float x * real_of_float (lapprox_posrat prec 2 3) - 1",
              auto simp add: max_def)
      qed
      finally have "?lb_horner (Float 1 (- 1)) + ?lb_horner ?max \<le> ln x"
        using ln_float_bounds(1)[of "Float 1 (- 1)" prec prec] add
        by (auto intro!: add_mono float_round_down_le)
      note float_round_down_le[OF this, of prec]
    }
    ultimately
    show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
      using \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True False by auto
  qed
next
  case False
  hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 (- 1)"
    using \<open>1 \<le> x\<close> by auto
  show ?thesis
  proof -
    define m where "m = mantissa x"
    define e where "e = exponent x"
    from Float_mantissa_exponent[of x] have Float: "x = Float m e"
      by (simp add: m_def e_def)
    let ?s = "Float (e + (bitlen m - 1)) 0"
    let ?x = "Float m (- (bitlen m - 1))"

    have "0 < m" and "m \<noteq> 0" using \<open>0 < x\<close> Float powr_gt_zero[of 2 e]
      by (auto simp add: zero_less_mult_iff)
    define bl where "bl = bitlen m - 1"
    then have bitlen: "bitlen m = bl + 1"
      by simp
    hence "bl \<ge> 0"
      using \<open>m > 0\<close> by (auto simp add: bitlen_alt_def)
    have "1 \<le> Float m e"
      using \<open>1 \<le> x\<close> Float unfolding less_eq_float_def by auto
    from bitlen_div[OF \<open>0 < m\<close>] float_gt1_scale[OF \<open>1 \<le> Float m e\<close>] \<open>bl \<ge> 0\<close>
    have x_bnds: "0 \<le> real_of_float (?x - 1)" "real_of_float (?x - 1) < 1"
      using abs_real_le_2_powr_bitlen [of m] \<open>m > 0\<close>
      by (simp_all add: bitlen powr_realpow [symmetric] powr_minus powr_add field_simps)
    {
      have "float_round_down prec (lb_ln2 prec * ?s) \<le> ln 2 * (e + (bitlen m - 1))"
          (is "real_of_float ?lb2 \<le> _")
        apply (rule float_round_down_le)
        unfolding nat_0 power_0 mult_1_right times_float.rep_eq
        using lb_ln2[of prec]
      proof (rule mult_mono)
        from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
        show "0 \<le> real_of_float (Float (e + (bitlen m - 1)) 0)" by simp
      qed auto
      moreover
      from ln_float_bounds(1)[OF x_bnds]
      have "float_round_down prec ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln ?x" (is "real_of_float ?lb_horner \<le> _")
        by (auto intro!: float_round_down_le)
      ultimately have "float_plus_down prec ?lb2 ?lb_horner \<le> ln x"
        unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e] by (auto intro!: float_plus_down_le)
    }
    moreover
    {
      from ln_float_bounds(2)[OF x_bnds]
      have "ln ?x \<le> float_round_up prec ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))"
          (is "_ \<le> real_of_float ?ub_horner")
        by (auto intro!: float_round_up_le)
      moreover
      have "ln 2 * (e + (bitlen m - 1)) \<le> float_round_up prec (ub_ln2 prec * ?s)"
          (is "_ \<le> real_of_float ?ub2")
        apply (rule float_round_up_le)
        unfolding nat_0 power_0 mult_1_right times_float.rep_eq
        using ub_ln2[of prec]
      proof (rule mult_mono)
        from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
        show "0 \<le> real_of_int (e + (bitlen m - 1))" by auto
        have "0 \<le> ln (2 :: real)" by simp
        thus "0 \<le> real_of_float (ub_ln2 prec)" using ub_ln2[of prec] by arith
      qed auto
      ultimately have "ln x \<le> float_plus_up prec ?ub2 ?ub_horner"
        unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e]
        by (auto intro!: float_plus_up_le)
    }
    ultimately show ?thesis
      unfolding lb_ln.simps
      unfolding ub_ln.simps
      unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_not_P[OF \<open>\<not> x < 1\<close>]
        if_not_P[OF False] if_not_P[OF \<open>\<not> x \<le> Float 3 (- 1)\<close>] Let_def
      unfolding plus_float.rep_eq e_def[symmetric] m_def[symmetric]
      by simp
  qed
qed

lemma ub_ln_lb_ln_bounds:
  assumes "0 < x"
  shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
    (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
proof (cases "x < 1")
  case False
  hence "1 \<le> x"
    unfolding less_float_def less_eq_float_def by auto
  show ?thesis
    using ub_ln_lb_ln_bounds'[OF \<open>1 \<le> x\<close>] .
next
  case True
  have "\<not> x \<le> 0" using \<open>0 < x\<close> by auto
  from True have "real_of_float x \<le> 1" "x \<le> 1"
    by simp_all
  have "0 < real_of_float x" and "real_of_float x \<noteq> 0"
    using \<open>0 < x\<close> by auto
  hence A: "0 < 1 / real_of_float x" by auto

  {
    let ?divl = "float_divl (max prec 1) 1 x"
    have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] by auto
    hence B: "0 < real_of_float ?divl" by auto

    have "ln ?divl \<le> ln (1 / x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
    hence "ln x \<le> - ln ?divl" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
    from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
    have "?ln \<le> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq by (rule order_trans)
  } moreover
  {
    let ?divr = "float_divr prec 1 x"
    have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close> \<open>x \<le> 1\<close>] unfolding less_eq_float_def less_float_def by auto
    hence B: "0 < real_of_float ?divr" by auto

    have "ln (1 / x) \<le> ln ?divr" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
    hence "- ln ?divr \<le> ln x" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
    from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
    have "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq by (rule order_trans)
  }
  ultimately show ?thesis unfolding lb_ln.simps[where x=x]  ub_ln.simps[where x=x]
    unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_P[OF True] by auto
qed

lemma lb_ln:
  assumes "Some y = lb_ln prec x"
  shows "y \<le> ln x" and "0 < real_of_float x"
proof -
  have "0 < x"
  proof (rule ccontr)
    assume "\<not> 0 < x"
    hence "x \<le> 0"
      unfolding less_eq_float_def less_float_def by auto
    thus False
      using assms by auto
  qed
  thus "0 < real_of_float x" by auto
  have "the (lb_ln prec x) \<le> ln x"
    using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
  thus "y \<le> ln x"
    unfolding assms[symmetric] by auto
qed

lemma ub_ln:
  assumes "Some y = ub_ln prec x"
  shows "ln x \<le> y" and "0 < real_of_float x"
proof -
  have "0 < x"
  proof (rule ccontr)
    assume "\<not> 0 < x"
    hence "x \<le> 0" by auto
    thus False
      using assms by auto
  qed
  thus "0 < real_of_float x" by auto
  have "ln x \<le> the (ub_ln prec x)"
    using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
  thus "ln x \<le> y"
    unfolding assms[symmetric] by auto
qed

lemma bnds_ln: "\<forall>(x::real) lx ux. (Some l, Some u) =
  (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> ln x \<and> ln x \<le> u"
proof (rule allI, rule allI, rule allI, rule impI)
  fix x :: real
  fix lx ux
  assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux}"
  hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}"
    by auto

  have "ln ux \<le> u" and "0 < real_of_float ux"
    using ub_ln u by auto
  have "l \<le> ln lx" and "0 < real_of_float lx" and "0 < x"
    using lb_ln[OF l] x by auto

  from ln_le_cancel_iff[OF \<open>0 < real_of_float lx\<close> \<open>0 < x\<close>] \<open>l \<le> ln lx\<close>
  have "l \<le> ln x"
    using x unfolding atLeastAtMost_iff by auto
  moreover
  from ln_le_cancel_iff[OF \<open>0 < x\<close> \<open>0 < real_of_float ux\<close>] \<open>ln ux \<le> real_of_float u\<close>
  have "ln x \<le> u"
    using x unfolding atLeastAtMost_iff by auto
  ultimately show "l \<le> ln x \<and> ln x \<le> u" ..
qed


section \<open>Real power function\<close>

definition bnds_powr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float \<times> float) option" where
  "bnds_powr prec l1 u1 l2 u2 = (
     if l1 = 0 \<and> u1 = 0 then
       Some (0, 0)
     else if l1 = 0 \<and> l2 \<ge> 1 then
       let uln = the (ub_ln prec u1)
       in  Some (0, ub_exp prec (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2))))
     else if l1 \<le> 0 then
       None
     else
       Some (map_bnds lb_exp ub_exp prec 
               (bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2)))"

lemmas [simp del] = lb_exp.simps ub_exp.simps

lemma mono_exp_real: "mono (exp :: real \<Rightarrow> real)"
  by (auto simp: mono_def)

lemma ub_exp_nonneg: "real_of_float (ub_exp prec x) \<ge> 0"
proof -
  have "0 \<le> exp (real_of_float x)" by simp
  also from exp_boundaries[of x prec] 
    have "\<dots> \<le> real_of_float (ub_exp prec x)" by simp
  finally show ?thesis .
qed

lemma bnds_powr:
  assumes lu: "Some (l, u) = bnds_powr prec l1 u1 l2 u2"
  assumes x: "x \<in> {real_of_float l1..real_of_float u1}"
  assumes y: "y \<in> {real_of_float l2..real_of_float u2}"
  shows   "x powr y \<in> {real_of_float l..real_of_float u}"
proof -
  consider "l1 = 0" "u1 = 0" | "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1" | 
           "l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))" | "l1 > 0" by force
  thus ?thesis
  proof cases
    assume "l1 = 0" "u1 = 0"
    with x lu show ?thesis by (auto simp: bnds_powr_def)
  next
    assume A: "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1"
    define uln where "uln = the (ub_ln prec u1)"
    show ?thesis
    proof (cases "x = 0")
      case False
      with A x y have "x powr y = exp (ln x * y)" by (simp add: powr_def)
      also {
        from A x False have "ln x \<le> ln (real_of_float u1)" by simp
        also from ub_ln_lb_ln_bounds[of u1 prec] A y x False
          have "ln (real_of_float u1) \<le> real_of_float uln" by (simp add: uln_def del: lb_ln.simps)
        also from A x y have "\<dots> * y \<le> real_of_float uln * (if uln \<ge> 0 then u2 else l2)"
          by (auto intro: mult_left_mono mult_left_mono_neg)
        also have "\<dots> \<le> real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))"
          by (simp add: float_round_up_le)
        finally have "ln x * y \<le> \<dots>" using A y by - simp
      }
      also have "exp (real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))) \<le>
                   real_of_float (ub_exp prec (float_round_up prec
                       (uln * (if uln \<ge> 0 then u2 else l2))))"
        using exp_boundaries by simp
      finally show ?thesis using A x y lu 
        by (simp add: bnds_powr_def uln_def Let_def del: lb_ln.simps ub_ln.simps)
    qed (insert x y lu A, simp_all add: bnds_powr_def Let_def ub_exp_nonneg
                                   del: lb_ln.simps ub_ln.simps)
  next
    assume "l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))"
    with lu show ?thesis by (simp add: bnds_powr_def split: if_split_asm)
  next
    assume l1: "l1 > 0"
    obtain lm um where lmum:
      "(lm, um) = bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2"
      by (cases "bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2") simp
    with l1 have "(l, u) = map_bnds lb_exp ub_exp prec (lm, um)"
      using lu by (simp add: bnds_powr_def del: lb_ln.simps ub_ln.simps split: if_split_asm)
    hence "exp (ln x * y) \<in> {real_of_float l..real_of_float u}"
    proof (rule map_bnds[OF _ mono_exp_real], goal_cases)
      case 1
      let ?lln = "the (lb_ln prec l1)" and ?uln = "the (ub_ln prec u1)"
      from ub_ln_lb_ln_bounds[of l1 prec] ub_ln_lb_ln_bounds[of u1 prec] x l1
        have "real_of_float ?lln \<le> ln (real_of_float l1) \<and> 
              ln (real_of_float u1) \<le> real_of_float ?uln"
        by (auto simp del: lb_ln.simps ub_ln.simps)
      moreover from l1 x have "ln (real_of_float l1) \<le> ln x \<and> ln x \<le> ln (real_of_float u1)"
        by auto
      ultimately have ln: "real_of_float ?lln \<le> ln x \<and> ln x \<le> real_of_float ?uln" by simp
      from lmum show ?case
        by (rule bnds_mult) (insert y ln, simp_all)
    qed (insert exp_boundaries[of lm prec] exp_boundaries[of um prec], simp_all)
    with x l1 show ?thesis
      by (simp add: powr_def mult_ac)
  qed
qed

end