Theory HOL-Library.Float

(*  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)
  by (metis (no_types, opaque_lifting) of_int_add powr_add mult.assoc mult.left_commute of_int_mult)

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 (cases x rule: linorder_cases[of 0]) (auto intro!: uminus_float)

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 (auto intro!: floatI[where m="m^b" and e = "e*b"]
      simp: power_mult_distrib powr_realpow[symmetric] powr_powr)
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"
proof (induct x)
  case One
  then show ?case by simp
qed (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 show ?thesis
      apply (rule_tac exI[of _ "k"])
      apply (rule_tac exI[of _ "e + int i"])
      apply (simp add: powr_add powr_realpow)
      done
  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"
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
      apply (cases "nat (e - exponent f)")
      apply auto
      done
    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 typreal, typnat, typint

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"
    apply (rule ceiling_eq)
    apply (auto simp flip: floor_divide_of_int_eq)
  proof -
    have "real_of_int real_of_int a / real_of_int b  real_of_int a / real_of_int b"
      by simp
    moreover have "real_of_int real_of_int a / real_of_int b  real_of_int a / real_of_int b"
      by (smt (verit) floor_divide_of_int_eq ne real_of_int_div_aux)
    ultimately show "real_of_int real_of_int a / real_of_int b < real_of_int a / real_of_int b" by arith
  qed
  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)
  with assms
  show ?thesis
    apply (auto simp: truncate_down_def round_down_def mult_powr_eq
      intro!: ge_one_powr_ge_zero mult_pos_pos)
    by linarith
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 of_nat_diff)
    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
    by (linarith |
      auto
        intro!: floor_eq2
        intro: powr_strict powr
        simp: powr_diff powr_add field_split_simps algebra_simps)+
  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 dvd_triv_left of_nat_dvd_iff)
      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 dvd_triv_left of_nat_dvd_iff)
      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