src/HOL/Library/Float.thy

countablility of finite subsets and rational numbers

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

header {* Floating-Point Numbers *}

theory Float
imports Complex_Main "~~/src/HOL/Library/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

defs (overloaded)
  real_of_float_def[code_unfold]: "real \<equiv> real_of_float"

lemma type_definition_float': "type_definition real float_of float"
  using type_definition_float unfolding real_of_float_def .

setup_lifting (no_code) type_definition_float'

lemmas float_of_inject[simp]

declare [[coercion "real :: float \<Rightarrow> real"]]

lemma real_of_float_eq:
  fixes f1 f2 :: float shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"
  unfolding real_of_float_def real_of_float_inject ..

lemma float_of_real[simp]: "float_of (real x) = x"
  unfolding real_of_float_def by (rule real_of_float_inverse)

lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
  unfolding real_of_float_def by (rule float_of_inverse)

subsection {* Real operations preserving the representation as floating point number *}

lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"
  by (auto simp: float_def)

lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)
lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp
lemma numeral_float[simp]: "numeral i \<in> float" by (intro floatI[of "numeral i" 0]) simp  
lemma neg_numeral_float[simp]: "neg_numeral i \<in> float" by (intro floatI[of "neg_numeral i" 0]) simp
lemma real_of_int_float[simp]: "real (x :: int) \<in> float" by (intro floatI[of x 0]) simp
lemma real_of_nat_float[simp]: "real (x :: nat) \<in> float" by (intro floatI[of x 0]) simp
lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float" by (intro floatI[of 1 i]) simp
lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) \<in> float" by (intro floatI[of 1 i]) simp
lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float" by (intro floatI[of 1 "-i"]) simp
lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float" by (intro floatI[of 1 "-i"]) simp
lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" by (intro floatI[of 1 "numeral i"]) simp
lemma two_powr_neg_numeral_float[simp]: "2 powr neg_numeral i \<in> float" by (intro floatI[of 1 "neg_numeral i"]) simp
lemma two_pow_float[simp]: "2 ^ n \<in> float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
lemma real_of_float_float[simp]: "real (f::float) \<in> float" by (cases f) simp

lemma plus_float[simp]: "r \<in> float \<Longrightarrow> p \<in> float \<Longrightarrow> r + p \<in> float"
  unfolding float_def
proof (safe, simp)
  fix e1 m1 e2 m2 :: int
  { fix e1 m1 e2 m2 :: int assume "e1 \<le> e2"
    then have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
      by (simp add: powr_realpow[symmetric] powr_divide2[symmetric] field_simps)
    then have "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
      by blast }
  note * = this
  show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
  proof (cases e1 e2 rule: linorder_le_cases)
    assume "e2 \<le> e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)
  qed (rule *)
qed

lemma uminus_float[simp]: "x \<in> float \<Longrightarrow> -x \<in> float"
  apply (auto simp: float_def)
  apply (rule_tac x="-x" in exI)
  apply (rule_tac x="xa" in exI)
  apply (simp add: field_simps)
  done

lemma times_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x * y \<in> float"
  apply (auto simp: float_def)
  apply (rule_tac x="x * xa" in exI)
  apply (rule_tac x="xb + xc" in exI)
  apply (simp add: powr_add)
  done

lemma minus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x - y \<in> float"
  unfolding ab_diff_minus by (intro uminus_float plus_float)

lemma abs_float[simp]: "x \<in> float \<Longrightarrow> abs x \<in> float"
  by (cases x rule: linorder_cases[of 0]) auto

lemma sgn_of_float[simp]: "x \<in> float \<Longrightarrow> sgn x \<in> float"
  by (cases x rule: linorder_cases[of 0]) (auto intro!: uminus_float)

lemma div_power_2_float[simp]: "x \<in> float \<Longrightarrow> x / 2^d \<in> float"
  apply (auto simp add: float_def)
  apply (rule_tac x="x" in exI)
  apply (rule_tac x="xa - d" in exI)
  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
  done

lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"
  apply (auto simp add: float_def)
  apply (rule_tac x="x" in exI)
  apply (rule_tac x="xa - d" in exI)
  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
  done

lemma div_numeral_Bit0_float[simp]:
  assumes x: "x / numeral n \<in> float" shows "x / (numeral (Num.Bit0 n)) \<in> float"
proof -
  have "(x / numeral n) / 2^1 \<in> float"
    by (intro x 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: "x / numeral n \<in> float" shows "x / (neg_numeral (Num.Bit0 n)) \<in> float"
proof -
  have "- (x / numeral (Num.Bit0 n)) \<in> float" using x by simp
  also have "- (x / numeral (Num.Bit0 n)) = x / neg_numeral (Num.Bit0 n)"
    unfolding neg_numeral_def by (simp del: minus_numeral)
  finally show ?thesis .
qed

lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e" by simp
declare Float.rep_eq[simp]

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

code_datatype Float

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 \<Rightarrow> float \<Rightarrow> float" is "op +" by simp
declare plus_float.rep_eq[simp]
lift_definition times_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op *" by simp
declare times_float.rep_eq[simp]
lift_definition minus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op -" by simp
declare minus_float.rep_eq[simp]
lift_definition uminus_float :: "float \<Rightarrow> float" is "uminus" by simp
declare uminus_float.rep_eq[simp]

lift_definition abs_float :: "float \<Rightarrow> float" is abs by simp
declare abs_float.rep_eq[simp]
lift_definition sgn_float :: "float \<Rightarrow> float" is sgn by simp
declare sgn_float.rep_eq[simp]

lift_definition equal_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op = :: real \<Rightarrow> real \<Rightarrow> bool" ..

lift_definition less_eq_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op \<le>" ..
declare less_eq_float.rep_eq[simp]
lift_definition less_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op <" ..
declare less_float.rep_eq[simp]

instance
  proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+
end

lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n"
  by (induct n) simp_all

lemma fixes x y::float 
  shows real_of_float_min: "real (min x y) = min (real x) (real y)"
    and real_of_float_max: "real (max x y) = max (real x) (real y)"
  by (simp_all add: min_def max_def)

instance float :: dense_linorder
proof
  fix a b :: float
  show "\<exists>c. a < c"
    apply (intro exI[of _ "a + 1"])
    apply transfer
    apply simp
    done
  show "\<exists>c. c < a"
    apply (intro exI[of _ "a - 1"])
    apply transfer
    apply simp
    done
  assume "a < b"
  then show "\<exists>c. a < c \<and> c < b"
    apply (intro exI[of _ "(a + b) * Float 1 -1"])
    apply transfer
    apply (simp add: powr_neg_numeral) 
    done
qed

instantiation float :: lattice_ab_group_add
begin

definition inf_float::"float\<Rightarrow>float\<Rightarrow>float"
where "inf_float a b = min a b"

definition sup_float::"float\<Rightarrow>float\<Rightarrow>float"
where "sup_float a b = max a b"

instance
  by default
     (transfer, simp_all add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+
end

lemma float_numeral[simp]: "real (numeral x :: float) = numeral x"
  apply (induct x)
  apply simp
  apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq real_float
                  plus_float.rep_eq one_float.rep_eq plus_float numeral_float one_float)
  done

lemma transfer_numeral [transfer_rule]: 
  "fun_rel (op =) cr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
  unfolding fun_rel_def cr_float_def by simp

lemma float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x"
  by (simp add: minus_numeral[symmetric] del: minus_numeral)

lemma transfer_neg_numeral [transfer_rule]: 
  "fun_rel (op =) cr_float (neg_numeral :: _ \<Rightarrow> real) (neg_numeral :: _ \<Rightarrow> float)"
  unfolding fun_rel_def cr_float_def by simp

lemma
  shows float_of_numeral[simp]: "numeral k = float_of (numeral k)"
    and float_of_neg_numeral[simp]: "neg_numeral k = float_of (neg_numeral k)"
  unfolding real_of_float_eq by simp_all

subsection {* Represent floats as unique mantissa and exponent *}

lemma int_induct_abs[case_names less]:
  fixes j :: int
  assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
  shows "P j"
proof (induct "nat \<bar>j\<bar>" 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 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
proof (induct n rule: int_induct_abs)
  case (less n)
  { fix m assume n: "n \<noteq> 0" "n = m * r"
    then have "\<bar>m \<bar> < \<bar>n\<bar>"
      by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7)
                dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le
                mult_eq_0_iff zdvd_mult_cancel1)
    from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
  then show ?case
    by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0)
qed

lemma mult_powr_eq_mult_powr_iff_asym:
  fixes m1 m2 e1 e2 :: int
  assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"
  shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
proof
  have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
  assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
  with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)"
    by (simp add: powr_divide2[symmetric] field_simps)
  also have "\<dots> = m2 * 2^nat (e2 - e1)"
    by (simp add: powr_realpow)
  finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
    unfolding real_of_int_inject .
  with m1 have "m1 = m2"
    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
  then show "m1 = m2 \<and> e1 = e2"
    using eq `m1 \<noteq> 0` by (simp add: powr_inj)
qed simp

lemma mult_powr_eq_mult_powr_iff:
  fixes m1 m2 e1 e2 :: int
  shows "\<not> 2 dvd m1 \<Longrightarrow> \<not> 2 dvd m2 \<Longrightarrow> m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
  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 \<in> float"
  obtains (zero) "x = 0"
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
proof atomize_elim
  { assume "x \<noteq> 0"
    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
    with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
      by auto
    with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
      by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
         (simp add: powr_add powr_realpow) }
  then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)"
    by blast
qed

lemma float_normed_cases:
  fixes f :: float
  obtains (zero) "f = 0"
   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 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 \<Rightarrow> int" where
  "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"

definition exponent :: "float \<Rightarrow> int" where
  "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"

lemma 
  shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
    and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
proof -
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
  then show ?E ?M
    by (auto simp add: mantissa_def exponent_def zero_float_def)
qed

lemma
  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
    and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
proof cases
  assume [simp]: "f \<noteq> (float_of 0)"
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
  proof (cases f rule: float_normed_cases)
    case (powr m e)
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
     \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
      by auto
    then show ?thesis
      unfolding exponent_def mantissa_def
      by (rule someI2_ex) (simp add: zero_float_def)
  qed (simp add: zero_float_def)
  then show ?E ?D by auto
qed simp

lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
  using mantissa_not_dvd[of f] by auto

lemma 
  fixes m e :: int
  defines "f \<equiv> float_of (m * 2 powr e)"
  assumes dvd: "\<not> 2 dvd m"
  shows mantissa_float: "mantissa f = m" (is "?M")
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
proof cases
  assume "m = 0" with dvd show "mantissa f = m" by auto
next
  assume "m \<noteq> 0"
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_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[case_names Float, 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 \<equiv> Float m e" and not_0: "f \<noteq> float_of 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_divide2[symmetric] field_simps)
  moreover
  have "e \<le> exponent f"
  proof (rule ccontr)
    assume "\<not> e \<le> exponent f"
    then have pos: "exponent f < e" by simp
    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
      by simp
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
      using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
      using eq by simp
    then have "mantissa f = m * 2^nat (e - exponent f)"
      unfolding real_of_int_inject by simp
    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 m = mantissa f * 2^nat (exponent f - e)"
    by (simp add: powr_realpow[symmetric])
  with `e \<le> exponent f`
  show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
    unfolding real_of_int_inject by auto
qed

lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
  by transfer simp
hide_fact (open) compute_float_zero

lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
  by transfer simp
hide_fact (open) compute_float_one

definition normfloat :: "float \<Rightarrow> float" where
  [simp]: "normfloat x = x"

lemma compute_normfloat[code]: "normfloat (Float m e) =
  (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
                           else if m = 0 then 0 else Float m e)"
  unfolding normfloat_def
  by transfer (auto simp add: powr_add zmod_eq_0_iff)
hide_fact (open) compute_normfloat

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

lemma compute_float_neg_numeral[code_abbrev]: "Float (neg_numeral k) 0 = neg_numeral k"
  by transfer simp
hide_fact (open) compute_float_neg_numeral

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

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

lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
  (if e1 \<le> 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_divide2[symmetric])
hide_fact (open) compute_float_plus

lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
  by simp
hide_fact (open) compute_float_minus

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_times)
hide_fact (open) compute_float_sgn

lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" ..

lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
hide_fact (open) compute_is_float_pos

lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
  by transfer (simp add: field_simps)
hide_fact (open) compute_float_less

lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" ..

lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
hide_fact (open) compute_is_float_nonneg

lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
  by transfer (simp add: field_simps)
hide_fact (open) compute_float_le

lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" by simp

lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
  by transfer (auto simp add: is_float_zero_def)
hide_fact (open) compute_is_float_zero

lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
  by transfer (simp add: abs_mult)
hide_fact (open) compute_float_abs

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

subsection {* Rounding Real numbers *}

definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"

definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"

lemma round_down_float[simp]: "round_down prec x \<in> float"
  unfolding round_down_def
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)

lemma round_up_float[simp]: "round_up prec x \<in> float"
  unfolding round_up_def
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)

lemma round_up: "x \<le> round_up prec x"
  by (simp add: powr_minus_divide le_divide_eq round_up_def)

lemma round_down: "round_down prec x \<le> 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 \<le> 2 powr -prec"
proof -
  have "round_up prec x - round_down prec x =
    (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
    by (simp add: round_up_def round_down_def field_simps)
  also have "\<dots> \<le> 1 * 2 powr -prec"
    by (rule mult_mono)
       (auto simp del: real_of_int_diff
             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff 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_divide2[symmetric])
    (simp add: powr_add[symmetric])

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_divide2[symmetric])
    (simp add: powr_add[symmetric])

subsection {* Rounding Floats *}

lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
declare float_up.rep_eq[simp]

lemma float_up_correct:
  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
unfolding atLeastAtMost_iff
proof
  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
  finally show "real (float_up e f) - real f \<le> 2 powr real (- e)"
    by simp
qed (simp add: algebra_simps round_up)

lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
declare float_down.rep_eq[simp]

lemma float_down_correct:
  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
unfolding atLeastAtMost_iff
proof
  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
  finally show "real f - real (float_down e f) \<le> 2 powr real (- e)"
    by simp
qed (simp add: algebra_simps round_down)

lemma compute_float_down[code]:
  "float_down p (Float m e) =
    (if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"
proof cases
  assume "p + e < 0"
  hence "real ((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 real_of_int_minus by (simp add: powr_add)
  finally show ?thesis
    using `p + e < 0`
    by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
next
  assume "\<not> p + e < 0"
  then have r: "real e + real p = real (nat (e + p))" by simp
  have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
    by (auto intro: exI[where x="m*2^nat (e+p)"]
             simp add: ac_simps powr_add[symmetric] r powr_realpow)
  with `\<not> p + e < 0` show ?thesis
    by transfer
       (auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide)
qed
hide_fact (open) compute_float_down

lemma ceil_divide_floor_conv:
assumes "b \<noteq> 0"
shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)"
proof cases
  assume "\<not> b dvd a"
  hence "a mod b \<noteq> 0" by auto
  hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
  proof -
    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
  qed
  thus ?thesis using `\<not> b dvd a` by simp
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
  floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)

lemma compute_float_up[code]:
  "float_up p (Float m e) =
    (let P = 2^nat (-(p + e)); r = m mod P in
      if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (-p) else Float m e)"
proof cases
  assume "p + e < 0"
  hence "real ((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 real_of_int_minus by (simp add: powr_add)
  finally have twopow_rewrite:
    "real ((2::int) ^ nat (- (p + e))) = 1 / 2 powr real p / 2 powr real e" .
  with `p + e < 0` have powr_rewrite:
    "2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat (- (p + e)))"
    unfolding powr_divide2 by simp
  show ?thesis
  proof cases
    assume "2^nat (-(p + e)) dvd m"
    with `p + e < 0` twopow_rewrite show ?thesis
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div dvd_eq_mod_eq_0)
  next
    assume ndvd: "\<not> 2 ^ nat (- (p + e)) dvd m"
    have one_div: "real m * (1 / real ((2::int) ^ nat (- (p + e)))) =
      real m / real ((2::int) ^ nat (- (p + e)))"
      by (simp add: field_simps)
    have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> =
      real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1"
      using ndvd unfolding powr_rewrite one_div
      by (subst ceil_divide_floor_conv) (auto simp: field_simps)
    thus ?thesis using `p + e < 0` twopow_rewrite
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div[symmetric])
  qed
next
  assume "\<not> p + e < 0"
  then have r1: "real e + real p = real (nat (e + p))" by simp
  have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p"
    by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow
      intro: exI[where x="m*2^nat (e+p)"])
  then show ?thesis using `\<not> p + e < 0`
    by transfer
       (simp add: round_up_def floor_divide_eq_div field_simps powr_add powr_minus inverse_eq_divide)
qed
hide_fact (open) compute_float_up

lemmas real_of_ints =
  real_of_int_zero
  real_of_one
  real_of_int_add
  real_of_int_minus
  real_of_int_diff
  real_of_int_mult
  real_of_int_power
  real_numeral
lemmas real_of_nats =
  real_of_nat_zero
  real_of_nat_one
  real_of_nat_1
  real_of_nat_add
  real_of_nat_mult
  real_of_nat_power

lemmas int_of_reals = real_of_ints[symmetric]
lemmas nat_of_reals = real_of_nats[symmetric]

lemma two_real_int: "(2::real) = real (2::int)" by simp
lemma two_real_nat: "(2::real) = real (2::nat)" by simp

lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp

subsection {* Compute bitlen of integers *}

definition bitlen :: "int \<Rightarrow> int" where
  "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"

lemma bitlen_nonneg: "0 \<le> bitlen x"
proof -
  {
    assume "0 > x"
    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
    also have "... < log 2 (-x)" using `0 > x` by auto
    finally have "-1 < log 2 (-x)" .
  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
qed

lemma bitlen_bounds:
  assumes "x > 0"
  shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
proof
  have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
    using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
    using real_nat_eq_real[of "floor (log 2 (real x))"]
    by simp
  also have "... \<le> 2 powr log 2 (real x)"
    by simp
  also have "... = real x"
    using `0 < x` by simp
  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
  thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
    by (simp add: bitlen_def)
next
  have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
    apply (simp add: powr_realpow[symmetric])
    using `x > 0` by simp
  finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
    by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)
qed

lemma bitlen_pow2[simp]:
  assumes "b > 0"
  shows "bitlen (b * 2 ^ c) = bitlen b + c"
proof -
  from assms have "b * 2 ^ c > 0" by (auto intro: mult_pos_pos)
  thus ?thesis
    using floor_add[of "log 2 b" c] assms
    by (auto simp add: log_mult log_nat_power bitlen_def)
qed

lemma bitlen_Float:
fixes m e
defines "f \<equiv> Float m e"
shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
proof cases
  assume "m \<noteq> 0"
  hence "f \<noteq> float_of 0"
    unfolding real_of_float_eq by (simp add: f_def)
  hence "mantissa f \<noteq> 0"
    by (simp add: mantissa_noteq_0)
  moreover
  from f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`] guess i .
  ultimately show ?thesis by (simp add: abs_mult)
qed (simp add: f_def bitlen_def Float_def)

lemma compute_bitlen[code]:
  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
proof -
  { assume "2 \<le> x"
    then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
      by (simp add: log_mult zmod_zdiv_equality')
    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
    proof cases
      assume "x mod 2 = 0" then show ?thesis by simp
    next
      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
      then have "0 \<le> n"
        using `2 \<le> x` by simp
      assume "x mod 2 \<noteq> 0"
      with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
      with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
      moreover
      { have "real (2^nat n :: int) = 2 powr (nat n)"
          by (simp add: powr_realpow)
        also have "\<dots> \<le> 2 powr (log 2 x)"
          using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
        finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
      ultimately have "2^nat n \<le> x - 1" by simp
      then have "2^nat n \<le> real (x - 1)"
        unfolding real_of_int_le_iff[symmetric] by simp
      { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
          using `0 \<le> n` by (simp add: log_nat_power)
        also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
          using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
        finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
      moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
        using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
      ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
        unfolding n_def `x mod 2 = 1` by auto
    qed
    finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
  moreover
  { assume "x < 2" "0 < x"
    then have "x = 1" by simp
    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
  ultimately show ?thesis
    unfolding bitlen_def
    by (auto simp: pos_imp_zdiv_pos_iff not_le)
qed
hide_fact (open) compute_bitlen

lemma float_gt1_scale: assumes "1 \<le> Float m e"
  shows "0 \<le> e + (bitlen m - 1)"
proof -
  have "0 < Float m e" using assms by auto
  hence "0 < m" using powr_gt_zero[of 2 e]
    by (auto simp: zero_less_mult_iff)
  hence "m \<noteq> 0" by auto
  show ?thesis
  proof (cases "0 \<le> e")
    case True thus ?thesis using `0 < m`  by (simp add: bitlen_def)
  next
    have "(1::int) < 2" by simp
    case False 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 inverse_eq_divide)
    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
      by (auto simp: powr_minus)
    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
    hence "?S \<le> real m" unfolding mult_assoc by auto
    hence "?S \<le> m" unfolding real_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)
    hence "-e < bitlen m" using False by auto
    thus ?thesis by auto
  qed
qed

lemma bitlen_div: assumes "0 < m" shows "1 \<le> real m / 2^nat (bitlen m - 1)" and "real m / 2^nat (bitlen m - 1) < 2"
proof -
  let ?B = "2^nat(bitlen m - 1)"

  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
  thus "1 \<le> real m / ?B" by auto

  have "m \<noteq> 0" using assms by auto
  have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)

  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
  thus "real m / ?B < 2" by auto
qed

subsection {* Approximation of positive rationals *}

lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"
by (simp add: zdiv_zmult2_eq)

lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)

lemma real_div_nat_eq_floor_of_divide:
  fixes a b::nat
  shows "a div b = real (floor (a/b))"
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)

definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"

lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp

lemma compute_lapprox_posrat[code]:
  fixes prec x y 
  shows "lapprox_posrat prec x y = 
   (let 
       l = rat_precision prec x y;
       d = if 0 \<le> 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 normfloat_def
    by transfer
       (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
             del: two_powr_minus_int_float)
hide_fact (open) compute_lapprox_posrat

lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp

(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *)
lemma compute_rapprox_posrat[code]:
  fixes prec x y
  defines "l \<equiv> rat_precision prec x y"
  shows "rapprox_posrat prec x y = (let
     l = l ;
     X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
     d = fst X div snd X ;
     m = fst X mod snd X
   in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
proof (cases "y = 0")
  assume "y = 0" thus ?thesis unfolding normfloat_def by transfer simp
next
  assume "y \<noteq> 0"
  show ?thesis
  proof (cases "0 \<le> l")
    assume "0 \<le> l"
    def x' == "x * 2 ^ nat l"
    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
    moreover have "real x * 2 powr real l = real x'"
      by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
    ultimately show ?thesis
      unfolding normfloat_def
      using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
        l_def[symmetric, THEN meta_eq_to_obj_eq]
      by transfer
         (simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def)
   next
    assume "\<not> 0 \<le> l"
    def y' == "y * 2 ^ nat (- l)"
    from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
    have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
      using `\<not> 0 \<le> l`
      by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps inverse_eq_divide)
    ultimately show ?thesis
      unfolding normfloat_def
      using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
        l_def[symmetric, THEN meta_eq_to_obj_eq]
      by transfer
         (simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0)
  qed
qed
hide_fact (open) compute_rapprox_posrat

lemma rat_precision_pos:
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
  shows "rat_precision n (int x) (int y) > 0"
proof -
  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
  hence "bitlen (int x) < bitlen (int y)" using assms
    by (simp add: bitlen_def del: floor_add_one)
      (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
  thus ?thesis
    using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
qed

lemma power_aux: assumes "x > 0" shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1"
proof -
  def y \<equiv> "nat (x - 1)" moreover
  have "(2::int) ^ y \<le> (2 ^ (y + 1)) - 1" by simp
  ultimately show ?thesis using assms by simp
qed

lemma rapprox_posrat_less1:
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
  shows "real (rapprox_posrat n x y) < 1"
proof -
  have powr1: "2 powr real (rat_precision n (int x) (int y)) = 
    2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms
    by (simp add: powr_realpow[symmetric])
  have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) *
     2 powr real (rat_precision n (int x) (int y))" by simp
  also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"
    apply (rule mult_strict_right_mono) by (insert assms) auto
  also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"
    by (simp add: powr_add diff_def powr_neg_numeral)
  also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"
    using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])
  also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"
    unfolding int_of_reals real_of_int_le_iff
    using rat_precision_pos[OF assms] by (rule power_aux)
  finally show ?thesis
    apply (transfer fixing: n x y)
    apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide powr1)
    unfolding int_of_reals real_of_int_less_iff
    apply (simp add: ceiling_less_eq)
    done
qed

lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp

lemma compute_lapprox_rat[code]:
  "lapprox_rat prec x y =
    (if y = 0 then 0
    else if 0 \<le> 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 (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hide_fact (open) compute_lapprox_rat

lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp

lemma compute_rapprox_rat[code]:
  "rapprox_rat prec x y =
    (if y = 0 then 0
    else if 0 \<le> x then
      (if 0 < y then rapprox_posrat prec (nat x) (nat y)
      else - (lapprox_posrat prec (nat x) (nat (-y)))) 
      else (if 0 < y
        then - (lapprox_posrat prec (nat (-x)) (nat y))
        else rapprox_posrat prec (nat (-x)) (nat (-y))))"
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hide_fact (open) compute_rapprox_rat

subsection {* Division *}

lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is
  "\<lambda>(prec::nat) a b. round_down (prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" by simp

lemma compute_float_divl[code]:
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
proof cases
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
  then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
    by (simp add: field_simps powr_divide2[symmetric])

  show ?thesis
    using not_0 
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift, simp add: field_simps)
qed (transfer, auto)
hide_fact (open) compute_float_divl

lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is
  "\<lambda>(prec::nat) a b. round_up (prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" by simp

lemma compute_float_divr[code]:
  "float_divr prec (Float m1 s1) (Float m2 s2) = rapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
proof cases
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
  then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
    by (simp add: field_simps powr_divide2[symmetric])

  show ?thesis
    using not_0 
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift, simp add: field_simps)
qed (transfer, auto)
hide_fact (open) compute_float_divr

subsection {* Lemmas needed by Approximate *}

lemma Float_num[simp]: shows
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
   "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
   "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of 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 1] powr_minus[of 2 2] powr_minus[of 2 3]
by auto

lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp

lemma float_zero[simp]: "real (Float 0 e) = 0" by simp

lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
by arith

lemma lapprox_rat:
  shows "real (lapprox_rat prec x y) \<le> real x / real y"
  using round_down by (simp add: lapprox_rat_def)

lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
proof -
  from zmod_zdiv_equality'[of a b]
  have "a = b * (a div b) + a mod b" by simp
  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
  using assms by simp
  finally show ?thesis by simp
qed

lemma lapprox_rat_nonneg:
  fixes n x y
  defines "p == int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))"
  assumes "0 \<le> x" "0 < y"
  shows "0 \<le> real (lapprox_rat n x y)"
using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric]
   powr_int[of 2, simplified]
  by (auto simp add: inverse_eq_divide intro!: mult_nonneg_nonneg divide_nonneg_pos mult_pos_pos)

lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
  using round_up by (simp add: rapprox_rat_def)

lemma rapprox_rat_le1:
  fixes n x y
  assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
  shows "real (rapprox_rat n x y) \<le> 1"
proof -
  have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
    using xy unfolding bitlen_def by (auto intro!: floor_mono)
  then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def)
  have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>
      \<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>"
    using xy by (auto intro!: ceiling_mono simp: field_simps)
  also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"
    using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>`
    by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int)
  finally show ?thesis
    by (simp add: rapprox_rat_def round_up_def)
       (simp add: powr_minus inverse_eq_divide)
qed

lemma rapprox_rat_nonneg_neg: 
  "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
  unfolding rapprox_rat_def round_up_def
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)

lemma rapprox_rat_neg:
  "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
  unfolding rapprox_rat_def round_up_def
  by (auto simp: field_simps mult_le_0_iff)

lemma rapprox_rat_nonpos_pos:
  "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
  unfolding rapprox_rat_def round_up_def
  by (auto simp: field_simps mult_le_0_iff)

lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
  by transfer (simp add: round_down)

lemma float_divl_lower_bound:
  "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
  by transfer (simp add: round_down_def zero_le_mult_iff zero_le_divide_iff)

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_def)

lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
proof
  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
  show "x = 0" by (simp add: zero_float_def z)
qed (simp add: zero_float_def)

lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
proof (cases "x = 0", simp)
  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
    by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]
      real_of_int_le_iff less_imp_le)
  finally show ?thesis by (simp add: powr_add)
qed

lemma float_divl_pos_less1_bound:
  "0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
proof transfer
  fix prec :: nat and x :: real assume x: "0 < x" "x < 1" "x \<in> float" and prec: "1 \<le> prec"
  def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>" 
  show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) "
  proof cases
    assume nonneg: "0 \<le> p"
    hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)"
      by (simp add: powr_int del: real_of_int_power) simp
    also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp
    also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le>
      floor (real ((2::int) ^ nat p) * (1 / x))"
      by (rule le_mult_floor) (auto simp: x prec less_imp_le)
    finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow)
    thus ?thesis unfolding p_def[symmetric]
      using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def)
  next
    assume neg: "\<not> 0 \<le> p"

    have "x = 2 powr (log 2 x)"
      using x by simp
    also have "2 powr (log 2 x) \<le> 2 powr p"
    proof (rule powr_mono)
      have "log 2 x \<le> \<lceil>log 2 x\<rceil>"
        by simp
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1"
        using ceiling_diff_floor_le_1[of "log 2 x"] by simp
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec"
        using prec by simp
      finally show "log 2 x \<le> real p"
        using x by (simp add: p_def)
    qed simp
    finally have x_le: "x \<le> 2 powr p" .

    from neg have "2 powr real p \<le> 2 powr 0"
      by (intro powr_mono) auto
    also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp
    also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff
      using x x_le by (intro floor_mono) (simp add:  pos_le_divide_eq mult_pos_pos)
    finally show ?thesis
      using prec x unfolding p_def[symmetric]
      by (simp add: round_down_def powr_minus_divide pos_le_divide_eq mult_pos_pos)
  qed
qed

lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
  using round_up by transfer simp

lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
proof -
  have "1 \<le> 1 / real x" using `0 < x` and `x < 1` by auto
  also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
  finally show ?thesis by auto
qed

lemma float_divr_nonpos_pos_upper_bound:
  "real x \<le> 0 \<Longrightarrow> 0 < real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"
  by transfer (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def)

lemma float_divr_nonneg_neg_upper_bound:
  "0 \<le> real x \<Longrightarrow> real y < 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"
  by transfer (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def)

lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is
  "\<lambda>(prec::nat) x. round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x" by simp

lemma float_round_up: "real x \<le> real (float_round_up prec x)"
  using round_up by transfer simp

lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is
  "\<lambda>(prec::nat) x. round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x" by simp

lemma float_round_down: "real (float_round_down prec x) \<le> real x"
  using round_down by transfer simp

lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>"
  using floor_add[of x i] by (simp del: floor_add add: ac_simps)

lemma compute_float_round_down[code]:
  "float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
    if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
             else Float m e)"
  using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong)
hide_fact (open) compute_float_round_down

lemma compute_float_round_up[code]:
  "float_round_up prec (Float m e) = (let d = (bitlen (abs m) - int prec) in
     if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P
                   in Float (n + (if r = 0 then 0 else 1)) (e + d)
              else Float m e)"
  using Float.compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
  unfolding Let_def
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong)
hide_fact (open) compute_float_round_up

lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
 apply (auto simp: zero_float_def mult_le_0_iff)
 using powr_gt_zero[of 2 b] by simp

lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real 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 (nprt a) = nprt (real a)"
  unfolding nprt_def inf_float_def min_def inf_real_def by auto

lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor by simp

lemma compute_int_floor_fl[code]:
  "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hide_fact (open) compute_int_floor_fl

lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp

lemma compute_floor_fl[code]:
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hide_fact (open) compute_floor_fl

lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp

lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp

lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
proof cases
  assume nzero: "floor_fl x \<noteq> float_of 0"
  have "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp
  from denormalize_shift[OF this[THEN eq_reflection] nzero] guess i . note i = this
  thus ?thesis by simp
qed (simp add: floor_fl_def)

end