src/HOL/Decision_Procs/Approximation.thy

Basic support for the SMT prover veriT.

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

header {* Prove Real Valued Inequalities by Computation *}

theory Approximation
imports
  Complex_Main
  "~~/src/HOL/Library/Float"
  Dense_Linear_Order
  "~~/src/HOL/Library/Code_Target_Numeral"
keywords "approximate" :: diag
begin

declare powr_one [simp]
declare powr_numeral [simp]
declare powr_neg_one [simp]
declare powr_neg_numeral [simp]

section "Horner Scheme"

subsection {* Define auxiliary helper @{text horner} function *}

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 setsum_right_distrib shift_pow uminus_add_conv_diff [symmetric] setsum_negf[symmetric]
    setsum_head_upt_Suc[OF zero_less_Suc]
    setsum.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 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 = lapprox_rat prec 1 k - 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 = rapprox_rat prec 1 k - 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'"] `0 \<le> real x`
    by (auto intro!: add_mono mult_left_mono simp add: lb_Suc ub_Suc field_simps f_Suc)
qed

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

text {*

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

*}

lemma horner_bounds:
  fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
  assumes "0 \<le> real 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 = lapprox_rat prec 1 k - 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 = rapprox_rat prec 1 k - 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 `0 \<le> real x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
    unfolding horner_schema[where f=f, OF f_Suc] .
  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 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 = lapprox_rat prec 1 k + 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 = rapprox_rat prec 1 k + 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 x ^ j)" (is "?lb") and
    "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
proof -
  { fix x y z :: float have "x - y * z = x + - y * z" by simp } note diff_mult_minus = this
  have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) =
    (\<Sum>j = 0..<n. -1 ^ j * (1 / (f (j' + j))) * real (- x) ^ j)"
    by (auto simp add: field_simps power_mult_distrib[symmetric])
  have "0 \<le> real (-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 refl ub_0 refl]
  show "?lb" and "?ub" unfolding minus_minus sum_eq
    by auto
qed

subsection {* Selectors for next even or odd number *}

text {*

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 get_odd} and @{term get_even}.

*}

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[unfolded even_mult_two_ex] .
lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto

section "Power function"

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

lemma float_power_bnds: "(l1, u1) = float_power_bnds n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}"
  by (auto simp: float_power_bnds_def max_def split: split_if_asm
           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 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 "Square root"

text {*

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.

*}

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 * (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 * (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 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 powr_gt_zero[of 2 e] 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 "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2]
        unfolding real_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 `0 \<le> ?E mod 2` this]
        show ?thesis by auto
      qed
      hence "sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)"
        by (auto simp del: real_sqrt_four)
      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 (simp add: int_of_reals del: real_of_ints)
      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 `0 < m`
      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 `0 < real x` by auto
  also have "\<dots> < real ?b" using Suc .
  finally have "sqrt x < (?b + x / ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < real x`] 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
  finally show ?case unfolding sqrt_iteration.simps Let_def distrib_left .
qed

lemma sqrt_iteration_lower_bound: assumes "0 < real x"
  shows "0 < real (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 x"
  shows "0 \<le> real (lb_sqrt prec x)"
proof (cases "0 < x")
  case True hence "0 < real x" and "0 \<le> x" using `0 \<le> real x` by auto
  hence "0 < sqrt_iteration prec prec x" using sqrt_iteration_lower_bound by auto
  hence "0 \<le> real (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \<le> x`] unfolding less_eq_float_def by auto
  thus ?thesis unfolding lb_sqrt.simps using True by auto
next
  case False with `0 \<le> real x` have "real 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 -
  { fix x :: float assume "0 < x"
    hence "0 < real x" and "0 \<le> real 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 `0 < real x`
               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 `0 \<le> real x`, symmetric] by auto
    finally have "lb_sqrt prec x \<le> sqrt x"
      unfolding lb_sqrt.simps if_P[OF `0 < x`] by auto }
  note lb = this

  { fix x :: float assume "0 < x"
    hence "0 < real x" by auto
    hence "0 < sqrt x" by auto
    hence "sqrt x < sqrt_iteration prec prec x"
      using sqrt_iteration_bound by auto
    hence "sqrt x \<le> ub_sqrt prec x"
      unfolding ub_sqrt.simps if_P[OF `0 < x`] by auto }
  note ub = this

  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 {*

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

*}

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 =
    (rapprox_rat prec 1 k) - 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 =
    (lapprox_rat prec 1 k) - x * (ub_arctan_horner prec n (k + 2) x)"

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

  have "0 \<le> real (x * x)" by auto
  from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto

  have "arctan x \<in> { ?S n .. ?S (Suc n) }"
  proof (cases "real x = 0")
    case False
    hence "0 < real x" using `0 \<le> real x` by auto
    hence prem: "0 < 1 / (0 * 2 + (1::nat)) * real x ^ (0 * 2 + 1)" by auto

    have "\<bar> real x \<bar> \<le> 1"  using `0 \<le> real x` `real x \<le> 1` 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 `2 * m = n`]
    show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_eq_plus1 atLeast0LessThan .
  qed auto
  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 `0 \<le> real (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]

  { have "(x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
      using bounds(1) `0 \<le> real x`
      unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
      unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
      by (auto intro!: mult_left_mono)
    also have "\<dots> \<le> arctan x" using arctan_bounds ..
    finally have "(x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan x" . }
  moreover
  { have "arctan x \<le> ?S (Suc n)" using arctan_bounds ..
    also have "\<dots> \<le> (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
      using bounds(2)[of "Suc n"] `0 \<le> real x`
      unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
      unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
      by (auto intro!: mult_left_mono)
    finally have "arctan x \<le> (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
  ultimately show ?thesis by auto
qed

lemma arctan_0_1_bounds: assumes "0 \<le> real x" "real x \<le> 1"
  shows "arctan x \<in> {(x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
  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)

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 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) -
                                                  B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (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 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) -
                                                  B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (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"
    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto

    have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
    have "real ?k \<le> 1" 
      by (rule rapprox_rat_le1, auto simp add: `0 < k` `1 \<le> k`)

    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 (?k * ?k))"
      using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
    finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k)" .
  } 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"
    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
    have "1 / k \<le> 1" using `1 < k` by auto
    have "\<And>n. 0 \<le> real ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`)
    have "\<And>n. real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / k \<le> 1`)

    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 (?k * ?k) \<le> arctan ?k"
      using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
    also have "\<dots> \<le> arctan (1 / k)" using `?k \<le> 1 / k` by (rule arctan_monotone')
    finally have "?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k) \<le> arctan (1 / k)" .
  } note lb_arctan = this

  have "pi \<le> ub_pi n \<and> lb_pi n \<le> pi"
    unfolding lb_pi_def ub_pi_def machin_pi Let_def unfolding Float_num
    using lb_arctan[of 5] ub_arctan[of 239] lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
    by (auto intro!: mult_left_mono add_mono simp add: uminus_add_conv_diff [symmetric] simp del: uminus_add_conv_diff)
  then 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. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
                           lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (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 (1 + ub_sqrt prec (1 + x * x)))
                          else (let inv = float_divr prec 1 x
                                in if inv > 1 then 0
                                              else lb_pi prec * Float 1 -1 - ub_horner inv)))"

| "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
                           ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (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 (1 + lb_sqrt prec (1 + x * x))
                               in if y > 1 then ub_pi prec * Float 1 -1
                                           else Float 1 1 * ub_horner y
                          else 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 x"
  shows "lb_arctan prec x \<le> arctan x"
proof -
  have "\<not> x < 0" and "0 \<le> x" using `0 \<le> real x` by auto
  let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
    and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"

  show ?thesis
  proof (cases "x \<le> Float 1 -1")
    case True hence "real x \<le> 1" by auto
    show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
      using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
  next
    case False hence "0 < real x" by auto
    let ?R = "1 + sqrt (1 + real x * real x)"
    let ?fR = "1 + ub_sqrt prec (1 + x * x)"
    let ?DIV = "float_divl prec x ?fR"

    have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)

    have "sqrt (1 + x * x) \<le> ub_sqrt prec (1 + x * x)"
      using bnds_sqrt'[of "1 + x * x"] by auto

    hence "?R \<le> ?fR" by auto
    hence "0 < ?fR" and "0 < real ?fR" using `0 < ?R` by auto

    have monotone: "(float_divl prec x ?fR) \<le> x / ?R"
    proof -
      have "?DIV \<le> real x / ?fR" by (rule float_divl)
      also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF `?R \<le> ?fR` `0 \<le> real x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> real ?fR`] 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 have "\<dots> \<le> (ub_sqrt prec (1 + x * x))"
        using bnds_sqrt'[of "1 + x * x"] by auto
      finally have "real x \<le> ?fR" by auto
      moreover have "?DIV \<le> real x / ?fR" by (rule float_divl)
      ultimately have "real ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < real ?fR`, symmetric] by auto

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

      have "(Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (float_divl prec x ?fR)"
        using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
      also have "\<dots> \<le> 2 * arctan (x / ?R)"
        using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
      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 `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] .
    next
      case False
      hence "2 < real x" by auto
      hence "1 \<le> real x" by auto

      let "?invx" = "float_divr prec 1 x"
      have "0 \<le> arctan x" using arctan_monotone'[OF `0 \<le> real x`] 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 `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] if_P[OF True]
          using `0 \<le> arctan x` by auto
      next
        case False
        hence "real ?invx \<le> 1" by auto
        have "0 \<le> real ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> real x`)

        have "1 / x \<noteq> 0" and "0 < 1 / x" using `0 < real x` 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[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
        finally have "pi / 2 - (?ub_horner ?invx) \<le> arctan x"
          using `0 \<le> arctan x` arctan_inverse[OF `1 / x \<noteq> 0`]
          unfolding real_sgn_pos[OF `0 < 1 / real x`] 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 `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False]
          by auto
      qed
    qed
  qed
qed

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

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

  show ?thesis
  proof (cases "x \<le> Float 1 -1")
    case True hence "real x \<le> 1" by auto
    show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
      using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
  next
    case False hence "0 < real x" by auto
    let ?R = "1 + sqrt (1 + real x * real x)"
    let ?fR = "1 + lb_sqrt prec (1 + x * x)"
    let ?DIV = "float_divr prec x ?fR"

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

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

    have "lb_sqrt prec (1 + x * x) \<le> sqrt (1 + x * x)"
      using bnds_sqrt'[of "1 + x * x"] by auto
    hence "?fR \<le> ?R" by auto
    have "0 < real ?fR" by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> real (1 + x*x)`])

    have monotone: "x / ?R \<le> (float_divr prec x ?fR)"
    proof -
      from divide_left_mono[OF `?fR \<le> ?R` `0 \<le> real x` mult_pos_pos[OF divisor_gt0 `0 < real ?fR`]]
      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 `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_P[OF True] .
      next
        case False
        hence "real ?DIV \<le> 1" by auto

        have "0 \<le> x / ?R" using `0 \<le> real x` `0 < ?R` unfolding zero_le_divide_iff by auto
        hence "0 \<le> real ?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[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
        finally show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_not_P[OF False] .
      qed
    next
      case False
      hence "2 < real x" by auto
      hence "1 \<le> real x" by auto
      hence "0 < real x" by auto
      hence "0 < x" by auto

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

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

      have "1 / x \<noteq> 0" and "0 < 1 / x" using `0 < real x` by auto

      have "(?lb_horner ?invx) \<le> arctan (?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
      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 `0 \<le> arctan x` arctan_inverse[OF `1 / x \<noteq> 0`]
        unfolding real_sgn_pos[OF `0 < 1 / x`] 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 `\<not> x < 0`]if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False]
        by auto
    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 x" by auto
  show ?thesis using ub_arctan_bound'[OF `0 \<le> real x`] lb_arctan_bound'[OF `0 \<le> real x`] unfolding atLeastAtMost_iff by auto
next
  let ?mx = "-x"
  case False hence "x < 0" and "0 \<le> real ?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 `0 \<le> real ?mx`] lb_arctan_bound'[OF `0 \<le> real ?mx`] 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 `x < 0`]
    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

  { 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 have "l \<le> arctan x" .
  } moreover
  { 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 have "arctan x \<le> u" .
  } ultimately show "l \<le> arctan x \<and> arctan x \<le> u" ..
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 =
    (rapprox_rat prec 1 k) - 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 =
    (lapprox_rat prec 1 k) - 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/real (fact (2 * i))) * x ^(2 * i))" (is "?lb")
  and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
proof -
  have "0 \<le> real (x * x)" by auto
  let "?f n" = "fact (2 * n)"

  { fix n
    have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)"
      unfolding F by auto } note f_eq = this

  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
    OF `0 \<le> real (x * x)` 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 x"])
qed

lemma cos_boundaries: assumes "0 \<le> real 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 x = 0")
  case False hence "real x \<noteq> 0" by auto
  hence "0 < x" and "0 < real x" using `0 \<le> real x` by auto
  have "0 < x * x" using `0 < x` by simp

  { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x ^ (2 * i))
    = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
  proof -
    have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
    also have "\<dots> =
      (\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (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) / (real (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) / (real (fact i)) else 0) * x ^ i)"
      by (rule setsum.cong) auto
    finally show ?thesis by assumption
  qed } note morph_to_if_power = this


  { fix n :: nat assume "0 < n"
    hence "0 < 2 * n" by auto
    obtain t where "0 < t" and "t < real x" and
      cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (real x) ^ i)
      + (cos (t + 1/2 * (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
      (is "_ = ?SUM + ?rest / ?fact * ?pow")
      using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`]
      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)"  using cos_add by auto
    also have "\<dots> = ?rest" by auto
    finally have "cos t * -1^n = ?rest" .
    moreover
    have "t \<le> pi / 2" using `t < real x` and `x \<le> pi / 2` by auto
    hence "0 \<le> cos t" using `0 < t` 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 `0 < real x` 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 `even n`] `0 < ?fact` `0 < ?pow`
        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 `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
        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 _ `0 < real x`[THEN less_imp_le]], auto)
    with `x \<le> pi / 2`
    show ?thesis unfolding `get_even n = 0` 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
  show ?thesis
  proof (cases "n = 0")
    case True
    thus ?thesis unfolding `n = 0` get_even_def get_odd_def
      using `real x = 0` lapprox_rat[where x="-1" and y=1]
      by (auto simp: Float.compute_lapprox_rat Float.compute_rapprox_rat)
  next
    case False with not0_implies_Suc obtain m where "n = Suc m" by blast
    thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
  qed
qed

lemma sin_aux: assumes "0 \<le> real x"
  shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb")
  and "(\<Sum> i=0..<n. -1^i * (1/real (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 (x * x)" by auto
  let "?f n" = "fact (2 * n + 1)"

  { fix n
    have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)"
      unfolding F by auto } note f_eq = this

  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
    OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
  show "?lb" and "?ub" using `0 \<le> real x`
    unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
    unfolding mult.commute[where 'a=real]
    by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
qed

lemma sin_boundaries: assumes "0 \<le> real 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 x = 0")
  case False hence "real x \<noteq> 0" by auto
  hence "0 < x" and "0 < real x" using `0 \<le> real x` by auto
  have "0 < x * x" using `0 < x` by simp

  { fix x n have "(\<Sum> j = 0 ..< n. -1 ^ (((2 * j + 1) - Suc 0) div 2) / (real (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))/(real (fact i))) * x ^ i)" (is "?SUM = _")
    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) / (real (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) / (real (fact i))) * x ^ i)"
        by (rule setsum.cong) auto
      finally show ?thesis by assumption
    qed } note setsum_morph = this

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

    have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add distrib_right distrib_left by auto
    moreover
    have "t \<le> pi / 2" using `t < real x` and `x \<le> pi / 2` by auto
    hence "0 \<le> cos t" using `0 < t` 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 `0 < real x` 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))/(real (fact i))) * (real x) ^ i)"
        using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
      also have "\<dots> \<le> ?SUM" by auto
      also have "\<dots> \<le> sin x"
      proof -
        from even[OF `even n`] `0 < ?fact` `0 < ?pow`
        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 `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
        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))/(real (fact i))) * (real x) ^ i)"
         by auto
      also have "\<dots> \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))"
        using sin_aux[OF `0 \<le> real x`] unfolding setsum_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 `x \<le> pi / 2` `0 \<le> real x`
    show ?thesis unfolding `get_even n = 0` 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 `n = 0` get_even_def get_odd_def using `real x = 0` 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 `n = Suc m` get_even_def get_odd_def using `real x = 0` 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 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 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 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 -
  { fix x :: real
    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 have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
  } note x_half = this[symmetric]

  have "\<not> x < 0" using `0 \<le> real x` 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 1 1 * x * x - 1"
  let "?lb_half x" = "if x < 0 then - 1 else 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 `\<not> x < 0`] if_P[OF `x < Float 1 -1`] Let_def
      using cos_boundaries[OF `0 \<le> real x` `x \<le> pi / 2`] .
  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 y" by auto
        from mult_mono[OF `y \<le> cos ?x2` `y \<le> cos ?x2` `0 \<le> cos ?x2` this]
        have "real y * real y \<le> cos ?x2 * cos ?x2" .
        hence "2 * real y * real y \<le> 2 * cos ?x2 * cos ?x2" by auto
        hence "2 * real y * real 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
      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 y" using `0 \<le> cos ?x2` ub by (rule order_trans)
        from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
        have "cos ?x2 * cos ?x2 \<le> real y * real y" .
        hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y" by auto
        hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real y * real y - 1" unfolding Float_num by auto
        thus ?thesis unfolding x_half Float_num by auto
      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] `0 \<le> real x` by (rule order_trans)

    show ?thesis
    proof (cases "x < 1")
      case True hence "real x \<le> 1" by auto
      have "0 \<le> real ?x2" and "?x2 \<le> pi / 2" using pi_ge_two `0 \<le> real x` 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 `-pi \<le> x` `x \<le> pi`]
        show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
      qed
      moreover have "?cos x \<le> (?ub x)"
      proof -
        from ub_half[OF ub `-pi \<le> x` `x \<le> pi`]
        show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
      qed
      ultimately show ?thesis by auto
    next
      case False
      have "0 \<le> real ?x4" and "?x4 \<le> pi / 2" using pi_ge_two `0 \<le> real x` `x \<le> pi` 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 transfer simp

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

lemma lb_cos_minus: assumes "-pi \<le> x" and "real x \<le> 0"
  shows "cos (real(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
proof -
  have "0 \<le> real (-x)" and "(-x) \<le> pi" using `-pi \<le> x` `real x \<le> 0` 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 = lx - k * 2 * (if k < 0 then lpi else upi) ;
    ux = 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 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 (Suc n)
  have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi"
    unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
  show ?case unfolding split_pi_off using Suc by auto
qed auto

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 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 ?lx = "lx - ?k * 2 * (if ?k < 0 then ?lpi else ?upi)"
  let ?ux = "ux - ?k * 2 * (if ?k < 0 then ?upi else ?lpi)"

  obtain k :: int where k: "k = real ?k" using 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 \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux"
    using x unfolding k[symmetric]
    by (cases "k = 0")
       (auto intro!: add_mono
                simp add: k [symmetric] uminus_add_conv_diff [symmetric]
                simp del: float_of_numeral uminus_add_conv_diff)
  note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
  hence lx_less_ux: "?lx \<le> real ?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 ?lx \<le> 0"
      by simp_all

    have "(lb_cos prec (- ?lx)) \<le> cos (real (- ?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 real_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 ?lx"
      by auto

    have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx"
      using cos_monotone_0_pi'[OF lx_0 lx pi_x]
      by (simp only: real_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 ?ux \<le> 0"
      by simp_all

    have "cos (x + (-k) * (2 * pi)) \<le> cos (real (- ?ux))"
      using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
      by (simp only: uminus_float.rep_eq real_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 ?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'[OF x_ge_0 ux pi_ux]
      by (simp only: real_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 False note 1 = this 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 False note 2 = this show ?thesis
  proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
    case True note Cond = this 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 False note 3 = this show ?thesis
  proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
    case True note Cond = this 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 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 (?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 real_of_int_minus real_of_one
          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 False note 4 = this show ?thesis
  proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
    case True note Cond = this 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 (?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'[OF lx_0 lx_le_x pi_x]
        by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
          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 = rapprox_rat prec 1 (int k) + 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 = lapprox_rat prec 1 (int k) + x * ub_exp_horner prec n (i + 1) (k * i) x"

lemma bnds_exp_horner: assumes "real 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 -
  { fix n
    have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" by (induct n, auto)
    have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^^ n) 1" unfolding F by auto } note f_eq = this

  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> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * real x ^ j)"
      using bounds(1) by auto
    also have "\<dots> \<le> exp x"
    proof -
      obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real x ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
        using Maclaurin_exp_le unfolding atLeast0LessThan by blast
      moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
        by (auto simp: zero_le_even_power)
      ultimately show ?thesis using get_odd exp_gt_zero by auto
    qed
    finally have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x" .
  } moreover
  {
    have x_less_zero: "real x ^ get_odd n \<le> 0"
    proof (cases "real 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 x < 0" using `real x \<le> 0` by auto
      show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq `real x < 0`)
    qed

    obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_odd n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n)"
      using Maclaurin_exp_le unfolding atLeast0LessThan by blast
    moreover have "exp t / real (fact (get_odd n)) * (real 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 / real (fact j) * real 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 have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x" .
  } ultimately show ?thesis by auto
qed

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 (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 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 1 (get_even 4) 1 1 (- 1)"
    unfolding get_even_def eq4
    by (auto simp add: Float.compute_lapprox_rat Float.compute_rapprox_rat
                  Float.compute_lapprox_posrat Float.compute_rapprox_posrat rat_precision_def Float.compute_bitlen)
  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: "\<And> x. 0 < ?horner x" unfolding Let_def by (cases "?lb_horner x \<le> 0", auto)
  moreover { fix x :: float fix num :: nat
    have "0 < real (?horner x) ^ num" using `0 < ?horner x` by simp
    also have "\<dots> = (?horner x) ^ num" by auto
    finally have "0 < real ((?horner x) ^ num)" .
  }
  ultimately show ?thesis
    unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
    by (cases "floor_fl x", cases "x < - 1", auto)
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 x \<le> 0" and "\<not> x > 0" using `x \<le> 0` by auto
  show ?thesis
  proof (cases "x < - 1")
    case False hence "- 1 \<le> real x" by auto
    show ?thesis
    proof (cases "?lb_exp_horner x \<le> 0")
      from `\<not> x < - 1` have "- 1 \<le> real 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 .
      moreover case True
      ultimately show ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
    next
      case False thus ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
    qed
  next
    case True

    let ?num = "nat (- int_floor_fl x)"

    have "real (int_floor_fl x) < - 1" using int_floor_fl[of x] `x < - 1`
      by simp
    hence "real (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 `0 < nat (- int_floor_fl x)` by auto
    have "0 < - int_floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] by simp
    hence "real (int_floor_fl x) < 0" unfolding less_float_def by auto
    have fl_eq: "real (- int_floor_fl x) = real (- floor_fl x)"
      by (simp add: floor_fl_def int_floor_fl_def)
    from `0 < - int_floor_fl x` have "0 < real (- floor_fl x)"
      by (simp add: floor_fl_def int_floor_fl_def)
    from `real (int_floor_fl x) < 0` have "real (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 (float_divr prec x (- floor_fl x)) \<le> 0"
        using float_divr_nonpos_pos_upper_bound[OF `real x \<le> 0` `0 < real (- floor_fl x)`]
        unfolding less_eq_float_def zero_float.rep_eq .

      have "exp x = exp (?num * (x / ?num))" using `real ?num \<noteq> 0` by auto
      also have "\<dots> = exp (x / ?num) ^ ?num" unfolding exp_real_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)
      finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] 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 ?horner" by auto

        have div_less_zero: "real (float_divl prec x (- floor_fl x)) \<le> 0"
          using `real (floor_fl x) < 0` `real x \<le> 0` 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 `0 \<le> real ?horner`[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_real_of_nat_mult ..
        also have "\<dots> = exp x" using `real ?num \<noteq> 0` by auto
        finally show ?thesis
          unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] int_floor_fl_def Let_def if_not_P[OF False] by auto
      next
        case True
        have "real (floor_fl x) \<noteq> 0" and "real (floor_fl x) \<le> 0" using `real (floor_fl x) < 0` by auto
        from divide_right_mono_neg[OF floor_fl[of x] `real (floor_fl x) \<le> 0`, unfolded divide_self[OF `real (floor_fl x) \<noteq> 0`]]
        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 (Float 1 -2) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
          by (auto intro!: power_mono)
        also have "\<dots> = exp x" unfolding num_eq fl_eq exp_real_of_nat_mult[symmetric] using `real (floor_fl x) \<noteq> 0` by auto
        finally show ?thesis
          unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] 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 `-x \<le> 0`]
      have ub_exp: "exp (- real 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 `0 < x` by auto

      from exp_boundaries'[OF `-x \<le> 0`]
      have lb_exp: "lb_exp prec (-x) \<le> exp (- real 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 `\<not> 0 < -x`, of prec]
        by (simp del: lb_exp.simps add: exp_minus inverse_eq_divide 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

  { 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 have "l \<le> exp x" .
  } moreover
  { 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 have "exp x \<le> u" .
  } ultimately show "l \<le> exp x \<and> exp x \<le> u" ..
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 = rapprox_rat prec 1 (int i) - 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 = lapprox_rat prec 1 (int i) - 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"] `0 \<le> x` `x < 1` by auto

  have "norm x < 1" using assms by auto
  have "?a ----> 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 `norm x < 1`]]] by auto
  { fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto simp: `0 \<le> x`) }
  { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
    proof (rule mult_mono)
      show "0 \<le> x ^ Suc (Suc n)" by (auto simp add: `0 \<le> x`)
      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 `x < 1`], auto simp: `0 \<le> x`)
      thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
    qed auto }
  from summable_Leibniz'(2,4)[OF `?a ----> 0` `\<And>n. 0 \<le> ?a n`, OF `\<And>n. ?a (Suc n) \<le> ?a n`, unfolded ln_eq]
  show "?lb" and "?ub" unfolding atLeast0LessThan by auto
qed

lemma ln_float_bounds:
  assumes "0 \<le> real x" and "real 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 x)^(Suc n)"

  have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding mult.commute[of "real 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 `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
    by (rule mult_right_mono)
  also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> real x` `real x < 1`] by auto
  finally show "?lb \<le> ?ln" .

  have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> real x` `real x < 1`] by auto
  also have "\<dots> \<le> ?ub" unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding mult.commute[of "real 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 `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
    by (rule mult_right_mono)
  finally show "?ln \<le> ?ub" .
qed

lemma ln_add: 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_divide_cancel_left[OF `x \<noteq> 0`] 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 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) +
                                           (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 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) +
                                           (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)"
    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 ?lthird < 1" by auto
  have lb3_lb: "0 \<le> real ?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 ?uthird" by auto

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

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

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

  show ?ub_ln2 unfolding ub_ln2_def Let_def plus_float.rep_eq ln2_sum Float_num(4)[symmetric]
  proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
    have "ln (1 / 3 + 1) \<le> ln (real ?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] .
    finally show "ln (1 / 3 + 1) \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird" .
  qed
  show ?lb_ln2 unfolding lb_ln2_def Let_def plus_float.rep_eq ln2_sum Float_num(4)[symmetric]
  proof (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 ?lthird + 1)"
      using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
    also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
    finally show "?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. 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 (horner (Float 1 -1) + horner (x * rapprox_rat prec 2 3 - 1))
                                   else let l = bitlen (mantissa x) - 1 in
                                        Some (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. 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 (horner (Float 1 -1) +
                                              horner (max (x * lapprox_rat prec 2 3 - 1) 0))
                                   else let l = bitlen (mantissa x) - 1 in
                                        Some (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 x \<le> 0" and "real x < 1" and "real (float_divl (max prec (Suc 0)) 1 x) < 1"
  hence "0 < real x" "1 \<le> max prec (Suc 0)" "real x < 1" by auto
  from float_divl_pos_less1_bound[OF `0 < real x` `real x < 1` `1 \<le> max prec (Suc 0)`]
  show False using `real (float_divl (max prec (Suc 0)) 1 x) < 1` by auto
next
  fix prec x assume "\<not> real x \<le> 0" and "real x < 1" and "real (float_divr prec 1 x) < 1"
  hence "0 < x" by auto
  from float_divr_pos_less1_lower_bound[OF `0 < x`, of prec] `real x < 1`
  show False using `real (float_divr prec 1 x) < 1` by auto
qed

lemma float_pos_eq_mantissa_pos:  "x > 0 \<longleftrightarrow> mantissa x > 0"
  apply (subst Float_mantissa_exponent[of x, symmetric])
  apply (auto simp add: zero_less_mult_iff zero_float_def powr_gt_zero[of 2 "exponent x"] dest: less_zeroE)
  using powr_gt_zero[of 2 "exponent x"]
  apply simp
  done

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 "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> float_of 0"
    unfolding zero_float_def[symmetric] using `0 < x` by auto
  from denormalize_shift[OF assms(1) this] guess i . note i = this

  have "2 powr (1 - (real (bitlen (mantissa x)) + real i)) =
    2 powr (1 - (real (bitlen (mantissa x)))) * inverse (2 powr (real i))"
    by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps)
  hence "real (mantissa x) * 2 powr (1 - real (bitlen (mantissa x))) =
    (real (mantissa x) * 2 ^ i) * 2 powr (1 - real (bitlen (mantissa x * 2 ^ i)))"
    using `mantissa x > 0` by (simp add: powr_realpow)
  then show ?th2
    unfolding i by transfer auto
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. 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 (horner (Float 1 -1) + horner (x * rapprox_rat prec 2 3 - 1))
                                   else let l = bitlen m - 1 in
                                        Some (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. 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 (horner (Float 1 -1) +
                                              horner (max (x * lapprox_rat prec 2 3 - 1) 0))
                                   else let l = bitlen m - 1 in
                                        Some (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_leE)
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)"
  def bl \<equiv> "bitlen m - 1"
  have "0 < real 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_def bl_def)
  show ?thesis
  proof (cases "0 \<le> e")
    case True 
    thus ?thesis
      unfolding bl_def[symmetric] using `0 < real m` `0 \<le> bl`
      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 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 `0 < real m` `0 \<le> bl`
      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 (x - 1) < 1" and "real x < 2" by auto
  have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` by auto
  hence "0 \<le> real (x - 1)" using `1 \<le> x` 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 `0 \<le> real (x - 1)` `real (x - 1) < 1`, of prec] `\<not> x \<le> 0` `\<not> x < 1` True
      by auto
  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 x * 2 / 3)"
      by (auto simp add: algebra_simps diff_divide_distrib)

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

    { have up: "real (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 (x * rapprox_rat prec 2 3 - 1)" by auto

      have "ln (real x * 2/3)
        \<le> ln (real (x * rapprox_rat prec 2 3 - 1) + 1)"
      proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
        show "real x * 2 / 3 \<le> real (x * rapprox_rat prec 2 3 - 1) + 1"
          using * low by auto
        show "0 < real x * 2 / 3" using * by simp
        show "0 < real (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 ln_float_bounds(2))
        from mult_less_le_imp_less[OF `real x < 2` up] low *
        show "real (x * rapprox_rat prec 2 3 - 1) < 1" by auto
        show "0 \<le> real (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 }
    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 (lapprox_rat prec 2 3)"
        using lapprox_rat_nonneg[of 2 3 prec] by simp

      have "?lb_horner ?max
        \<le> ln (real ?max + 1)"
      proof (rule ln_float_bounds(1))
        from mult_less_le_imp_less[OF `real x < 2` up] * low
        show "real ?max < 1" by (cases "real (lapprox_rat prec 2 3) = 0",
          auto simp add: real_of_float_max)
        show "0 \<le> real ?max" by (auto simp add: real_of_float_max)
      qed
      also have "\<dots> \<le> ln (real x * 2/3)"
      proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
        show "0 < real ?max + 1" by (auto simp add: real_of_float_max)
        show "0 < real x * 2/3" using * by auto
        show "real ?max + 1 \<le> real x * 2/3" using * up
          by (cases "0 < real x * real (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 }
    ultimately
    show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
      using `\<not> x \<le> 0` `\<not> x < 1` 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 `1 \<le> x` by auto
  show ?thesis
  proof -
    def m \<equiv> "mantissa x"
    def e \<equiv> "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 `0 < x` Float powr_gt_zero[of 2 e]
      by (auto simp: zero_less_mult_iff)
    def bl \<equiv> "bitlen m - 1" hence "bl \<ge> 0" using `m > 0` by (simp add: bitlen_def)
    have "1 \<le> Float m e" using `1 \<le> x` Float unfolding less_eq_float_def by auto
    from bitlen_div[OF `0 < m`] float_gt1_scale[OF `1 \<le> Float m e`] `bl \<ge> 0`
    have x_bnds: "0 \<le> real (?x - 1)" "real (?x - 1) < 1"
      unfolding bl_def[symmetric]
      by (auto simp: powr_realpow[symmetric] field_simps inverse_eq_divide)
         (auto simp : powr_minus field_simps inverse_eq_divide)

    {
      have "lb_ln2 prec * ?s \<le> ln 2 * (e + (bitlen m - 1))" (is "?lb2 \<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 `1 \<le> Float m e`]
        show "0 \<le> real (Float (e + (bitlen m - 1)) 0)" by simp
      qed auto
      moreover
      from ln_float_bounds(1)[OF x_bnds]
      have "(?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1) \<le> ln ?x" (is "?lb_horner \<le> _") by auto
      ultimately have "?lb2 + ?lb_horner \<le> ln x"
        unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
    }
    moreover
    {
      from ln_float_bounds(2)[OF x_bnds]
      have "ln ?x \<le> (?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1)" (is "_ \<le> ?ub_horner") by auto
      moreover
      have "ln 2 * (e + (bitlen m - 1)) \<le> ub_ln2 prec * ?s" (is "_ \<le> ?ub2")
        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 `1 \<le> Float m e`]
        show "0 \<le> real (e + (bitlen m - 1))" by auto
      next
        have "0 \<le> ln 2" by simp
        thus "0 \<le> real (ub_ln2 prec)" using ub_ln2[of prec] by arith
      qed auto
      ultimately have "ln x \<le> ?ub2 + ?ub_horner"
        unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
    }
    ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
      unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] if_not_P[OF `\<not> x \<le> Float 3 -1`] 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 `1 \<le> x`] .
next
  case True have "\<not> x \<le> 0" using `0 < x` by auto
  from True have "real x < 1" by simp
  have "0 < real x" and "real x \<noteq> 0" using `0 < x` by auto
  hence A: "0 < 1 / real x" by auto

  {
    let ?divl = "float_divl (max prec 1) 1 x"
    have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF `0 < real x` `real x < 1`] by auto
    hence B: "0 < real ?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 `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] 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 `0 < x` `x < 1`] unfolding less_eq_float_def less_float_def by auto
    hence B: "0 < real ?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 `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] 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 `\<not> x \<le> 0`] 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 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 x" by auto
  have "the (lb_ln prec x) \<le> ln x" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
  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 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 x" by auto
  have "ln x \<le> the (ub_ln prec x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
  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 and 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 ux" using ub_ln u by auto
  have "l \<le> ln lx" and "0 < real lx" and "0 < x" using lb_ln[OF l] x by auto

  from ln_le_cancel_iff[OF `0 < real lx` `0 < x`] `l \<le> ln lx`
  have "l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
  moreover
  from ln_le_cancel_iff[OF `0 < x` `0 < real ux`] `ln ux \<le> real u`
  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 "Implement floatarith"

subsection "Define syntax and semantics"

datatype floatarith
  = Add floatarith floatarith
  | Minus floatarith
  | Mult floatarith floatarith
  | Inverse floatarith
  | Cos floatarith
  | Arctan floatarith
  | Abs floatarith
  | Max floatarith floatarith
  | Min floatarith floatarith
  | Pi
  | Sqrt floatarith
  | Exp floatarith
  | Ln floatarith
  | Power floatarith nat
  | Var nat
  | Num float

fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real" where
"interpret_floatarith (Add a b) vs   = (interpret_floatarith a vs) + (interpret_floatarith b vs)" |
"interpret_floatarith (Minus a) vs    = - (interpret_floatarith a vs)" |
"interpret_floatarith (Mult a b) vs   = (interpret_floatarith a vs) * (interpret_floatarith b vs)" |
"interpret_floatarith (Inverse a) vs  = inverse (interpret_floatarith a vs)" |
"interpret_floatarith (Cos a) vs      = cos (interpret_floatarith a vs)" |
"interpret_floatarith (Arctan a) vs   = arctan (interpret_floatarith a vs)" |
"interpret_floatarith (Min a b) vs    = min (interpret_floatarith a vs) (interpret_floatarith b vs)" |
"interpret_floatarith (Max a b) vs    = max (interpret_floatarith a vs) (interpret_floatarith b vs)" |
"interpret_floatarith (Abs a) vs      = abs (interpret_floatarith a vs)" |
"interpret_floatarith Pi vs           = pi" |
"interpret_floatarith (Sqrt a) vs     = sqrt (interpret_floatarith a vs)" |
"interpret_floatarith (Exp a) vs      = exp (interpret_floatarith a vs)" |
"interpret_floatarith (Ln a) vs       = ln (interpret_floatarith a vs)" |
"interpret_floatarith (Power a n) vs  = (interpret_floatarith a vs)^n" |
"interpret_floatarith (Num f) vs      = f" |
"interpret_floatarith (Var n) vs     = vs ! n"

lemma interpret_floatarith_divide: "interpret_floatarith (Mult a (Inverse b)) vs = (interpret_floatarith a vs) / (interpret_floatarith b vs)"
  unfolding divide_inverse interpret_floatarith.simps ..

lemma interpret_floatarith_diff: "interpret_floatarith (Add a (Minus b)) vs = (interpret_floatarith a vs) - (interpret_floatarith b vs)"
  unfolding interpret_floatarith.simps by simp

lemma interpret_floatarith_sin: "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) vs =
  sin (interpret_floatarith a vs)"
  unfolding sin_cos_eq interpret_floatarith.simps
            interpret_floatarith_divide interpret_floatarith_diff
  by auto

lemma interpret_floatarith_tan:
  "interpret_floatarith (Mult (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) (Inverse (Cos a))) vs =
   tan (interpret_floatarith a vs)"
  unfolding interpret_floatarith.simps(3,4) interpret_floatarith_sin tan_def divide_inverse
  by auto

lemma interpret_floatarith_powr: "interpret_floatarith (Exp (Mult b (Ln a))) vs = (interpret_floatarith a vs) powr (interpret_floatarith b vs)"
  unfolding powr_def interpret_floatarith.simps ..

lemma interpret_floatarith_log: "interpret_floatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (interpret_floatarith b vs) (interpret_floatarith x vs)"
  unfolding log_def interpret_floatarith.simps divide_inverse ..

lemma interpret_floatarith_num:
  shows "interpret_floatarith (Num (Float 0 0)) vs = 0"
  and "interpret_floatarith (Num (Float 1 0)) vs = 1"
  and "interpret_floatarith (Num (Float (- 1) 0)) vs = - 1"
  and "interpret_floatarith (Num (Float (numeral a) 0)) vs = numeral a"
  and "interpret_floatarith (Num (Float (- numeral a) 0)) vs = - numeral a" by auto

subsection "Implement approximation function"

fun lift_bin' :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
"lift_bin' a b f = None"

fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
                                             | t \<Rightarrow> None)" |
"lift_un b f = None"

fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
"lift_un' b f = None"

definition
"bounded_by xs vs \<longleftrightarrow>
  (\<forall> i < length vs. case vs ! i of None \<Rightarrow> True
         | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u })"

lemma bounded_byE:
  assumes "bounded_by xs vs"
  shows "\<And> i. i < length vs \<Longrightarrow> case vs ! i of None \<Rightarrow> True
         | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u }"
  using assms bounded_by_def by blast

lemma bounded_by_update:
  assumes "bounded_by xs vs"
  and bnd: "xs ! i \<in> { real l .. real u }"
  shows "bounded_by xs (vs[i := Some (l,u)])"
proof -
{ fix j
  let ?vs = "vs[i := Some (l,u)]"
  assume "j < length ?vs" hence [simp]: "j < length vs" by simp
  have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real l .. real u }"
  proof (cases "?vs ! j")
    case (Some b)
    thus ?thesis
    proof (cases "i = j")
      case True
      thus ?thesis using `?vs ! j = Some b` and bnd by auto
    next
      case False
      thus ?thesis using `bounded_by xs vs` unfolding bounded_by_def by auto
    qed
  qed auto }
  thus ?thesis unfolding bounded_by_def by auto
qed

lemma bounded_by_None:
  shows "bounded_by xs (replicate (length xs) None)"
  unfolding bounded_by_def by auto

fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option" where
"approx' prec a bs          = (case (approx prec a bs) of Some (l, u) \<Rightarrow> Some (float_round_down prec l, float_round_up prec u) | None \<Rightarrow> None)" |
"approx prec (Add a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (l1 + l2, u1 + u2))" |
"approx prec (Minus a) bs   = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
"approx prec (Mult a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs)
                                    (\<lambda> a1 a2 b1 b2. (nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1,
                                                     pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1))" |
"approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\<lambda> l u. if (0 < l \<or> u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" |
"approx prec (Cos a) bs     = lift_un' (approx' prec a bs) (bnds_cos prec)" |
"approx prec Pi bs          = Some (lb_pi prec, ub_pi prec)" |
"approx prec (Min a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (min l1 l2, min u1 u2))" |
"approx prec (Max a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (max l1 l2, max u1 u2))" |
"approx prec (Abs a) bs     = lift_un' (approx' prec a bs) (\<lambda>l u. (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))" |
"approx prec (Arctan a) bs  = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
"approx prec (Sqrt a) bs    = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
"approx prec (Exp a) bs     = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
"approx prec (Ln a) bs      = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
"approx prec (Num f) bs     = Some (f, f)" |
"approx prec (Var i) bs    = (if i < length bs then bs ! i else None)"

lemma lift_bin'_ex:
  assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
  shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
proof (cases a)
  case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
  thus ?thesis using lift_bin'_Some by auto
next
  case (Some a')
  show ?thesis
  proof (cases b)
    case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
    thus ?thesis using lift_bin'_Some by auto
  next
    case (Some b')
    obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
    obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
    thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
  qed
qed

lemma lift_bin'_f:
  assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b"
  shows "\<exists> l1 u1 l2 u2. P l1 u1 a \<and> P l2 u2 b \<and> l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
proof -
  obtain l1 u1 l2 u2
    where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
  have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto
  have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
  thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto
qed

lemma approx_approx':
  assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
  and approx': "Some (l, u) = approx' prec a vs"
  shows "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
proof -
  obtain l' u' where S: "Some (l', u') = approx prec a vs"
    using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
  have l': "l = float_round_down prec l'" and u': "u = float_round_up prec u'"
    using approx' unfolding approx'.simps S[symmetric] by auto
  show ?thesis unfolding l' u'
    using order_trans[OF Pa[OF S, THEN conjunct2] float_round_up[of u']]
    using order_trans[OF float_round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
qed

lemma lift_bin':
  assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
  and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u"
  shows "\<exists> l1 u1 l2 u2. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and>
                        (l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u2) \<and>
                        l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
proof -
  { fix l u assume "Some (l, u) = approx' prec a bs"
    with approx_approx'[of prec a bs, OF _ this] Pa
    have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this
  { fix l u assume "Some (l, u) = approx' prec b bs"
    with approx_approx'[of prec b bs, OF _ this] Pb
    have "l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u" by auto } note Pb = this

  from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
  show ?thesis by auto
qed

lemma lift_un'_ex:
  assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
  shows "\<exists> l u. Some (l, u) = a"
proof (cases a)
  case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
  thus ?thesis using lift_un'_Some by auto
next
  case (Some a')
  obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
  thus ?thesis unfolding `a = Some a'` a' by auto
qed

lemma lift_un'_f:
  assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
  shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
proof -
  obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
  have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
  have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
  thus ?thesis using Pa[OF Sa] by auto
qed

lemma lift_un':
  assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
  shows "\<exists> l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and>
                        l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
proof -
  { fix l u assume "Some (l, u) = approx' prec a bs"
    with approx_approx'[of prec a bs, OF _ this] Pa
    have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this
  from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
  show ?thesis by auto
qed

lemma lift_un'_bnds:
  assumes bnds: "\<forall> (x::real) lx ux. (l, u) = f lx ux \<and> x \<in> { lx .. ux } \<longrightarrow> l \<le> f' x \<and> f' x \<le> u"
  and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
  shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
proof -
  from lift_un'[OF lift_un'_Some Pa]
  obtain l1 u1 where "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast
  hence "(l, u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" by auto
  thus ?thesis using bnds by auto
qed

lemma lift_un_ex:
  assumes lift_un_Some: "Some (l, u) = lift_un a f"
  shows "\<exists> l u. Some (l, u) = a"
proof (cases a)
  case None hence "None = lift_un a f" unfolding None lift_un.simps ..
  thus ?thesis using lift_un_Some by auto
next
  case (Some a')
  obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
  thus ?thesis unfolding `a = Some a'` a' by auto
qed

lemma lift_un_f:
  assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
  shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
proof -
  obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
  have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
  proof (rule ccontr)
    assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
    hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
    hence "lift_un (g a) f = None"
    proof (cases "fst (f l1 u1) = None")
      case True
      then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
      thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
    next
      case False hence "snd (f l1 u1) = None" using or by auto
      with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
      thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
    qed
    thus False using lift_un_Some by auto
  qed
  then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
  from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
  have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
  thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
qed

lemma lift_un:
  assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
  shows "\<exists> l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and>
                  Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
proof -
  { fix l u assume "Some (l, u) = approx' prec a bs"
    with approx_approx'[of prec a bs, OF _ this] Pa
    have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this
  from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
  show ?thesis by auto
qed

lemma lift_un_bnds:
  assumes bnds: "\<forall> (x::real) lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { lx .. ux } \<longrightarrow> l \<le> f' x \<and> f' x \<le> u"
  and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
  and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
  shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
proof -
  from lift_un[OF lift_un_Some Pa]
  obtain l1 u1 where "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast
  hence "(Some l, Some u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" by auto
  thus ?thesis using bnds by auto
qed

lemma approx:
  assumes "bounded_by xs vs"
  and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
  shows "l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> u" (is "?P l u arith")
  using `Some (l, u) = approx prec arith vs`
proof (induct arith arbitrary: l u)
  case (Add a b)
  from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
  obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
    "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1"
    "l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast
  thus ?case unfolding interpret_floatarith.simps by auto
next
  case (Minus a)
  from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
  obtain l1 u1 where "l = -u1" and "u = -l1"
    "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1" unfolding fst_conv snd_conv by blast
  thus ?case unfolding interpret_floatarith.simps using minus_float.rep_eq by auto
next
  case (Mult a b)
  from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
  obtain l1 u1 l2 u2
    where l: "l = nprt l1 * pprt u2 + nprt u1 * nprt u2 + pprt l1 * pprt l2 + pprt u1 * nprt l2"
    and u: "u = pprt u1 * pprt u2 + pprt l1 * nprt u2 + nprt u1 * pprt l2 + nprt l1 * nprt l2"
    and "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1"
    and "l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast
  thus ?case unfolding interpret_floatarith.simps l u
    using mult_le_prts mult_ge_prts by auto
next
  case (Inverse a)
  from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps
  obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)"
    and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
    and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1" by blast
  have either: "0 < l1 \<or> u1 < 0" proof (rule ccontr) assume P: "\<not> (0 < l1 \<or> u1 < 0)" show False using l' unfolding if_not_P[OF P] by auto qed
  moreover have l1_le_u1: "real l1 \<le> real u1" using l1 u1 by auto
  ultimately have "real l1 \<noteq> 0" and "real u1 \<noteq> 0" by auto

  have inv: "inverse u1 \<le> inverse (interpret_floatarith a xs)
           \<and> inverse (interpret_floatarith a xs) \<le> inverse l1"
  proof (cases "0 < l1")
    case True hence "0 < real u1" and "0 < real l1" "0 < interpret_floatarith a xs"
      using l1_le_u1 l1 by auto
    show ?thesis
      unfolding inverse_le_iff_le[OF `0 < real u1` `0 < interpret_floatarith a xs`]
        inverse_le_iff_le[OF `0 < interpret_floatarith a xs` `0 < real l1`]
      using l1 u1 by auto
  next
    case False hence "u1 < 0" using either by blast
    hence "real u1 < 0" and "real l1 < 0" "interpret_floatarith a xs < 0"
      using l1_le_u1 u1 by auto
    show ?thesis
      unfolding inverse_le_iff_le_neg[OF `real u1 < 0` `interpret_floatarith a xs < 0`]
        inverse_le_iff_le_neg[OF `interpret_floatarith a xs < 0` `real l1 < 0`]
      using l1 u1 by auto
  qed

  from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
  hence "l \<le> inverse u1" unfolding nonzero_inverse_eq_divide[OF `real u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto
  also have "\<dots> \<le> inverse (interpret_floatarith a xs)" using inv by auto
  finally have "l \<le> inverse (interpret_floatarith a xs)" .
  moreover
  from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
  hence "inverse l1 \<le> u" unfolding nonzero_inverse_eq_divide[OF `real l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto
  hence "inverse (interpret_floatarith a xs) \<le> u" by (rule order_trans[OF inv[THEN conjunct2]])
  ultimately show ?case unfolding interpret_floatarith.simps using l1 u1 by auto
next
  case (Abs x)
  from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
  obtain l1 u1 where l': "l = (if l1 < 0 \<and> 0 < u1 then 0 else min \<bar>l1\<bar> \<bar>u1\<bar>)" and u': "u = max \<bar>l1\<bar> \<bar>u1\<bar>"
    and l1: "l1 \<le> interpret_floatarith x xs" and u1: "interpret_floatarith x xs \<le> u1" by blast
  thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: real_of_float_min real_of_float_max)
next
  case (Min a b)
  from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
  obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
    and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1"
    and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" by blast
  thus ?case unfolding l' u' by (auto simp add: real_of_float_min)
next
  case (Max a b)
  from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
  obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
    and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1"
    and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" by blast
  thus ?case unfolding l' u' by (auto simp add: real_of_float_max)
next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
next case Pi with pi_boundaries show ?case by auto
next case (Sqrt a) with lift_un'_bnds[OF bnds_sqrt] show ?case by auto
next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
next case (Num f) thus ?case by auto
next
  case (Var n)
  from this[symmetric] `bounded_by xs vs`[THEN bounded_byE, of n]
  show ?case by (cases "n < length vs", auto)
qed

datatype form = Bound floatarith floatarith floatarith form
              | Assign floatarith floatarith form
              | Less floatarith floatarith
              | LessEqual floatarith floatarith
              | AtLeastAtMost floatarith floatarith floatarith

fun interpret_form :: "form \<Rightarrow> real list \<Rightarrow> bool" where
"interpret_form (Bound x a b f) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs } \<longrightarrow> interpret_form f vs)" |
"interpret_form (Assign x a f) vs  = (interpret_floatarith x vs = interpret_floatarith a vs \<longrightarrow> interpret_form f vs)" |
"interpret_form (Less a b) vs      = (interpret_floatarith a vs < interpret_floatarith b vs)" |
"interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)" |
"interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs })"

fun approx_form' and approx_form :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> nat list \<Rightarrow> bool" where
"approx_form' prec f 0 n l u bs ss = approx_form prec f (bs[n := Some (l, u)]) ss" |
"approx_form' prec f (Suc s) n l u bs ss =
  (let m = (l + u) * Float 1 -1
   in (if approx_form' prec f s n l m bs ss then approx_form' prec f s n m u bs ss else False))" |
"approx_form prec (Bound (Var n) a b f) bs ss =
   (case (approx prec a bs, approx prec b bs)
   of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
    | _ \<Rightarrow> False)" |
"approx_form prec (Assign (Var n) a f) bs ss =
   (case (approx prec a bs)
   of (Some (l, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
    | _ \<Rightarrow> False)" |
"approx_form prec (Less a b) bs ss =
   (case (approx prec a bs, approx prec b bs)
   of (Some (l, u), Some (l', u')) \<Rightarrow> u < l'
    | _ \<Rightarrow> False)" |
"approx_form prec (LessEqual a b) bs ss =
   (case (approx prec a bs, approx prec b bs)
   of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l'
    | _ \<Rightarrow> False)" |
"approx_form prec (AtLeastAtMost x a b) bs ss =
   (case (approx prec x bs, approx prec a bs, approx prec b bs)
   of (Some (lx, ux), Some (l, u), Some (l', u')) \<Rightarrow> u \<le> lx \<and> ux \<le> l'
    | _ \<Rightarrow> False)" |
"approx_form _ _ _ _ = False"

lemma lazy_conj: "(if A then B else False) = (A \<and> B)" by simp

lemma approx_form_approx_form':
  assumes "approx_form' prec f s n l u bs ss" and "(x::real) \<in> { l .. u }"
  obtains l' u' where "x \<in> { l' .. u' }"
    and "approx_form prec f (bs[n := Some (l', u')]) ss"
using assms proof (induct s arbitrary: l u)
  case 0
  from this(1)[of l u] this(2,3)
  show thesis by auto
next
  case (Suc s)

  let ?m = "(l + u) * Float 1 -1"
  have "real l \<le> ?m" and "?m \<le> real u"
    unfolding less_eq_float_def using Suc.prems by auto

  with `x \<in> { l .. u }`
  have "x \<in> { l .. ?m} \<or> x \<in> { ?m .. u }" by auto
  thus thesis
  proof (rule disjE)
    assume *: "x \<in> { l .. ?m }"
    with Suc.hyps[OF _ _ *] Suc.prems
    show thesis by (simp add: Let_def lazy_conj)
  next
    assume *: "x \<in> { ?m .. u }"
    with Suc.hyps[OF _ _ *] Suc.prems
    show thesis by (simp add: Let_def lazy_conj)
  qed
qed

lemma approx_form_aux:
  assumes "approx_form prec f vs ss"
    and "bounded_by xs vs"
  shows "interpret_form f xs"
using assms proof (induct f arbitrary: vs)
  case (Bound x a b f)
  then obtain n
    where x_eq: "x = Var n" by (cases x) auto

  with Bound.prems obtain l u' l' u
    where l_eq: "Some (l, u') = approx prec a vs"
    and u_eq: "Some (l', u) = approx prec b vs"
    and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
    by (cases "approx prec a vs", simp) (cases "approx prec b vs", auto)

  { assume "xs ! n \<in> { interpret_floatarith a xs .. interpret_floatarith b xs }"
    with approx[OF Bound.prems(2) l_eq] and approx[OF Bound.prems(2) u_eq]
    have "xs ! n \<in> { l .. u}" by auto

    from approx_form_approx_form'[OF approx_form' this]
    obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }"
      and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .

    from `bounded_by xs vs` bnds
    have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
    with Bound.hyps[OF approx_form]
    have "interpret_form f xs" by blast }
  thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
next
  case (Assign x a f)
  then obtain n
    where x_eq: "x = Var n" by (cases x) auto

  with Assign.prems obtain l u
    where bnd_eq: "Some (l, u) = approx prec a vs"
    and x_eq: "x = Var n"
    and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
    by (cases "approx prec a vs") auto

  { assume bnds: "xs ! n = interpret_floatarith a xs"
    with approx[OF Assign.prems(2) bnd_eq]
    have "xs ! n \<in> { l .. u}" by auto
    from approx_form_approx_form'[OF approx_form' this]
    obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }"
      and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .

    from `bounded_by xs vs` bnds
    have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
    with Assign.hyps[OF approx_form]
    have "interpret_form f xs" by blast }
  thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
next
  case (Less a b)
  then obtain l u l' u'
    where l_eq: "Some (l, u) = approx prec a vs"
      and u_eq: "Some (l', u') = approx prec b vs"
      and inequality: "u < l'"
    by (cases "approx prec a vs", auto,
      cases "approx prec b vs", auto)
  from inequality approx[OF Less.prems(2) l_eq] approx[OF Less.prems(2) u_eq]
  show ?case by auto
next
  case (LessEqual a b)
  then obtain l u l' u'
    where l_eq: "Some (l, u) = approx prec a vs"
      and u_eq: "Some (l', u') = approx prec b vs"
      and inequality: "u \<le> l'"
    by (cases "approx prec a vs", auto,
      cases "approx prec b vs", auto)
  from inequality approx[OF LessEqual.prems(2) l_eq] approx[OF LessEqual.prems(2) u_eq]
  show ?case by auto
next
  case (AtLeastAtMost x a b)
  then obtain lx ux l u l' u'
    where x_eq: "Some (lx, ux) = approx prec x vs"
    and l_eq: "Some (l, u) = approx prec a vs"
    and u_eq: "Some (l', u') = approx prec b vs"
    and inequality: "u \<le> lx \<and> ux \<le> l'"
    by (cases "approx prec x vs", auto,
      cases "approx prec a vs", auto,
      cases "approx prec b vs", auto)
  from inequality approx[OF AtLeastAtMost.prems(2) l_eq] approx[OF AtLeastAtMost.prems(2) u_eq] approx[OF AtLeastAtMost.prems(2) x_eq]
  show ?case by auto
qed

lemma approx_form:
  assumes "n = length xs"
  assumes "approx_form prec f (replicate n None) ss"
  shows "interpret_form f xs"
  using approx_form_aux[OF _ bounded_by_None] assms by auto

subsection {* Implementing Taylor series expansion *}

fun isDERIV :: "nat \<Rightarrow> floatarith \<Rightarrow> real list \<Rightarrow> bool" where
"isDERIV x (Add a b) vs         = (isDERIV x a vs \<and> isDERIV x b vs)" |
"isDERIV x (Mult a b) vs        = (isDERIV x a vs \<and> isDERIV x b vs)" |
"isDERIV x (Minus a) vs         = isDERIV x a vs" |
"isDERIV x (Inverse a) vs       = (isDERIV x a vs \<and> interpret_floatarith a vs \<noteq> 0)" |
"isDERIV x (Cos a) vs           = isDERIV x a vs" |
"isDERIV x (Arctan a) vs        = isDERIV x a vs" |
"isDERIV x (Min a b) vs         = False" |
"isDERIV x (Max a b) vs         = False" |
"isDERIV x (Abs a) vs           = False" |
"isDERIV x Pi vs                = True" |
"isDERIV x (Sqrt a) vs          = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
"isDERIV x (Exp a) vs           = isDERIV x a vs" |
"isDERIV x (Ln a) vs            = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
"isDERIV x (Power a 0) vs       = True" |
"isDERIV x (Power a (Suc n)) vs = isDERIV x a vs" |
"isDERIV x (Num f) vs           = True" |
"isDERIV x (Var n) vs          = True"

fun DERIV_floatarith :: "nat \<Rightarrow> floatarith \<Rightarrow> floatarith" where
"DERIV_floatarith x (Add a b)         = Add (DERIV_floatarith x a) (DERIV_floatarith x b)" |
"DERIV_floatarith x (Mult a b)        = Add (Mult a (DERIV_floatarith x b)) (Mult (DERIV_floatarith x a) b)" |
"DERIV_floatarith x (Minus a)         = Minus (DERIV_floatarith x a)" |
"DERIV_floatarith x (Inverse a)       = Minus (Mult (DERIV_floatarith x a) (Inverse (Power a 2)))" |
"DERIV_floatarith x (Cos a)           = Minus (Mult (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Arctan a)        = Mult (Inverse (Add (Num 1) (Power a 2))) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Min a b)         = Num 0" |
"DERIV_floatarith x (Max a b)         = Num 0" |
"DERIV_floatarith x (Abs a)           = Num 0" |
"DERIV_floatarith x Pi                = Num 0" |
"DERIV_floatarith x (Sqrt a)          = (Mult (Inverse (Mult (Sqrt a) (Num 2))) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Exp a)           = Mult (Exp a) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Ln a)            = Mult (Inverse a) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Power a 0)       = Num 0" |
"DERIV_floatarith x (Power a (Suc n)) = Mult (Num (Float (int (Suc n)) 0)) (Mult (Power a n) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Num f)           = Num 0" |
"DERIV_floatarith x (Var n)          = (if x = n then Num 1 else Num 0)"

lemma DERIV_floatarith:
  assumes "n < length vs"
  assumes isDERIV: "isDERIV n f (vs[n := x])"
  shows "DERIV (\<lambda> x'. interpret_floatarith f (vs[n := x'])) x :>
               interpret_floatarith (DERIV_floatarith n f) (vs[n := x])"
   (is "DERIV (?i f) x :> _")
using isDERIV
proof (induct f arbitrary: x)
  case (Inverse a)
  thus ?case
    by (auto intro!: derivative_eq_intros simp add: algebra_simps power2_eq_square)
next
  case (Cos a)
  thus ?case
    by (auto intro!: derivative_eq_intros
           simp del: interpret_floatarith.simps(5)
           simp add: interpret_floatarith_sin interpret_floatarith.simps(5)[of a])
next
  case (Power a n)
  thus ?case
    by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc simp add: real_of_nat_def)
next
  case (Ln a)
  thus ?case by (auto intro!: derivative_eq_intros simp add: divide_inverse)
next
  case (Var i)
  thus ?case using `n < length vs` by auto
qed (auto intro!: derivative_eq_intros)

declare approx.simps[simp del]

fun isDERIV_approx :: "nat \<Rightarrow> nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> bool" where
"isDERIV_approx prec x (Add a b) vs         = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
"isDERIV_approx prec x (Mult a b) vs        = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
"isDERIV_approx prec x (Minus a) vs         = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Inverse a) vs       =
  (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l \<or> u < 0 | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Cos a) vs           = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Arctan a) vs        = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Min a b) vs         = False" |
"isDERIV_approx prec x (Max a b) vs         = False" |
"isDERIV_approx prec x (Abs a) vs           = False" |
"isDERIV_approx prec x Pi vs                = True" |
"isDERIV_approx prec x (Sqrt a) vs          =
  (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Exp a) vs           = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Ln a) vs            =
  (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Power a 0) vs       = True" |
"isDERIV_approx prec x (Power a (Suc n)) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Num f) vs           = True" |
"isDERIV_approx prec x (Var n) vs          = True"

lemma isDERIV_approx:
  assumes "bounded_by xs vs"
    and isDERIV_approx: "isDERIV_approx prec x f vs"
  shows "isDERIV x f xs"
  using isDERIV_approx
proof (induct f)
  case (Inverse a)
  then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
    and *: "0 < l \<or> u < 0"
    by (cases "approx prec a vs") auto
  with approx[OF `bounded_by xs vs` approx_Some]
  have "interpret_floatarith a xs \<noteq> 0" by auto
  thus ?case using Inverse by auto
next
  case (Ln a)
  then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
    and *: "0 < l"
    by (cases "approx prec a vs") auto
  with approx[OF `bounded_by xs vs` approx_Some]
  have "0 < interpret_floatarith a xs" by auto
  thus ?case using Ln by auto
next
  case (Sqrt a)
  then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
    and *: "0 < l"
    by (cases "approx prec a vs") auto
  with approx[OF `bounded_by xs vs` approx_Some]
  have "0 < interpret_floatarith a xs" by auto
  thus ?case using Sqrt by auto
next
  case (Power a n) thus ?case by (cases n) auto
qed auto

lemma bounded_by_update_var:
  assumes "bounded_by xs vs" and "vs ! i = Some (l, u)"
    and bnd: "x \<in> { real l .. real u }"
  shows "bounded_by (xs[i := x]) vs"
proof (cases "i < length xs")
  case False
  thus ?thesis using `bounded_by xs vs` by auto
next
  let ?xs = "xs[i := x]"
  case True hence "i < length ?xs" by auto
  {
    fix j
    assume "j < length vs"
    have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> { real l .. real u }"
    proof (cases "vs ! j")
      case (Some b)
      thus ?thesis
      proof (cases "i = j")
        case True
        thus ?thesis using `vs ! i = Some (l, u)` Some and bnd `i < length ?xs`
          by auto
      next
        case False
        thus ?thesis
          using `bounded_by xs vs`[THEN bounded_byE, OF `j < length vs`] Some by auto
      qed
    qed auto
  }
  thus ?thesis unfolding bounded_by_def by auto
qed

lemma isDERIV_approx':
  assumes "bounded_by xs vs"
    and vs_x: "vs ! x = Some (l, u)" and X_in: "X \<in> { real l .. real u }"
    and approx: "isDERIV_approx prec x f vs"
  shows "isDERIV x f (xs[x := X])"
proof -
  note bounded_by_update_var[OF `bounded_by xs vs` vs_x X_in] approx
  thus ?thesis by (rule isDERIV_approx)
qed

lemma DERIV_approx:
  assumes "n < length xs" and bnd: "bounded_by xs vs"
    and isD: "isDERIV_approx prec n f vs"
    and app: "Some (l, u) = approx prec (DERIV_floatarith n f) vs" (is "_ = approx _ ?D _")
  shows "\<exists>(x::real). l \<le> x \<and> x \<le> u \<and>
             DERIV (\<lambda> x. interpret_floatarith f (xs[n := x])) (xs!n) :> x"
         (is "\<exists> x. _ \<and> _ \<and> DERIV (?i f) _ :> _")
proof (rule exI[of _ "?i ?D (xs!n)"], rule conjI[OF _ conjI])
  let "?i f x" = "interpret_floatarith f (xs[n := x])"
  from approx[OF bnd app]
  show "l \<le> ?i ?D (xs!n)" and "?i ?D (xs!n) \<le> u"
    using `n < length xs` by auto
  from DERIV_floatarith[OF `n < length xs`, of f "xs!n"] isDERIV_approx[OF bnd isD]
  show "DERIV (?i f) (xs!n) :> (?i ?D (xs!n))" by simp
qed

fun lift_bin :: "(float * float) option \<Rightarrow>
    (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float) option) \<Rightarrow>
    (float * float) option" where
  "lift_bin (Some (l1, u1)) (Some (l2, u2)) f = f l1 u1 l2 u2"
| "lift_bin a b f = None"

lemma lift_bin:
  assumes lift_bin_Some: "Some (l, u) = lift_bin a b f"
  obtains l1 u1 l2 u2
  where "a = Some (l1, u1)"
    and "b = Some (l2, u2)"
    and "f l1 u1 l2 u2 = Some (l, u)"
  using assms by (cases a, simp, cases b, simp, auto)

fun approx_tse where
"approx_tse prec n 0 c k f bs = approx prec f bs" |
"approx_tse prec n (Suc s) c k f bs =
  (if isDERIV_approx prec n f bs then
    lift_bin (approx prec f (bs[n := Some (c,c)]))
             (approx_tse prec n s c (Suc k) (DERIV_floatarith n f) bs)
             (\<lambda> l1 u1 l2 u2. approx prec
                 (Add (Var 0)
                      (Mult (Inverse (Num (Float (int k) 0)))
                                 (Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
                                       (Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), bs!n])
  else approx prec f bs)"

lemma bounded_by_Cons:
  assumes bnd: "bounded_by xs vs"
    and x: "x \<in> { real l .. real u }"
  shows "bounded_by (x#xs) ((Some (l, u))#vs)"
proof -
  {
    fix i assume *: "i < length ((Some (l, u))#vs)"
    have "case ((Some (l,u))#vs) ! i of Some (l, u) \<Rightarrow> (x#xs)!i \<in> { real l .. real u } | None \<Rightarrow> True"
    proof (cases i)
      case 0 with x show ?thesis by auto
    next
      case (Suc i) with * have "i < length vs" by auto
      from bnd[THEN bounded_byE, OF this]
      show ?thesis unfolding Suc nth_Cons_Suc .
    qed
  }
  thus ?thesis by (auto simp add: bounded_by_def)
qed

lemma approx_tse_generic:
  assumes "bounded_by xs vs"
    and bnd_c: "bounded_by (xs[x := c]) vs" and "x < length vs" and "x < length xs"
    and bnd_x: "vs ! x = Some (lx, ux)"
    and ate: "Some (l, u) = approx_tse prec x s c k f vs"
  shows "\<exists> n. (\<forall> m < n. \<forall> (z::real) \<in> {lx .. ux}.
      DERIV (\<lambda> y. interpret_floatarith ((DERIV_floatarith x ^^ m) f) (xs[x := y])) z :>
            (interpret_floatarith ((DERIV_floatarith x ^^ (Suc m)) f) (xs[x := z])))
   \<and> (\<forall> (t::real) \<in> {lx .. ux}.  (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) *
                  interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := c]) *
                  (xs!x - c)^i) +
      inverse (real (\<Prod> j \<in> {k..<k+n}. j)) *
      interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) *
      (xs!x - c)^n \<in> {l .. u})" (is "\<exists> n. ?taylor f k l u n")
using ate proof (induct s arbitrary: k f l u)
  case 0
  {
    fix t::real assume "t \<in> {lx .. ux}"
    note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
    from approx[OF this 0[unfolded approx_tse.simps]]
    have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}"
      by (auto simp add: algebra_simps)
  }
  thus ?case by (auto intro!: exI[of _ 0])
next
  case (Suc s)
  show ?case
  proof (cases "isDERIV_approx prec x f vs")
    case False
    note ap = Suc.prems[unfolded approx_tse.simps if_not_P[OF False]]
    {
      fix t::real assume "t \<in> {lx .. ux}"
      note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
      from approx[OF this ap]
      have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}"
        by (auto simp add: algebra_simps)
    }
    thus ?thesis by (auto intro!: exI[of _ 0])
  next
    case True
    with Suc.prems
    obtain l1 u1 l2 u2
      where a: "Some (l1, u1) = approx prec f (vs[x := Some (c,c)])"
        and ate: "Some (l2, u2) = approx_tse prec x s c (Suc k) (DERIV_floatarith x f) vs"
        and final: "Some (l, u) = approx prec
          (Add (Var 0)
               (Mult (Inverse (Num (Float (int k) 0)))
                     (Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
                           (Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), vs!x]"
      by (auto elim!: lift_bin)

    from bnd_c `x < length xs`
    have bnd: "bounded_by (xs[x:=c]) (vs[x:= Some (c,c)])"
      by (auto intro!: bounded_by_update)

    from approx[OF this a]
    have f_c: "interpret_floatarith ((DERIV_floatarith x ^^ 0) f) (xs[x := c]) \<in> { l1 .. u1 }"
              (is "?f 0 (real c) \<in> _")
      by auto

    {
      fix f :: "'a \<Rightarrow> 'a" fix n :: nat fix x :: 'a
      have "(f ^^ Suc n) x = (f ^^ n) (f x)"
        by (induct n) auto
    }
    note funpow_Suc = this[symmetric]
    from Suc.hyps[OF ate, unfolded this]
    obtain n
      where DERIV_hyp: "\<And> m z. \<lbrakk> m < n ; (z::real) \<in> { lx .. ux } \<rbrakk> \<Longrightarrow> DERIV (?f (Suc m)) z :> ?f (Suc (Suc m)) z"
      and hyp: "\<forall> t \<in> {real lx .. real ux}. (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {Suc k..<Suc k + i}. j)) * ?f (Suc i) c * (xs!x - c)^i) +
           inverse (real (\<Prod> j \<in> {Suc k..<Suc k + n}. j)) * ?f (Suc n) t * (xs!x - c)^n \<in> {l2 .. u2}"
          (is "\<forall> t \<in> _. ?X (Suc k) f n t \<in> _")
      by blast

    {
      fix m and z::real
      assume "m < Suc n" and bnd_z: "z \<in> { lx .. ux }"
      have "DERIV (?f m) z :> ?f (Suc m) z"
      proof (cases m)
        case 0
        with DERIV_floatarith[OF `x < length xs` isDERIV_approx'[OF `bounded_by xs vs` bnd_x bnd_z True]]
        show ?thesis by simp
      next
        case (Suc m')
        hence "m' < n" using `m < Suc n` by auto
        from DERIV_hyp[OF this bnd_z]
        show ?thesis using Suc by simp
      qed
    } note DERIV = this

    have "\<And> k i. k < i \<Longrightarrow> {k ..< i} = insert k {Suc k ..< i}" by auto
    hence setprod_head_Suc: "\<And> k i. \<Prod> {k ..< k + Suc i} = k * \<Prod> {Suc k ..< Suc k + i}" by auto
    have setsum_move0: "\<And> k F. setsum F {0..<Suc k} = F 0 + setsum (\<lambda> k. F (Suc k)) {0..<k}"
      unfolding setsum_shift_bounds_Suc_ivl[symmetric]
      unfolding setsum_head_upt_Suc[OF zero_less_Suc] ..
    def C \<equiv> "xs!x - c"

    {
      fix t::real assume t: "t \<in> {lx .. ux}"
      hence "bounded_by [xs!x] [vs!x]"
        using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`]
        by (cases "vs!x", auto simp add: bounded_by_def)

      with hyp[THEN bspec, OF t] f_c
      have "bounded_by [?f 0 c, ?X (Suc k) f n t, xs!x] [Some (l1, u1), Some (l2, u2), vs!x]"
        by (auto intro!: bounded_by_Cons)
      from approx[OF this final, unfolded atLeastAtMost_iff[symmetric]]
      have "?X (Suc k) f n t * (xs!x - real c) * inverse k + ?f 0 c \<in> {l .. u}"
        by (auto simp add: algebra_simps)
      also have "?X (Suc k) f n t * (xs!x - real c) * inverse (real k) + ?f 0 c =
               (\<Sum> i = 0..<Suc n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * ?f i c * (xs!x - c)^i) +
               inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T")
        unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc
        by (auto simp add: algebra_simps)
          (simp only: mult.left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric])
      finally have "?T \<in> {l .. u}" .
    }
    thus ?thesis using DERIV by blast
  qed
qed

lemma setprod_fact: "\<Prod> {1..<1 + k} = fact (k :: nat)"
proof (induct k)
  case 0
  show ?case by simp
next
  case (Suc k)
  have "{ 1 ..< Suc (Suc k) } = insert (Suc k) { 1 ..< Suc k }" by auto
  hence "\<Prod> { 1 ..< Suc (Suc k) } = (Suc k) * \<Prod> { 1 ..< Suc k }" by auto
  thus ?case using Suc by auto
qed

lemma approx_tse:
  assumes "bounded_by xs vs"
    and bnd_x: "vs ! x = Some (lx, ux)" and bnd_c: "real c \<in> {lx .. ux}"
    and "x < length vs" and "x < length xs"
    and ate: "Some (l, u) = approx_tse prec x s c 1 f vs"
  shows "interpret_floatarith f xs \<in> { l .. u }"
proof -
  def F \<equiv> "\<lambda> n z. interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])"
  hence F0: "F 0 = (\<lambda> z. interpret_floatarith f (xs[x := z]))" by auto

  hence "bounded_by (xs[x := c]) vs" and "x < length vs" "x < length xs"
    using `bounded_by xs vs` bnd_x bnd_c `x < length vs` `x < length xs`
    by (auto intro!: bounded_by_update_var)

  from approx_tse_generic[OF `bounded_by xs vs` this bnd_x ate]
  obtain n
    where DERIV: "\<forall> m z. m < n \<and> real lx \<le> z \<and> z \<le> real ux \<longrightarrow> DERIV (F m) z :> F (Suc m) z"
    and hyp: "\<And> (t::real). t \<in> {lx .. ux} \<Longrightarrow>
           (\<Sum> j = 0..<n. inverse (real (fact j)) * F j c * (xs!x - c)^j) +
             inverse (real (fact n)) * F n t * (xs!x - c)^n
             \<in> {l .. u}" (is "\<And> t. _ \<Longrightarrow> ?taylor t \<in> _")
    unfolding F_def atLeastAtMost_iff[symmetric] setprod_fact by blast

  have bnd_xs: "xs ! x \<in> { lx .. ux }"
    using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto

  show ?thesis
  proof (cases n)
    case 0 thus ?thesis using hyp[OF bnd_xs] unfolding F_def by auto
  next
    case (Suc n')
    show ?thesis
    proof (cases "xs ! x = c")
      case True
      from True[symmetric] hyp[OF bnd_xs] Suc show ?thesis
        unfolding F_def Suc setsum_head_upt_Suc[OF zero_less_Suc] setsum_shift_bounds_Suc_ivl by auto
    next
      case False

      have "lx \<le> real c" "real c \<le> ux" "lx \<le> xs!x" "xs!x \<le> ux"
        using Suc bnd_c `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto
      from Taylor.taylor[OF zero_less_Suc, of F, OF F0 DERIV[unfolded Suc] this False]
      obtain t::real where t_bnd: "if xs ! x < c then xs ! x < t \<and> t < c else c < t \<and> t < xs ! x"
        and fl_eq: "interpret_floatarith f (xs[x := xs ! x]) =
           (\<Sum>m = 0..<Suc n'. F m c / real (fact m) * (xs ! x - c) ^ m) +
           F (Suc n') t / real (fact (Suc n')) * (xs ! x - c) ^ Suc n'"
        unfolding atLeast0LessThan by blast

      from t_bnd bnd_xs bnd_c have *: "t \<in> {lx .. ux}"
        by (cases "xs ! x < c", auto)

      have "interpret_floatarith f (xs[x := xs ! x]) = ?taylor t"
        unfolding fl_eq Suc by (auto simp add: algebra_simps divide_inverse)
      also have "\<dots> \<in> {l .. u}" using * by (rule hyp)
      finally show ?thesis by simp
    qed
  qed
qed

fun approx_tse_form' where
"approx_tse_form' prec t f 0 l u cmp =
  (case approx_tse prec 0 t ((l + u) * Float 1 -1) 1 f [Some (l, u)]
     of Some (l, u) \<Rightarrow> cmp l u | None \<Rightarrow> False)" |
"approx_tse_form' prec t f (Suc s) l u cmp =
  (let m = (l + u) * Float 1 -1
   in (if approx_tse_form' prec t f s l m cmp then
      approx_tse_form' prec t f s m u cmp else False))"

lemma approx_tse_form':
  fixes x :: real
  assumes "approx_tse_form' prec t f s l u cmp" and "x \<in> {l .. u}"
  shows "\<exists> l' u' ly uy. x \<in> { l' .. u' } \<and> real l \<le> l' \<and> u' \<le> real u \<and> cmp ly uy \<and>
                  approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 f [Some (l', u')] = Some (ly, uy)"
using assms proof (induct s arbitrary: l u)
  case 0
  then obtain ly uy
    where *: "approx_tse prec 0 t ((l + u) * Float 1 -1) 1 f [Some (l, u)] = Some (ly, uy)"
    and **: "cmp ly uy" by (auto elim!: case_optionE)
  with 0 show ?case by auto
next
  case (Suc s)
  let ?m = "(l + u) * Float 1 -1"
  from Suc.prems
  have l: "approx_tse_form' prec t f s l ?m cmp"
    and u: "approx_tse_form' prec t f s ?m u cmp"
    by (auto simp add: Let_def lazy_conj)

  have m_l: "real l \<le> ?m" and m_u: "?m \<le> real u"
    unfolding less_eq_float_def using Suc.prems by auto

  with `x \<in> { l .. u }`
  have "x \<in> { l .. ?m} \<or> x \<in> { ?m .. u }" by auto
  thus ?case
  proof (rule disjE)
    assume "x \<in> { l .. ?m}"
    from Suc.hyps[OF l this]
    obtain l' u' ly uy
      where "x \<in> { l' .. u' } \<and> real l \<le> l' \<and> real u' \<le> ?m \<and> cmp ly uy \<and>
                  approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 f [Some (l', u')] = Some (ly, uy)" by blast
    with m_u show ?thesis by (auto intro!: exI)
  next
    assume "x \<in> { ?m .. u }"
    from Suc.hyps[OF u this]
    obtain l' u' ly uy
      where "x \<in> { l' .. u' } \<and> ?m \<le> real l' \<and> u' \<le> real u \<and> cmp ly uy \<and>
                  approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 f [Some (l', u')] = Some (ly, uy)" by blast
    with m_u show ?thesis by (auto intro!: exI)
  qed
qed

lemma approx_tse_form'_less:
  fixes x :: real
  assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 < l)"
  and x: "x \<in> {l .. u}"
  shows "interpret_floatarith b [x] < interpret_floatarith a [x]"
proof -
  from approx_tse_form'[OF tse x]
  obtain l' u' ly uy
    where x': "x \<in> { l' .. u' }" and "l \<le> real l'"
    and "real u' \<le> u" and "0 < ly"
    and tse: "approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
    by blast

  hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)

  from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
  have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
    by auto
  from order_less_le_trans[OF _ this, of 0] `0 < ly`
  show ?thesis by auto
qed

lemma approx_tse_form'_le:
  fixes x :: real
  assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 \<le> l)"
  and x: "x \<in> {l .. u}"
  shows "interpret_floatarith b [x] \<le> interpret_floatarith a [x]"
proof -
  from approx_tse_form'[OF tse x]
  obtain l' u' ly uy
    where x': "x \<in> { l' .. u' }" and "l \<le> real l'"
    and "real u' \<le> u" and "0 \<le> ly"
    and tse: "approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
    by blast

  hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)

  from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
  have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
    by auto
  from order_trans[OF _ this, of 0] `0 \<le> ly`
  show ?thesis by auto
qed

definition
"approx_tse_form prec t s f =
  (case f
   of (Bound x a b f) \<Rightarrow> x = Var 0 \<and>
     (case (approx prec a [None], approx prec b [None])
      of (Some (l, u), Some (l', u')) \<Rightarrow>
        (case f
         of Less lf rt \<Rightarrow> approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)
          | LessEqual lf rt \<Rightarrow> approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)
          | AtLeastAtMost x lf rt \<Rightarrow>
            (if approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l) then
            approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l) else False)
          | _ \<Rightarrow> False)
       | _ \<Rightarrow> False)
   | _ \<Rightarrow> False)"

lemma approx_tse_form:
  assumes "approx_tse_form prec t s f"
  shows "interpret_form f [x]"
proof (cases f)
  case (Bound i a b f') note f_def = this
  with assms obtain l u l' u'
    where a: "approx prec a [None] = Some (l, u)"
    and b: "approx prec b [None] = Some (l', u')"
    unfolding approx_tse_form_def by (auto elim!: case_optionE)

  from Bound assms have "i = Var 0" unfolding approx_tse_form_def by auto
  hence i: "interpret_floatarith i [x] = x" by auto

  { let "?f z" = "interpret_floatarith z [x]"
    assume "?f i \<in> { ?f a .. ?f b }"
    with approx[OF _ a[symmetric], of "[x]"] approx[OF _ b[symmetric], of "[x]"]
    have bnd: "x \<in> { l .. u'}" unfolding bounded_by_def i by auto

    have "interpret_form f' [x]"
    proof (cases f')
      case (Less lf rt)
      with Bound a b assms
      have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)"
        unfolding approx_tse_form_def by auto
      from approx_tse_form'_less[OF this bnd]
      show ?thesis using Less by auto
    next
      case (LessEqual lf rt)
      with Bound a b assms
      have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
        unfolding approx_tse_form_def by auto
      from approx_tse_form'_le[OF this bnd]
      show ?thesis using LessEqual by auto
    next
      case (AtLeastAtMost x lf rt)
      with Bound a b assms
      have "approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l)"
        and "approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
        unfolding approx_tse_form_def lazy_conj by auto
      from approx_tse_form'_le[OF this(1) bnd] approx_tse_form'_le[OF this(2) bnd]
      show ?thesis using AtLeastAtMost by auto
    next
      case (Bound x a b f') with assms
      show ?thesis by (auto elim!: case_optionE simp add: f_def approx_tse_form_def)
    next
      case (Assign x a f') with assms
      show ?thesis by (auto elim!: case_optionE simp add: f_def approx_tse_form_def)
    qed } thus ?thesis unfolding f_def by auto
next
  case Assign
  with assms show ?thesis by (auto simp add: approx_tse_form_def)
next
  case LessEqual
  with assms show ?thesis by (auto simp add: approx_tse_form_def)
next
  case Less
  with assms show ?thesis by (auto simp add: approx_tse_form_def)
next
  case AtLeastAtMost
  with assms show ?thesis by (auto simp add: approx_tse_form_def)
qed

text {* @{term approx_form_eval} is only used for the {\tt value}-command. *}

fun approx_form_eval :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option list" where
"approx_form_eval prec (Bound (Var n) a b f) bs =
   (case (approx prec a bs, approx prec b bs)
   of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
    | _ \<Rightarrow> bs)" |
"approx_form_eval prec (Assign (Var n) a f) bs =
   (case (approx prec a bs)
   of (Some (l, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
    | _ \<Rightarrow> bs)" |
"approx_form_eval prec (Less a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
"approx_form_eval prec (LessEqual a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
"approx_form_eval prec (AtLeastAtMost x a b) bs =
   bs @ [approx prec x bs, approx prec a bs, approx prec b bs]" |
"approx_form_eval _ _ bs = bs"

subsection {* Implement proof method \texttt{approximation} *}

lemmas interpret_form_equations = interpret_form.simps interpret_floatarith.simps interpret_floatarith_num
  interpret_floatarith_divide interpret_floatarith_diff interpret_floatarith_tan interpret_floatarith_powr interpret_floatarith_log
  interpret_floatarith_sin

oracle approximation_oracle = {* fn (thy, t) =>
let
  fun bad t = error ("Bad term: " ^ Syntax.string_of_term_global thy t);

  fun term_of_bool true = @{term True}
    | term_of_bool false = @{term False};

  val mk_int = HOLogic.mk_number @{typ int} o @{code integer_of_int};
  val dest_int = @{code int_of_integer} o snd o HOLogic.dest_number;

  fun term_of_float (@{code Float} (k, l)) =
    @{term Float} $ mk_int k $ mk_int l;

  fun term_of_float_float_option NONE = @{term "None :: (float \<times> float) option"}
    | term_of_float_float_option (SOME ff) = @{term "Some :: float \<times> float \<Rightarrow> _"}
        $ HOLogic.mk_prod (pairself term_of_float ff);

  val term_of_float_float_option_list =
    HOLogic.mk_list @{typ "(float \<times> float) option"} o map term_of_float_float_option;

  fun nat_of_term t = @{code nat_of_integer}
    (HOLogic.dest_nat t handle TERM _ => snd (HOLogic.dest_number t));

  fun float_of_term (@{term Float} $ k $ l) =
        @{code Float} (dest_int k, dest_int l)
    | float_of_term t = bad t;

  fun floatarith_of_term (@{term Add} $ a $ b) = @{code Add} (floatarith_of_term a, floatarith_of_term b)
    | floatarith_of_term (@{term Minus} $ a) = @{code Minus} (floatarith_of_term a)
    | floatarith_of_term (@{term Mult} $ a $ b) = @{code Mult} (floatarith_of_term a, floatarith_of_term b)
    | floatarith_of_term (@{term Inverse} $ a) = @{code Inverse} (floatarith_of_term a)
    | floatarith_of_term (@{term Cos} $ a) = @{code Cos} (floatarith_of_term a)
    | floatarith_of_term (@{term Arctan} $ a) = @{code Arctan} (floatarith_of_term a)
    | floatarith_of_term (@{term Abs} $ a) = @{code Abs} (floatarith_of_term a)
    | floatarith_of_term (@{term Max} $ a $ b) = @{code Max} (floatarith_of_term a, floatarith_of_term b)
    | floatarith_of_term (@{term Min} $ a $ b) = @{code Min} (floatarith_of_term a, floatarith_of_term b)
    | floatarith_of_term @{term Pi} = @{code Pi}
    | floatarith_of_term (@{term Sqrt} $ a) = @{code Sqrt} (floatarith_of_term a)
    | floatarith_of_term (@{term Exp} $ a) = @{code Exp} (floatarith_of_term a)
    | floatarith_of_term (@{term Ln} $ a) = @{code Ln} (floatarith_of_term a)
    | floatarith_of_term (@{term Power} $ a $ n) =
        @{code Power} (floatarith_of_term a, nat_of_term n)
    | floatarith_of_term (@{term Var} $ n) = @{code Var} (nat_of_term n)
    | floatarith_of_term (@{term Num} $ m) = @{code Num} (float_of_term m)
    | floatarith_of_term t = bad t;

  fun form_of_term (@{term Bound} $ a $ b $ c $ p) = @{code Bound}
        (floatarith_of_term a, floatarith_of_term b, floatarith_of_term c, form_of_term p)
    | form_of_term (@{term Assign} $ a $ b $ p) = @{code Assign}
        (floatarith_of_term a, floatarith_of_term b, form_of_term p)
    | form_of_term (@{term Less} $ a $ b) = @{code Less}
        (floatarith_of_term a, floatarith_of_term b)
    | form_of_term (@{term LessEqual} $ a $ b) = @{code LessEqual}
        (floatarith_of_term a, floatarith_of_term b)
    | form_of_term (@{term AtLeastAtMost} $ a $ b $ c) = @{code AtLeastAtMost}
        (floatarith_of_term a, floatarith_of_term b, floatarith_of_term c)
    | form_of_term t = bad t;

  fun float_float_option_of_term @{term "None :: (float \<times> float) option"} = NONE
    | float_float_option_of_term (@{term "Some :: float \<times> float \<Rightarrow> _"} $ ff) =
        SOME (pairself float_of_term (HOLogic.dest_prod ff))
    | float_float_option_of_term (@{term approx'} $ n $ a $ ffs) = @{code approx'}
        (nat_of_term n) (floatarith_of_term a) (float_float_option_list_of_term ffs)
    | float_float_option_of_term t = bad t
  and float_float_option_list_of_term
        (@{term "replicate :: _ \<Rightarrow> (float \<times> float) option \<Rightarrow> _"} $ n $ @{term "None :: (float \<times> float) option"}) =
          @{code replicate} (nat_of_term n) NONE
    | float_float_option_list_of_term (@{term approx_form_eval} $ n $ p $ ffs) =
        @{code approx_form_eval} (nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs)
    | float_float_option_list_of_term t = map float_float_option_of_term
        (HOLogic.dest_list t);

  val nat_list_of_term = map nat_of_term o HOLogic.dest_list ;

  fun bool_of_term (@{term approx_form} $ n $ p $ ffs $ ms) = @{code approx_form}
        (nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs) (nat_list_of_term ms)
    | bool_of_term (@{term approx_tse_form} $ m $ n $ q $ p) =
        @{code approx_tse_form} (nat_of_term m) (nat_of_term n) (nat_of_term q) (form_of_term p)
    | bool_of_term t = bad t;

  fun eval t = case fastype_of t
   of @{typ bool} =>
        (term_of_bool o bool_of_term) t
    | @{typ "(float \<times> float) option"} =>
        (term_of_float_float_option o float_float_option_of_term) t
    | @{typ "(float \<times> float) option list"} =>
        (term_of_float_float_option_list o float_float_option_list_of_term) t
    | _ => bad t;

  val normalize = eval o Envir.beta_norm o Envir.eta_long [];

in Thm.cterm_of thy (Logic.mk_equals (t, normalize t)) end
*}

ML {*
  fun reorder_bounds_tac prems i =
    let
      fun variable_of_bound (Const (@{const_name Trueprop}, _) $
                             (Const (@{const_name Set.member}, _) $
                              Free (name, _) $ _)) = name
        | variable_of_bound (Const (@{const_name Trueprop}, _) $
                             (Const (@{const_name HOL.eq}, _) $
                              Free (name, _) $ _)) = name
        | variable_of_bound t = raise TERM ("variable_of_bound", [t])

      val variable_bounds
        = map (` (variable_of_bound o prop_of)) prems

      fun add_deps (name, bnds)
        = Graph.add_deps_acyclic (name,
            remove (op =) name (Term.add_free_names (prop_of bnds) []))

      val order = Graph.empty
                  |> fold Graph.new_node variable_bounds
                  |> fold add_deps variable_bounds
                  |> Graph.strong_conn |> map the_single |> rev
                  |> map_filter (AList.lookup (op =) variable_bounds)

      fun prepend_prem th tac
        = tac THEN rtac (th RSN (2, @{thm mp})) i
    in
      fold prepend_prem order all_tac
    end

  fun approximation_conv ctxt ct =
    approximation_oracle (Proof_Context.theory_of ctxt, Thm.term_of ct |> tap (tracing o Syntax.string_of_term ctxt));

  fun approximate ctxt t =
    approximation_oracle (Proof_Context.theory_of ctxt, t)
    |> Thm.prop_of |> Logic.dest_equals |> snd;

  (* Should be in HOL.thy ? *)
  fun gen_eval_tac conv ctxt = CONVERSION
    (Object_Logic.judgment_conv (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt))
    THEN' rtac TrueI

  val form_equations = @{thms interpret_form_equations};

  fun rewrite_interpret_form_tac ctxt prec splitting taylor i st = let
      fun lookup_splitting (Free (name, _))
        = case AList.lookup (op =) splitting name
          of SOME s => HOLogic.mk_number @{typ nat} s
           | NONE => @{term "0 :: nat"}
      val vs = nth (prems_of st) (i - 1)
               |> Logic.strip_imp_concl
               |> HOLogic.dest_Trueprop
               |> Term.strip_comb |> snd |> List.last
               |> HOLogic.dest_list
      val p = prec
              |> HOLogic.mk_number @{typ nat}
              |> Thm.cterm_of (Proof_Context.theory_of ctxt)
    in case taylor
    of NONE => let
         val n = vs |> length
                 |> HOLogic.mk_number @{typ nat}
                 |> Thm.cterm_of (Proof_Context.theory_of ctxt)
         val s = vs
                 |> map lookup_splitting
                 |> HOLogic.mk_list @{typ nat}
                 |> Thm.cterm_of (Proof_Context.theory_of ctxt)
       in
         (rtac (Thm.instantiate ([], [(@{cpat "?n::nat"}, n),
                                     (@{cpat "?prec::nat"}, p),
                                     (@{cpat "?ss::nat list"}, s)])
              @{thm "approx_form"}) i
          THEN simp_tac (put_simpset (simpset_of @{context}) ctxt) i) st
       end

     | SOME t => if length vs <> 1 then raise (TERM ("More than one variable used for taylor series expansion", [prop_of st]))
       else let
         val t = t
              |> HOLogic.mk_number @{typ nat}
              |> Thm.cterm_of (Proof_Context.theory_of ctxt)
         val s = vs |> map lookup_splitting |> hd
              |> Thm.cterm_of (Proof_Context.theory_of ctxt)
       in
         rtac (Thm.instantiate ([], [(@{cpat "?s::nat"}, s),
                                     (@{cpat "?t::nat"}, t),
                                     (@{cpat "?prec::nat"}, p)])
              @{thm "approx_tse_form"}) i st
       end
    end

  val free = Args.context -- Args.term >> (fn (_, Free (n, _)) => n | (ctxt, t) =>
    error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
*}

lemma intervalE: "a \<le> x \<and> x \<le> b \<Longrightarrow> \<lbrakk> x \<in> { a .. b } \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
  by auto

lemma meta_eqE: "x \<equiv> a \<Longrightarrow> \<lbrakk> x = a \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
  by auto

method_setup approximation = {*
  Scan.lift Parse.nat
  --
  Scan.optional (Scan.lift (Args.$$$ "splitting" |-- Args.colon)
    |-- Parse.and_list' (free --| Scan.lift (Args.$$$ "=") -- Scan.lift Parse.nat)) []
  --
  Scan.option (Scan.lift (Args.$$$ "taylor" |-- Args.colon)
    |-- (free |-- Scan.lift (Args.$$$ "=") |-- Scan.lift Parse.nat))
  >>
  (fn ((prec, splitting), taylor) => fn ctxt =>
    SIMPLE_METHOD' (fn i =>
      REPEAT (FIRST' [etac @{thm intervalE},
                      etac @{thm meta_eqE},
                      rtac @{thm impI}] i)
      THEN Subgoal.FOCUS (fn {prems, ...} => reorder_bounds_tac prems i) ctxt i
      THEN DETERM (TRY (filter_prems_tac (K false) i))
      THEN DETERM (Reification.tac ctxt form_equations NONE i)
      THEN rewrite_interpret_form_tac ctxt prec splitting taylor i
      THEN gen_eval_tac (approximation_conv ctxt) ctxt i))
*} "real number approximation"

ML_file "approximation.ML"

end