Quelle  Float.thy

(*  Title:      HOL/Library/Float.thy
  Author: Johannes Hölzl, Fabian Immler
  Copyright 2012 TU München
*)

section ‹Floating-Point Numbers›

theory Float
imports Log_Nat Lattice_Algebras
begin

definition "float = {m * 2 powr e | (m :: int) (e :: int). True}"

typedef float = float
  morphisms real_of_float float_of
  unfolding float_def by auto

setup_lifting type_definition_float

declare real_of_float [code_unfold]

lemmas float_of_inject[simp]

declare [[coercion "real_of_float :: float ==> real"]]

lemma real_of_float_eq: "f1 = f2 ⟷ real_of_float f1 = real_of_float f2" for f1 f2 :: float
  unfolding real_of_float_inject ..

declare real_of_float_inverse[simp] float_of_inverse [simp]
declare real_of_float [simp]


subsection ‹Real operations preserving the representation as floating point number›

lemma floatI: "m * 2 powr e = x ==> x ∈ float" for m e :: int
  by (auto simp: float_def)

lemma zero_float[simp]: "0 ∈ float"
  by (auto simp: float_def)

lemma one_float[simp]: "1 ∈ float"
  by (intro floatI[of 1 0]) simp

lemma numeral_float[simp]: "numeral i ∈ float"
  by (intro floatI[of "numeral i" 0]) simp

lemma neg_numeral_float[simp]: "- numeral i ∈ float"
  by (intro floatI[of "- numeral i" 0]) simp

lemma real_of_int_float[simp]: "real_of_int x ∈ float" for x :: int
  by (intro floatI[of x 0]) simp

lemma real_of_nat_float[simp]: "real x ∈ float" for x :: nat
  by (intro floatI[of x 0]) simp

lemma two_powr_int_float[simp]: "2 powr (real_of_int i) ∈ float" for i :: int
  by (intro floatI[of 1 i]) simp

lemma two_powr_nat_float[simp]: "2 powr (real i) ∈ float" for i :: nat
  by (intro floatI[of 1 i]) simp

lemma two_powr_minus_int_float[simp]: "2 powr - (real_of_int i) ∈ float" for i :: int
  by (intro floatI[of 1 "-i"]) simp

lemma two_powr_minus_nat_float[simp]: "2 powr - (real i) ∈ float" for i :: nat
  by (intro floatI[of 1 "-i"]) simp

lemma two_powr_numeral_float[simp]: "2 powr numeral i ∈ float"
  by (intro floatI[of 1 "numeral i"]) simp

lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i ∈ float"
  by (intro floatI[of 1 "- numeral i"]) simp

lemma two_pow_float[simp]: "2 ^ n ∈ float"
  by (intro floatI[of 1 n]) (simp add: powr_realpow)


lemma plus_float[simp]: "r ∈ float ==> p ∈ float ==> r + p ∈ float"
  unfolding float_def
proof (safe, simp)
  have *: "∃(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
    if "e1 ≤ e2" for e1 m1 e2 m2 :: int
  proof -
    from that have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
      by (simp add: powr_diff field_simps flip: powr_realpow)
    then show ?thesis
      by blast
  qed
  fix e1 m1 e2 m2 :: int
  consider "e2 ≤ e1" | "e1 ≤ e2" by (rule linorder_le_cases)
  then show "∃(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
  proof cases
    case 1
    from *[OF this, of m2 m1] show ?thesis
      by (simp add: ac_simps)
  next
    case 2
    then show ?thesis by (rule *)
  qed
qed

lemma uminus_float[simp]: "x ∈ float ==> -x ∈ float"
  by (simp add: float_def) (metis mult_minus_left of_int_minus)

lemma times_float[simp]: "x ∈ float ==> y ∈ float ==> x * y ∈ float"
  apply (clarsimp simp: float_def mult_ac)
  by (metis mult.assoc of_int_mult of_int_add powr_add)

lemma minus_float[simp]: "x ∈ float ==> y ∈ float ==> x - y ∈ float"
  using plus_float [of x "- y"] by simp

lemma abs_float[simp]: "x ∈ float ==> ∣x∣ ∈ float"
  by (cases x rule: linorder_cases[of 0]) auto

lemma sgn_of_float[simp]: "x ∈ float ==> sgn x ∈ float"
  by (simp add: sgn_real_def)

lemma div_power_2_float[simp]: "x ∈ float ==> x / 2^d ∈ float" 
  by (simp add: float_def) (metis of_int_diff of_int_of_nat_eq powr_diff powr_realpow zero_less_numeral times_divide_eq_right)

lemma div_power_2_int_float[simp]: "x ∈ float ==> x / (2::int)^d ∈ float"
  by simp

lemma div_numeral_Bit0_float[simp]:
  assumes "x / numeral n ∈ float"
  shows "x / (numeral (Num.Bit0 n)) ∈ float"
proof -
  have "(x / numeral n) / 2^1 ∈ float"
    by (intro assms div_power_2_float)
  also have "(x / numeral n) / 2^1 = x / (numeral (Num.Bit0 n))"
    by (induct n) auto
  finally show ?thesis .
qed

lemma div_neg_numeral_Bit0_float[simp]:
  assumes "x / numeral n ∈ float"
  shows "x / (- numeral (Num.Bit0 n)) ∈ float"
  using assms by force

lemma power_float[simp]:
  assumes "a ∈ float"
  shows "a ^ b ∈ float"
proof -
  from assms obtain m e :: int where "a = m * 2 powr e"
    by (auto simp: float_def)
  then show ?thesis
    by (intro floatI[where m="m^b" and e = "e*b"])
       (auto simp: powr_powr power_mult_distrib simp flip: powr_realpow)
qed

lift_definition Float :: "int ==> int ==> float" is "λ(m::int) (e::int). m * 2 powr e"
  by simp
declare Float.rep_eq[simp]

code_datatype Float

lemma compute_real_of_float[code]:
  "real_of_float (Float m e) = (if e ≥ 0 then m * 2 ^ nat e else m / 2 ^ (nat (-e)))"
  by (simp add: powr_int)


subsection ‹Arithmetic operations on floating point numbers›

instantiation float :: "{ring_1,linorder,linordered_ring,linordered_idom,numeral,equal}"
begin

lift_definition zero_float :: float is 0 by simp
declare zero_float.rep_eq[simp]

lift_definition one_float :: float is 1 by simp
declare one_float.rep_eq[simp]

lift_definition plus_float :: "float ==> float ==> float" is "(+)" by simp
declare plus_float.rep_eq[simp]

lift_definition times_float :: "float ==> float ==> float" is "(*)" by simp
declare times_float.rep_eq[simp]

lift_definition minus_float :: "float ==> float ==> float" is "(-)" by simp
declare minus_float.rep_eq[simp]

lift_definition uminus_float :: "float ==> float" is "uminus" by simp
declare uminus_float.rep_eq[simp]

lift_definition abs_float :: "float ==> float" is abs by simp
declare abs_float.rep_eq[simp]

lift_definition sgn_float :: "float ==> float" is sgn by simp
declare sgn_float.rep_eq[simp]

lift_definition equal_float :: "float ==> float ==> bool" is "(=) :: real ==> real ==> bool" .

lift_definition less_eq_float :: "float ==> float ==> bool" is "(≤)" .
declare less_eq_float.rep_eq[simp]

lift_definition less_float :: "float ==> float ==> bool" is "(<)" .
declare less_float.rep_eq[simp]

instance
  by standard (transfer; fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+

end

lemma real_of_float [simp]: "real_of_float (of_nat n) = of_nat n"
  by (induct n) simp_all

lemma real_of_float_of_int_eq [simp]: "real_of_float (of_int z) = of_int z"
  by (cases z rule: int_diff_cases) (simp_all add: of_rat_diff)

lemma Float_0_eq_0[simp]: "Float 0 e = 0"
  by transfer simp

lemma real_of_float_power[simp]: "real_of_float (f^n) = real_of_float f^n" for f :: float
  by (induct n) simp_all

lemma real_of_float_min: "real_of_float (min x y) = min (real_of_float x) (real_of_float y)"
  and real_of_float_max: "real_of_float (max x y) = max (real_of_float x) (real_of_float y)"
  for x y :: float
  by (simp_all add: min_def max_def)

instance float :: unbounded_dense_linorder
proof
  fix a b :: float
  show "∃c. a < c"
    by (metis Float.real_of_float less_float.rep_eq reals_Archimedean2)
  show "∃c. c < a"
    by (metis add_0 add_strict_right_mono neg_less_0_iff_less zero_less_one)
  show "∃c. a < c ∧ c < b" if "a < b"
    apply (rule exI[of _ "(a + b) * Float 1 (- 1)"])
    using that
    apply transfer
    apply (simp add: powr_minus)
    done
qed

instantiation float :: lattice_ab_group_add
begin

definition inf_float :: "float ==> float ==> float"
  where "inf_float a b = min a b"

definition sup_float :: "float ==> float ==> float"
  where "sup_float a b = max a b"

instance
  by standard (transfer; simp add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+

end

lemma float_numeral[simp]: "real_of_float (numeral x :: float) = numeral x"
  by (metis of_int_numeral real_of_float_of_int_eq)

lemma transfer_numeral [transfer_rule]:
  "rel_fun (=) pcr_float (numeral :: _ ==> real) (numeral :: _ ==> float)"
  by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)

lemma float_neg_numeral[simp]: "real_of_float (- numeral x :: float) = - numeral x"
  by simp

lemma transfer_neg_numeral [transfer_rule]:
  "rel_fun (=) pcr_float (- numeral :: _ ==> real) (- numeral :: _ ==> float)"
  by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)

lemma float_of_numeral: "numeral k = float_of (numeral k)"
  and float_of_neg_numeral: "- numeral k = float_of (- numeral k)"
  unfolding real_of_float_eq by simp_all


subsection ‹Quickcheck›

instantiation float :: exhaustive
begin

definition exhaustive_float where
  "exhaustive_float f d =
    Quickcheck_Exhaustive.exhaustive (λx. Quickcheck_Exhaustive.exhaustive (λy. f (Float x y)) d) d"

instance ..

end

context
  includes term_syntax
begin

definition [code_unfold]:
  "valtermify_float x y = Code_Evaluation.valtermify Float {⋅} x {⋅} y"

end

instantiation float :: full_exhaustive
begin

definition
  "full_exhaustive_float f d =
    Quickcheck_Exhaustive.full_exhaustive
      (λx. Quickcheck_Exhaustive.full_exhaustive (λy. f (valtermify_float x y)) d) d"

instance ..

end

instantiation float :: random
begin

definition "Quickcheck_Random.random i =
  scomp (Quickcheck_Random.random (2 ^ nat_of_natural i))
    (λman. scomp (Quickcheck_Random.random i) (λexp. Pair (valtermify_float man exp)))"

instance ..

end


subsection ‹Represent floats as unique mantissa and exponent›

lemma int_induct_abs[case_names less]:
  fixes j :: int
  assumes H: "∧n. (∧i. ∣i∣ < ∣n∣ ==> P i) ==> P n"
  shows "P j"
proof (induct "nat ∣j∣" arbitrary: j rule: less_induct)
  case less
  show ?case by (rule H[OF less]) simp
qed

lemma int_cancel_factors:
  fixes n :: int
  assumes "1 < r"
  shows "n = 0 ∨ (∃k i. n = k * r ^ i ∧ ¬ r dvd k)"
proof (induct n rule: int_induct_abs)
  case (less n)
  have "∃k i. n = k * r ^ Suc i ∧ ¬ r dvd k" if "n ≠ 0" "n = m * r" for m
  proof -
    from that have "∣m ∣ < ∣n∣"
      using ‹1 < r› by (simp add: abs_mult)
    from less[OF this] that show ?thesis by auto
  qed
  then show ?case
    by (metis dvd_def monoid_mult_class.mult.right_neutral mult.commute power_0)
qed

lemma mult_powr_eq_mult_powr_iff_asym:
  fixes m1 m2 e1 e2 :: int
  assumes m1: "¬ 2 dvd m1"
    and "e1 ≤ e2"
  shows "m1 * 2 powr e1 = m2 * 2 powr e2 ⟷ m1 = m2 ∧ e1 = e2"
  (is "?lhs ⟷ ?rhs")
proof
  show ?rhs if eq: ?lhs
  proof -
    have "m1 ≠ 0"
      using m1 unfolding dvd_def by auto
    from ‹e1 ≤ e2› eq have "m1 = m2 * 2 powr nat (e2 - e1)"
      by (simp add: powr_diff field_simps)
    also have "… = m2 * 2^nat (e2 - e1)"
      by (simp add: powr_realpow)
    finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
      by linarith
    with m1 have "m1 = m2"
      by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
    then show ?thesis
      using eq ‹m1 ≠ 0› by (simp add: powr_inj)
  qed
  show ?lhs if ?rhs
    using that by simp
qed

lemma mult_powr_eq_mult_powr_iff:
  "¬ 2 dvd m1 ==> ¬ 2 dvd m2 ==> m1 * 2 powr e1 = m2 * 2 powr e2 ⟷ m1 = m2 ∧ e1 = e2"
  for m1 m2 e1 e2 :: int
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
  by (cases e1 e2 rule: linorder_le_cases) auto

lemma floatE_normed:
  assumes x: "x ∈ float"
  obtains (zero) "x = 0"
   | (powr) m e :: int where "x = m * 2 powr e" "¬ 2 dvd m" "x ≠ 0"
proof -
  have "∃(m::int) (e::int). x = m * 2 powr e ∧ ¬ (2::int) dvd m" if "x ≠ 0"
  proof -
    from x obtain m e :: int where x: "x = m * 2 powr e"
      by (auto simp: float_def)
    with ‹x ≠ 0› int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "¬ 2 dvd k"
      by auto
    with ‹¬ 2 dvd k› x have "x = real_of_int k * 2 powr real_of_int (e + int i) ∧ odd k"
      by (simp add: powr_add powr_realpow)
    then show ?thesis
      by blast
  qed
  with that show thesis by blast
qed

lemma float_normed_cases:
  fixes f :: float
  obtains (zero) "f = 0"
   | (powr) m e :: int where "real_of_float f = m * 2 powr e" "¬ 2 dvd m" "f ≠ 0"
proof (atomize_elim, induct f)
  case (float_of y)
  then show ?case
    by (cases rule: floatE_normed) (auto simp: zero_float_def)
qed

definition mantissa :: "float ==> int"
  where "mantissa f =
    fst (SOME p::int × int. (f = 0 ∧ fst p = 0 ∧ snd p = 0) ∨
      (f ≠ 0 ∧ real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) ∧ ¬ 2 dvd fst p))"

definition exponent :: "float ==> int"
  where "exponent f =
    snd (SOME p::int × int. (f = 0 ∧ fst p = 0 ∧ snd p = 0) ∨
      (f ≠ 0 ∧ real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) ∧ ¬ 2 dvd fst p))"

lemma exponent_0[simp]: "exponent 0 = 0" (is ?E)
  and mantissa_0[simp]: "mantissa 0 = 0" (is ?M)
proof -
  have "∧p::int × int. fst p = 0 ∧ snd p = 0 ⟷ p = (0, 0)"
    by auto
  then show ?E ?M
    by (auto simp add: mantissa_def exponent_def zero_float_def)
qed

lemma mantissa_exponent: "real_of_float f = mantissa f * 2 powr exponent f" (is ?E)
  and mantissa_not_dvd: "f ≠ 0 ==> ¬ 2 dvd mantissa f" (is "_ ==> ?D")
proof cases
  assume [simp]: "f ≠ 0"
  have "f = mantissa f * 2 powr exponent f ∧ ¬ 2 dvd mantissa f"
  proof (cases f rule: float_normed_cases)
    case zero
    then show ?thesis by simp
  next
    case (powr m e)
    then have "∃p::int × int. (f = 0 ∧ fst p = 0 ∧ snd p = 0) ∨
      (f ≠ 0 ∧ real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) ∧ ¬ 2 dvd fst p)"
      by auto
    then show ?thesis
      unfolding exponent_def mantissa_def
      by (rule someI2_ex) simp
  qed
  then show ?E ?D by auto
qed simp

lemma mantissa_noteq_0: "f ≠ 0 ==> mantissa f ≠ 0"
  using mantissa_not_dvd[of f] by auto

lemma mantissa_eq_zero_iff: "mantissa x = 0 ⟷ x = 0"
  (is "?lhs ⟷ ?rhs")
proof
  show ?rhs if ?lhs
  proof -
    from that have z: "0 = real_of_float x"
      using mantissa_exponent by simp
    show ?thesis
      by (simp add: zero_float_def z)
  qed
  show ?lhs if ?rhs
    using that by simp
qed

lemma mantissa_pos_iff: "0 < mantissa x ⟷ 0 < x"
  by (auto simp: mantissa_exponent algebra_split_simps)

lemma mantissa_nonneg_iff: "0 ≤ mantissa x ⟷ 0 ≤ x"
  by (auto simp: mantissa_exponent algebra_split_simps)

lemma mantissa_neg_iff: "0 > mantissa x ⟷ 0 > x"
  by (auto simp: mantissa_exponent algebra_split_simps)

lemma
  fixes m e :: int
  defines "f ≡ float_of (m * 2 powr e)"
  assumes dvd: "¬ 2 dvd m"
  shows mantissa_float: "mantissa f = m" (is "?M")
    and exponent_float: "m ≠ 0 ==> exponent f = e" (is "_ ==> ?E")
proof cases
  assume "m = 0"
  with dvd show "mantissa f = m" by auto
next
  assume "m ≠ 0"
  then have f_not_0: "f ≠ 0" by (simp add: f_def zero_float_def)
  from mantissa_exponent[of f] have "m * 2 powr e = mantissa f * 2 powr exponent f"
    by (auto simp add: f_def)
  then show ?M ?E
    using mantissa_not_dvd[OF f_not_0] dvd
    by (auto simp: mult_powr_eq_mult_powr_iff)
qed


subsection ‹Compute arithmetic operations›

lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
  unfolding real_of_float_eq mantissa_exponent[of f] by simp

lemma Float_cases [cases type: float]:
  fixes f :: float
  obtains (Float) m e :: int where "f = Float m e"
  using Float_mantissa_exponent[symmetric]
  by (atomize_elim) auto

lemma denormalize_shift:
  assumes f_def: "f = Float m e"
    and not_0: "f ≠ 0"
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
proof
  from mantissa_exponent[of f] f_def
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
    by simp
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
    by (simp add: powr_diff field_simps)
  moreover
  have "e ≤ exponent f"
  proof (rule ccontr)
    assume "¬ e ≤ exponent f"
    then have pos: "exponent f < e" by simp
    then have "2 powr (exponent f - e) = 2 powr - real_of_int (e - exponent f)"
      by simp
    also have "… = 1 / 2^nat (e - exponent f)"
      using pos by (simp flip: powr_realpow add: powr_diff)
    finally have "m * 2^nat (e - exponent f) = real_of_int (mantissa f)"
      using eq by simp
    then have "mantissa f = m * 2^nat (e - exponent f)"
      by linarith
    with ‹exponent f < e› have "2 dvd mantissa f"
      by (force intro: dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
    then show False using mantissa_not_dvd[OF not_0] by simp
  qed
  ultimately have "real_of_int m = mantissa f * 2^nat (exponent f - e)"
    by (simp flip: powr_realpow)
  with ‹e ≤ exponent f›
  show "m = mantissa f * 2 ^ nat (exponent f - e)"
    by linarith
  show "e = exponent f - nat (exponent f - e)"
    using ‹e ≤ exponent f› by auto
qed

context
begin

qualified lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
  by transfer simp

qualified lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
  by transfer simp

lift_definition normfloat :: "float ==> float" is "λx. x" .
lemma normloat_id[simp]: "normfloat x = x" by transfer rule

qualified lemma compute_normfloat[code]:
  "normfloat (Float m e) =
    (if m mod 2 = 0 ∧ m ≠ 0 then normfloat (Float (m div 2) (e + 1))
     else if m = 0 then 0 else Float m e)"
  by transfer (auto simp add: powr_add zmod_eq_0_iff)

qualified lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
  by transfer simp

qualified lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
  by transfer simp

qualified lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
  by transfer simp

qualified lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
  by transfer (simp add: field_simps powr_add)

qualified lemma compute_float_plus[code]:
  "Float m1 e1 + Float m2 e2 =
    (if m1 = 0 then Float m2 e2
     else if m2 = 0 then Float m1 e1
     else if e1 ≤ e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
     else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
  by transfer (simp add: field_simps powr_realpow[symmetric] powr_diff)

qualified lemma compute_float_minus[code]: "f - g = f + (-g)" for f g :: float
  by simp

qualified lemma compute_float_sgn[code]:
  "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
  by transfer (simp add: sgn_mult)

lift_definition is_float_pos :: "float ==> bool" is "(<) 0 :: real ==> bool" .

qualified lemma compute_is_float_pos[code]: "is_float_pos (Float m e) ⟷ 0 < m"
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])

lift_definition is_float_nonneg :: "float ==> bool" is "(≤) 0 :: real ==> bool" .

qualified lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) ⟷ 0 ≤ m"
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])

lift_definition is_float_zero :: "float ==> bool" is "(=) 0 :: real ==> bool" .

qualified lemma compute_is_float_zero[code]: "is_float_zero (Float m e) ⟷ 0 = m"
  by transfer (auto simp add: is_float_zero_def)

qualified lemma compute_float_abs[code]: "∣Float m e∣ = Float ∣m∣ e"
  by transfer (simp add: abs_mult)

qualified lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
  by transfer simp

end


subsection ‹Lemmas for types 🍋‹real›, 🍋‹nat›, 🍋‹int›\
 
 lemmas real_of_ints =
  of_int_add
  of_int_minus
  of_int_diff
  of_int_mult
  of_int_power
  of_int_numeral of_int_neg_numeral
 
 lemmas int_of_reals = real_of_ints[symmetric]
 
 
 subsection ‹Rounding Real Numbers›
 
 definition round_down :: "int ==> real ==> real"
  where "round_down prec x = ⌊x * 2 powr prec⌋ * 2 powr -prec"
 
 definition round_up :: "int ==> real ==> real"
  where "round_up prec x = ⌈x * 2 powr prec⌉ * 2 powr -prec"
 
 lemma round_down_float[simp]: "round_down prec x ∈ float"
  unfolding round_down_def
  by (auto intro!: times_float simp flip: of_int_minus)
 
 lemma round_up_float[simp]: "round_up prec x ∈ float"
  unfolding round_up_def
  by (auto intro!: times_float simp flip: of_int_minus)
 
 lemma round_up: "x ≤ round_up prec x"
  by (simp add: powr_minus_divide le_divide_eq round_up_def ceiling_correct)
 
 lemma round_down: "round_down prec x ≤ x"
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
 
 lemma round_up_0[simp]: "round_up p 0 = 0"
  unfolding round_up_def by simp
 
 lemma round_down_0[simp]: "round_down p 0 = 0"
  unfolding round_down_def by simp
 
 lemma round_up_diff_round_down: "round_up prec x - round_down prec x ≤ 2 powr -prec"
 proof -
  have "round_up prec x - round_down prec x = (⌈x * 2 powr prec⌉ - ⌊x * 2 powr prec⌋) * 2 powr -prec"
  by (simp add: round_up_def round_down_def field_simps)
  also have "… ≤ 1 * 2 powr -prec"
  by (rule mult_mono)
  (auto simp flip: of_int_diff simp: ceiling_diff_floor_le_1)
  finally show ?thesis by simp
 qed
 
 lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
  unfolding round_down_def
  by (simp add: powr_add powr_mult field_simps powr_diff)
  (simp flip: powr_add)
 
 lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
  unfolding round_up_def
  by (simp add: powr_add powr_mult field_simps powr_diff)
  (simp flip: powr_add)
 
 lemma round_up_uminus_eq: "round_up p (-x) = - round_down p x"
  and round_down_uminus_eq: "round_down p (-x) = - round_up p x"
  by (auto simp: round_up_def round_down_def ceiling_def)
 
 lemma round_up_mono: "x ≤ y ==> round_up p x ≤ round_up p y"
  by (auto intro!: ceiling_mono simp: round_up_def)
 
 lemma round_up_le1:
  assumes "x ≤ 1" "prec ≥ 0"
  shows "round_up prec x ≤ 1"
 proof -
  have "real_of_int ⌈x * 2 powr prec⌉ ≤ real_of_int ⌈2 powr real_of_int prec⌉"
  using assms by (auto intro!: ceiling_mono)
  also have "… = 2 powr prec" using assms by (auto simp: powr_int intro!: exI[where x="2^nat prec"])
  finally show ?thesis
  by (simp add: round_up_def) (simp add: powr_minus inverse_eq_divide)
 qed
 
 lemma round_up_less1:
  assumes "x < 1 / 2" "p > 0"
  shows "round_up p x < 1"
 proof -
  have "x * 2 powr p < 1 / 2 * 2 powr p"
  using assms by simp
  also have "… ≤ 2 powr p - 1" using ‹p > 0›
  by (auto simp: powr_diff powr_int field_simps self_le_power)
  finally show ?thesis using ‹p > 0›
  by (simp add: round_up_def field_simps powr_minus powr_int ceiling_less_iff)
 qed
 
 lemma round_down_ge1:
  assumes x: "x ≥ 1"
  assumes prec: "p ≥ - log 2 x"
  shows "1 ≤ round_down p x"
 proof cases
  assume nonneg: "0 ≤ p"
  have "2 powr p = real_of_int ⌊2 powr real_of_int p⌋"
  using nonneg by (auto simp: powr_int)
  also have "… ≤ real_of_int ⌊x * 2 powr p⌋"
  using assms by (auto intro!: floor_mono)
  finally show ?thesis
  by (simp add: round_down_def) (simp add: powr_minus inverse_eq_divide)
 next
  assume neg: "¬ 0 ≤ p"
  have "x = 2 powr (log 2 x)"
  using x by simp
  also have "2 powr (log 2 x) ≥ 2 powr - p"
  using prec by auto
  finally have x_le: "x ≥ 2 powr -p" .
 
  from neg have "2 powr real_of_int p ≤ 2 powr 0"
  by (intro powr_mono) auto
  also have "… ≤ ⌊2 powr 0::real⌋" by simp
  also have "… ≤ ⌊x * 2 powr (real_of_int p)⌋"
  unfolding of_int_le_iff
  using x x_le by (intro floor_mono) (simp add: powr_minus_divide field_simps)
  finally show ?thesis
  using prec x
  by (simp add: round_down_def powr_minus_divide pos_le_divide_eq)
 qed
 
 lemma round_up_le0: "x ≤ 0 ==> round_up p x ≤ 0"
  unfolding round_up_def
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
 
 
 subsection ‹Rounding Floats›
 
 definition div_twopow :: "int ==> nat ==> int"
  where [simp]: "div_twopow x n = x div (2 ^ n)"
 
 definition mod_twopow :: "int ==> nat ==> int"
  where [simp]: "mod_twopow x n = x mod (2 ^ n)"
 
 lemma compute_div_twopow[code]:
  "div_twopow x n = (if x = 0 ∨ x = -1 ∨ n = 0 then x else div_twopow (x div 2) (n - 1))"
  by (cases n) (auto simp: zdiv_zmult2_eq div_eq_minus1)
 
 lemma compute_mod_twopow[code]:
  "mod_twopow x n = (if n = 0 then 0 else x mod 2 + 2 * mod_twopow (x div 2) (n - 1))"
  by (cases n) (auto simp: zmod_zmult2_eq)
 
 lift_definition float_up :: "int ==> float ==> float" is round_up by simp
 declare float_up.rep_eq[simp]
 
 lemma round_up_correct: "round_up e f - f ∈ {0..2 powr -e}"
  unfolding atLeastAtMost_iff
 proof
  have "round_up e f - f ≤ round_up e f - round_down e f"
  using round_down by simp
  also have "… ≤ 2 powr -e"
  using round_up_diff_round_down by simp
  finally show "round_up e f - f ≤ 2 powr - (real_of_int e)"
  by simp
 qed (simp add: algebra_simps round_up)
 
 lemma float_up_correct: "real_of_float (float_up e f) - real_of_float f ∈ {0..2 powr -e}"
  by transfer (rule round_up_correct)
 
 lift_definition float_down :: "int ==> float ==> float" is round_down by simp
 declare float_down.rep_eq[simp]
 
 lemma round_down_correct: "f - (round_down e f) ∈ {0..2 powr -e}"
  unfolding atLeastAtMost_iff
 proof
  have "f - round_down e f ≤ round_up e f - round_down e f"
  using round_up by simp
  also have "… ≤ 2 powr -e"
  using round_up_diff_round_down by simp
  finally show "f - round_down e f ≤ 2 powr - (real_of_int e)"
  by simp
 qed (simp add: algebra_simps round_down)
 
 lemma float_down_correct: "real_of_float f - real_of_float (float_down e f) ∈ {0..2 powr -e}"
  by transfer (rule round_down_correct)
 
 context
 begin
 
 qualified lemma compute_float_down[code]:
  "float_down p (Float m e) =
  (if p + e < 0 then Float (div_twopow m (nat (-(p + e)))) (-p) else Float m e)"
 proof (cases "p + e < 0")
  case True
  then have "real_of_int ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
  using powr_realpow[of 2 "nat (-(p + e))"] by simp
  also have "… = 1 / 2 powr p / 2 powr e"
  unfolding powr_minus_divide of_int_minus by (simp add: powr_add)
  finally show ?thesis
  using ‹p + e < 0›
  apply transfer
  apply (simp add: round_down_def field_simps flip: floor_divide_of_int_eq powr_add)
  apply (metis (no_types, opaque_lifting) Float.rep_eq
  add.inverse_inverse compute_real_of_float diff_minus_eq_add
  floor_divide_of_int_eq int_of_reals(1) linorder_not_le
  minus_add_distrib of_int_eq_numeral_power_cancel_iff)
  done
 next
  case False
  then have r: "real_of_int e + real_of_int p = real (nat (e + p))"
  by simp
  have r: "⌊(m * 2 powr e) * 2 powr real_of_int p⌋ = (m * 2 powr e) * 2 powr real_of_int p"
  by (auto intro: exI[where x="m*2^nat (e+p)"]
  simp add: ac_simps powr_add[symmetric] r powr_realpow)
  with ‹¬ p + e < 0› show ?thesis
  by transfer (auto simp add: round_down_def field_simps powr_add powr_minus)
 qed
 
 lemma abs_round_down_le: "∣f - (round_down e f)∣ ≤ 2 powr -e"
  using round_down_correct[of f e] by simp
 
 lemma abs_round_up_le: "∣f - (round_up e f)∣ ≤ 2 powr -e"
  using round_up_correct[of e f] by simp
 
 lemma round_down_nonneg: "0 ≤ s ==> 0 ≤ round_down p s"
  by (auto simp: round_down_def)
 
 lemma ceil_divide_floor_conv:
  assumes "b ≠ 0"
  shows "⌈real_of_int a / real_of_int b⌉ =
  (if b dvd a then a div b else ⌊real_of_int a / real_of_int b⌋ + 1)"
 proof (cases "b dvd a")
  case True
  then show ?thesis
  by (simp add: ceiling_def floor_divide_of_int_eq dvd_neg_div
  flip: of_int_minus divide_minus_left)
 next
  case False
  then have "a mod b ≠ 0"
  by auto
  then have ne: "real_of_int (a mod b) / real_of_int b ≠ 0"
  using ‹b ≠ 0› by auto
  have "⌈real_of_int a / real_of_int b⌉ = ⌊real_of_int a / real_of_int b⌋ + 1"
  by (metis add_cancel_left_right ceiling_altdef floor_divide_of_int_eq ne of_int_div_aux)
  then show ?thesis
  using ‹¬ b dvd a› by simp
 qed
 
 qualified lemma compute_float_up[code]: "float_up p x = - float_down p (-x)"
  by transfer (simp add: round_down_uminus_eq)
 
 end
 
 
 lemma bitlen_Float:
  fixes m e
  defines [THEN meta_eq_to_obj_eq]: "f ≡ Float m e"
  shows "bitlen ∣mantissa f∣ + exponent f = (if m = 0 then 0 else bitlen ∣m∣ + e)"
 proof (cases "m = 0")
  case True
  then show ?thesis by (simp add: f_def bitlen_alt_def)
 next
  case False
  then have "f ≠ 0"
  unfolding real_of_float_eq by (simp add: f_def)
  then have "mantissa f ≠ 0"
  by (simp add: mantissa_eq_zero_iff)
  moreover
  obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
  by (rule f_def[THEN denormalize_shift, OF ‹f ≠ 0›])
  ultimately show ?thesis by (simp add: abs_mult)
 qed
 
 lemma float_gt1_scale:
  assumes "1 ≤ Float m e"
  shows "0 ≤ e + (bitlen m - 1)"
 proof -
  have "0 < Float m e" using assms by auto
  then have "0 < m" using powr_gt_zero[of 2 e]
  by (auto simp: zero_less_mult_iff)
  then have "m ≠ 0" by auto
  show ?thesis
  proof (cases "0 ≤ e")
  case True
  then show ?thesis
  using ‹0 < m› by (simp add: bitlen_alt_def)
  next
  case False
  have "(1::int) < 2" by simp
  let ?S = "2^(nat (-e))"
  have "inverse (2 ^ nat (- e)) = 2 powr e"
  using assms False powr_realpow[of 2 "nat (-e)"]
  by (auto simp: powr_minus field_simps)
  then have "1 ≤ real_of_int m * inverse ?S"
  using assms False powr_realpow[of 2 "nat (-e)"]
  by (auto simp: powr_minus)
  then have "1 * ?S ≤ real_of_int m * inverse ?S * ?S"
  by (rule mult_right_mono) auto
  then have "?S ≤ real_of_int m"
  unfolding mult.assoc by auto
  then have "?S ≤ m"
  unfolding of_int_le_iff[symmetric] by auto
  from this bitlen_bounds[OF ‹0 < m›, THEN conjunct2]
  have "nat (-e) < (nat (bitlen m))"
  unfolding power_strict_increasing_iff[OF ‹1 < 2›, symmetric]
  by (rule order_le_less_trans)
  then have "-e < bitlen m"
  using False by auto
  then show ?thesis
  by auto
  qed
 qed
 
 
 subsection ‹Truncating Real Numbers›
 
 definition truncate_down::"nat ==> real ==> real"
  where "truncate_down prec x = round_down (prec - ⌊log 2 ∣x∣⌋) x"
 
 lemma truncate_down: "truncate_down prec x ≤ x"
  using round_down by (simp add: truncate_down_def)
 
 lemma truncate_down_le: "x ≤ y ==> truncate_down prec x ≤ y"
  by (rule order_trans[OF truncate_down])
 
 lemma truncate_down_zero[simp]: "truncate_down prec 0 = 0"
  by (simp add: truncate_down_def)
 
 lemma truncate_down_float[simp]: "truncate_down p x ∈ float"
  by (auto simp: truncate_down_def)
 
 definition truncate_up::"nat ==> real ==> real"
  where "truncate_up prec x = round_up (prec - ⌊log 2 ∣x∣⌋) x"
 
 lemma truncate_up: "x ≤ truncate_up prec x"
  using round_up by (simp add: truncate_up_def)
 
 lemma truncate_up_le: "x ≤ y ==> x ≤ truncate_up prec y"
  by (rule order_trans[OF _ truncate_up])
 
 lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0"
  by (simp add: truncate_up_def)
 
 lemma truncate_up_uminus_eq: "truncate_up prec (-x) = - truncate_down prec x"
  and truncate_down_uminus_eq: "truncate_down prec (-x) = - truncate_up prec x"
  by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
 
 lemma truncate_up_float[simp]: "truncate_up p x ∈ float"
  by (auto simp: truncate_up_def)
 
 lemma mult_powr_eq: "0 < b ==> b ≠ 1 ==> 0 < x ==> x * b powr y = b powr (y + log b x)"
  by (simp_all add: powr_add)
 
 lemma truncate_down_pos:
  assumes "x > 0"
  shows "truncate_down p x > 0"
 proof -
  have "0 ≤ log 2 x - real_of_int ⌊log 2 x⌋"
  by (simp add: algebra_simps)
  moreover have "0 ≤ real p - real_of_int ⌊log 2 x⌋ + log 2 x"
  by linarith
  ultimately show ?thesis
  using assms
  by (auto simp: truncate_down_def round_down_def mult_powr_eq
  intro!: ge_one_powr_ge_zero mult_pos_pos)
 qed
 
 lemma truncate_down_nonneg: "0 ≤ y ==> 0 ≤ truncate_down prec y"
  by (auto simp: truncate_down_def round_down_def)
 
 lemma truncate_down_ge1: "1 ≤ x ==> 1 ≤ truncate_down p x"
  apply (auto simp: truncate_down_def algebra_simps intro!: round_down_ge1)
  apply linarith
  done
 
 lemma truncate_up_nonpos: "x ≤ 0 ==> truncate_up prec x ≤ 0"
  by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)
 
 lemma truncate_up_le1:
  assumes "x ≤ 1"
  shows "truncate_up p x ≤ 1"
 proof -
  consider "x ≤ 0" | "x > 0"
  by arith
  then show ?thesis
  proof cases
  case 1
  with truncate_up_nonpos[OF this, of p] show ?thesis
  by simp
  next
  case 2
  then have le: "⌊log 2 ∣x∣⌋ ≤ 0"
  using assms by (auto simp: log_less_iff)
  from assms have "0 ≤ int p" by simp
  from add_mono[OF this le]
  show ?thesis
  using assms by (simp add: truncate_up_def round_up_le1 add_mono)
  qed
 qed
 
 lemma truncate_down_shift_int:
  "truncate_down p (x * 2 powr real_of_int k) = truncate_down p x * 2 powr k"
  by (cases "x = 0")
  (simp_all add: algebra_simps abs_mult log_mult truncate_down_def
  round_down_shift[of _ _ k, simplified])
 
 lemma truncate_down_shift_nat: "truncate_down p (x * 2 powr real k) = truncate_down p x * 2 powr k"
  by (metis of_int_of_nat_eq truncate_down_shift_int)
 
 lemma truncate_up_shift_int: "truncate_up p (x * 2 powr real_of_int k) = truncate_up p x * 2 powr k"
  by (cases "x = 0")
  (simp_all add: algebra_simps abs_mult log_mult truncate_up_def
  round_up_shift[of _ _ k, simplified])
 
 lemma truncate_up_shift_nat: "truncate_up p (x * 2 powr real k) = truncate_up p x * 2 powr k"
  by (metis of_int_of_nat_eq truncate_up_shift_int)
 
 
 subsection ‹Truncating Floats›
 
 lift_definition float_round_up :: "nat ==> float ==> float" is truncate_up
  by (simp add: truncate_up_def)
 
 lemma float_round_up: "real_of_float x ≤ real_of_float (float_round_up prec x)"
  using truncate_up by transfer simp
 
 lemma float_round_up_zero[simp]: "float_round_up prec 0 = 0"
  by transfer simp
 
 lift_definition float_round_down :: "nat ==> float ==> float" is truncate_down
  by (simp add: truncate_down_def)
 
 lemma float_round_down: "real_of_float (float_round_down prec x) ≤ real_of_float x"
  using truncate_down by transfer simp
 
 lemma float_round_down_zero[simp]: "float_round_down prec 0 = 0"
  by transfer simp
 
 lemmas float_round_up_le = order_trans[OF _ float_round_up]
  and float_round_down_le = order_trans[OF float_round_down]
 
 lemma minus_float_round_up_eq: "- float_round_up prec x = float_round_down prec (- x)"
  and minus_float_round_down_eq: "- float_round_down prec x = float_round_up prec (- x)"
  by (transfer; simp add: truncate_down_uminus_eq truncate_up_uminus_eq)+
 
 context
 begin
 
 qualified lemma compute_float_round_down[code]:
  "float_round_down prec (Float m e) =
  (let d = bitlen ∣m∣ - int prec - 1 in
  if 0 < d then Float (div_twopow m (nat d)) (e + d)
  else Float m e)"
  using Float.compute_float_down[of "Suc prec - bitlen ∣m∣ - e" m e, symmetric]
  by transfer
  (simp add: field_simps abs_mult log_mult bitlen_alt_def truncate_down_def
  cong del: if_weak_cong)
 
 qualified lemma compute_float_round_up[code]:
  "float_round_up prec x = - float_round_down prec (-x)"
  by transfer (simp add: truncate_down_uminus_eq)
 
 end
 
 lemma truncate_up_nonneg_mono:
  assumes "0 ≤ x" "x ≤ y"
  shows "truncate_up prec x ≤ truncate_up prec y"
 proof -
  consider "⌊log 2 x⌋ = ⌊log 2 y⌋" | "⌊log 2 x⌋ ≠ ⌊log 2 y⌋" "0 < x" | "x ≤ 0"
  by arith
  then show ?thesis
  proof cases
  case 1
  then show ?thesis
  using assms
  by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
  next
  case 2
  from assms ‹0 < x› have "log 2 x ≤ log 2 y"
  by auto
  with ‹⌊log 2 x⌋ ≠ ⌊log 2 y⌋›
  have logless: "log 2 x < log 2 y"
  by linarith
  have flogless: "⌊log 2 x⌋ < ⌊log 2 y⌋"
  using ‹⌊log 2 x⌋ ≠ ⌊log 2 y⌋› ‹log 2 x ≤ log 2 y› by linarith
  have "truncate_up prec x =
  real_of_int ⌈x * 2 powr real_of_int (int prec - ⌊log 2 x⌋ )⌉ * 2 powr - real_of_int (int prec - ⌊log 2 x⌋)"
  using assms by (simp add: truncate_up_def round_up_def)
  also have "⌈x * 2 powr real_of_int (int prec - ⌊log 2 x⌋)⌉ ≤ (2 ^ (Suc prec))"
  proof (simp only: ceiling_le_iff)
  have "x * 2 powr real_of_int (int prec - ⌊log 2 x⌋) ≤
  x * (2 powr real (Suc prec) / (2 powr log 2 x))"
  using real_of_int_floor_add_one_ge[of "log 2 x"] assms
  by (auto simp: algebra_simps simp flip: powr_diff intro!: mult_left_mono)
  then show "x * 2 powr real_of_int (int prec - ⌊log 2 x⌋) ≤ real_of_int ((2::int) ^ (Suc prec))"
  using ‹0 < x› by (simp add: powr_realpow powr_add)
  qed
  then have "real_of_int ⌈x * 2 powr real_of_int (int prec - ⌊log 2 x⌋)⌉ ≤ 2 powr int (Suc prec)"
  by (auto simp: powr_realpow powr_add)
  (metis power_Suc of_int_le_numeral_power_cancel_iff)
  also
  have "2 powr - real_of_int (int prec - ⌊log 2 x⌋) ≤ 2 powr - real_of_int (int prec - ⌊log 2 y⌋ + 1)"
  using logless flogless by (auto intro!: floor_mono)
  also have "2 powr real_of_int (int (Suc prec)) ≤
  2 powr (log 2 y + real_of_int (int prec - ⌊log 2 y⌋ + 1))"
  using assms ‹0 < x›
  by (auto simp: algebra_simps)
  finally have "truncate_up prec x ≤
  2 powr (log 2 y + real_of_int (int prec - ⌊log 2 y⌋ + 1)) * 2 powr - real_of_int (int prec - ⌊log 2 y⌋ + 1)"
  by simp
  also have "… = 2 powr (log 2 y + real_of_int (int prec - ⌊log 2 y⌋) - real_of_int (int prec - ⌊log 2 y⌋))"
  by (subst powr_add[symmetric]) simp
  also have "… = y"
  using ‹0 < x› assms
  by (simp add: powr_add)
  also have "… ≤ truncate_up prec y"
  by (rule truncate_up)
  finally show ?thesis .
  next
  case 3
  then show ?thesis
  using assms
  by (auto intro!: truncate_up_le)
  qed
 qed
 
 lemma truncate_up_switch_sign_mono:
  assumes "x ≤ 0" "0 ≤ y"
  shows "truncate_up prec x ≤ truncate_up prec y"
 proof -
  note truncate_up_nonpos[OF ‹x ≤ 0›]
  also note truncate_up_le[OF ‹0 ≤ y›]
  finally show ?thesis .
 qed
 
 lemma truncate_down_switch_sign_mono:
  assumes "x ≤ 0"
  and "0 ≤ y"
  and "x ≤ y"
  shows "truncate_down prec x ≤ truncate_down prec y"
 proof -
  note truncate_down_le[OF ‹x ≤ 0›]
  also note truncate_down_nonneg[OF ‹0 ≤ y›]
  finally show ?thesis .
 qed
 
 lemma truncate_down_nonneg_mono:
  assumes "0 ≤ x" "x ≤ y"
  shows "truncate_down prec x ≤ truncate_down prec y"
 proof -
  consider "x ≤ 0" | "⌊log 2 ∣x∣⌋ = ⌊log 2 ∣y∣⌋" |
  "0 < x" "⌊log 2 ∣x∣⌋ ≠ ⌊log 2 ∣y∣⌋"
  by arith
  then show ?thesis
  proof cases
  case 1
  with assms have "x = 0" "0 ≤ y" by simp_all
  then show ?thesis
  by (auto intro!: truncate_down_nonneg)
  next
  case 2
  then show ?thesis
  using assms
  by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
  next
  case 3
  from ‹0 < x› have "log 2 x ≤ log 2 y" "0 < y" "0 ≤ y"
  using assms by auto
  with ‹⌊log 2 ∣x∣⌋ ≠ ⌊log 2 ∣y∣⌋›
  have logless: "log 2 x < log 2 y" and flogless: "⌊log 2 x⌋ < ⌊log 2 y⌋"
  unfolding atomize_conj abs_of_pos[OF ‹0 < x›] abs_of_pos[OF ‹0 < y›]
  by (metis floor_less_cancel linorder_cases not_le)
  have "2 powr prec ≤ y * 2 powr real prec / (2 powr log 2 y)"
  using ‹0 < y› by simp
  also have "… ≤ y * 2 powr real (Suc prec) / (2 powr (real_of_int ⌊log 2 y⌋ + 1))"
  using ‹0 ≤ y› ‹0 ≤ x› assms(2)
  by (auto intro!: powr_mono divide_left_mono
  simp: of_nat_diff powr_add powr_diff)
  also have "… = y * 2 powr real (Suc prec) / (2 powr real_of_int ⌊log 2 y⌋ * 2)"
  by (auto simp: powr_add)
  finally have "(2 ^ prec) ≤ ⌊y * 2 powr real_of_int (int (Suc prec) - ⌊log 2 ∣y∣⌋ - 1)⌋"
  using ‹0 ≤ y›
  by (auto simp: powr_diff le_floor_iff powr_realpow powr_add)
  then have "(2 ^ (prec)) * 2 powr - real_of_int (int prec - ⌊log 2 ∣y∣⌋) ≤ truncate_down prec y"
  by (auto simp: truncate_down_def round_down_def)
  moreover have "x ≤ (2 ^ prec) * 2 powr - real_of_int (int prec - ⌊log 2 ∣y∣⌋)"
  proof -
  have "x = 2 powr (log 2 ∣x∣)" using ‹0 < x› by simp
  also have "… ≤ (2 ^ (Suc prec )) * 2 powr - real_of_int (int prec - ⌊log 2 ∣x∣⌋)"
  using real_of_int_floor_add_one_ge[of "log 2 ∣x∣"] ‹0 < x›
  by (auto simp flip: powr_realpow powr_add simp: algebra_simps powr_mult_base le_powr_iff)
  also
  have "2 powr - real_of_int (int prec - ⌊log 2 ∣x∣⌋) ≤ 2 powr - real_of_int (int prec - ⌊log 2 ∣y∣⌋ + 1)"
  using logless flogless ‹x > 0› ‹y > 0›
  by (auto intro!: floor_mono)
  finally show ?thesis
  by (auto simp flip: powr_realpow simp: powr_diff assms)
  qed
  ultimately show ?thesis
  by (metis dual_order.trans truncate_down)
  qed
 qed
 
 lemma truncate_down_eq_truncate_up: "truncate_down p x = - truncate_up p (-x)"
  and truncate_up_eq_truncate_down: "truncate_up p x = - truncate_down p (-x)"
  by (auto simp: truncate_up_uminus_eq truncate_down_uminus_eq)
 
 lemma truncate_down_mono: "x ≤ y ==> truncate_down p x ≤ truncate_down p y"
  by (smt (verit) truncate_down_nonneg_mono truncate_up_nonneg_mono truncate_up_uminus_eq)
 
 lemma truncate_up_mono: "x ≤ y ==> truncate_up p x ≤ truncate_up p y"
  by (simp add: truncate_up_eq_truncate_down truncate_down_mono)
 
 lemma truncate_up_nonneg: "0 ≤ truncate_up p x" if "0 ≤ x"
  by (simp add: that truncate_up_le)
 
 lemma truncate_up_pos: "0 < truncate_up p x" if "0 < x"
  by (meson less_le_trans that truncate_up)
 
 lemma truncate_up_less_zero_iff[simp]: "truncate_up p x < 0 ⟷ x < 0"
  by (smt (verit) truncate_down_pos truncate_down_uminus_eq truncate_up_nonneg)
 
 lemma truncate_up_nonneg_iff[simp]: "truncate_up p x ≥ 0 ⟷ x ≥ 0"
  using truncate_up_less_zero_iff[of p x] truncate_up_nonneg[of x]
  by linarith
 
 lemma truncate_down_less_zero_iff[simp]: "truncate_down p x < 0 ⟷ x < 0"
  by (metis le_less_trans not_less_iff_gr_or_eq truncate_down truncate_down_pos truncate_down_zero)
 
 lemma truncate_down_nonneg_iff[simp]: "truncate_down p x ≥ 0 ⟷ x ≥ 0"
  using truncate_down_less_zero_iff[of p x] truncate_down_nonneg[of x p]
  by linarith
 
 lemma truncate_down_eq_zero_iff[simp]: "truncate_down prec x = 0 ⟷ x = 0"
  by (metis not_less_iff_gr_or_eq truncate_down_less_zero_iff truncate_down_pos truncate_down_zero)
 
 lemma truncate_up_eq_zero_iff[simp]: "truncate_up prec x = 0 ⟷ x = 0"
  by (metis not_less_iff_gr_or_eq truncate_up_less_zero_iff truncate_up_pos truncate_up_zero)
 
 
 subsection ‹Approximation of positive rationals›
 
 lemma div_mult_twopow_eq: "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)" for a b :: nat
  by (cases "b = 0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
 
 lemma real_div_nat_eq_floor_of_divide: "a div b = real_of_int ⌊a / b⌋" for a b :: nat
  by (simp add: floor_divide_of_nat_eq [of a b])
 
 definition "rat_precision prec x y =
  (let d = bitlen x - bitlen y
  in int prec - d + (if Float (abs x) 0 < Float (abs y) d then 1 else 0))"
 
 lemma floor_log_divide_eq:
  assumes "i > 0" "j > 0" "p > 1"
  shows "⌊log p (i / j)⌋ = floor (log p i) - floor (log p j) -
  (if i ≥ j * p powr (floor (log p i) - floor (log p j)) then 0 else 1)"
 proof -
  let ?l = "log p"
  let ?fl = "λx. floor (?l x)"
  have "⌊?l (i / j)⌋ = ⌊?l i - ?l j⌋" using assms
  by (auto simp: log_divide)
  also have "… = floor (real_of_int (?fl i - ?fl j) + (?l i - ?fl i - (?l j - ?fl j)))"
  (is "_ = floor (_ + ?r)")
  by (simp add: algebra_simps)
  also note floor_add2
  also note ‹p > 1›
  note powr = powr_le_cancel_iff[symmetric, OF ‹1 < p›, THEN iffD2]
  note powr_strict = powr_less_cancel_iff[symmetric, OF ‹1 < p›, THEN iffD2]
  have "floor ?r = (if i ≥ j * p powr (?fl i - ?fl j) then 0 else -1)" (is "_ = ?if")
  using assms
  apply simp
  by (smt (verit, ccfv_SIG) floor_less_iff floor_uminus_of_int le_log_iff mult_powr_eq
  of_int_1 real_of_int_floor_add_one_gt zero_le_floor)
  finally
  show ?thesis by simp
 qed
 
 lemma truncate_down_rat_precision:
  "truncate_down prec (real x / real y) = round_down (rat_precision prec x y) (real x / real y)"
  and truncate_up_rat_precision:
  "truncate_up prec (real x / real y) = round_up (rat_precision prec x y) (real x / real y)"
  unfolding truncate_down_def truncate_up_def rat_precision_def
  by (cases x; cases y) (auto simp: floor_log_divide_eq algebra_simps bitlen_alt_def)
 
 lift_definition lapprox_posrat :: "nat ==> nat ==> nat ==> float"
  is "λprec (x::nat) (y::nat). truncate_down prec (x / y)"
  by simp
 
 context
 begin
 
 qualified lemma compute_lapprox_posrat[code]:
  "lapprox_posrat prec x y =
  (let
  l = rat_precision prec x y;
  d = if 0 ≤ l then x * 2^nat l div y else x div 2^nat (- l) div y
  in normfloat (Float d (- l)))"
  unfolding div_mult_twopow_eq
  by transfer
  (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
  truncate_down_rat_precision del: two_powr_minus_int_float)
 
 end
 
 lift_definition rapprox_posrat :: "nat ==> nat ==> nat ==> float"
  is "λprec (x::nat) (y::nat). truncate_up prec (x / y)"
  by simp
 
 context
 begin
 
 qualified lemma compute_rapprox_posrat[code]:
  fixes prec x y
  defines "l ≡ rat_precision prec x y"
  shows "rapprox_posrat prec x y =
  (let
  l = l;
  (r, s) = if 0 ≤ l then (x * 2^nat l, y) else (x, y * 2^nat(-l));
  d = r div s;
  m = r mod s
  in normfloat (Float (d + (if m = 0 ∨ y = 0 then 0 else 1)) (- l)))"
 proof (cases "y = 0")
  assume "y = 0"
  then show ?thesis by transfer simp
 next
  assume "y ≠ 0"
  show ?thesis
  proof (cases "0 ≤ l")
  case True
  define x' where "x' = x * 2 ^ nat l"
  have "int x * 2 ^ nat l = x'"
  by (simp add: x'_def)
  moreover have "real x * 2 powr l = real x'"
  by (simp flip: powr_realpow add: ‹0 ≤ l› x'_def)
  ultimately show ?thesis
  using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] ‹0 ≤ l› ‹y ≠ 0›
  l_def[symmetric, THEN meta_eq_to_obj_eq]
  apply transfer
  apply (auto simp add: round_up_def truncate_up_rat_precision)
  apply (metis floor_divide_of_int_eq of_int_of_nat_eq)
  done
  next
  case False
  define y' where "y' = y * 2 ^ nat (- l)"
  from ‹y ≠ 0› have "y' ≠ 0" by (simp add: y'_def)
  have "int y * 2 ^ nat (- l) = y'"
  by (simp add: y'_def)
  moreover have "real x * real_of_int (2::int) powr real_of_int l / real y = x / real y'"
  using ‹¬ 0 ≤ l› by (simp flip: powr_realpow add: powr_minus y'_def field_simps)
  ultimately show ?thesis
  using ceil_divide_floor_conv[of y' x] ‹¬ 0 ≤ l› ‹y' ≠ 0› ‹y ≠ 0›
  l_def[symmetric, THEN meta_eq_to_obj_eq]
  apply transfer
  apply (auto simp add: round_up_def ceil_divide_floor_conv truncate_up_rat_precision)
  apply (metis floor_divide_of_int_eq of_int_of_nat_eq)
  done
  qed
 qed
 
 end
 
 lemma rat_precision_pos:
  assumes "0 ≤ x"
  and "0 < y"
  and "2 * x < y"
  shows "rat_precision n (int x) (int y) > 0"
 proof -
  have "0 < x ==> log 2 x + 1 = log 2 (2 * x)"
  by (simp add: log_mult)
  then have "bitlen (int x) < bitlen (int y)"
  using assms
  by (simp add: bitlen_alt_def)
  (auto intro!: floor_mono simp add: one_add_floor)
  then show ?thesis
  using assms
  by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
 qed
 
 lemma rapprox_posrat_less1:
  "0 ≤ x ==> 0 < y ==> 2 * x < y ==> real_of_float (rapprox_posrat n x y) < 1"
  by transfer (simp add: rat_precision_pos round_up_less1 truncate_up_rat_precision)
 
 lift_definition lapprox_rat :: "nat ==> int ==> int ==> float" is
  "λprec (x::int) (y::int). truncate_down prec (x / y)"
  by simp
 
 context
 begin
 
 qualified lemma compute_lapprox_rat[code]:
  "lapprox_rat prec x y =
  (if y = 0 then 0
  else if 0 ≤ x then
  (if 0 < y then lapprox_posrat prec (nat x) (nat y)
  else - (rapprox_posrat prec (nat x) (nat (-y))))
  else
  (if 0 < y
  then - (rapprox_posrat prec (nat (-x)) (nat y))
  else lapprox_posrat prec (nat (-x)) (nat (-y))))"
  by transfer (simp add: truncate_up_uminus_eq)
 
 lift_definition rapprox_rat :: "nat ==> int ==> int ==> float" is
  "λprec (x::int) (y::int). truncate_up prec (x / y)"
  by simp
 
 lemma "rapprox_rat = rapprox_posrat"
  by transfer auto
 
 lemma "lapprox_rat = lapprox_posrat"
  by transfer auto
 
 qualified lemma compute_rapprox_rat[code]:
  "rapprox_rat prec x y = - lapprox_rat prec (-x) y"
  by transfer (simp add: truncate_down_uminus_eq)
 
 qualified lemma compute_truncate_down[code]:
  "truncate_down p (Ratreal r) = (let (a, b) = quotient_of r in lapprox_rat p a b)"
  by transfer (auto split: prod.split simp: of_rat_divide dest!: quotient_of_div)
 
 qualified lemma compute_truncate_up[code]:
  "truncate_up p (Ratreal r) = (let (a, b) = quotient_of r in rapprox_rat p a b)"
  by transfer (auto split: prod.split simp: of_rat_divide dest!: quotient_of_div)
 
 end
 
 
 subsection ‹Division›
 
 definition "real_divl prec a b = truncate_down prec (a / b)"
 
 definition "real_divr prec a b = truncate_up prec (a / b)"
 
 lift_definition float_divl :: "nat ==> float ==> float ==> float" is real_divl
  by (simp add: real_divl_def)
 
 context
 begin
 
 qualified lemma compute_float_divl[code]:
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
  apply transfer
  unfolding real_divl_def of_int_1 mult_1 truncate_down_shift_int[symmetric]
  apply (simp add: powr_diff powr_minus)
  done
 
 lift_definition float_divr :: "nat ==> float ==> float ==> float" is real_divr
  by (simp add: real_divr_def)
 
 qualified lemma compute_float_divr[code]:
  "float_divr prec x y = - float_divl prec (-x) y"
  by transfer (simp add: real_divr_def real_divl_def truncate_down_uminus_eq)
 
 end
 
 
 subsection ‹Approximate Addition›
 
 definition "plus_down prec x y = truncate_down prec (x + y)"
 
 definition "plus_up prec x y = truncate_up prec (x + y)"
 
 lemma float_plus_down_float[intro, simp]: "x ∈ float ==> y ∈ float ==> plus_down p x y ∈ float"
  by (simp add: plus_down_def)
 
 lemma float_plus_up_float[intro, simp]: "x ∈ float ==> y ∈ float ==> plus_up p x y ∈ float"
  by (simp add: plus_up_def)
 
 lift_definition float_plus_down :: "nat ==> float ==> float ==> float" is plus_down ..
 
 lift_definition float_plus_up :: "nat ==> float ==> float ==> float" is plus_up ..
 
 lemma plus_down: "plus_down prec x y ≤ x + y"
  and plus_up: "x + y ≤ plus_up prec x y"
  by (auto simp: plus_down_def truncate_down plus_up_def truncate_up)
 
 lemma float_plus_down: "real_of_float (float_plus_down prec x y) ≤ x + y"
  and float_plus_up: "x + y ≤ real_of_float (float_plus_up prec x y)"
  by (transfer; rule plus_down plus_up)+
 
 lemmas plus_down_le = order_trans[OF plus_down]
  and plus_up_le = order_trans[OF _ plus_up]
  and float_plus_down_le = order_trans[OF float_plus_down]
  and float_plus_up_le = order_trans[OF _ float_plus_up]
 
 lemma compute_plus_up[code]: "plus_up p x y = - plus_down p (-x) (-y)"
  using truncate_down_uminus_eq[of p "x + y"]
  by (auto simp: plus_down_def plus_up_def)
 
 lemma truncate_down_log2_eqI:
  assumes "⌊log 2 ∣x∣⌋ = ⌊log 2 ∣y∣⌋"
  assumes "⌊x * 2 powr (p - ⌊log 2 ∣x∣⌋)⌋ = ⌊y * 2 powr (p - ⌊log 2 ∣x∣⌋)⌋"
  shows "truncate_down p x = truncate_down p y"
  using assms by (auto simp: truncate_down_def round_down_def)
 
 lemma sum_neq_zeroI:
  "∣a∣ ≥ k ==> ∣b∣ < k ==> a + b ≠ 0"
  "∣a∣ > k ==> ∣b∣ ≤ k ==> a + b ≠ 0"
  for a k :: real
  by auto
 
 lemma abs_real_le_2_powr_bitlen[simp]: "∣real_of_int m2∣ < 2 powr real_of_int (bitlen ∣m2∣)"
 proof (cases "m2 = 0")
  case True
  then show ?thesis by simp
 next
  case False
  then have "∣m2∣ < 2 ^ nat (bitlen ∣m2∣)"
  using bitlen_bounds[of "∣m2∣"]
  by (auto simp: powr_add bitlen_nonneg)
  then show ?thesis
  by (metis bitlen_nonneg powr_int of_int_abs of_int_less_numeral_power_cancel_iff
  zero_less_numeral)
 qed
 
 lemma floor_sum_times_2_powr_sgn_eq:
  fixes ai p q :: int
  and a b :: real
  assumes "a * 2 powr p = ai"
  and b_le_1: "∣b * 2 powr (p + 1)∣ ≤ 1"
  and leqp: "q ≤ p"
  shows "⌊(a + b) * 2 powr q⌋ = ⌊(2 * ai + sgn b) * 2 powr (q - p - 1)⌋"
 proof -
  consider "b = 0" | "b > 0" | "b < 0" by arith
  then show ?thesis
  proof cases
  case 1
  then show ?thesis
  by (simp flip: assms(1) powr_add add: algebra_simps powr_mult_base)
  next
  case 2
  then have "b * 2 powr p < ∣b * 2 powr (p + 1)∣"
  by simp
  also note b_le_1
  finally have b_less_1: "b * 2 powr real_of_int p < 1" .
 
  from b_less_1 ‹b > 0› have floor_eq: "⌊b * 2 powr real_of_int p⌋ = 0" "⌊sgn b / 2⌋ = 0"
  by (simp_all add: floor_eq_iff)
 
  have "⌊(a + b) * 2 powr q⌋ = ⌊(a + b) * 2 powr p * 2 powr (q - p)⌋"
  by (simp add: algebra_simps flip: powr_realpow powr_add)
  also have "… = ⌊(ai + b * 2 powr p) * 2 powr (q - p)⌋"
  by (simp add: assms algebra_simps)
  also have "… = ⌊(ai + b * 2 powr p) / real_of_int ((2::int) ^ nat (p - q))⌋"
  using assms
  by (simp add: algebra_simps divide_powr_uminus flip: powr_realpow powr_add)
  also have "… = ⌊ai / real_of_int ((2::int) ^ nat (p - q))⌋"
  by (simp del: of_int_power add: floor_divide_real_eq_div floor_eq)
  finally have "⌊(a + b) * 2 powr real_of_int q⌋ = ⌊real_of_int ai / real_of_int ((2::int) ^ nat (p - q))⌋" .
  moreover
  have "⌊(2 * ai + (sgn b)) * 2 powr (real_of_int (q - p) - 1)⌋ =
  ⌊real_of_int ai / real_of_int ((2::int) ^ nat (p - q))⌋"
  proof -
  have "⌊(2 * ai + sgn b) * 2 powr (real_of_int (q - p) - 1)⌋ = ⌊(ai + sgn b / 2) * 2 powr (q - p)⌋"
  by (subst powr_diff) (simp add: field_simps)
  also have "… = ⌊(ai + sgn b / 2) / real_of_int ((2::int) ^ nat (p - q))⌋"
  using leqp by (simp flip: powr_realpow add: powr_diff)
  also have "… = ⌊ai / real_of_int ((2::int) ^ nat (p - q))⌋"
  by (simp del: of_int_power add: floor_divide_real_eq_div floor_eq)
  finally show ?thesis .
  qed
  ultimately show ?thesis by simp
  next
  case 3
  then have floor_eq: "⌊b * 2 powr (real_of_int p + 1)⌋ = -1"
  using b_le_1
  by (auto simp: floor_eq_iff algebra_simps pos_divide_le_eq[symmetric] abs_if divide_powr_uminus
  intro!: mult_neg_pos split: if_split_asm)
  have "⌊(a + b) * 2 powr q⌋ = ⌊(2*a + 2*b) * 2 powr p * 2 powr (q - p - 1)⌋"
  by (simp add: algebra_simps powr_mult_base flip: powr_realpow powr_add)
  also have "… = ⌊(2 * (a * 2 powr p) + 2 * b * 2 powr p) * 2 powr (q - p - 1)⌋"
  by (simp add: algebra_simps)
  also have "… = ⌊(2 * ai + b * 2 powr (p + 1)) / 2 powr (1 - q + p)⌋"
  using assms by (simp add: algebra_simps powr_mult_base divide_powr_uminus)
  also have "… = ⌊(2 * ai + b * 2 powr (p + 1)) / real_of_int ((2::int) ^ nat (p - q + 1))⌋"
  using assms by (simp add: algebra_simps flip: powr_realpow)
  also have "… = ⌊(2 * ai - 1) / real_of_int ((2::int) ^ nat (p - q + 1))⌋"
  using ‹b < 0› assms
  by (simp add: floor_divide_of_int_eq floor_eq floor_divide_real_eq_div
  del: of_int_mult of_int_power of_int_diff)
  also have "… = ⌊(2 * ai - 1) * 2 powr (q - p - 1)⌋"
  using assms by (simp add: algebra_simps divide_powr_uminus flip: powr_realpow)
  finally show ?thesis
  using ‹b < 0› by simp
  qed
 qed
 
 lemma log2_abs_int_add_less_half_sgn_eq:
  fixes ai :: int
  and b :: real
  assumes "∣b∣ ≤ 1/2"
  and "ai ≠ 0"
  shows "⌊log 2 ∣real_of_int ai + b∣⌋ = ⌊log 2 ∣ai + sgn b / 2∣⌋"
 proof (cases "b = 0")
  case True
  then show ?thesis by simp
 next
  case False
  define k where "k = ⌊log 2 ∣ai∣⌋"
  then have "⌊log 2 ∣ai∣⌋ = k"
  by simp
  then have k: "2 powr k ≤ ∣ai∣" "∣ai∣ < 2 powr (k + 1)"
  by (simp_all add: floor_log_eq_powr_iff ‹ai ≠ 0›)
  have "k ≥ 0"
  using assms by (auto simp: k_def)
  define r where "r = ∣ai∣ - 2 ^ nat k"
  have r: "0 ≤ r" "r < 2 powr k"
  using ‹k ≥ 0› k
  by (auto simp: r_def k_def algebra_simps powr_add abs_if powr_int)
  then have "r ≤ (2::int) ^ nat k - 1"
  using ‹k ≥ 0› by (auto simp: powr_int)
  from this[simplified of_int_le_iff[symmetric]] ‹0 ≤ k›
  have r_le: "r ≤ 2 powr k - 1"
  by (auto simp: algebra_simps powr_int)
  (metis of_int_1 of_int_add of_int_le_numeral_power_cancel_iff)
 
  have "∣ai∣ = 2 powr k + r"
  using ‹k ≥ 0› by (auto simp: k_def r_def simp flip: powr_realpow)
 
  have pos: "∣b∣ < 1 ==> 0 < 2 powr k + (r + b)" for b :: real
  using ‹0 ≤ k› ‹ai ≠ 0›
  by (auto simp add: r_def powr_realpow[symmetric] abs_if sgn_if algebra_simps
  split: if_split_asm)
  have less: "∣sgn ai * b∣ < 1"
  and less': "∣sgn (sgn ai * b) / 2∣ < 1"
  using ‹∣b∣ ≤ _› by (auto simp: abs_if sgn_if split: if_split_asm)
 
  have floor_eq: "∧b::real. ∣b∣ ≤ 1 / 2 ==>
  ⌊log 2 (1 + (r + b) / 2 powr k)⌋ = (if r = 0 ∧ b < 0 then -1 else 0)"
  using ‹k ≥ 0› r r_le
  by (auto simp: floor_log_eq_powr_iff powr_minus_divide field_simps sgn_if)
 
  from ‹real_of_int ∣ai∣ = _› have "∣ai + b∣ = 2 powr k + (r + sgn ai * b)"
  using ‹∣b∣ ≤ _› ‹0 ≤ k› r
  by (auto simp add: sgn_if abs_if)
  also have "⌊log 2 …⌋ = ⌊log 2 (2 powr k + r + sgn (sgn ai * b) / 2)⌋"
  proof -
  have "2 powr k + (r + (sgn ai) * b) = 2 powr k * (1 + (r + sgn ai * b) / 2 powr k)"
  by (simp add: field_simps)
  also have "⌊log 2 …⌋ = k + ⌊log 2 (1 + (r + sgn ai * b) / 2 powr k)⌋"
  using pos[OF less]
  by (subst log_mult) (simp_all add: log_mult powr_mult field_simps)
  also
  let ?if = "if r = 0 ∧ sgn ai * b < 0 then -1 else 0"
  have "⌊log 2 (1 + (r + sgn ai * b) / 2 powr k)⌋ = ?if"
  using ‹∣b∣ ≤ _›
  by (intro floor_eq) (auto simp: abs_mult sgn_if)
  also
  have "… = ⌊log 2 (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k)⌋"
  by (subst floor_eq) (auto simp: sgn_if)
  also have "k + … = ⌊log 2 (2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k))⌋"
  unfolding int_add_floor
  using pos[OF less'] ‹∣b∣ ≤ _›
  by (simp add: field_simps add_log_eq_powr del: floor_add2)
  also have "2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k) =
  2 powr k + r + sgn (sgn ai * b) / 2"
  by (simp add: sgn_if field_simps)
  finally show ?thesis .
  qed
  also have "2 powr k + r + sgn (sgn ai * b) / 2 = ∣ai + sgn b / 2∣"
  unfolding ‹real_of_int ∣ai∣ = _›[symmetric] using ‹ai ≠ 0›
  by (auto simp: abs_if sgn_if algebra_simps)
  finally show ?thesis .
 qed
 
 context
 begin
 
 qualified lemma compute_far_float_plus_down:
  fixes m1 e1 m2 e2 :: int
  and p :: nat
  defines "k1 ≡ Suc p - nat (bitlen ∣m1∣)"
  assumes H: "bitlen ∣m2∣ ≤ e1 - e2 - k1 - 2" "m1 ≠ 0" "m2 ≠ 0" "e1 ≥ e2"
  shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
  float_round_down p (Float (m1 * 2 ^ (Suc (Suc k1)) + sgn m2) (e1 - int k1 - 2))"
 proof -
  let ?a = "real_of_float (Float m1 e1)"
  let ?b = "real_of_float (Float m2 e2)"
  let ?sum = "?a + ?b"
  let ?shift = "real_of_int e2 - real_of_int e1 + real k1 + 1"
  let ?m1 = "m1 * 2 ^ Suc k1"
  let ?m2 = "m2 * 2 powr ?shift"
  let ?m2' = "sgn m2 / 2"
  let ?e = "e1 - int k1 - 1"
 
  have sum_eq: "?sum = (?m1 + ?m2) * 2 powr ?e"
  by (auto simp flip: powr_add powr_mult powr_realpow simp: powr_mult_base algebra_simps)
 
  have "∣?m2∣ * 2 < 2 powr (bitlen ∣m2∣ + ?shift + 1)"
  by (auto simp: field_simps powr_add powr_mult_base powr_diff abs_mult)
  also have "… ≤ 2 powr 0"
  using H by (intro powr_mono) auto
  finally have abs_m2_less_half: "∣?m2∣ < 1 / 2"
  by simp
 
  then have "∣real_of_int m2∣ < 2 powr -(?shift + 1)"
  unfolding powr_minus_divide by (auto simp: bitlen_alt_def field_simps powr_mult_base abs_mult)
  also have "… ≤ 2 powr real_of_int (e1 - e2 - 2)"
  by simp
  finally have b_less_quarter: "∣?b∣ < 1/4 * 2 powr real_of_int e1"
  by (simp add: powr_add field_simps powr_diff abs_mult)
  also have "1/4 < ∣real_of_int m1∣ / 2" using ‹m1 ≠ 0› by simp
  finally have b_less_half_a: "∣?b∣ < 1/2 * ∣?a∣"
  by (simp add: algebra_simps powr_mult_base abs_mult)
  then have a_half_less_sum: "∣?a∣ / 2 < ∣?sum∣"
  by (auto simp: field_simps abs_if split: if_split_asm)
 
  from b_less_half_a have "∣?b∣ < ∣?a∣" "∣?b∣ ≤ ∣?a∣"
  by simp_all
 
  have "∣real_of_float (Float m1 e1)∣ ≥ 1/4 * 2 powr real_of_int e1"
  using ‹m1 ≠ 0›
  by (auto simp: powr_add powr_int bitlen_nonneg divide_right_mono abs_mult)
  then have "?sum ≠ 0" using b_less_quarter
  by (rule sum_neq_zeroI)
  then have "?m1 + ?m2 ≠ 0"
  unfolding sum_eq by (simp add: abs_mult zero_less_mult_iff)
 
  have "∣real_of_int ?m1∣ ≥ 2 ^ Suc k1" "∣?m2'∣ < 2 ^ Suc k1"
  using ‹m1 ≠ 0› ‹m2 ≠ 0› by (auto simp: sgn_if less_1_mult abs_mult simp del: power.simps)
  then have sum'_nz: "?m1 + ?m2' ≠ 0"
  by (intro sum_neq_zeroI)
 
  have "⌊log 2 ∣real_of_float (Float m1 e1) + real_of_float (Float m2 e2)∣⌋ = ⌊log 2 ∣?m1 + ?m2∣⌋ + ?e"
  using ‹?m1 + ?m2 ≠ 0›
  unfolding floor_add[symmetric] sum_eq
  by (simp add: abs_mult log_mult) linarith
  also have "⌊log 2 ∣?m1 + ?m2∣⌋ = ⌊log 2 ∣?m1 + sgn (real_of_int m2 * 2 powr ?shift) / 2∣⌋"
  using abs_m2_less_half ‹m1 ≠ 0›
  by (intro log2_abs_int_add_less_half_sgn_eq) (auto simp: abs_mult)
  also have "sgn (real_of_int m2 * 2 powr ?shift) = sgn m2"
  by (auto simp: sgn_if zero_less_mult_iff less_not_sym)
  also
  have "∣?m1 + ?m2'∣ * 2 powr ?e = ∣?m1 * 2 + sgn m2∣ * 2 powr (?e - 1)"
  by (auto simp: field_simps powr_minus[symmetric] powr_diff powr_mult_base)
  then have "⌊log 2 ∣?m1 + ?m2'∣⌋ + ?e = ⌊log 2 ∣real_of_float (Float (?m1 * 2 + sgn m2) (?e - 1))∣⌋"
  using ‹?m1 + ?m2' ≠ 0›
  unfolding floor_add_int
  by (simp add: log_add_eq_powr abs_mult_pos del: floor_add2)
  finally
  have "⌊log 2 ∣?sum∣⌋ = ⌊log 2 ∣real_of_float (Float (?m1*2 + sgn m2) (?e - 1))∣⌋" .
  then have "plus_down p (Float m1 e1) (Float m2 e2) =
  truncate_down p (Float (?m1*2 + sgn m2) (?e - 1))"
  unfolding plus_down_def
  proof (rule truncate_down_log2_eqI)
  let ?f = "(int p - ⌊log 2 ∣real_of_float (Float m1 e1) + real_of_float (Float m2 e2)∣⌋)"
  let ?ai = "m1 * 2 ^ (Suc k1)"
  have "⌊(?a + ?b) * 2 powr real_of_int ?f⌋ = ⌊(real_of_int (2 * ?ai) + sgn ?b) * 2 powr real_of_int (?f - - ?e - 1)⌋"
  proof (rule floor_sum_times_2_powr_sgn_eq)
  show "?a * 2 powr real_of_int (-?e) = real_of_int ?ai"
  by (simp add: powr_add powr_realpow[symmetric] powr_diff)
  show "∣?b * 2 powr real_of_int (-?e + 1)∣ ≤ 1"
  using abs_m2_less_half
  by (simp add: abs_mult powr_add[symmetric] algebra_simps powr_mult_base)
  next
  have "e1 + ⌊log 2 ∣real_of_int m1∣⌋ - 1 = ⌊log 2 ∣?a∣⌋ - 1"
  using ‹m1 ≠ 0›
  by (simp add: int_add_floor algebra_simps log_mult abs_mult del: floor_add2)
  also have "… ≤ ⌊log 2 ∣?a + ?b∣⌋"
  using a_half_less_sum ‹m1 ≠ 0› ‹?sum ≠ 0›
  unfolding floor_diff_of_int[symmetric]
  by (auto simp add: log_minus_eq_powr powr_minus_divide intro!: floor_mono)
  finally
  have "int p - ⌊log 2 ∣?a + ?b∣⌋ ≤ p - (bitlen ∣m1∣) - e1 + 2"
  by (auto simp: algebra_simps bitlen_alt_def ‹m1 ≠ 0›)
  also have "… ≤ - ?e"
  using bitlen_nonneg[of "∣m1∣"] by (simp add: k1_def)
  finally show "?f ≤ - ?e" by simp
  qed
  also have "sgn ?b = sgn m2"
  using powr_gt_zero[of 2 e2]
  by (auto simp add: sgn_if zero_less_mult_iff simp del: powr_gt_zero)
  also have "⌊(real_of_int (2 * ?m1) + real_of_int (sgn m2)) * 2 powr real_of_int (?f - - ?e - 1)⌋ =
  ⌊Float (?m1 * 2 + sgn m2) (?e - 1) * 2 powr ?f⌋"
  by (simp flip: powr_add powr_realpow add: algebra_simps)
  finally
  show "⌊(?a + ?b) * 2 powr ?f⌋ = ⌊real_of_float (Float (?m1 * 2 + sgn m2) (?e - 1)) * 2 powr ?f⌋" .
  qed
  then show ?thesis
  by transfer (simp add: plus_down_def ac_simps Let_def)
 qed
 
 lemma compute_float_plus_down_naive: "float_plus_down p x y = float_round_down p (x + y)"
  by transfer (auto simp: plus_down_def)
 
 qualified lemma compute_float_plus_down[code]:
  fixes p::nat and m1 e1 m2 e2::int
  shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
  (if m1 = 0 then float_round_down p (Float m2 e2)
  else if m2 = 0 then float_round_down p (Float m1 e1)
  else
  (if e1 ≥ e2 then
  (let k1 = Suc p - nat (bitlen ∣m1∣) in
  if bitlen ∣m2∣ > e1 - e2 - k1 - 2
  then float_round_down p ((Float m1 e1) + (Float m2 e2))
  else float_round_down p (Float (m1 * 2 ^ (Suc (Suc k1)) + sgn m2) (e1 - int k1 - 2)))
  else float_plus_down p (Float m2 e2) (Float m1 e1)))"
 proof -
  {
  assume "bitlen ∣m2∣ ≤ e1 - e2 - (Suc p - nat (bitlen ∣m1∣)) - 2" "m1 ≠ 0" "m2 ≠ 0" "e1 ≥ e2"
  note compute_far_float_plus_down[OF this]
  }
  then show ?thesis
  by transfer (simp add: Let_def plus_down_def ac_simps)
 qed
 
 qualified lemma compute_float_plus_up[code]: "float_plus_up p x y = - float_plus_down p (-x) (-y)"
  using truncate_down_uminus_eq[of p "x + y"]
  by transfer (simp add: plus_down_def plus_up_def ac_simps)
 
 lemma mantissa_zero: "mantissa 0 = 0"
  by (fact mantissa_0)
 
 qualified lemma compute_float_less[code]: "a < b ⟷ is_float_pos (float_plus_down 0 b (- a))"
  using truncate_down[of 0 "b - a"] truncate_down_pos[of "b - a" 0]
  by transfer (auto simp: plus_down_def)
 
 qualified lemma compute_float_le[code]: "a ≤ b ⟷ is_float_nonneg (float_plus_down 0 b (- a))"
  using truncate_down[of 0 "b - a"] truncate_down_nonneg[of "b - a" 0]
  by transfer (auto simp: plus_down_def)
 
 end
 
 lemma plus_down_mono: "plus_down p a b ≤ plus_down p c d" if "a + b ≤ c + d"
  by (auto simp: plus_down_def intro!: truncate_down_mono that)
 
 lemma plus_up_mono: "plus_up p a b ≤ plus_up p c d" if "a + b ≤ c + d"
  by (auto simp: plus_up_def intro!: truncate_up_mono that)
 
 subsection ‹Approximate Multiplication›
 
 lemma mult_mono_nonpos_nonneg: "a * b ≤ c * d"
  if "a ≤ c" "a ≤ 0" "0 ≤ d" "d ≤ b" for a b c d::"'a::ordered_ring"
  by (meson dual_order.trans mult_left_mono_neg mult_right_mono that)
 
 lemma mult_mono_nonneg_nonpos: "b * a ≤ d * c"
  if "a ≤ c" "c ≤ 0" "0 ≤ d" "d ≤ b" for a b c d::"'a::ordered_ring"
  by (meson dual_order.trans mult_right_mono_neg mult_left_mono that)
 
 lemma mult_mono_nonpos_nonpos: "a * b ≤ c * d"
  if "a ≥ c" "a ≤ 0" "b ≥ d" "d ≤ 0" for a b c d::real
  by (meson dual_order.trans mult_left_mono_neg mult_right_mono_neg that)
 
 lemma mult_float_mono1:
  shows "a ≤ b ==> ab ≤ bb ==>
  aa ≤ a ==>
  b ≤ ba ==>
  ac ≤ ab ==>
  bb ≤ bc ==>
  plus_down prec (nprt aa * pprt bc)
  (plus_down prec (nprt ba * nprt bc)
  (plus_down prec (pprt aa * pprt ac)
  (pprt ba * nprt ac)))
  ≤ plus_down prec (nprt a * pprt bb)
  (plus_down prec (nprt b * nprt bb)
  (plus_down prec (pprt a * pprt ab)
  (pprt b * nprt ab)))"
  by (smt (verit, best) mult_mono plus_down_mono add_mono nprt_mono nprt_le_zero zero_le_pprt
 pprt_mono mult_mono_nonpos_nonneg mult_mono_nonpos_nonpos mult_mono_nonneg_nonpos)
 
 lemma mult_float_mono2:
  shows "a ≤ b ==>
  ab ≤ bb ==>
  aa ≤ a ==>
  b ≤ ba ==>
  ac ≤ ab ==>
  bb ≤ bc ==>
  plus_up prec (pprt b * pprt bb)
  (plus_up prec (pprt a * nprt bb)
  (plus_up prec (nprt b * pprt ab)
  (nprt a * nprt ab)))
  ≤ plus_up prec (pprt ba * pprt bc)
  (plus_up prec (pprt aa * nprt bc)
  (plus_up prec (nprt ba * pprt ac)
  (nprt aa * nprt ac)))"
  by (smt (verit, best) plus_up_mono add_mono mult_mono nprt_mono nprt_le_zero zero_le_pprt pprt_mono
  mult_mono_nonpos_nonneg mult_mono_nonpos_nonpos mult_mono_nonneg_nonpos)
 
 
 subsection ‹Approximate Power›
 
 lemma div2_less_self[termination_simp]: "odd n ==> n div 2 < n" for n :: nat
  by (simp add: odd_pos)
 
 fun power_down :: "nat ==> real ==> nat ==> real"
 where
  "power_down p x 0 = 1"
 | "power_down p x (Suc n) =
  (if odd n then truncate_down (Suc p) ((power_down p x (Suc n div 2))🪙2)
  else truncate_down (Suc p) (x * power_down p x n))"
 
 fun power_up :: "nat ==> real ==> nat ==> real"
 where
  "power_up p x 0 = 1"
 | "power_up p x (Suc n) =
  (if odd n then truncate_up p ((power_up p x (Suc n div 2))🪙2)
  else truncate_up p (x * power_up p x n))"
 
 lift_definition power_up_fl :: "nat ==> float ==> nat ==> float" is power_up
  by (induct_tac rule: power_up.induct) simp_all
 
 lift_definition power_down_fl :: "nat ==> float ==> nat ==> float" is power_down
  by (induct_tac rule: power_down.induct) simp_all
 
 lemma power_float_transfer[transfer_rule]:
  "(rel_fun pcr_float (rel_fun (=) pcr_float)) (^) (^)"
  unfolding power_def
  by transfer_prover
 
 lemma compute_power_up_fl[code]:
  "power_up_fl p x 0 = 1"
  "power_up_fl p x (Suc n) =
  (if odd n then float_round_up p ((power_up_fl p x (Suc n div 2))🪙2)
  else float_round_up p (x * power_up_fl p x n))"
  and compute_power_down_fl[code]:
  "power_down_fl p x 0 = 1"
  "power_down_fl p x (Suc n) =
  (if odd n then float_round_down (Suc p) ((power_down_fl p x (Suc n div 2))🪙2)
  else float_round_down (Suc p) (x * power_down_fl p x n))"
  unfolding atomize_conj by transfer simp
 
 lemma power_down_pos: "0 < x ==> 0 < power_down p x n"
  by (induct p x n rule: power_down.induct)
  (auto simp del: odd_Suc_div_two intro!: truncate_down_pos)
 
 lemma power_down_nonneg: "0 ≤ x ==> 0 ≤ power_down p x n"
  by (induct p x n rule: power_down.induct)
  (auto simp del: odd_Suc_div_two intro!: truncate_down_nonneg mult_nonneg_nonneg)
 
 lemma power_down: "0 ≤ x ==> power_down p x n ≤ x ^ n"
 proof (induct p x n rule: power_down.induct)
  case (2 p x n)
  have ?case if "odd n"
  proof -
  from that 2 have "(power_down p x (Suc n div 2)) ^ 2 ≤ (x ^ (Suc n div 2)) ^ 2"
  by (auto intro: power_mono power_down_nonneg simp del: odd_Suc_div_two)
  also have "… = x ^ (Suc n div 2 * 2)"
  by (simp flip: power_mult)
  also have "Suc n div 2 * 2 = Suc n"
  using ‹odd n› by presburger
  finally show ?thesis
  using that by (auto intro!: truncate_down_le simp del: odd_Suc_div_two)
  qed
  then show ?case
  by (auto intro!: truncate_down_le mult_left_mono 2 mult_nonneg_nonneg power_down_nonneg)
 qed simp
 
 lemma power_up: "0 ≤ x ==> x ^ n ≤ power_up p x n"
 proof (induct p x n rule: power_up.induct)
  case (2 p x n)
  have ?case if "odd n"
  proof -
  from that even_Suc have "Suc n = Suc n div 2 * 2"
  by presburger
  then have "x ^ Suc n ≤ (x ^ (Suc n div 2))🪙2"
  by (simp flip: power_mult)
  also from that 2 have "… ≤ (power_up p x (Suc n div 2))🪙2"
  by (auto intro: power_mono simp del: odd_Suc_div_two)
  finally show ?thesis
  using that by (auto intro!: truncate_up_le simp del: odd_Suc_div_two)
  qed
  then show ?case
  by (auto intro!: truncate_up_le mult_left_mono 2)
 qed simp
 
 lemmas power_up_le = order_trans[OF _ power_up]
  and power_up_less = less_le_trans[OF _ power_up]
  and power_down_le = order_trans[OF power_down]
 
 lemma power_down_fl: "0 ≤ x ==> power_down_fl p x n ≤ x ^ n"
  by transfer (rule power_down)
 
 lemma power_up_fl: "0 ≤ x ==> x ^ n ≤ power_up_fl p x n"
  by transfer (rule power_up)
 
 lemma real_power_up_fl: "real_of_float (power_up_fl p x n) = power_up p x n"
  by transfer simp
 
 lemma real_power_down_fl: "real_of_float (power_down_fl p x n) = power_down p x n"
  by transfer simp
 
 lemmas [simp del] = power_down.simps(2) power_up.simps(2)
 
 lemmas power_down_simp = power_down.simps(2)
 lemmas power_up_simp = power_up.simps(2)
 
 lemma power_down_even_nonneg: "even n ==> 0 ≤ power_down p x n"
  by (induct p x n rule: power_down.induct)
  (auto simp: power_down_simp simp del: odd_Suc_div_two intro!: truncate_down_nonneg )
 
 lemma power_down_eq_zero_iff[simp]: "power_down prec b n = 0 ⟷ b = 0 ∧ n ≠ 0"
 proof (induction n arbitrary: b rule: less_induct)
  case (less x)
  then show ?case
  using power_down_simp[of _ _ "x - 1"]
  by (cases x) (auto simp add: div2_less_self)
 qed
 
 lemma power_down_nonneg_iff[simp]:
  "power_down prec b n ≥ 0 ⟷ even n ∨ b ≥ 0"
 proof (induction n arbitrary: b rule: less_induct)
  case (less x)
  show ?case
  using less(1)[of "x - 1" b] power_down_simp[of _ _ "x - 1"]
  by (cases x) (auto simp: algebra_split_simps zero_le_mult_iff)
 qed
 
 lemma power_down_neg_iff[simp]:
  "power_down prec b n < 0 ⟷
  b < 0 ∧ odd n"
  using power_down_nonneg_iff[of prec b n] by (auto simp del: power_down_nonneg_iff)
 
 lemma power_down_nonpos_iff[simp]:
  notes [simp del] = power_down_neg_iff power_down_eq_zero_iff
  shows "power_down prec b n ≤ 0 ⟷ b < 0 ∧ odd n ∨ b = 0 ∧ n ≠ 0"
  using power_down_neg_iff[of prec b n] power_down_eq_zero_iff[of prec b n]
  by auto
 
 lemma power_down_mono:
  "power_down prec a n ≤ power_down prec b n"
  if "((0 ≤ a ∧ a ≤ b)∨(odd n ∧ a ≤ b) ∨ (even n ∧ a ≤ 0 ∧ b ≤ a))"
  using that
 proof (induction n arbitrary: a b rule: less_induct)
  case (less i)
  show ?case
  proof (cases i)
  case j: (Suc j)
  note IH = less[unfolded j even_Suc not_not]
  note [simp del] = power_down.simps
  show ?thesis
  proof cases
  assume [simp]: "even j"
  have "a * power_down prec a j ≤ b * power_down prec b j"
  by (metis IH(1) IH(2) ‹even j› lessI linear mult_mono mult_mono' mult_mono_nonpos_nonneg power_down_even_nonneg)
  then have "truncate_down (Suc prec) (a * power_down prec a j) ≤ truncate_down (Suc prec) (b * power_down prec b j)"
  by (auto intro!: truncate_down_mono simp: abs_le_square_iff[symmetric] abs_real_def)
  then show ?thesis
  unfolding j
  by (simp add: power_down_simp)
  next
  assume [simp]: "odd j"
  have "power_down prec 0 (Suc (j div 2)) ≤ - power_down prec b (Suc (j div 2))"
  if "b < 0" "even (j div 2)"
  by (metis even_Suc le_minus_iff Suc_neq_Zero neg_equal_zero power_down_eq_zero_iff
  power_down_nonpos_iff that)
  then have "truncate_down (Suc prec) ((power_down prec a (Suc (j div 2)))🪙2)
  ≤ truncate_down (Suc prec) ((power_down prec b (Suc (j div 2)))🪙2)"
  by (smt (verit) IH Suc_less_eq ‹odd j› div2_less_self mult_mono_nonpos_nonpos
  Suc_neq_Zero power2_eq_square power_down_neg_iff power_down_nonpos_iff power_mono truncate_down_mono)
  then show ?thesis
  unfolding j by (simp add: power_down_simp)
  qed
  qed simp
 qed
 
 lemma power_up_even_nonneg: "even n ==> 0 ≤ power_up p x n"
  by (induct p x n rule: power_up.induct)
  (auto simp: power_up.simps simp del: odd_Suc_div_two)
 
 lemma power_up_eq_zero_iff[simp]: "power_up prec b n = 0 ⟷ b = 0 ∧ n ≠ 0"
 proof (induction n arbitrary: b rule: less_induct)
  case (less x)
  then show ?case
  using power_up_simp[of _ _ "x - 1"]
  by (cases x) (auto simp: algebra_split_simps zero_le_mult_iff div2_less_self)
 qed
 
 lemma power_up_nonneg_iff[simp]:
  "power_up prec b n ≥ 0 ⟷ even n ∨ b ≥ 0"
 proof (induction n arbitrary: b rule: less_induct)
  case (less x)
  show ?case
  using less(1)[of "x - 1" b] power_up_simp[of _ _ "x - 1"]
  by (cases x) (auto simp: algebra_split_simps zero_le_mult_iff)
 qed
 
 lemma power_up_neg_iff[simp]:
  "power_up prec b n < 0 ⟷ b < 0 ∧ odd n"
  using power_up_nonneg_iff[of prec b n] by (auto simp del: power_up_nonneg_iff)
 
 lemma power_up_nonpos_iff[simp]:
  notes [simp del] = power_up_neg_iff power_up_eq_zero_iff
  shows "power_up prec b n ≤ 0 ⟷ b < 0 ∧ odd n ∨ b = 0 ∧ n ≠ 0"
  using power_up_neg_iff[of prec b n] power_up_eq_zero_iff[of prec b n]
  by auto
 
 lemma power_up_mono:
  "power_up prec a n ≤ power_up prec b n"
  if "((0 ≤ a ∧ a ≤ b)∨(odd n ∧ a ≤ b) ∨ (even n ∧ a ≤ 0 ∧ b ≤ a))"
  using that
 proof (induction n arbitrary: a b rule: less_induct)
  case (less i)
  show ?case
  proof (cases i)
  case j: (Suc j)
  note IH = less[unfolded j even_Suc not_not]
  note [simp del] = power_up.simps
  show ?thesis
  proof cases
  assume [simp]: "even j"
  have "a * power_up prec a j ≤ b * power_up prec b j"
  by (metis IH(1) IH(2) ‹even j› lessI linear mult_mono mult_mono' mult_mono_nonpos_nonneg power_up_even_nonneg)
  then have "truncate_up prec (a * power_up prec a j) ≤ truncate_up prec (b * power_up prec b j)"
  by (auto intro!: truncate_up_mono simp: abs_le_square_iff[symmetric] abs_real_def)
  then show ?thesis
  unfolding j
  by (simp add: power_up_simp)
  next
  assume [simp]: "odd j"
  have "power_up prec 0 (Suc (j div 2)) ≤ - power_up prec b (Suc (j div 2))"
  if "b < 0" "even (j div 2)"
  by (metis Suc_neq_Zero even_Suc neg_0_le_iff_le power_up_eq_zero_iff power_up_nonpos_iff that)
  then have "truncate_up prec ((power_up prec a (Suc (j div 2)))🪙2)
  ≤ truncate_up prec ((power_up prec b (Suc (j div 2)))🪙2)"
  using IH
  by (auto intro!: truncate_up_mono intro: order_trans[where y=0]
  simp: abs_le_square_iff[symmetric] abs_real_def
  div2_less_self)
  then show ?thesis
  unfolding j
  by (simp add: power_up_simp)
  qed
  qed simp
 qed
 
 
 subsection ‹Lemmas needed by Approximate›
 
 lemma Float_num[simp]:
  "real_of_float (Float 1 0) = 1"
  "real_of_float (Float 1 1) = 2"
  "real_of_float (Float 1 2) = 4"
  "real_of_float (Float 1 (- 1)) = 1/2"
  "real_of_float (Float 1 (- 2)) = 1/4"
  "real_of_float (Float 1 (- 3)) = 1/8"
  "real_of_float (Float (- 1) 0) = -1"
  "real_of_float (Float (numeral n) 0) = numeral n"
  "real_of_float (Float (- numeral n) 0) = - numeral n"
  using two_powr_int_float[of 2] two_powr_int_float[of "-1"] two_powr_int_float[of "-2"]
  two_powr_int_float[of "-3"]
  using powr_realpow[of 2 2] powr_realpow[of 2 3]
  using powr_minus[of "2::real" 1] powr_minus[of "2::real" 2] powr_minus[of "2::real" 3]
  by auto
 
 lemma real_of_Float_int[simp]: "real_of_float (Float n 0) = real n"
  by simp
 
 lemma float_zero[simp]: "real_of_float (Float 0 e) = 0"
  by simp
 
 lemma abs_div_2_less: "a ≠ 0 ==> a ≠ -1 ==> ∣(a::int) div 2∣ < ∣a∣"
  by arith
 
 lemma lapprox_rat: "real_of_float (lapprox_rat prec x y) ≤ real_of_int x / real_of_int y"
  by (simp add: lapprox_rat.rep_eq truncate_down)
 
 lemma mult_div_le:
  fixes a b :: int
  assumes "b > 0"
  shows "a ≥ b * (a div b)"
  by (smt (verit, ccfv_threshold) assms minus_div_mult_eq_mod mod_int_pos_iff mult.commute)
 
 lemma lapprox_rat_nonneg:
  assumes "0 ≤ x" and "0 ≤ y"
  shows "0 ≤ real_of_float (lapprox_rat n x y)"
  using assms
  by transfer (simp add: truncate_down_nonneg)
 
 lemma rapprox_rat: "real_of_int x / real_of_int y ≤ real_of_float (rapprox_rat prec x y)"
  by (simp add: rapprox_rat.rep_eq truncate_up)
 
 lemma rapprox_rat_le1:
  assumes "0 ≤ x" "0 < y" "x ≤ y"
  shows "real_of_float (rapprox_rat n x y) ≤ 1"
  using assms
  by transfer (simp add: truncate_up_le1)
 
 lemma rapprox_rat_nonneg_nonpos: "0 ≤ x ==> y ≤ 0 ==> real_of_float (rapprox_rat n x y) ≤ 0"
  by transfer (simp add: truncate_up_nonpos divide_nonneg_nonpos)
 
 lemma rapprox_rat_nonpos_nonneg: "x ≤ 0 ==> 0 ≤ y ==> real_of_float (rapprox_rat n x y) ≤ 0"
  by transfer (simp add: truncate_up_nonpos divide_nonpos_nonneg)
 
 lemma real_divl: "real_divl prec x y ≤ x / y"
  by (simp add: real_divl_def truncate_down)
 
 lemma real_divr: "x / y ≤ real_divr prec x y"
  by (simp add: real_divr_def truncate_up)
 
 lemma float_divl: "real_of_float (float_divl prec x y) ≤ x / y"
  by transfer (rule real_divl)
 
 lemma real_divl_lower_bound: "0 ≤ x ==> 0 ≤ y ==> 0 ≤ real_divl prec x y"
  by (simp add: real_divl_def truncate_down_nonneg)
 
 lemma float_divl_lower_bound: "0 ≤ x ==> 0 ≤ y ==> 0 ≤ real_of_float (float_divl prec x y)"
  by transfer (rule real_divl_lower_bound)
 
 lemma exponent_1: "exponent 1 = 0"
  using exponent_float[of 1 0] by (simp add: one_float_def)
 
 lemma mantissa_1: "mantissa 1 = 1"
  using mantissa_float[of 1 0] by (simp add: one_float_def)
 
 lemma bitlen_1: "bitlen 1 = 1"
  by (simp add: bitlen_alt_def)
 
 lemma float_upper_bound: "x ≤ 2 powr (bitlen ∣mantissa x∣ + exponent x)"
 proof (cases "x = 0")
  case True
  then show ?thesis by simp
 next
  case False
  then have "mantissa x ≠ 0"
  using mantissa_eq_zero_iff by auto
  have "x = mantissa x * 2 powr (exponent x)"
  by (rule mantissa_exponent)
  also have "mantissa x ≤ ∣mantissa x∣"
  by simp
  also have "… ≤ 2 powr (bitlen ∣mantissa x∣)"
  using bitlen_bounds[of "∣mantissa x∣"] bitlen_nonneg ‹mantissa x ≠ 0›
  by (auto simp del: of_int_abs simp add: powr_int)
  finally show ?thesis by (simp add: powr_add)
 qed
 
 lemma real_divl_pos_less1_bound:
  assumes "0 < x" "x ≤ 1"
  shows "1 ≤ real_divl prec 1 x"
  using assms
  by (auto intro!: truncate_down_ge1 simp: real_divl_def)
 
 lemma float_divl_pos_less1_bound:
  "0 < real_of_float x ==> real_of_float x ≤ 1 ==> prec ≥ 1 ==>
  1 ≤ real_of_float (float_divl prec 1 x)"
  by transfer (rule real_divl_pos_less1_bound)
 
 lemma float_divr: "real_of_float x / real_of_float y ≤ real_of_float (float_divr prec x y)"
  by (simp add: float_divr.rep_eq real_divr)
 
 lemma real_divr_pos_less1_lower_bound:
  assumes "0 < x"
  and "x ≤ 1"
  shows "1 ≤ real_divr prec 1 x"
 proof -
  have "1 ≤ 1 / x"
  using ‹0 < x› and ‹x ≤ 1› by auto
  also have "… ≤ real_divr prec 1 x"
  using real_divr[where x = 1 and y = x] by auto
  finally show ?thesis by auto
 qed
 
 lemma float_divr_pos_less1_lower_bound: "0 < x ==> x ≤ 1 ==> 1 ≤ float_divr prec 1 x"
  by transfer (rule real_divr_pos_less1_lower_bound)
 
 lemma real_divr_nonpos_pos_upper_bound: "x ≤ 0 ==> 0 ≤ y ==> real_divr prec x y ≤ 0"
  by (simp add: real_divr_def truncate_up_nonpos divide_le_0_iff)
 
 lemma float_divr_nonpos_pos_upper_bound:
  "real_of_float x ≤ 0 ==> 0 ≤ real_of_float y ==> real_of_float (float_divr prec x y) ≤ 0"
  by transfer (rule real_divr_nonpos_pos_upper_bound)
 
 lemma real_divr_nonneg_neg_upper_bound: "0 ≤ x ==> y ≤ 0 ==> real_divr prec x y ≤ 0"
  by (simp add: real_divr_def truncate_up_nonpos divide_le_0_iff)
 
 lemma float_divr_nonneg_neg_upper_bound:
  "0 ≤ real_of_float x ==> real_of_float y ≤ 0 ==> real_of_float (float_divr prec x y) ≤ 0"
  by transfer (rule real_divr_nonneg_neg_upper_bound)
 
 lemma Float_le_zero_iff: "Float a b ≤ 0 ⟷ a ≤ 0"
  by (auto simp: zero_float_def mult_le_0_iff)
 
 lemma real_of_float_pprt[simp]:
  fixes a :: float
  shows "real_of_float (pprt a) = pprt (real_of_float a)"
  unfolding pprt_def sup_float_def max_def sup_real_def by auto
 
 lemma real_of_float_nprt[simp]:
  fixes a :: float
  shows "real_of_float (nprt a) = nprt (real_of_float a)"
  unfolding nprt_def inf_float_def min_def inf_real_def by auto
 
 context
 begin
 
 lift_definition int_floor_fl :: "float ==> int" is floor .
 
 qualified lemma compute_int_floor_fl[code]:
  "int_floor_fl (Float m e) = (if 0 ≤ e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
  apply transfer
  by (smt (verit, ccfv_threshold) Float.rep_eq compute_real_of_float floor_divide_of_int_eq
  floor_of_int of_int_1 of_int_add of_int_mult of_int_power)
 
 lift_definition floor_fl :: "float ==> float" is "λx. real_of_int ⌊x⌋"
  by simp
 
 qualified lemma compute_floor_fl[code]:
  "floor_fl (Float m e) = (if 0 ≤ e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
  apply transfer
  using compute_int_floor_fl int_floor_fl.rep_eq powr_int by auto
 
 end
 
 lemma floor_fl: "real_of_float (floor_fl x) ≤ real_of_float x"
  by transfer simp
 
 lemma int_floor_fl: "real_of_int (int_floor_fl x) ≤ real_of_float x"
  by transfer simp
 
 lemma floor_pos_exp: "exponent (floor_fl x) ≥ 0"
 proof (cases "floor_fl x = 0")
  case True
  then show ?thesis
  by (simp add: floor_fl_def)
 next
  case False
  have eq: "floor_fl x = Float ⌊real_of_float x⌋ 0"
  by transfer simp
  obtain i where "⌊real_of_float x⌋ = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
  by (rule denormalize_shift[OF eq False])
  then show ?thesis
  by simp
 qed
 
 lemma compute_mantissa[code]:
  "mantissa (Float m e) =
  (if m = 0 then 0 else if 2 dvd m then mantissa (normfloat (Float m e)) else m)"
  by (auto simp: mantissa_float Float.abs_eq simp flip: zero_float_def)
 
 lemma compute_exponent[code]:
  "exponent (Float m e) =
  (if m = 0 then 0 else if 2 dvd m then exponent (normfloat (Float m e)) else e)"
  by (auto simp: exponent_float Float.abs_eq simp flip: zero_float_def)
 
 lifting_update Float.float.lifting
 lifting_forget Float.float.lifting
 
 end