src/HOL/Library/Float.thy
author haftmann
Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
changeset 66010 2f7d39285a1a
parent 65583 8d53b3bebab4
child 66912 a99a7cbf0fb5
permissions -rw-r--r--
executable domain membership checks
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(*  Title:      HOL/Library/Float.thy
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    Author:     Johannes Hölzl, Fabian Immler
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    Copyright   2012  TU München
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*)
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section \<open>Floating-Point Numbers\<close>
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theory Float
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imports Log_Nat Lattice_Algebras
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begin
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definition "float = {m * 2 powr e | (m :: int) (e :: int). True}"
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typedef float = float
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  morphisms real_of_float float_of
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  unfolding float_def by auto
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setup_lifting type_definition_float
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declare real_of_float [code_unfold]
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lemmas float_of_inject[simp]
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declare [[coercion "real_of_float :: float \<Rightarrow> real"]]
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lemma real_of_float_eq: "f1 = f2 \<longleftrightarrow> real_of_float f1 = real_of_float f2" for f1 f2 :: float
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  unfolding real_of_float_inject ..
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declare real_of_float_inverse[simp] float_of_inverse [simp]
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declare real_of_float [simp]
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subsection \<open>Real operations preserving the representation as floating point number\<close>
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lemma floatI: "m * 2 powr e = x \<Longrightarrow> x \<in> float" for m e :: int
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  by (auto simp: float_def)
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lemma zero_float[simp]: "0 \<in> float"
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  by (auto simp: float_def)
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lemma one_float[simp]: "1 \<in> float"
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  by (intro floatI[of 1 0]) simp
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lemma numeral_float[simp]: "numeral i \<in> float"
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  by (intro floatI[of "numeral i" 0]) simp
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lemma neg_numeral_float[simp]: "- numeral i \<in> float"
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  by (intro floatI[of "- numeral i" 0]) simp
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lemma real_of_int_float[simp]: "real_of_int x \<in> float" for x :: int
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  by (intro floatI[of x 0]) simp
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lemma real_of_nat_float[simp]: "real x \<in> float" for x :: nat
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  by (intro floatI[of x 0]) simp
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lemma two_powr_int_float[simp]: "2 powr (real_of_int i) \<in> float" for i :: int
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  by (intro floatI[of 1 i]) simp
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lemma two_powr_nat_float[simp]: "2 powr (real i) \<in> float" for i :: nat
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  by (intro floatI[of 1 i]) simp
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lemma two_powr_minus_int_float[simp]: "2 powr - (real_of_int i) \<in> float" for i :: int
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  by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_minus_nat_float[simp]: "2 powr - (real i) \<in> float" for i :: nat
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  by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float"
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  by (intro floatI[of 1 "numeral i"]) simp
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lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i \<in> float"
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  by (intro floatI[of 1 "- numeral i"]) simp
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lemma two_pow_float[simp]: "2 ^ n \<in> float"
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  by (intro floatI[of 1 n]) (simp add: powr_realpow)
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lemma plus_float[simp]: "r \<in> float \<Longrightarrow> p \<in> float \<Longrightarrow> r + p \<in> float"
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  unfolding float_def
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proof (safe, simp)
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  have *: "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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    if "e1 \<le> e2" for e1 m1 e2 m2 :: int
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  proof -
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    from that have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
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      by (simp add: powr_realpow[symmetric] powr_diff field_simps)
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    then show ?thesis
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      by blast
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  qed
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  fix e1 m1 e2 m2 :: int
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  consider "e2 \<le> e1" | "e1 \<le> e2" by (rule linorder_le_cases)
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  then show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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  proof cases
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    case 1
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    from *[OF this, of m2 m1] show ?thesis
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      by (simp add: ac_simps)
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  next
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    case 2
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    then show ?thesis by (rule *)
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  qed
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qed
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lemma uminus_float[simp]: "x \<in> float \<Longrightarrow> -x \<in> float"
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  apply (auto simp: float_def)
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  apply hypsubst_thin
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  apply (rename_tac m e)
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  apply (rule_tac x="-m" in exI)
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  apply (rule_tac x="e" in exI)
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  apply (simp add: field_simps)
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  done
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lemma times_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x * y \<in> float"
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  apply (auto simp: float_def)
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  apply hypsubst_thin
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  apply (rename_tac mx my ex ey)
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  apply (rule_tac x="mx * my" in exI)
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  apply (rule_tac x="ex + ey" in exI)
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  apply (simp add: powr_add)
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  done
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lemma minus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x - y \<in> float"
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  using plus_float [of x "- y"] by simp
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lemma abs_float[simp]: "x \<in> float \<Longrightarrow> \<bar>x\<bar> \<in> float"
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  by (cases x rule: linorder_cases[of 0]) auto
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lemma sgn_of_float[simp]: "x \<in> float \<Longrightarrow> sgn x \<in> float"
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  by (cases x rule: linorder_cases[of 0]) (auto intro!: uminus_float)
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lemma div_power_2_float[simp]: "x \<in> float \<Longrightarrow> x / 2^d \<in> float"
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  apply (auto simp add: float_def)
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  apply hypsubst_thin
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  apply (rename_tac m e)
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  apply (rule_tac x="m" in exI)
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  apply (rule_tac x="e - d" in exI)
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  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
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  done
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lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"
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  apply (auto simp add: float_def)
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  apply hypsubst_thin
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  apply (rename_tac m e)
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  apply (rule_tac x="m" in exI)
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  apply (rule_tac x="e - d" in exI)
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  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
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  done
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lemma div_numeral_Bit0_float[simp]:
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  assumes "x / numeral n \<in> float"
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  shows "x / (numeral (Num.Bit0 n)) \<in> float"
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proof -
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  have "(x / numeral n) / 2^1 \<in> float"
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    by (intro assms div_power_2_float)
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  also have "(x / numeral n) / 2^1 = x / (numeral (Num.Bit0 n))"
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    by (induct n) auto
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  finally show ?thesis .
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qed
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lemma div_neg_numeral_Bit0_float[simp]:
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  assumes "x / numeral n \<in> float"
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  shows "x / (- numeral (Num.Bit0 n)) \<in> float"
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proof -
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  have "- (x / numeral (Num.Bit0 n)) \<in> float"
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    using assms by simp
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  also have "- (x / numeral (Num.Bit0 n)) = x / - numeral (Num.Bit0 n)"
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    by simp
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  finally show ?thesis .
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qed
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lemma power_float[simp]:
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  assumes "a \<in> float"
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  shows "a ^ b \<in> float"
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proof -
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  from assms obtain m e :: int where "a = m * 2 powr e"
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    by (auto simp: float_def)
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  then show ?thesis
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    by (auto intro!: floatI[where m="m^b" and e = "e*b"]
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      simp: power_mult_distrib powr_realpow[symmetric] powr_powr)
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qed
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lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e"
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  by simp
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declare Float.rep_eq[simp]
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code_datatype Float
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lemma compute_real_of_float[code]:
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  "real_of_float (Float m e) = (if e \<ge> 0 then m * 2 ^ nat e else m / 2 ^ (nat (-e)))"
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  by (simp add: powr_int)
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subsection \<open>Arithmetic operations on floating point numbers\<close>
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instantiation float :: "{ring_1,linorder,linordered_ring,linordered_idom,numeral,equal}"
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begin
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lift_definition zero_float :: float is 0 by simp
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declare zero_float.rep_eq[simp]
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lift_definition one_float :: float is 1 by simp
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declare one_float.rep_eq[simp]
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lift_definition plus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op +" by simp
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declare plus_float.rep_eq[simp]
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lift_definition times_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op *" by simp
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declare times_float.rep_eq[simp]
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lift_definition minus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op -" by simp
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declare minus_float.rep_eq[simp]
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lift_definition uminus_float :: "float \<Rightarrow> float" is "uminus" by simp
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declare uminus_float.rep_eq[simp]
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lift_definition abs_float :: "float \<Rightarrow> float" is abs by simp
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declare abs_float.rep_eq[simp]
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lift_definition sgn_float :: "float \<Rightarrow> float" is sgn by simp
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declare sgn_float.rep_eq[simp]
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lift_definition equal_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op = :: real \<Rightarrow> real \<Rightarrow> bool" .
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lift_definition less_eq_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op \<le>" .
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declare less_eq_float.rep_eq[simp]
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lift_definition less_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op <" .
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declare less_float.rep_eq[simp]
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instance
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  by standard (transfer; fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+
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end
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lemma real_of_float [simp]: "real_of_float (of_nat n) = of_nat n"
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  by (induct n) simp_all
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lemma real_of_float_of_int_eq [simp]: "real_of_float (of_int z) = of_int z"
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  by (cases z rule: int_diff_cases) (simp_all add: of_rat_diff)
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lemma Float_0_eq_0[simp]: "Float 0 e = 0"
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  by transfer simp
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lemma real_of_float_power[simp]: "real_of_float (f^n) = real_of_float f^n" for f :: float
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  by (induct n) simp_all
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lemma real_of_float_min: "real_of_float (min x y) = min (real_of_float x) (real_of_float y)"
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  and real_of_float_max: "real_of_float (max x y) = max (real_of_float x) (real_of_float y)"
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  for x y :: float
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  by (simp_all add: min_def max_def)
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instance float :: unbounded_dense_linorder
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proof
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  fix a b :: float
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  show "\<exists>c. a < c"
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    apply (intro exI[of _ "a + 1"])
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    apply transfer
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    apply simp
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    done
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  show "\<exists>c. c < a"
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    apply (intro exI[of _ "a - 1"])
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    apply transfer
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    apply simp
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    done
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  show "\<exists>c. a < c \<and> c < b" if "a < b"
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    apply (rule exI[of _ "(a + b) * Float 1 (- 1)"])
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    using that
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    apply transfer
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    apply (simp add: powr_minus)
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    done
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qed
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instantiation float :: lattice_ab_group_add
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begin
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definition inf_float :: "float \<Rightarrow> float \<Rightarrow> float"
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  where "inf_float a b = min a b"
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definition sup_float :: "float \<Rightarrow> float \<Rightarrow> float"
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  where "sup_float a b = max a b"
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instance
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  by standard (transfer; simp add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+
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end
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lemma float_numeral[simp]: "real_of_float (numeral x :: float) = numeral x"
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  apply (induct x)
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  apply simp
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  apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq float_of_inverse
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          plus_float.rep_eq one_float.rep_eq plus_float numeral_float one_float)
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  done
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lemma transfer_numeral [transfer_rule]:
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  "rel_fun (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
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  by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)
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lemma float_neg_numeral[simp]: "real_of_float (- numeral x :: float) = - numeral x"
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  by simp
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lemma transfer_neg_numeral [transfer_rule]:
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  "rel_fun (op =) pcr_float (- numeral :: _ \<Rightarrow> real) (- numeral :: _ \<Rightarrow> float)"
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  by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)
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lemma float_of_numeral[simp]: "numeral k = float_of (numeral k)"
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  and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)"
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  unfolding real_of_float_eq by simp_all
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subsection \<open>Quickcheck\<close>
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instantiation float :: exhaustive
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begin
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definition exhaustive_float where
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  "exhaustive_float f d =
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    Quickcheck_Exhaustive.exhaustive (\<lambda>x. Quickcheck_Exhaustive.exhaustive (\<lambda>y. f (Float x y)) d) d"
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instance ..
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end
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   321
definition (in term_syntax) [code_unfold]:
immler@58987
   322
  "valtermify_float x y = Code_Evaluation.valtermify Float {\<cdot>} x {\<cdot>} y"
immler@58987
   323
immler@58987
   324
instantiation float :: full_exhaustive
immler@58987
   325
begin
immler@58987
   326
wenzelm@63356
   327
definition
immler@58987
   328
  "full_exhaustive_float f d =
immler@58987
   329
    Quickcheck_Exhaustive.full_exhaustive
immler@58987
   330
      (\<lambda>x. Quickcheck_Exhaustive.full_exhaustive (\<lambda>y. f (valtermify_float x y)) d) d"
immler@58987
   331
immler@58987
   332
instance ..
immler@58987
   333
immler@58987
   334
end
immler@58987
   335
immler@58987
   336
instantiation float :: random
immler@58987
   337
begin
immler@58987
   338
immler@58987
   339
definition "Quickcheck_Random.random i =
immler@58987
   340
  scomp (Quickcheck_Random.random (2 ^ nat_of_natural i))
immler@58987
   341
    (\<lambda>man. scomp (Quickcheck_Random.random i) (\<lambda>exp. Pair (valtermify_float man exp)))"
immler@58987
   342
immler@58987
   343
instance ..
immler@58987
   344
immler@58987
   345
end
immler@58987
   346
immler@58987
   347
wenzelm@60500
   348
subsection \<open>Represent floats as unique mantissa and exponent\<close>
huffman@47108
   349
hoelzl@47599
   350
lemma int_induct_abs[case_names less]:
hoelzl@47599
   351
  fixes j :: int
hoelzl@47599
   352
  assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
hoelzl@47599
   353
  shows "P j"
hoelzl@47599
   354
proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)
wenzelm@60698
   355
  case less
wenzelm@60698
   356
  show ?case by (rule H[OF less]) simp
hoelzl@47599
   357
qed
hoelzl@47599
   358
hoelzl@47599
   359
lemma int_cancel_factors:
wenzelm@60698
   360
  fixes n :: int
wenzelm@60698
   361
  assumes "1 < r"
wenzelm@60698
   362
  shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
hoelzl@47599
   363
proof (induct n rule: int_induct_abs)
hoelzl@47599
   364
  case (less n)
wenzelm@60698
   365
  have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" if "n \<noteq> 0" "n = m * r" for m
wenzelm@60698
   366
  proof -
wenzelm@60698
   367
    from that have "\<bar>m \<bar> < \<bar>n\<bar>"
wenzelm@60500
   368
      using \<open>1 < r\<close> by (simp add: abs_mult)
wenzelm@60698
   369
    from less[OF this] that show ?thesis by auto
wenzelm@60698
   370
  qed
hoelzl@47599
   371
  then show ?case
haftmann@59554
   372
    by (metis dvd_def monoid_mult_class.mult.right_neutral mult.commute power_0)
hoelzl@47599
   373
qed
hoelzl@47599
   374
hoelzl@47599
   375
lemma mult_powr_eq_mult_powr_iff_asym:
hoelzl@47599
   376
  fixes m1 m2 e1 e2 :: int
wenzelm@60698
   377
  assumes m1: "\<not> 2 dvd m1"
wenzelm@60698
   378
    and "e1 \<le> e2"
hoelzl@47599
   379
  shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
wenzelm@60698
   380
  (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@47599
   381
proof
wenzelm@60698
   382
  show ?rhs if eq: ?lhs
wenzelm@60698
   383
  proof -
wenzelm@60698
   384
    have "m1 \<noteq> 0"
wenzelm@60698
   385
      using m1 unfolding dvd_def by auto
wenzelm@60698
   386
    from \<open>e1 \<le> e2\<close> eq have "m1 = m2 * 2 powr nat (e2 - e1)"
lp15@65583
   387
      by (simp add: powr_diff field_simps)
wenzelm@60698
   388
    also have "\<dots> = m2 * 2^nat (e2 - e1)"
wenzelm@60698
   389
      by (simp add: powr_realpow)
wenzelm@60698
   390
    finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
lp15@61649
   391
      by linarith
wenzelm@60698
   392
    with m1 have "m1 = m2"
wenzelm@60698
   393
      by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
wenzelm@60698
   394
    then show ?thesis
wenzelm@60698
   395
      using eq \<open>m1 \<noteq> 0\<close> by (simp add: powr_inj)
wenzelm@60698
   396
  qed
wenzelm@60698
   397
  show ?lhs if ?rhs
wenzelm@60698
   398
    using that by simp
wenzelm@60698
   399
qed
hoelzl@47599
   400
hoelzl@47599
   401
lemma mult_powr_eq_mult_powr_iff:
wenzelm@63356
   402
  "\<not> 2 dvd m1 \<Longrightarrow> \<not> 2 dvd m2 \<Longrightarrow> m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
wenzelm@63356
   403
  for m1 m2 e1 e2 :: int
hoelzl@47599
   404
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
hoelzl@47599
   405
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
hoelzl@47599
   406
  by (cases e1 e2 rule: linorder_le_cases) auto
hoelzl@47599
   407
hoelzl@47599
   408
lemma floatE_normed:
hoelzl@47599
   409
  assumes x: "x \<in> float"
hoelzl@47599
   410
  obtains (zero) "x = 0"
hoelzl@47599
   411
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
wenzelm@60698
   412
proof -
wenzelm@63356
   413
  have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m" if "x \<noteq> 0"
wenzelm@63356
   414
  proof -
wenzelm@60698
   415
    from x obtain m e :: int where x: "x = m * 2 powr e"
wenzelm@60698
   416
      by (auto simp: float_def)
wenzelm@60500
   417
    with \<open>x \<noteq> 0\<close> int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
hoelzl@47599
   418
      by auto
wenzelm@63356
   419
    with \<open>\<not> 2 dvd k\<close> x show ?thesis
wenzelm@63356
   420
      apply (rule_tac exI[of _ "k"])
wenzelm@63356
   421
      apply (rule_tac exI[of _ "e + int i"])
wenzelm@63356
   422
      apply (simp add: powr_add powr_realpow)
wenzelm@63356
   423
      done
wenzelm@63356
   424
  qed
wenzelm@60698
   425
  with that show thesis by blast
hoelzl@47599
   426
qed
hoelzl@47599
   427
hoelzl@47599
   428
lemma float_normed_cases:
hoelzl@47599
   429
  fixes f :: float
hoelzl@47599
   430
  obtains (zero) "f = 0"
lp15@61609
   431
   | (powr) m e :: int where "real_of_float f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
hoelzl@47599
   432
proof (atomize_elim, induct f)
wenzelm@60698
   433
  case (float_of y)
wenzelm@60698
   434
  then show ?case
hoelzl@47600
   435
    by (cases rule: floatE_normed) (auto simp: zero_float_def)
hoelzl@47599
   436
qed
hoelzl@47599
   437
wenzelm@63356
   438
definition mantissa :: "float \<Rightarrow> int"
wenzelm@63356
   439
  where "mantissa f =
wenzelm@63356
   440
    fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) \<or>
wenzelm@63356
   441
      (f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   442
wenzelm@63356
   443
definition exponent :: "float \<Rightarrow> int"
wenzelm@63356
   444
  where "exponent f =
wenzelm@63356
   445
    snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) \<or>
wenzelm@63356
   446
      (f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   447
wenzelm@63356
   448
lemma exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
wenzelm@63356
   449
  and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
hoelzl@47599
   450
proof -
wenzelm@60698
   451
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)"
wenzelm@60698
   452
    by auto
hoelzl@47599
   453
  then show ?E ?M
hoelzl@47600
   454
    by (auto simp add: mantissa_def exponent_def zero_float_def)
hoelzl@29804
   455
qed
hoelzl@29804
   456
wenzelm@63356
   457
lemma mantissa_exponent: "real_of_float f = mantissa f * 2 powr exponent f" (is ?E)
wenzelm@63356
   458
  and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
hoelzl@47599
   459
proof cases
wenzelm@60698
   460
  assume [simp]: "f \<noteq> float_of 0"
hoelzl@47599
   461
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
hoelzl@47599
   462
  proof (cases f rule: float_normed_cases)
wenzelm@60698
   463
    case zero
wenzelm@63356
   464
    then show ?thesis by (simp add: zero_float_def)
wenzelm@60698
   465
  next
hoelzl@47599
   466
    case (powr m e)
wenzelm@60698
   467
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) \<or>
lp15@61609
   468
      (f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> \<not> 2 dvd fst p)"
hoelzl@47599
   469
      by auto
hoelzl@47599
   470
    then show ?thesis
hoelzl@47599
   471
      unfolding exponent_def mantissa_def
hoelzl@47600
   472
      by (rule someI2_ex) (simp add: zero_float_def)
wenzelm@60698
   473
  qed
hoelzl@47599
   474
  then show ?E ?D by auto
hoelzl@47599
   475
qed simp
hoelzl@47599
   476
hoelzl@47599
   477
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
hoelzl@47599
   478
  using mantissa_not_dvd[of f] by auto
hoelzl@47599
   479
wenzelm@53381
   480
lemma
hoelzl@47599
   481
  fixes m e :: int
hoelzl@47599
   482
  defines "f \<equiv> float_of (m * 2 powr e)"
hoelzl@47599
   483
  assumes dvd: "\<not> 2 dvd m"
hoelzl@47599
   484
  shows mantissa_float: "mantissa f = m" (is "?M")
hoelzl@47599
   485
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
hoelzl@47599
   486
proof cases
wenzelm@60698
   487
  assume "m = 0"
wenzelm@60698
   488
  with dvd show "mantissa f = m" by auto
hoelzl@47599
   489
next
hoelzl@47599
   490
  assume "m \<noteq> 0"
hoelzl@47599
   491
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
wenzelm@60698
   492
  from mantissa_exponent[of f] have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   493
    by (auto simp add: f_def)
wenzelm@63356
   494
  then show ?M ?E
hoelzl@47599
   495
    using mantissa_not_dvd[OF f_not_0] dvd
hoelzl@47599
   496
    by (auto simp: mult_powr_eq_mult_powr_iff)
hoelzl@47599
   497
qed
hoelzl@47599
   498
wenzelm@60698
   499
wenzelm@60500
   500
subsection \<open>Compute arithmetic operations\<close>
hoelzl@47600
   501
hoelzl@47600
   502
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
hoelzl@47600
   503
  unfolding real_of_float_eq mantissa_exponent[of f] by simp
hoelzl@47600
   504
wenzelm@60698
   505
lemma Float_cases [cases type: float]:
hoelzl@47600
   506
  fixes f :: float
hoelzl@47600
   507
  obtains (Float) m e :: int where "f = Float m e"
hoelzl@47600
   508
  using Float_mantissa_exponent[symmetric]
hoelzl@47600
   509
  by (atomize_elim) auto
hoelzl@47600
   510
hoelzl@47599
   511
lemma denormalize_shift:
wenzelm@60698
   512
  assumes f_def: "f \<equiv> Float m e"
wenzelm@60698
   513
    and not_0: "f \<noteq> float_of 0"
hoelzl@47599
   514
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
hoelzl@47599
   515
proof
hoelzl@47599
   516
  from mantissa_exponent[of f] f_def
hoelzl@47599
   517
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   518
    by simp
hoelzl@47599
   519
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
lp15@65583
   520
    by (simp add: powr_diff field_simps)
hoelzl@47599
   521
  moreover
hoelzl@47599
   522
  have "e \<le> exponent f"
hoelzl@47599
   523
  proof (rule ccontr)
hoelzl@47599
   524
    assume "\<not> e \<le> exponent f"
hoelzl@47599
   525
    then have pos: "exponent f < e" by simp
lp15@61609
   526
    then have "2 powr (exponent f - e) = 2 powr - real_of_int (e - exponent f)"
hoelzl@47599
   527
      by simp
hoelzl@47599
   528
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
lp15@65583
   529
      using pos by (simp add: powr_realpow[symmetric] powr_diff)
lp15@61609
   530
    finally have "m * 2^nat (e - exponent f) = real_of_int (mantissa f)"
hoelzl@47599
   531
      using eq by simp
hoelzl@47599
   532
    then have "mantissa f = m * 2^nat (e - exponent f)"
lp15@61609
   533
      by linarith
wenzelm@60500
   534
    with \<open>exponent f < e\<close> have "2 dvd mantissa f"
hoelzl@47599
   535
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
hoelzl@47599
   536
      apply (cases "nat (e - exponent f)")
hoelzl@47599
   537
      apply auto
hoelzl@47599
   538
      done
hoelzl@47599
   539
    then show False using mantissa_not_dvd[OF not_0] by simp
hoelzl@47599
   540
  qed
lp15@61609
   541
  ultimately have "real_of_int m = mantissa f * 2^nat (exponent f - e)"
hoelzl@47599
   542
    by (simp add: powr_realpow[symmetric])
wenzelm@60500
   543
  with \<open>e \<le> exponent f\<close>
wenzelm@63356
   544
  show "m = mantissa f * 2 ^ nat (exponent f - e)"
lp15@61649
   545
    by linarith
lp15@61649
   546
  show "e = exponent f - nat (exponent f - e)"
wenzelm@61799
   547
    using \<open>e \<le> exponent f\<close> by auto
hoelzl@29804
   548
qed
hoelzl@29804
   549
wenzelm@60698
   550
context
wenzelm@60698
   551
begin
hoelzl@47600
   552
wenzelm@60698
   553
qualified lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
hoelzl@47600
   554
  by transfer simp
wenzelm@60698
   555
wenzelm@60698
   556
qualified lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
wenzelm@60698
   557
  by transfer simp
hoelzl@47600
   558
immler@58982
   559
lift_definition normfloat :: "float \<Rightarrow> float" is "\<lambda>x. x" .
immler@58982
   560
lemma normloat_id[simp]: "normfloat x = x" by transfer rule
hoelzl@47600
   561
wenzelm@63356
   562
qualified lemma compute_normfloat[code]:
wenzelm@63356
   563
  "normfloat (Float m e) =
wenzelm@63356
   564
    (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
wenzelm@63356
   565
     else if m = 0 then 0 else Float m e)"
hoelzl@47600
   566
  by transfer (auto simp add: powr_add zmod_eq_0_iff)
hoelzl@47599
   567
wenzelm@60698
   568
qualified lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
hoelzl@47600
   569
  by transfer simp
hoelzl@47599
   570
wenzelm@60698
   571
qualified lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
hoelzl@47600
   572
  by transfer simp
hoelzl@47599
   573
wenzelm@60698
   574
qualified lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
hoelzl@47600
   575
  by transfer simp
hoelzl@47599
   576
wenzelm@60698
   577
qualified lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
hoelzl@47600
   578
  by transfer (simp add: field_simps powr_add)
hoelzl@47599
   579
wenzelm@63356
   580
qualified lemma compute_float_plus[code]:
wenzelm@63356
   581
  "Float m1 e1 + Float m2 e2 =
wenzelm@63356
   582
    (if m1 = 0 then Float m2 e2
wenzelm@63356
   583
     else if m2 = 0 then Float m1 e1
wenzelm@63356
   584
     else if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
wenzelm@63356
   585
     else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
lp15@65583
   586
  by transfer (simp add: field_simps powr_realpow[symmetric] powr_diff)
hoelzl@47599
   587
wenzelm@63356
   588
qualified lemma compute_float_minus[code]: "f - g = f + (-g)" for f g :: float
hoelzl@47600
   589
  by simp
hoelzl@47599
   590
wenzelm@63356
   591
qualified lemma compute_float_sgn[code]:
wenzelm@63356
   592
  "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
haftmann@64240
   593
  by transfer (simp add: sgn_mult)
hoelzl@47599
   594
kuncar@55565
   595
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   596
wenzelm@60698
   597
qualified lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
hoelzl@47600
   598
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
hoelzl@47599
   599
kuncar@55565
   600
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   601
wenzelm@60698
   602
qualified lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
hoelzl@47600
   603
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
hoelzl@47599
   604
kuncar@55565
   605
lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   606
wenzelm@60698
   607
qualified lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
hoelzl@47600
   608
  by transfer (auto simp add: is_float_zero_def)
hoelzl@47599
   609
wenzelm@61945
   610
qualified lemma compute_float_abs[code]: "\<bar>Float m e\<bar> = Float \<bar>m\<bar> e"
hoelzl@47600
   611
  by transfer (simp add: abs_mult)
hoelzl@47599
   612
wenzelm@60698
   613
qualified lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
hoelzl@47600
   614
  by transfer simp
wenzelm@60698
   615
wenzelm@60698
   616
end
hoelzl@47599
   617
immler@58982
   618
wenzelm@60500
   619
subsection \<open>Lemmas for types @{typ real}, @{typ nat}, @{typ int}\<close>
immler@58982
   620
immler@58982
   621
lemmas real_of_ints =
lp15@61609
   622
  of_int_add
lp15@61609
   623
  of_int_minus
lp15@61609
   624
  of_int_diff
lp15@61609
   625
  of_int_mult
lp15@61609
   626
  of_int_power
lp15@61609
   627
  of_int_numeral of_int_neg_numeral
immler@58982
   628
immler@58982
   629
lemmas int_of_reals = real_of_ints[symmetric]
immler@58982
   630
immler@58982
   631
wenzelm@60500
   632
subsection \<open>Rounding Real Numbers\<close>
hoelzl@47599
   633
wenzelm@60698
   634
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real"
wenzelm@61942
   635
  where "round_down prec x = \<lfloor>x * 2 powr prec\<rfloor> * 2 powr -prec"
hoelzl@47599
   636
wenzelm@60698
   637
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real"
wenzelm@61942
   638
  where "round_up prec x = \<lceil>x * 2 powr prec\<rceil> * 2 powr -prec"
hoelzl@47599
   639
hoelzl@47599
   640
lemma round_down_float[simp]: "round_down prec x \<in> float"
hoelzl@47599
   641
  unfolding round_down_def
lp15@61609
   642
  by (auto intro!: times_float simp: of_int_minus[symmetric] simp del: of_int_minus)
hoelzl@47599
   643
hoelzl@47599
   644
lemma round_up_float[simp]: "round_up prec x \<in> float"
hoelzl@47599
   645
  unfolding round_up_def
lp15@61609
   646
  by (auto intro!: times_float simp: of_int_minus[symmetric] simp del: of_int_minus)
hoelzl@47599
   647
hoelzl@47599
   648
lemma round_up: "x \<le> round_up prec x"
lp15@61609
   649
  by (simp add: powr_minus_divide le_divide_eq round_up_def ceiling_correct)
hoelzl@47599
   650
hoelzl@47599
   651
lemma round_down: "round_down prec x \<le> x"
hoelzl@47599
   652
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
hoelzl@47599
   653
hoelzl@47599
   654
lemma round_up_0[simp]: "round_up p 0 = 0"
hoelzl@47599
   655
  unfolding round_up_def by simp
hoelzl@47599
   656
hoelzl@47599
   657
lemma round_down_0[simp]: "round_down p 0 = 0"
hoelzl@47599
   658
  unfolding round_down_def by simp
hoelzl@47599
   659
wenzelm@63356
   660
lemma round_up_diff_round_down: "round_up prec x - round_down prec x \<le> 2 powr -prec"
hoelzl@47599
   661
proof -
wenzelm@63356
   662
  have "round_up prec x - round_down prec x = (\<lceil>x * 2 powr prec\<rceil> - \<lfloor>x * 2 powr prec\<rfloor>) * 2 powr -prec"
hoelzl@47599
   663
    by (simp add: round_up_def round_down_def field_simps)
hoelzl@47599
   664
  also have "\<dots> \<le> 1 * 2 powr -prec"
hoelzl@47599
   665
    by (rule mult_mono)
wenzelm@63356
   666
      (auto simp del: of_int_diff simp: of_int_diff[symmetric] ceiling_diff_floor_le_1)
hoelzl@47599
   667
  finally show ?thesis by simp
hoelzl@29804
   668
qed
hoelzl@29804
   669
hoelzl@47599
   670
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
hoelzl@47599
   671
  unfolding round_down_def
lp15@65583
   672
  by (simp add: powr_add powr_mult field_simps powr_diff)
hoelzl@47599
   673
    (simp add: powr_add[symmetric])
hoelzl@29804
   674
hoelzl@47599
   675
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
hoelzl@47599
   676
  unfolding round_up_def
lp15@65583
   677
  by (simp add: powr_add powr_mult field_simps powr_diff)
hoelzl@47599
   678
    (simp add: powr_add[symmetric])
hoelzl@47599
   679
immler@58982
   680
lemma round_up_uminus_eq: "round_up p (-x) = - round_down p x"
immler@58982
   681
  and round_down_uminus_eq: "round_down p (-x) = - round_up p x"
immler@58982
   682
  by (auto simp: round_up_def round_down_def ceiling_def)
immler@58982
   683
immler@58982
   684
lemma round_up_mono: "x \<le> y \<Longrightarrow> round_up p x \<le> round_up p y"
immler@58982
   685
  by (auto intro!: ceiling_mono simp: round_up_def)
immler@58982
   686
immler@58982
   687
lemma round_up_le1:
immler@58982
   688
  assumes "x \<le> 1" "prec \<ge> 0"
immler@58982
   689
  shows "round_up prec x \<le> 1"
immler@58982
   690
proof -
lp15@61609
   691
  have "real_of_int \<lceil>x * 2 powr prec\<rceil> \<le> real_of_int \<lceil>2 powr real_of_int prec\<rceil>"
immler@58982
   692
    using assms by (auto intro!: ceiling_mono)
immler@58982
   693
  also have "\<dots> = 2 powr prec" using assms by (auto simp: powr_int intro!: exI[where x="2^nat prec"])
immler@58982
   694
  finally show ?thesis
immler@58982
   695
    by (simp add: round_up_def) (simp add: powr_minus inverse_eq_divide)
immler@58982
   696
qed
immler@58982
   697
immler@58982
   698
lemma round_up_less1:
immler@58982
   699
  assumes "x < 1 / 2" "p > 0"
immler@58982
   700
  shows "round_up p x < 1"
immler@58982
   701
proof -
immler@58982
   702
  have "x * 2 powr p < 1 / 2 * 2 powr p"
immler@58982
   703
    using assms by simp
wenzelm@60500
   704
  also have "\<dots> \<le> 2 powr p - 1" using \<open>p > 0\<close>
lp15@65583
   705
    by (auto simp: powr_diff powr_int field_simps self_le_power)
wenzelm@60500
   706
  finally show ?thesis using \<open>p > 0\<close>
lp15@61609
   707
    by (simp add: round_up_def field_simps powr_minus powr_int ceiling_less_iff)
immler@58982
   708
qed
immler@58982
   709
immler@58982
   710
lemma round_down_ge1:
immler@58982
   711
  assumes x: "x \<ge> 1"
immler@58982
   712
  assumes prec: "p \<ge> - log 2 x"
immler@58982
   713
  shows "1 \<le> round_down p x"
immler@58982
   714
proof cases
immler@58982
   715
  assume nonneg: "0 \<le> p"
lp15@61609
   716
  have "2 powr p = real_of_int \<lfloor>2 powr real_of_int p\<rfloor>"
immler@58985
   717
    using nonneg by (auto simp: powr_int)
lp15@61609
   718
  also have "\<dots> \<le> real_of_int \<lfloor>x * 2 powr p\<rfloor>"
immler@58985
   719
    using assms by (auto intro!: floor_mono)
immler@58985
   720
  finally show ?thesis
immler@58985
   721
    by (simp add: round_down_def) (simp add: powr_minus inverse_eq_divide)
immler@58982
   722
next
immler@58982
   723
  assume neg: "\<not> 0 \<le> p"
immler@58982
   724
  have "x = 2 powr (log 2 x)"
immler@58982
   725
    using x by simp
immler@58982
   726
  also have "2 powr (log 2 x) \<ge> 2 powr - p"
immler@58982
   727
    using prec by auto
immler@58982
   728
  finally have x_le: "x \<ge> 2 powr -p" .
immler@58982
   729
lp15@61609
   730
  from neg have "2 powr real_of_int p \<le> 2 powr 0"
immler@58982
   731
    by (intro powr_mono) auto
lp15@60017
   732
  also have "\<dots> \<le> \<lfloor>2 powr 0::real\<rfloor>" by simp
lp15@61609
   733
  also have "\<dots> \<le> \<lfloor>x * 2 powr (real_of_int p)\<rfloor>"
lp15@61609
   734
    unfolding of_int_le_iff
immler@58982
   735
    using x x_le by (intro floor_mono) (simp add: powr_minus_divide field_simps)
immler@58982
   736
  finally show ?thesis
immler@58982
   737
    using prec x
immler@58982
   738
    by (simp add: round_down_def powr_minus_divide pos_le_divide_eq)
immler@58982
   739
qed
immler@58982
   740
immler@58982
   741
lemma round_up_le0: "x \<le> 0 \<Longrightarrow> round_up p x \<le> 0"
immler@58982
   742
  unfolding round_up_def
immler@58982
   743
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
immler@58982
   744
immler@58982
   745
wenzelm@60500
   746
subsection \<open>Rounding Floats\<close>
hoelzl@29804
   747
wenzelm@60698
   748
definition div_twopow :: "int \<Rightarrow> nat \<Rightarrow> int"
wenzelm@60698
   749
  where [simp]: "div_twopow x n = x div (2 ^ n)"
immler@58985
   750
wenzelm@60698
   751
definition mod_twopow :: "int \<Rightarrow> nat \<Rightarrow> int"
wenzelm@60698
   752
  where [simp]: "mod_twopow x n = x mod (2 ^ n)"
immler@58985
   753
immler@58985
   754
lemma compute_div_twopow[code]:
immler@58985
   755
  "div_twopow x n = (if x = 0 \<or> x = -1 \<or> n = 0 then x else div_twopow (x div 2) (n - 1))"
immler@58985
   756
  by (cases n) (auto simp: zdiv_zmult2_eq div_eq_minus1)
immler@58985
   757
immler@58985
   758
lemma compute_mod_twopow[code]:
immler@58985
   759
  "mod_twopow x n = (if n = 0 then 0 else x mod 2 + 2 * mod_twopow (x div 2) (n - 1))"
immler@58985
   760
  by (cases n) (auto simp: zmod_zmult2_eq)
immler@58985
   761
hoelzl@47600
   762
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
hoelzl@47601
   763
declare float_up.rep_eq[simp]
hoelzl@29804
   764
wenzelm@60698
   765
lemma round_up_correct: "round_up e f - f \<in> {0..2 powr -e}"
wenzelm@60698
   766
  unfolding atLeastAtMost_iff
hoelzl@47599
   767
proof
wenzelm@60698
   768
  have "round_up e f - f \<le> round_up e f - round_down e f"
wenzelm@60698
   769
    using round_down by simp
wenzelm@60698
   770
  also have "\<dots> \<le> 2 powr -e"
wenzelm@60698
   771
    using round_up_diff_round_down by simp
lp15@61609
   772
  finally show "round_up e f - f \<le> 2 powr - (real_of_int e)"
hoelzl@47600
   773
    by simp
hoelzl@47600
   774
qed (simp add: algebra_simps round_up)
hoelzl@29804
   775
lp15@61609
   776
lemma float_up_correct: "real_of_float (float_up e f) - real_of_float f \<in> {0..2 powr -e}"
immler@54782
   777
  by transfer (rule round_up_correct)
immler@54782
   778
hoelzl@47600
   779
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
hoelzl@47601
   780
declare float_down.rep_eq[simp]
obua@16782
   781
wenzelm@60698
   782
lemma round_down_correct: "f - (round_down e f) \<in> {0..2 powr -e}"
wenzelm@60698
   783
  unfolding atLeastAtMost_iff
hoelzl@47599
   784
proof
wenzelm@60698
   785
  have "f - round_down e f \<le> round_up e f - round_down e f"
wenzelm@60698
   786
    using round_up by simp
wenzelm@60698
   787
  also have "\<dots> \<le> 2 powr -e"
wenzelm@60698
   788
    using round_up_diff_round_down by simp
lp15@61609
   789
  finally show "f - round_down e f \<le> 2 powr - (real_of_int e)"
hoelzl@47600
   790
    by simp
hoelzl@47600
   791
qed (simp add: algebra_simps round_down)
obua@24301
   792
lp15@61609
   793
lemma float_down_correct: "real_of_float f - real_of_float (float_down e f) \<in> {0..2 powr -e}"
immler@54782
   794
  by transfer (rule round_down_correct)
immler@54782
   795
wenzelm@60698
   796
context
wenzelm@60698
   797
begin
wenzelm@60698
   798
wenzelm@60698
   799
qualified lemma compute_float_down[code]:
hoelzl@47599
   800
  "float_down p (Float m e) =
immler@58985
   801
    (if p + e < 0 then Float (div_twopow m (nat (-(p + e)))) (-p) else Float m e)"
wenzelm@60698
   802
proof (cases "p + e < 0")
wenzelm@60698
   803
  case True
lp15@61609
   804
  then have "real_of_int ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   805
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
wenzelm@60698
   806
  also have "\<dots> = 1 / 2 powr p / 2 powr e"
lp15@61609
   807
    unfolding powr_minus_divide of_int_minus by (simp add: powr_add)
hoelzl@47599
   808
  finally show ?thesis
wenzelm@60500
   809
    using \<open>p + e < 0\<close>
lp15@61609
   810
    apply transfer
wenzelm@63356
   811
    apply (simp add: ac_simps round_down_def floor_divide_of_int_eq[symmetric])
lp15@61609
   812
    proof - (*FIXME*)
lp15@61609
   813
      fix pa :: int and ea :: int and ma :: int
lp15@61609
   814
      assume a1: "2 ^ nat (- pa - ea) = 1 / (2 powr real_of_int pa * 2 powr real_of_int ea)"
lp15@61609
   815
      assume "pa + ea < 0"
wenzelm@63356
   816
      have "\<lfloor>real_of_int ma / real_of_int (int 2 ^ nat (- (pa + ea)))\<rfloor> =
wenzelm@63356
   817
          \<lfloor>real_of_float (Float ma (pa + ea))\<rfloor>"
lp15@61609
   818
        using a1 by (simp add: powr_add)
wenzelm@63356
   819
      then show "\<lfloor>real_of_int ma * (2 powr real_of_int pa * 2 powr real_of_int ea)\<rfloor> =
wenzelm@63356
   820
          ma div 2 ^ nat (- pa - ea)"
wenzelm@63356
   821
        by (metis Float.rep_eq add_uminus_conv_diff floor_divide_of_int_eq
wenzelm@63356
   822
            minus_add_distrib of_int_simps(3) of_nat_numeral powr_add)
lp15@61609
   823
    qed
hoelzl@47599
   824
next
wenzelm@60698
   825
  case False
wenzelm@63356
   826
  then have r: "real_of_int e + real_of_int p = real (nat (e + p))"
wenzelm@63356
   827
    by simp
lp15@61609
   828
  have r: "\<lfloor>(m * 2 powr e) * 2 powr real_of_int p\<rfloor> = (m * 2 powr e) * 2 powr real_of_int p"
hoelzl@47600
   829
    by (auto intro: exI[where x="m*2^nat (e+p)"]
wenzelm@63356
   830
        simp add: ac_simps powr_add[symmetric] r powr_realpow)
wenzelm@60500
   831
  with \<open>\<not> p + e < 0\<close> show ?thesis
wenzelm@57862
   832
    by transfer (auto simp add: round_down_def field_simps powr_add powr_minus)
hoelzl@47599
   833
qed
obua@24301
   834
immler@54782
   835
lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"
immler@54782
   836
  using round_down_correct[of f e] by simp
immler@54782
   837
immler@54782
   838
lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e"
immler@54782
   839
  using round_up_correct[of e f] by simp
immler@54782
   840
immler@54782
   841
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s"
nipkow@56536
   842
  by (auto simp: round_down_def)
immler@54782
   843
hoelzl@47599
   844
lemma ceil_divide_floor_conv:
wenzelm@60698
   845
  assumes "b \<noteq> 0"
wenzelm@63356
   846
  shows "\<lceil>real_of_int a / real_of_int b\<rceil> =
wenzelm@63356
   847
    (if b dvd a then a div b else \<lfloor>real_of_int a / real_of_int b\<rfloor> + 1)"
wenzelm@60698
   848
proof (cases "b dvd a")
wenzelm@60698
   849
  case True
wenzelm@60698
   850
  then show ?thesis
lp15@61609
   851
    by (simp add: ceiling_def of_int_minus[symmetric] divide_minus_left[symmetric]
lp15@61609
   852
      floor_divide_of_int_eq dvd_neg_div del: divide_minus_left of_int_minus)
wenzelm@60698
   853
next
wenzelm@60698
   854
  case False
wenzelm@60698
   855
  then have "a mod b \<noteq> 0"
wenzelm@60698
   856
    by auto
lp15@61609
   857
  then have ne: "real_of_int (a mod b) / real_of_int b \<noteq> 0"
wenzelm@60698
   858
    using \<open>b \<noteq> 0\<close> by auto
lp15@61609
   859
  have "\<lceil>real_of_int a / real_of_int b\<rceil> = \<lfloor>real_of_int a / real_of_int b\<rfloor> + 1"
wenzelm@60698
   860
    apply (rule ceiling_eq)
lp15@61609
   861
    apply (auto simp: floor_divide_of_int_eq[symmetric])
hoelzl@47599
   862
  proof -
lp15@61609
   863
    have "real_of_int \<lfloor>real_of_int a / real_of_int b\<rfloor> \<le> real_of_int a / real_of_int b"
wenzelm@60698
   864
      by simp
lp15@61609
   865
    moreover have "real_of_int \<lfloor>real_of_int a / real_of_int b\<rfloor> \<noteq> real_of_int a / real_of_int b"
wenzelm@60698
   866
      apply (subst (2) real_of_int_div_aux)
lp15@61609
   867
      unfolding floor_divide_of_int_eq
wenzelm@60698
   868
      using ne \<open>b \<noteq> 0\<close> apply auto
wenzelm@60698
   869
      done
lp15@61609
   870
    ultimately show "real_of_int \<lfloor>real_of_int a / real_of_int b\<rfloor> < real_of_int a / real_of_int b" by arith
hoelzl@47599
   871
  qed
wenzelm@60698
   872
  then show ?thesis
wenzelm@60698
   873
    using \<open>\<not> b dvd a\<close> by simp
wenzelm@60698
   874
qed
wenzelm@19765
   875
wenzelm@60698
   876
qualified lemma compute_float_up[code]: "float_up p x = - float_down p (-x)"
immler@58982
   877
  by transfer (simp add: round_down_uminus_eq)
wenzelm@60698
   878
wenzelm@60698
   879
end
hoelzl@29804
   880
hoelzl@29804
   881
nipkow@63664
   882
lemma bitlen_Float:
nipkow@63664
   883
  fixes m e
nipkow@63664
   884
  defines "f \<equiv> Float m e"
nipkow@63664
   885
  shows "bitlen \<bar>mantissa f\<bar> + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
nipkow@63664
   886
proof (cases "m = 0")
nipkow@63664
   887
  case True
nipkow@63664
   888
  then show ?thesis by (simp add: f_def bitlen_alt_def Float_def)
nipkow@63664
   889
next
nipkow@63664
   890
  case False
nipkow@63664
   891
  then have "f \<noteq> float_of 0"
nipkow@63664
   892
    unfolding real_of_float_eq by (simp add: f_def)
nipkow@63664
   893
  then have "mantissa f \<noteq> 0"
nipkow@63664
   894
    by (simp add: mantissa_noteq_0)
nipkow@63664
   895
  moreover
nipkow@63664
   896
  obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
nipkow@63664
   897
    by (rule f_def[THEN denormalize_shift, OF \<open>f \<noteq> float_of 0\<close>])
nipkow@63664
   898
  ultimately show ?thesis by (simp add: abs_mult)
nipkow@63664
   899
qed
nipkow@63664
   900
wenzelm@63356
   901
lemma float_gt1_scale:
wenzelm@63356
   902
  assumes "1 \<le> Float m e"
hoelzl@47599
   903
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@47599
   904
proof -
hoelzl@47599
   905
  have "0 < Float m e" using assms by auto
wenzelm@60698
   906
  then have "0 < m" using powr_gt_zero[of 2 e]
lp15@60017
   907
    apply (auto simp: zero_less_mult_iff)
wenzelm@63356
   908
    using not_le powr_ge_pzero
wenzelm@63356
   909
    apply blast
wenzelm@60698
   910
    done
wenzelm@60698
   911
  then have "m \<noteq> 0" by auto
hoelzl@47599
   912
  show ?thesis
hoelzl@47599
   913
  proof (cases "0 \<le> e")
wenzelm@60698
   914
    case True
wenzelm@60698
   915
    then show ?thesis
immler@63248
   916
      using \<open>0 < m\<close> by (simp add: bitlen_alt_def)
hoelzl@29804
   917
  next
wenzelm@60698
   918
    case False
hoelzl@47599
   919
    have "(1::int) < 2" by simp
wenzelm@60698
   920
    let ?S = "2^(nat (-e))"
wenzelm@60698
   921
    have "inverse (2 ^ nat (- e)) = 2 powr e"
wenzelm@60698
   922
      using assms False powr_realpow[of 2 "nat (-e)"]
wenzelm@57862
   923
      by (auto simp: powr_minus field_simps)
lp15@61609
   924
    then have "1 \<le> real_of_int m * inverse ?S"
wenzelm@60698
   925
      using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   926
      by (auto simp: powr_minus)
lp15@61609
   927
    then have "1 * ?S \<le> real_of_int m * inverse ?S * ?S"
wenzelm@60698
   928
      by (rule mult_right_mono) auto
lp15@61609
   929
    then have "?S \<le> real_of_int m"
wenzelm@60698
   930
      unfolding mult.assoc by auto
wenzelm@60698
   931
    then have "?S \<le> m"
lp15@61609
   932
      unfolding of_int_le_iff[symmetric] by auto
wenzelm@60500
   933
    from this bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
wenzelm@60698
   934
    have "nat (-e) < (nat (bitlen m))"
wenzelm@60698
   935
      unfolding power_strict_increasing_iff[OF \<open>1 < 2\<close>, symmetric]
immler@58985
   936
      by (rule order_le_less_trans)
wenzelm@60698
   937
    then have "-e < bitlen m"
wenzelm@60698
   938
      using False by auto
wenzelm@60698
   939
    then show ?thesis
wenzelm@60698
   940
      by auto
hoelzl@29804
   941
  qed
hoelzl@47599
   942
qed
hoelzl@29804
   943
wenzelm@60698
   944
wenzelm@60500
   945
subsection \<open>Truncating Real Numbers\<close>
immler@58985
   946
wenzelm@60698
   947
definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real"
immler@62420
   948
  where "truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>) x"
immler@58985
   949
immler@58985
   950
lemma truncate_down: "truncate_down prec x \<le> x"
immler@58985
   951
  using round_down by (simp add: truncate_down_def)
immler@58985
   952
immler@58985
   953
lemma truncate_down_le: "x \<le> y \<Longrightarrow> truncate_down prec x \<le> y"
immler@58985
   954
  by (rule order_trans[OF truncate_down])
immler@58985
   955
immler@58985
   956
lemma truncate_down_zero[simp]: "truncate_down prec 0 = 0"
immler@58985
   957
  by (simp add: truncate_down_def)
immler@58985
   958
immler@58985
   959
lemma truncate_down_float[simp]: "truncate_down p x \<in> float"
immler@58985
   960
  by (auto simp: truncate_down_def)
immler@58985
   961
wenzelm@60698
   962
definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real"
immler@62420
   963
  where "truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>) x"
immler@58985
   964
immler@58985
   965
lemma truncate_up: "x \<le> truncate_up prec x"
immler@58985
   966
  using round_up by (simp add: truncate_up_def)
immler@58985
   967
immler@58985
   968
lemma truncate_up_le: "x \<le> y \<Longrightarrow> x \<le> truncate_up prec y"
immler@58985
   969
  by (rule order_trans[OF _ truncate_up])
immler@58985
   970
immler@58985
   971
lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0"
immler@58985
   972
  by (simp add: truncate_up_def)
immler@58985
   973
immler@58985
   974
lemma truncate_up_uminus_eq: "truncate_up prec (-x) = - truncate_down prec x"
immler@58985
   975
  and truncate_down_uminus_eq: "truncate_down prec (-x) = - truncate_up prec x"
immler@58985
   976
  by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
immler@58985
   977
immler@58985
   978
lemma truncate_up_float[simp]: "truncate_up p x \<in> float"
immler@58985
   979
  by (auto simp: truncate_up_def)
immler@58985
   980
immler@58985
   981
lemma mult_powr_eq: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> x * b powr y = b powr (y + log b x)"
immler@58985
   982
  by (simp_all add: powr_add)
immler@58985
   983
immler@58985
   984
lemma truncate_down_pos:
immler@62420
   985
  assumes "x > 0"
immler@58985
   986
  shows "truncate_down p x > 0"
immler@58985
   987
proof -
lp15@61609
   988
  have "0 \<le> log 2 x - real_of_int \<lfloor>log 2 x\<rfloor>"
immler@58985
   989
    by (simp add: algebra_simps)
lp15@61762
   990
  with assms
immler@58985
   991
  show ?thesis
wenzelm@63356
   992
    apply (auto simp: truncate_down_def round_down_def mult_powr_eq
immler@58985
   993
      intro!: ge_one_powr_ge_zero mult_pos_pos)
lp15@61762
   994
    by linarith
immler@58985
   995
qed
immler@58985
   996
immler@58985
   997
lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y"
immler@58985
   998
  by (auto simp: truncate_down_def round_down_def)
immler@58985
   999
immler@62420
  1000
lemma truncate_down_ge1: "1 \<le> x \<Longrightarrow> 1 \<le> truncate_down p x"
immler@62420
  1001
  apply (auto simp: truncate_down_def algebra_simps intro!: round_down_ge1)
immler@62420
  1002
  apply linarith
immler@62420
  1003
  done
immler@58985
  1004
immler@58985
  1005
lemma truncate_up_nonpos: "x \<le> 0 \<Longrightarrow> truncate_up prec x \<le> 0"
immler@58985
  1006
  by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)
hoelzl@47599
  1007
immler@58985
  1008
lemma truncate_up_le1:
immler@62420
  1009
  assumes "x \<le> 1"
wenzelm@60698
  1010
  shows "truncate_up p x \<le> 1"
immler@58985
  1011
proof -
wenzelm@60698
  1012
  consider "x \<le> 0" | "x > 0"
wenzelm@60698
  1013
    by arith
wenzelm@60698
  1014
  then show ?thesis
wenzelm@60698
  1015
  proof cases
wenzelm@60698
  1016
    case 1
wenzelm@60698
  1017
    with truncate_up_nonpos[OF this, of p] show ?thesis
wenzelm@60698
  1018
      by simp
wenzelm@60698
  1019
  next
wenzelm@60698
  1020
    case 2
wenzelm@60698
  1021
    then have le: "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<le> 0"
immler@58985
  1022
      using assms by (auto simp: log_less_iff)
immler@62420
  1023
    from assms have "0 \<le> int p" by simp
immler@58985
  1024
    from add_mono[OF this le]
wenzelm@60698
  1025
    show ?thesis
wenzelm@60698
  1026
      using assms by (simp add: truncate_up_def round_up_le1 add_mono)
wenzelm@60698
  1027
  qed
immler@58985
  1028
qed
immler@58985
  1029
wenzelm@63356
  1030
lemma truncate_down_shift_int:
wenzelm@63356
  1031
  "truncate_down p (x * 2 powr real_of_int k) = truncate_down p x * 2 powr k"
immler@62420
  1032
  by (cases "x = 0")
wenzelm@63356
  1033
    (simp_all add: algebra_simps abs_mult log_mult truncate_down_def
wenzelm@63356
  1034
      round_down_shift[of _ _ k, simplified])
immler@62420
  1035
immler@62420
  1036
lemma truncate_down_shift_nat: "truncate_down p (x * 2 powr real k) = truncate_down p x * 2 powr k"
immler@62420
  1037
  by (metis of_int_of_nat_eq truncate_down_shift_int)
immler@62420
  1038
immler@62420
  1039
lemma truncate_up_shift_int: "truncate_up p (x * 2 powr real_of_int k) = truncate_up p x * 2 powr k"
immler@62420
  1040
  by (cases "x = 0")
wenzelm@63356
  1041
    (simp_all add: algebra_simps abs_mult log_mult truncate_up_def
wenzelm@63356
  1042
      round_up_shift[of _ _ k, simplified])
immler@62420
  1043
immler@62420
  1044
lemma truncate_up_shift_nat: "truncate_up p (x * 2 powr real k) = truncate_up p x * 2 powr k"
immler@62420
  1045
  by (metis of_int_of_nat_eq truncate_up_shift_int)
immler@62420
  1046
wenzelm@60698
  1047
wenzelm@60500
  1048
subsection \<open>Truncating Floats\<close>
immler@58985
  1049
immler@58985
  1050
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
immler@58985
  1051
  by (simp add: truncate_up_def)
immler@58985
  1052
lp15@61609
  1053
lemma float_round_up: "real_of_float x \<le> real_of_float (float_round_up prec x)"
immler@58985
  1054
  using truncate_up by transfer simp
immler@58985
  1055
immler@58985
  1056
lemma float_round_up_zero[simp]: "float_round_up prec 0 = 0"
immler@58985
  1057
  by transfer simp
immler@58985
  1058
immler@58985
  1059
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down
immler@58985
  1060
  by (simp add: truncate_down_def)
immler@58985
  1061
lp15@61609
  1062
lemma float_round_down: "real_of_float (float_round_down prec x) \<le> real_of_float x"
immler@58985
  1063
  using truncate_down by transfer simp
immler@58985
  1064
immler@58985
  1065
lemma float_round_down_zero[simp]: "float_round_down prec 0 = 0"
immler@58985
  1066
  by transfer simp
immler@58985
  1067
immler@58985
  1068
lemmas float_round_up_le = order_trans[OF _ float_round_up]
immler@58985
  1069
  and float_round_down_le = order_trans[OF float_round_down]
immler@58985
  1070
immler@58985
  1071
lemma minus_float_round_up_eq: "- float_round_up prec x = float_round_down prec (- x)"
immler@58985
  1072
  and minus_float_round_down_eq: "- float_round_down prec x = float_round_up prec (- x)"
wenzelm@63356
  1073
  by (transfer; simp add: truncate_down_uminus_eq truncate_up_uminus_eq)+
immler@58985
  1074
wenzelm@60698
  1075
context
wenzelm@60698
  1076
begin
wenzelm@60698
  1077
wenzelm@60698
  1078
qualified lemma compute_float_round_down[code]:
wenzelm@63356
  1079
  "float_round_down prec (Float m e) =
wenzelm@63356
  1080
    (let d = bitlen \<bar>m\<bar> - int prec - 1 in
wenzelm@63356
  1081
      if 0 < d then Float (div_twopow m (nat d)) (e + d)
wenzelm@63356
  1082
      else Float m e)"
immler@62420
  1083
  using Float.compute_float_down[of "Suc prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
immler@62420
  1084
  by transfer
immler@63248
  1085
    (simp add: field_simps abs_mult log_mult bitlen_alt_def truncate_down_def
immler@62420
  1086
      cong del: if_weak_cong)
immler@58985
  1087
wenzelm@60698
  1088
qualified lemma compute_float_round_up[code]:
immler@58985
  1089
  "float_round_up prec x = - float_round_down prec (-x)"
immler@58985
  1090
  by transfer (simp add: truncate_down_uminus_eq)
wenzelm@60698
  1091
wenzelm@60698
  1092
end
immler@58985
  1093
immler@58985
  1094
wenzelm@60500
  1095
subsection \<open>Approximation of positive rationals\<close>
hoelzl@29804
  1096
wenzelm@63356
  1097
lemma div_mult_twopow_eq: "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)" for a b :: nat
wenzelm@60698
  1098
  by (cases "b = 0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
hoelzl@29804
  1099
wenzelm@63356
  1100
lemma real_div_nat_eq_floor_of_divide: "a div b = real_of_int \<lfloor>a / b\<rfloor>" for a b :: nat
lp15@61609
  1101
  by (simp add: floor_divide_of_nat_eq [of a b])
hoelzl@29804
  1102
immler@62420
  1103
definition "rat_precision prec x y =
wenzelm@63356
  1104
  (let d = bitlen x - bitlen y
wenzelm@63356
  1105
   in int prec - d + (if Float (abs x) 0 < Float (abs y) d then 1 else 0))"
immler@62420
  1106
immler@62420
  1107
lemma floor_log_divide_eq:
immler@62420
  1108
  assumes "i > 0" "j > 0" "p > 1"
immler@62420
  1109
  shows "\<lfloor>log p (i / j)\<rfloor> = floor (log p i) - floor (log p j) -
wenzelm@63356
  1110
    (if i \<ge> j * p powr (floor (log p i) - floor (log p j)) then 0 else 1)"
immler@62420
  1111
proof -
immler@62420
  1112
  let ?l = "log p"
immler@62420
  1113
  let ?fl = "\<lambda>x. floor (?l x)"
immler@62420
  1114
  have "\<lfloor>?l (i / j)\<rfloor> = \<lfloor>?l i - ?l j\<rfloor>" using assms
immler@62420
  1115
    by (auto simp: log_divide)
immler@62420
  1116
  also have "\<dots> = floor (real_of_int (?fl i - ?fl j) + (?l i - ?fl i - (?l j - ?fl j)))"
immler@62420
  1117
    (is "_ = floor (_ + ?r)")
immler@62420
  1118
    by (simp add: algebra_simps)
immler@62420
  1119
  also note floor_add2
immler@62420
  1120
  also note \<open>p > 1\<close>
immler@62420
  1121
  note powr = powr_le_cancel_iff[symmetric, OF \<open>1 < p\<close>, THEN iffD2]
immler@62420
  1122
  note powr_strict = powr_less_cancel_iff[symmetric, OF \<open>1 < p\<close>, THEN iffD2]
immler@62420
  1123
  have "floor ?r = (if i \<ge> j * p powr (?fl i - ?fl j) then 0 else -1)" (is "_ = ?if")
immler@62420
  1124
    using assms
immler@62420
  1125
    by (linarith |
immler@62420
  1126
      auto
immler@62420
  1127
        intro!: floor_eq2
immler@62420
  1128
        intro: powr_strict powr
lp15@65583
  1129
        simp: powr_diff powr_add divide_simps algebra_simps)+
immler@62420
  1130
  finally
immler@62420
  1131
  show ?thesis by simp
immler@62420
  1132
qed
immler@62420
  1133
immler@62420
  1134
lemma truncate_down_rat_precision:
immler@62420
  1135
    "truncate_down prec (real x / real y) = round_down (rat_precision prec x y) (real x / real y)"
immler@62420
  1136
  and truncate_up_rat_precision:
immler@62420
  1137
    "truncate_up prec (real x / real y) = round_up (rat_precision prec x y) (real x / real y)"
immler@62420
  1138
  unfolding truncate_down_def truncate_up_def rat_precision_def
immler@63248
  1139
  by (cases x; cases y) (auto simp: floor_log_divide_eq algebra_simps bitlen_alt_def)
hoelzl@29804
  1140
hoelzl@47600
  1141
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
immler@62420
  1142
  is "\<lambda>prec (x::nat) (y::nat). truncate_down prec (x / y)"
wenzelm@60698
  1143
  by simp
obua@16782
  1144
wenzelm@60698
  1145
context
wenzelm@60698
  1146
begin
wenzelm@60698
  1147
wenzelm@60698
  1148
qualified lemma compute_lapprox_posrat[code]:
wenzelm@63356
  1149
  "lapprox_posrat prec x y =
wenzelm@53381
  1150
   (let
wenzelm@60698
  1151
      l = rat_precision prec x y;
wenzelm@60698
  1152
      d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
hoelzl@47599
  1153
    in normfloat (Float d (- l)))"
immler@58982
  1154
    unfolding div_mult_twopow_eq
hoelzl@47600
  1155
    by transfer
immler@62420
  1156
      (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
immler@62420
  1157
        truncate_down_rat_precision del: two_powr_minus_int_float)
wenzelm@60698
  1158
wenzelm@60698
  1159
end
hoelzl@29804
  1160
hoelzl@47600
  1161
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
immler@62420
  1162
  is "\<lambda>prec (x::nat) (y::nat). truncate_up prec (x / y)"
immler@62420
  1163
  by simp
hoelzl@29804
  1164
wenzelm@60376
  1165
context
wenzelm@60376
  1166
begin
wenzelm@60376
  1167
wenzelm@60376
  1168
qualified lemma compute_rapprox_posrat[code]:
hoelzl@47599
  1169
  fixes prec x y
hoelzl@47599
  1170
  defines "l \<equiv> rat_precision prec x y"
wenzelm@63356
  1171
  shows "rapprox_posrat prec x y =
wenzelm@63356
  1172
   (let
wenzelm@63356
  1173
      l = l;
wenzelm@63356
  1174
      (r, s) = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l));
wenzelm@63356
  1175
      d = r div s;
wenzelm@63356
  1176
      m = r mod s
wenzelm@63356
  1177
    in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
hoelzl@47599
  1178
proof (cases "y = 0")
wenzelm@60698
  1179
  assume "y = 0"
wenzelm@60698
  1180
  then show ?thesis by transfer simp
hoelzl@47599
  1181
next
hoelzl@47599
  1182
  assume "y \<noteq> 0"
hoelzl@29804
  1183
  show ?thesis
hoelzl@47599
  1184
  proof (cases "0 \<le> l")
wenzelm@60698
  1185
    case True
wenzelm@63040
  1186
    define x' where "x' = x * 2 ^ nat l"
wenzelm@60698
  1187
    have "int x * 2 ^ nat l = x'"
wenzelm@63356
  1188
      by (simp add: x'_def)
lp15@61609
  1189
    moreover have "real x * 2 powr l = real x'"
wenzelm@60500
  1190
      by (simp add: powr_realpow[symmetric] \<open>0 \<le> l\<close> x'_def)
hoelzl@47599
  1191
    ultimately show ?thesis
wenzelm@60500
  1192
      using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] \<open>0 \<le> l\<close> \<open>y \<noteq> 0\<close>
hoelzl@47600
  1193
        l_def[symmetric, THEN meta_eq_to_obj_eq]
lp15@61609
  1194
      apply transfer
immler@62420
  1195
      apply (auto simp add: round_up_def truncate_up_rat_precision)
wenzelm@63356
  1196
      apply (metis floor_divide_of_int_eq of_int_of_nat_eq)
wenzelm@63356
  1197
      done
hoelzl@47599
  1198
   next
wenzelm@60698
  1199
    case False
wenzelm@63040
  1200
    define y' where "y' = y * 2 ^ nat (- l)"
wenzelm@60500
  1201
    from \<open>y \<noteq> 0\<close> have "y' \<noteq> 0" by (simp add: y'_def)
wenzelm@63356
  1202
    have "int y * 2 ^ nat (- l) = y'"
wenzelm@63356
  1203
      by (simp add: y'_def)
lp15@61609
  1204
    moreover have "real x * real_of_int (2::int) powr real_of_int l / real y = x / real y'"
wenzelm@63356
  1205
      using \<open>\<not> 0 \<le> l\<close> by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps)
hoelzl@47599
  1206
    ultimately show ?thesis
wenzelm@60500
  1207
      using ceil_divide_floor_conv[of y' x] \<open>\<not> 0 \<le> l\<close> \<open>y' \<noteq> 0\<close> \<open>y \<noteq> 0\<close>
hoelzl@47600
  1208
        l_def[symmetric, THEN meta_eq_to_obj_eq]
lp15@61609
  1209
      apply transfer
immler@62420
  1210
      apply (auto simp add: round_up_def ceil_divide_floor_conv truncate_up_rat_precision)
wenzelm@63356
  1211
      apply (metis floor_divide_of_int_eq of_int_of_nat_eq)
wenzelm@63356
  1212
      done
hoelzl@29804
  1213
  qed
hoelzl@29804
  1214
qed
wenzelm@60376
  1215
wenzelm@60376
  1216
end
hoelzl@29804
  1217
hoelzl@47599
  1218
lemma rat_precision_pos:
wenzelm@60698
  1219
  assumes "0 \<le> x"
wenzelm@60698
  1220
    and "0 < y"
wenzelm@60698
  1221
    and "2 * x < y"
hoelzl@47599
  1222
  shows "rat_precision n (int x) (int y) > 0"
hoelzl@29804
  1223
proof -
wenzelm@60698
  1224
  have "0 < x \<Longrightarrow> log 2 x + 1 = log 2 (2 * x)"
wenzelm@60698
  1225
    by (simp add: log_mult)
wenzelm@60698
  1226
  then have "bitlen (int x) < bitlen (int y)"
wenzelm@60698
  1227
    using assms
nipkow@63599
  1228
    by (simp add: bitlen_alt_def)
nipkow@63599
  1229
      (auto intro!: floor_mono simp add: one_add_floor)
wenzelm@60698
  1230
  then show ?thesis
wenzelm@60698
  1231
    using assms
wenzelm@60698
  1232
    by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
hoelzl@29804
  1233
qed
obua@16782
  1234
hoelzl@47601
  1235
lemma rapprox_posrat_less1:
immler@62420
  1236
  "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 2 * x < y \<Longrightarrow> real_of_float (rapprox_posrat n x y) < 1"
immler@62420
  1237
  by transfer (simp add: rat_precision_pos round_up_less1 truncate_up_rat_precision)
hoelzl@29804
  1238
hoelzl@47600
  1239
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
immler@62420
  1240
  "\<lambda>prec (x::int) (y::int). truncate_down prec (x / y)"
wenzelm@60698
  1241
  by simp
obua@16782
  1242
wenzelm@60698
  1243
context
wenzelm@60698
  1244
begin
wenzelm@60698
  1245
wenzelm@60698
  1246
qualified lemma compute_lapprox_rat[code]:
hoelzl@47599
  1247
  "lapprox_rat prec x y =
wenzelm@60698
  1248
   (if y = 0 then 0
hoelzl@47599
  1249
    else if 0 \<le> x then
wenzelm@60698
  1250
     (if 0 < y then lapprox_posrat prec (nat x) (nat y)
wenzelm@53381
  1251
      else - (rapprox_posrat prec (nat x) (nat (-y))))
wenzelm@63356
  1252
      else
wenzelm@63356
  1253
       (if 0 < y
hoelzl@47599
  1254
        then - (rapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
  1255
        else lapprox_posrat prec (nat (-x)) (nat (-y))))"
immler@62420
  1256
  by transfer (simp add: truncate_up_uminus_eq)
hoelzl@47599
  1257
hoelzl@47600
  1258
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
immler@62420
  1259
  "\<lambda>prec (x::int) (y::int). truncate_up prec (x / y)"
wenzelm@60698
  1260
  by simp
hoelzl@47599
  1261
immler@58982
  1262
lemma "rapprox_rat = rapprox_posrat"
immler@58982
  1263
  by transfer auto
immler@58982
  1264
immler@58982
  1265
lemma "lapprox_rat = lapprox_posrat"
immler@58982
  1266
  by transfer auto
immler@58982
  1267
wenzelm@60698
  1268
qualified lemma compute_rapprox_rat[code]:
immler@58982
  1269
  "rapprox_rat prec x y = - lapprox_rat prec (-x) y"
immler@62420
  1270
  by transfer (simp add: truncate_down_uminus_eq)
immler@62420
  1271
wenzelm@63356
  1272
qualified lemma compute_truncate_down[code]:
wenzelm@63356
  1273
  "truncate_down p (Ratreal r) = (let (a, b) = quotient_of r in lapprox_rat p a b)"
immler@62420
  1274
  by transfer (auto split: prod.split simp: of_rat_divide dest!: quotient_of_div)
immler@62420
  1275
wenzelm@63356
  1276
qualified lemma compute_truncate_up[code]:
wenzelm@63356
  1277
  "truncate_up p (Ratreal r) = (let (a, b) = quotient_of r in rapprox_rat p a b)"
immler@62420
  1278
  by transfer (auto split: prod.split simp:  of_rat_divide dest!: quotient_of_div)
wenzelm@60698
  1279
wenzelm@60698
  1280
end
wenzelm@60698
  1281
hoelzl@47599
  1282
wenzelm@60500
  1283
subsection \<open>Division\<close>
hoelzl@47599
  1284
immler@62420
  1285
definition "real_divl prec a b = truncate_down prec (a / b)"
immler@54782
  1286
immler@62420
  1287
definition "real_divr prec a b = truncate_up prec (a / b)"
immler@54782
  1288
immler@54782
  1289
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl
immler@54782
  1290
  by (simp add: real_divl_def)
hoelzl@47599
  1291
wenzelm@60698
  1292
context
wenzelm@60698
  1293
begin
wenzelm@60698
  1294
wenzelm@60698
  1295
qualified lemma compute_float_divl[code]:
hoelzl@47600
  1296
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
immler@62420
  1297
  apply transfer
immler@62420
  1298
  unfolding real_divl_def of_int_1 mult_1 truncate_down_shift_int[symmetric]
lp15@65583
  1299
  apply (simp add: powr_diff powr_minus)
immler@62420
  1300
  done
hoelzl@47600
  1301
immler@54782
  1302
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr
immler@54782
  1303
  by (simp add: real_divr_def)
hoelzl@47599
  1304
wenzelm@60698
  1305
qualified lemma compute_float_divr[code]:
immler@58982
  1306
  "float_divr prec x y = - float_divl prec (-x) y"
immler@62420
  1307
  by transfer (simp add: real_divr_def real_divl_def truncate_down_uminus_eq)
wenzelm@60698
  1308
wenzelm@60698
  1309
end
hoelzl@47600
  1310
obua@16782
  1311
wenzelm@60500
  1312
subsection \<open>Approximate Power\<close>
immler@58985
  1313
wenzelm@63356
  1314
lemma div2_less_self[termination_simp]: "odd n \<Longrightarrow> n div 2 < n" for n :: nat
immler@58985
  1315
  by (simp add: odd_pos)
immler@58985
  1316
wenzelm@60698
  1317
fun power_down :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real"
wenzelm@60698
  1318
where
immler@58985
  1319
  "power_down p x 0 = 1"
immler@58985
  1320
| "power_down p x (Suc n) =
wenzelm@60698
  1321
    (if odd n then truncate_down (Suc p) ((power_down p x (Suc n div 2))\<^sup>2)
wenzelm@60698
  1322
     else truncate_down (Suc p) (x * power_down p x n))"
immler@58985
  1323
wenzelm@60698
  1324
fun power_up :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real"
wenzelm@60698
  1325
where
immler@58985
  1326
  "power_up p x 0 = 1"
immler@58985
  1327
| "power_up p x (Suc n) =
wenzelm@60698
  1328
    (if odd n then truncate_up p ((power_up p x (Suc n div 2))\<^sup>2)
wenzelm@60698
  1329
     else truncate_up p (x * power_up p x n))"
immler@58985
  1330
immler@58985
  1331
lift_definition power_up_fl :: "nat \<Rightarrow> float \<Rightarrow> nat \<Rightarrow> float" is power_up
immler@58985
  1332
  by (induct_tac rule: power_up.induct) simp_all
immler@58985
  1333
immler@58985
  1334
lift_definition power_down_fl :: "nat \<Rightarrow> float \<Rightarrow> nat \<Rightarrow> float" is power_down
immler@58985
  1335
  by (induct_tac rule: power_down.induct) simp_all
immler@58985
  1336
immler@58985
  1337
lemma power_float_transfer[transfer_rule]:
immler@58985
  1338
  "(rel_fun pcr_float (rel_fun op = pcr_float)) op ^ op ^"
immler@58985
  1339
  unfolding power_def
immler@58985
  1340
  by transfer_prover
immler@58985
  1341
immler@58985
  1342
lemma compute_power_up_fl[code]:
wenzelm@63356
  1343
    "power_up_fl p x 0 = 1"
wenzelm@63356
  1344
    "power_up_fl p x (Suc n) =
wenzelm@63356
  1345
      (if odd n then float_round_up p ((power_up_fl p x (Suc n div 2))\<^sup>2)
wenzelm@63356
  1346
       else float_round_up p (x * power_up_fl p x n))"
immler@58985
  1347
  and compute_power_down_fl[code]:
wenzelm@63356
  1348
    "power_down_fl p x 0 = 1"
wenzelm@63356
  1349
    "power_down_fl p x (Suc n) =
wenzelm@63356
  1350
      (if odd n then float_round_down (Suc p) ((power_down_fl p x (Suc n div 2))\<^sup>2)
wenzelm@63356
  1351
       else float_round_down (Suc p) (x * power_down_fl p x n))"
wenzelm@63356
  1352
  unfolding atomize_conj by transfer simp
immler@58985
  1353
immler@58985
  1354
lemma power_down_pos: "0 < x \<Longrightarrow> 0 < power_down p x n"
immler@58985
  1355
  by (induct p x n rule: power_down.induct)
immler@58985
  1356
    (auto simp del: odd_Suc_div_two intro!: truncate_down_pos)
immler@58985
  1357
immler@58985
  1358
lemma power_down_nonneg: "0 \<le> x \<Longrightarrow> 0 \<le> power_down p x n"
immler@58985
  1359
  by (induct p x n rule: power_down.induct)
immler@58985
  1360
    (auto simp del: odd_Suc_div_two intro!: truncate_down_nonneg mult_nonneg_nonneg)
immler@58985
  1361
immler@58985
  1362
lemma power_down: "0 \<le> x \<Longrightarrow> power_down p x n \<le> x ^ n"
immler@58985
  1363
proof (induct p x n rule: power_down.induct)
immler@58985
  1364
  case (2 p x n)
wenzelm@63356
  1365
  have ?case if "odd n"
wenzelm@63356
  1366
  proof -
wenzelm@63356
  1367
    from that 2 have "(power_down p x (Suc n div 2)) ^ 2 \<le> (x ^ (Suc n div 2)) ^ 2"
immler@58985
  1368
      by (auto intro: power_mono power_down_nonneg simp del: odd_Suc_div_two)
immler@58985
  1369
    also have "\<dots> = x ^ (Suc n div 2 * 2)"
immler@58985
  1370
      by (simp add: power_mult[symmetric])
immler@58985
  1371
    also have "Suc n div 2 * 2 = Suc n"
wenzelm@60500
  1372
      using \<open>odd n\<close> by presburger
wenzelm@63356
  1373
    finally show ?thesis
wenzelm@63356
  1374
      using that by (auto intro!: truncate_down_le simp del: odd_Suc_div_two)
wenzelm@63356
  1375
  qed
wenzelm@60698
  1376
  then show ?case
immler@58985
  1377
    by (auto intro!: truncate_down_le mult_left_mono 2 mult_nonneg_nonneg power_down_nonneg)
immler@58985
  1378
qed simp
immler@58985
  1379
immler@58985
  1380
lemma power_up: "0 \<le> x \<Longrightarrow> x ^ n \<le> power_up p x n"
immler@58985
  1381
proof (induct p x n rule: power_up.induct)
immler@58985
  1382
  case (2 p x n)
wenzelm@63356
  1383
  have ?case if "odd n"
wenzelm@63356
  1384
  proof -
wenzelm@63356
  1385
    from that even_Suc have "Suc n = Suc n div 2 * 2"
wenzelm@63356
  1386
      by presburger
wenzelm@60698
  1387
    then have "x ^ Suc n \<le> (x ^ (Suc n div 2))\<^sup>2"
immler@58985
  1388
      by (simp add: power_mult[symmetric])
wenzelm@63356
  1389
    also from that 2 have "\<dots> \<le> (power_up p x (Suc n div 2))\<^sup>2"
wenzelm@63356
  1390
      by (auto intro: power_mono simp del: odd_Suc_div_two)
wenzelm@63356
  1391
    finally show ?thesis
wenzelm@63356
  1392
      using that by (auto intro!: truncate_up_le simp del: odd_Suc_div_two)
wenzelm@63356
  1393
  qed
wenzelm@60698
  1394
  then show ?case
immler@58985
  1395
    by (auto intro!: truncate_up_le mult_left_mono 2)
immler@58985
  1396
qed simp
immler@58985
  1397
immler@58985
  1398
lemmas power_up_le = order_trans[OF _ power_up]
immler@58985
  1399
  and power_up_less = less_le_trans[OF _ power_up]
immler@58985
  1400
  and power_down_le = order_trans[OF power_down]
immler@58985
  1401
immler@58985
  1402
lemma power_down_fl: "0 \<le> x \<Longrightarrow> power_down_fl p x n \<le> x ^ n"
immler@58985
  1403
  by transfer (rule power_down)
immler@58985
  1404
immler@58985
  1405
lemma power_up_fl: "0 \<le> x \<Longrightarrow> x ^ n \<le> power_up_fl p x n"
immler@58985
  1406
  by transfer (rule power_up)
immler@58985
  1407
lp15@61609
  1408
lemma real_power_up_fl: "real_of_float (power_up_fl p x n) = power_up p x n"
immler@58985
  1409
  by transfer simp
immler@58985
  1410
lp15@61609
  1411
lemma real_power_down_fl: "real_of_float (power_down_fl p x n) = power_down p x n"
immler@58985
  1412
  by transfer simp
immler@58985
  1413
immler@58985
  1414
wenzelm@60500
  1415
subsection \<open>Approximate Addition\<close>
immler@58985
  1416
immler@58985
  1417
definition "plus_down prec x y = truncate_down prec (x + y)"
immler@58985
  1418
immler@58985
  1419
definition "plus_up prec x y = truncate_up prec (x + y)"
immler@58985
  1420
immler@58985
  1421
lemma float_plus_down_float[intro, simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> plus_down p x y \<in> float"
immler@58985
  1422
  by (simp add: plus_down_def)
immler@58985
  1423
immler@58985
  1424
lemma float_plus_up_float[intro, simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> plus_up p x y \<in> float"
immler@58985
  1425
  by (simp add: plus_up_def)
immler@58985
  1426
wenzelm@63356
  1427
lift_definition float_plus_down :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is plus_down ..
immler@58985
  1428
wenzelm@63356
  1429
lift_definition float_plus_up :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is plus_up ..
immler@58985
  1430
immler@58985
  1431
lemma plus_down: "plus_down prec x y \<le> x + y"
immler@58985
  1432
  and plus_up: "x + y \<le> plus_up prec x y"
immler@58985
  1433
  by (auto simp: plus_down_def truncate_down plus_up_def truncate_up)
immler@58985
  1434
lp15@61609
  1435
lemma float_plus_down: "real_of_float (float_plus_down prec x y) \<le> x + y"
lp15@61609
  1436
  and float_plus_up: "x + y \<le> real_of_float (float_plus_up prec x y)"
wenzelm@63356
  1437
  by (transfer; rule plus_down plus_up)+
immler@58985
  1438
immler@58985
  1439
lemmas plus_down_le = order_trans[OF plus_down]
immler@58985
  1440
  and plus_up_le = order_trans[OF _ plus_up]
immler@58985
  1441
  and float_plus_down_le = order_trans[OF float_plus_down]
immler@58985
  1442
  and float_plus_up_le = order_trans[OF _ float_plus_up]
immler@58985
  1443
immler@58985
  1444
lemma compute_plus_up[code]: "plus_up p x y = - plus_down p (-x) (-y)"
immler@58985
  1445
  using truncate_down_uminus_eq[of p "x + y"]
immler@58985
  1446
  by (auto simp: plus_down_def plus_up_def)
immler@58985
  1447
wenzelm@60698
  1448
lemma truncate_down_log2_eqI:
immler@58985
  1449
  assumes "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@62420
  1450
  assumes "\<lfloor>x * 2 powr (p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>)\<rfloor> = \<lfloor>y * 2 powr (p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>)\<rfloor>"
immler@58985
  1451
  shows "truncate_down p x = truncate_down p y"
immler@58985
  1452
  using assms by (auto simp: truncate_down_def round_down_def)
immler@58985
  1453
wenzelm@60698
  1454
lemma sum_neq_zeroI:
wenzelm@63356
  1455
  "\<bar>a\<bar> \<ge> k \<Longrightarrow> \<bar>b\<bar> < k \<Longrightarrow> a + b \<noteq> 0"
wenzelm@63356
  1456
  "\<bar>a\<bar> > k \<Longrightarrow> \<bar>b\<bar> \<le> k \<Longrightarrow> a + b \<noteq> 0"
wenzelm@63356
  1457
  for a k :: real
immler@58985
  1458
  by auto
immler@58985
  1459
lp15@61609
  1460
lemma abs_real_le_2_powr_bitlen[simp]: "\<bar>real_of_int m2\<bar> < 2 powr real_of_int (bitlen \<bar>m2\<bar>)"
wenzelm@60698
  1461
proof (cases "m2 = 0")
wenzelm@60698
  1462
  case True
wenzelm@60698
  1463
  then show ?thesis by simp
wenzelm@60698
  1464
next
wenzelm@60698
  1465
  case False
wenzelm@60698
  1466
  then have "\<bar>m2\<bar> < 2 ^ nat (bitlen \<bar>m2\<bar>)"
immler@58985
  1467
    using bitlen_bounds[of "\<bar>m2\<bar>"]
immler@58985
  1468
    by (auto simp: powr_add bitlen_nonneg)
wenzelm@60698
  1469
  then show ?thesis
wenzelm@63356
  1470
    by (metis bitlen_nonneg powr_int of_int_abs real_of_int_less_numeral_power_cancel_iff
wenzelm@63356
  1471
      zero_less_numeral)
wenzelm@60698
  1472
qed
immler@58985
  1473
immler@58985
  1474
lemma floor_sum_times_2_powr_sgn_eq:
wenzelm@60698
  1475
  fixes ai p q :: int
wenzelm@60698
  1476
    and a b :: real
immler@58985
  1477
  assumes "a * 2 powr p = ai"
wenzelm@61945
  1478
    and b_le_1: "\<bar>b * 2 powr (p + 1)\<bar> \<le> 1"
wenzelm@60698
  1479
    and leqp: "q \<le> p"
immler@58985
  1480
  shows "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(2 * ai + sgn b) * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1481
proof -
wenzelm@60698
  1482
  consider "b = 0" | "b > 0" | "b < 0" by arith
wenzelm@60698
  1483
  then show ?thesis
wenzelm@60698
  1484
  proof cases
wenzelm@60698
  1485
    case 1
wenzelm@60698
  1486
    then show ?thesis
immler@58985
  1487
      by (simp add: assms(1)[symmetric] powr_add[symmetric] algebra_simps powr_mult_base)
wenzelm@60698
  1488
  next
wenzelm@60698
  1489
    case 2
wenzelm@61945
  1490
    then have "b * 2 powr p < \<bar>b * 2 powr (p + 1)\<bar>"
wenzelm@60698
  1491
      by simp
immler@58985
  1492
    also note b_le_1
lp15@61609
  1493
    finally have b_less_1: "b * 2 powr real_of_int p < 1" .
immler@58985
  1494
lp15@61609
  1495
    from b_less_1 \<open>b > 0\<close> have floor_eq: "\<lfloor>b * 2 powr real_of_int p\<rfloor> = 0" "\<lfloor>sgn b / 2\<rfloor> = 0"
immler@58985
  1496
      by (simp_all add: floor_eq_iff)
immler@58985
  1497
immler@58985
  1498
    have "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(a + b) * 2 powr p * 2 powr (q - p)\<rfloor>"
immler@58985
  1499
      by (simp add: algebra_simps powr_realpow[symmetric] powr_add[symmetric])
immler@58985
  1500
    also have "\<dots> = \<lfloor>(ai + b * 2 powr p) * 2 powr (q - p)\<rfloor>"
immler@58985
  1501
      by (simp add: assms algebra_simps)
lp15@61609
  1502
    also have "\<dots> = \<lfloor>(ai + b * 2 powr p) / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
immler@58985
  1503
      using assms
immler@58985
  1504
      by (simp add: algebra_simps powr_realpow[symmetric] divide_powr_uminus powr_add[symmetric])
lp15@61609
  1505
    also have "\<dots> = \<lfloor>ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
lp15@61609
  1506
      by (simp del: of_int_power add: floor_divide_real_eq_div floor_eq)
lp15@61609
  1507
    finally have "\<lfloor>(a + b) * 2 powr real_of_int q\<rfloor> = \<lfloor>real_of_int ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>" .
immler@58985
  1508
    moreover
wenzelm@63356
  1509
    have "\<lfloor>(2 * ai + (sgn b)) * 2 powr (real_of_int (q - p) - 1)\<rfloor> =
wenzelm@63356
  1510
        \<lfloor>real_of_int ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
wenzelm@63356
  1511
    proof -
lp15@61609
  1512
      have "\<lfloor>(2 * ai + sgn b) * 2 powr (real_of_int (q - p) - 1)\<rfloor> = \<lfloor>(ai + sgn b / 2) * 2 powr (q - p)\<rfloor>"
lp15@65583
  1513
        by (subst powr_diff) (simp add: field_simps)
lp15@61609
  1514
      also have "\<dots> = \<lfloor>(ai + sgn b / 2) / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
lp15@65583
  1515
        using leqp by (simp add: powr_realpow[symmetric] powr_diff)
lp15@61609
  1516
      also have "\<dots> = \<lfloor>ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
lp15@61609
  1517
        by (simp del: of_int_power add: floor_divide_real_eq_div floor_eq)
wenzelm@63356
  1518
      finally show ?thesis .
wenzelm@63356
  1519
    qed
wenzelm@60698
  1520
    ultimately show ?thesis by simp
wenzelm@60698
  1521
  next
wenzelm@60698
  1522
    case 3
lp15@61609
  1523
    then have floor_eq: "\<lfloor>b * 2 powr (real_of_int p + 1)\<rfloor> = -1"
immler@58985
  1524
      using b_le_1
immler@58985
  1525
      by (auto simp: floor_eq_iff algebra_simps pos_divide_le_eq[symmetric] abs_if divide_powr_uminus
nipkow@62390
  1526
        intro!: mult_neg_pos split: if_split_asm)
immler@58985
  1527
    have "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(2*a + 2*b) * 2 powr p * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1528
      by (simp add: algebra_simps powr_realpow[symmetric] powr_add[symmetric] powr_mult_base)
immler@58985
  1529
    also have "\<dots> = \<lfloor>(2 * (a * 2 powr p) + 2 * b * 2 powr p) * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1530
      by (simp add: algebra_simps)
immler@58985
  1531
    also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / 2 powr (1 - q + p)\<rfloor>"
immler@58985
  1532
      using assms by (simp add: algebra_simps powr_mult_base divide_powr_uminus)
lp15@61609
  1533
    also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / real_of_int ((2::int) ^ nat (p - q + 1))\<rfloor>"
immler@58985
  1534
      using assms by (simp add: algebra_simps powr_realpow[symmetric])
lp15@61609
  1535
    also have "\<dots> = \<lfloor>(2 * ai - 1) / real_of_int ((2::int) ^ nat (p - q + 1))\<rfloor>"
wenzelm@60500
  1536
      using \<open>b < 0\<close> assms
lp15@61609
  1537
      by (simp add: floor_divide_of_int_eq floor_eq floor_divide_real_eq_div
lp15@61609
  1538
        del: of_int_mult of_int_power of_int_diff)
immler@58985
  1539
    also have "\<dots> = \<lfloor>(2 * ai - 1) * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1540
      using assms by (simp add: algebra_simps divide_powr_uminus powr_realpow[symmetric])
wenzelm@60698
  1541
    finally show ?thesis
wenzelm@60698
  1542
      using \<open>b < 0\<close> by simp
wenzelm@60698
  1543
  qed
immler@58985
  1544
qed
immler@58985
  1545
wenzelm@60698
  1546
lemma log2_abs_int_add_less_half_sgn_eq:
wenzelm@60698
  1547
  fixes ai :: int
wenzelm@60698
  1548
    and b :: real
wenzelm@61945
  1549
  assumes "\<bar>b\<bar> \<le> 1/2"
wenzelm@60698
  1550
    and "ai \<noteq> 0"
lp15@61609
  1551
  shows "\<lfloor>log 2 \<bar>real_of_int ai + b\<bar>\<rfloor> = \<lfloor>log 2 \<bar>ai + sgn b / 2\<bar>\<rfloor>"
wenzelm@60698
  1552
proof (cases "b = 0")
wenzelm@60698
  1553
  case True
wenzelm@60698
  1554
  then show ?thesis by simp
immler@58985
  1555
next
wenzelm@60698
  1556
  case False
wenzelm@63040
  1557
  define k where "k = \<lfloor>log 2 \<bar>ai\<bar>\<rfloor>"
wenzelm@60698
  1558
  then have "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor> = k"
wenzelm@60698
  1559
    by simp
wenzelm@60698
  1560
  then have k: "2 powr k \<le> \<bar>ai\<bar>" "\<bar>ai\<bar> < 2 powr (k + 1)"
wenzelm@60500
  1561
    by (simp_all add: floor_log_eq_powr_iff \<open>ai \<noteq> 0\<close>)
immler@58985
  1562
  have "k \<ge> 0"
immler@58985
  1563
    using assms by (auto simp: k_def)
wenzelm@63040
  1564
  define r where "r = \<bar>ai\<bar> - 2 ^ nat k"
immler@58985
  1565
  have r: "0 \<le> r" "r < 2 powr k"
wenzelm@60500
  1566
    using \<open>k \<ge> 0\<close> k
immler@58985
  1567
    by (auto simp: r_def k_def algebra_simps powr_add abs_if powr_int)
wenzelm@60698
  1568
  then have "r \<le> (2::int) ^ nat k - 1"
wenzelm@60500
  1569
    using \<open>k \<ge> 0\<close> by (auto simp: powr_int)
lp15@61609
  1570
  from this[simplified of_int_le_iff[symmetric]] \<open>0 \<le> k\<close>
immler@58985
  1571
  have r_le: "r \<le> 2 powr k - 1"
wenzelm@63356
  1572
    by (auto simp: algebra_simps powr_int)
wenzelm@63356
  1573
      (metis of_int_1 of_int_add real_of_int_le_numeral_power_cancel_iff)
immler@58985
  1574
immler@58985
  1575
  have "\<bar>ai\<bar> = 2 powr k + r"
wenzelm@60500
  1576
    using \<open>k \<ge> 0\<close> by (auto simp: k_def r_def powr_realpow[symmetric])
immler@58985
  1577
wenzelm@61945
  1578
  have pos: "\<bar>b\<bar> < 1 \<Longrightarrow> 0 < 2 powr k + (r + b)" for b :: real
wenzelm@60500
  1579
    using \<open>0 \<le> k\<close> \<open>ai \<noteq> 0\<close>
immler@58985
  1580
    by (auto simp add: r_def powr_realpow[symmetric] abs_if sgn_if algebra_simps
nipkow@62390
  1581
      split: if_split_asm)
immler@58985
  1582
  have less: "\<bar>sgn ai * b\<bar> < 1"
immler@58985
  1583
    and less': "\<bar>sgn (sgn ai * b) / 2\<bar> < 1"
nipkow@62390
  1584
    using \<open>\<bar>b\<bar> \<le> _\<close> by (auto simp: abs_if sgn_if split: if_split_asm)
immler@58985
  1585
wenzelm@61945
  1586
  have floor_eq: "\<And>b::real. \<bar>b\<bar> \<le> 1 / 2 \<Longrightarrow>
immler@58985
  1587
      \<lfloor>log 2 (1 + (r + b) / 2 powr k)\<rfloor> = (if r = 0 \<and> b < 0 then -1 else 0)"
wenzelm@60500
  1588
    using \<open>k \<ge> 0\<close> r r_le
immler@58985
  1589
    by (auto simp: floor_log_eq_powr_iff powr_minus_divide field_simps sgn_if)
immler@58985
  1590
lp15@61609
  1591
  from \<open>real_of_int \<bar>ai\<bar> = _\<close> have "\<bar>ai + b\<bar> = 2 powr k + (r + sgn ai * b)"
wenzelm@63356
  1592
    using \<open>\<bar>b\<bar> \<le> _\<close> \<open>0 \<le> k\<close> r
immler@58985
  1593
    by (auto simp add: sgn_if abs_if)
immler@58985
  1594
  also have "\<lfloor>log 2 \<dots>\<rfloor> = \<lfloor>log 2 (2 powr k + r + sgn (sgn ai * b) / 2)\<rfloor>"
immler@58985
  1595
  proof -
immler@58985
  1596
    have "2 powr k + (r + (sgn ai) * b) = 2 powr k * (1 + (r + sgn ai * b) / 2 powr k)"
immler@58985
  1597
      by (simp add: field_simps)
immler@58985
  1598
    also have "\<lfloor>log 2 \<dots>\<rfloor> = k + \<lfloor>log 2 (1 + (r + sgn ai * b) / 2 powr k)\<rfloor>"
immler@58985
  1599
      using pos[OF less]
immler@58985
  1600
      by (subst log_mult) (simp_all add: log_mult powr_mult field_simps)
immler@58985
  1601
    also
immler@58985
  1602
    let ?if = "if r = 0 \<and> sgn ai * b < 0 then -1 else 0"
immler@58985
  1603
    have "\<lfloor>log 2 (1 + (r + sgn ai * b) / 2 powr k)\<rfloor> = ?if"
wenzelm@63356
  1604
      using \<open>\<bar>b\<bar> \<le> _\<close>
immler@58985
  1605
      by (intro floor_eq) (auto simp: abs_mult sgn_if)
immler@58985
  1606
    also
immler@58985
  1607
    have "\<dots> = \<lfloor>log 2 (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k)\<rfloor>"
immler@58985
  1608
      by (subst floor_eq) (auto simp: sgn_if)
immler@58985
  1609
    also have "k + \<dots> = \<lfloor>log 2 (2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k))\<rfloor>"
nipkow@63599
  1610
      unfolding int_add_floor
wenzelm@61945
  1611
      using pos[OF less'] \<open>\<bar>b\<bar> \<le> _\<close>
nipkow@63599
  1612
      by (simp add: field_simps add_log_eq_powr del: floor_add2)
immler@58985
  1613
    also have "2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k) =
immler@58985
  1614
        2 powr k + r + sgn (sgn ai * b) / 2"
immler@58985
  1615
      by (simp add: sgn_if field_simps)
immler@58985
  1616
    finally show ?thesis .
immler@58985
  1617
  qed
immler@58985
  1618
  also have "2 powr k + r + sgn (sgn ai * b) / 2 = \<bar>ai + sgn b / 2\<bar>"
lp15@61609
  1619
    unfolding \<open>real_of_int \<bar>ai\<bar> = _\<close>[symmetric] using \<open>ai \<noteq> 0\<close>
immler@58985
  1620
    by (auto simp: abs_if sgn_if algebra_simps)
immler@58985
  1621
  finally show ?thesis .
immler@58985
  1622
qed
immler@58985
  1623
wenzelm@60698
  1624
context
wenzelm@60698
  1625
begin
wenzelm@60698
  1626
wenzelm@60698
  1627
qualified lemma compute_far_float_plus_down:
wenzelm@60698
  1628
  fixes m1 e1 m2 e2 :: int
wenzelm@60698
  1629
    and p :: nat
immler@62420
  1630
  defines "k1 \<equiv> Suc p - nat (bitlen \<bar>m1\<bar>)"
immler@58985
  1631
  assumes H: "bitlen \<bar>m2\<bar> \<le> e1 - e2 - k1 - 2" "m1 \<noteq> 0" "m2 \<noteq> 0" "e1 \<ge> e2"
immler@58985
  1632
  shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
immler@58985
  1633
    float_round_down p (Float (m1 * 2 ^ (Suc (Suc k1)) + sgn m2) (e1 - int k1 - 2))"
immler@58985
  1634
proof -
lp15@61609
  1635
  let ?a = "real_of_float (Float m1 e1)"
lp15@61609
  1636
  let ?b = "real_of_float (Float m2 e2)"
immler@58985
  1637
  let ?sum = "?a + ?b"
lp15@61609
  1638
  let ?shift = "real_of_int e2 - real_of_int e1 + real k1 + 1"
immler@58985
  1639
  let ?m1 = "m1 * 2 ^ Suc k1"
immler@58985
  1640
  let ?m2 = "m2 * 2 powr ?shift"
immler@58985
  1641
  let ?m2' = "sgn m2 / 2"
immler@58985
  1642
  let ?e = "e1 - int k1 - 1"
immler@58985
  1643
immler@58985
  1644
  have sum_eq: "?sum = (?m1 + ?m2) * 2 powr ?e"
immler@58985
  1645
    by (auto simp: powr_add[symmetric] powr_mult[symmetric] algebra_simps
immler@58985
  1646
      powr_realpow[symmetric] powr_mult_base)
immler@58985
  1647
immler@58985
  1648
  have "\<bar>?m2\<bar> * 2 < 2 powr (bitlen \<bar>m2\<bar> + ?shift + 1)"
lp15@65583
  1649
    by (auto simp: field_simps powr_add powr_mult_base powr_diff abs_mult)
immler@58985
  1650
  also have "\<dots> \<le> 2 powr 0"
immler@58985
  1651
    using H by (intro powr_mono) auto
immler@58985
  1652
  finally have abs_m2_less_half: "\<bar>?m2\<bar> < 1 / 2"
immler@58985
  1653
    by simp
immler@58985
  1654
lp15@61609
  1655
  then have "\<bar>real_of_int m2\<bar> < 2 powr -(?shift + 1)"
immler@63248
  1656
    unfolding powr_minus_divide by (auto simp: bitlen_alt_def field_simps powr_mult_base abs_mult)
lp15@61609
  1657
  also have "\<dots> \<le> 2 powr real_of_int (e1 - e2 - 2)"
immler@58985
  1658
    by simp
lp15@61609
  1659
  finally have b_less_quarter: "\<bar>?b\<bar> < 1/4 * 2 powr real_of_int e1"
lp15@65583
  1660
    by (simp add: powr_add field_simps powr_diff abs_mult)
lp15@61609
  1661
  also have "1/4 < \<bar>real_of_int m1\<bar> / 2" using \<open>m1 \<noteq> 0\<close> by simp
immler@58985
  1662
  finally have b_less_half_a: "\<bar>?b\<bar> < 1/2 * \<bar>?a\<bar>"
immler@58985
  1663
    by (simp add: algebra_simps powr_mult_base abs_mult)
wenzelm@60698
  1664
  then have a_half_less_sum: "\<bar>?a\<bar> / 2 < \<bar>?sum\<bar>"
nipkow@62390
  1665
    by (auto simp: field_simps abs_if split: if_split_asm)
immler@58985
  1666
immler@58985
  1667
  from b_less_half_a have "\<bar>?b\<bar> < \<bar>?a\<bar>" "\<bar>?b\<bar> \<le> \<bar>?a\<bar>"
immler@58985
  1668
    by simp_all
immler@58985
  1669
lp15@61609
  1670
  have "\<bar>real_of_float (Float m1 e1)\<bar> \<ge> 1/4 * 2 powr real_of_int e1"
wenzelm@60500
  1671
    using \<open>m1 \<noteq> 0\<close>
immler@58985
  1672
    by (auto simp: powr_add powr_int bitlen_nonneg divide_right_mono abs_mult)
wenzelm@60698
  1673
  then have "?sum \<noteq> 0" using b_less_quarter
immler@58985
  1674
    by (rule sum_neq_zeroI)
wenzelm@60698
  1675
  then have "?m1 + ?m2 \<noteq> 0"
immler@58985
  1676
    unfolding sum_eq by (simp add: abs_mult zero_less_mult_iff)
immler@58985
  1677
lp15@61609
  1678
  have "\<bar>real_of_int ?m1\<bar> \<ge> 2 ^ Suc k1" "\<bar>?m2'\<bar> < 2 ^ Suc k1"
wenzelm@60500
  1679
    using \<open>m1 \<noteq> 0\<close> \<open>m2 \<noteq> 0\<close> by (auto simp: sgn_if less_1_mult abs_mult simp del: power.simps)
wenzelm@60698
  1680
  then have sum'_nz: "?m1 + ?m2' \<noteq> 0"
immler@58985
  1681
    by (intro sum_neq_zeroI)
immler@58985
  1682
lp15@61609
  1683
  have "\<lfloor>log 2 \<bar>real_of_float (Float m1 e1) + real_of_float (Float m2 e2)\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> + ?e"
wenzelm@60500
  1684
    using \<open>?m1 + ?m2 \<noteq> 0\<close>
immler@58985
  1685
    unfolding floor_add[symmetric] sum_eq
lp15@61609
  1686
    by (simp add: abs_mult log_mult) linarith
lp15@61609
  1687
  also have "\<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + sgn (real_of_int m2 * 2 powr ?shift) / 2\<bar>\<rfloor>"
wenzelm@60500
  1688
    using abs_m2_less_half \<open>m1 \<noteq> 0\<close>
immler@58985
  1689
    by (intro log2_abs_int_add_less_half_sgn_eq) (auto simp: abs_mult)
lp15@61609
  1690
  also have "sgn (real_of_int m2 * 2 powr ?shift) = sgn m2"
immler@58985
  1691
    by (auto simp: sgn_if zero_less_mult_iff less_not_sym)
immler@58985
  1692
  also
immler@58985
  1693
  have "\<bar>?m1 + ?m2'\<bar> * 2 powr ?e = \<bar>?m1 * 2 + sgn m2\<bar> * 2 powr (?e - 1)"
lp15@65583
  1694
    by (auto simp: field_simps powr_minus[symmetric] powr_diff powr_mult_base)
lp15@61609
  1695
  then have "\<lfloor>log 2 \<bar>?m1 + ?m2'\<bar>\<rfloor> + ?e = \<lfloor>log 2 \<bar>real_of_float (Float (?m1 * 2 + sgn m2) (?e - 1))\<bar>\<rfloor>"
wenzelm@60500
  1696
    using \<open>?m1 + ?m2' \<noteq> 0\<close>
nipkow@63599
  1697
    unfolding floor_add_int
nipkow@63599
  1698
    by (simp add: log_add_eq_powr abs_mult_pos del: floor_add2)
immler@58985
  1699
  finally
lp15@61609
  1700
  have "\<lfloor>log 2 \<bar>?sum\<bar>\<rfloor> = \<lfloor>log 2 \<bar>real_of_float (Float (?m1*2 + sgn m2) (?e - 1))\<bar>\<rfloor>" .
wenzelm@60698
  1701
  then have "plus_down p (Float m1 e1) (Float m2 e2) =
immler@58985
  1702
      truncate_down p (Float (?m1*2 + sgn m2) (?e - 1))"
immler@58985
  1703
    unfolding plus_down_def
immler@58985
  1704
  proof (rule truncate_down_log2_eqI)
immler@62420
  1705
    let ?f = "(int p - \<lfloor>log 2 \<bar>real_of_float (Float m1 e1) + real_of_float (Float m2 e2)\<bar>\<rfloor>)"
immler@58985
  1706
    let ?ai = "m1 * 2 ^ (Suc k1)"
lp15@61609
  1707
    have "\<lfloor>(?a + ?b) * 2 powr real_of_int ?f\<rfloor> = \<lfloor>(real_of_int (2 * ?ai) + sgn ?b) * 2 powr real_of_int (?f - - ?e - 1)\<rfloor>"
immler@58985
  1708
    proof (rule floor_sum_times_2_powr_sgn_eq)
lp15@61609
  1709
      show "?a * 2 powr real_of_int (-?e) = real_of_int ?ai"
lp15@65583
  1710
        by (simp add: powr_add powr_realpow[symmetric] powr_diff)
lp15@61609
  1711
      show "\<bar>?b * 2 powr real_of_int (-?e + 1)\<bar> \<le> 1"
immler@58985
  1712
        using abs_m2_less_half
immler@58985
  1713
        by (simp add: abs_mult powr_add[symmetric] algebra_simps powr_mult_base)
immler@58985
  1714
    next
lp15@61609
  1715
      have "e1 + \<lfloor>log 2 \<bar>real_of_int m1\<bar>\<rfloor> - 1 = \<lfloor>log 2 \<bar>?a\<bar>\<rfloor> - 1"
wenzelm@60500
  1716
        using \<open>m1 \<noteq> 0\<close>
nipkow@63599
  1717
        by (simp add: int_add_floor algebra_simps log_mult abs_mult del: floor_add2)
immler@58985
  1718
      also have "\<dots> \<le> \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor>"
wenzelm@60500
  1719
        using a_half_less_sum \<open>m1 \<noteq> 0\<close> \<open>?sum \<noteq> 0\<close>
lp15@61609
  1720
        unfolding floor_diff_of_int[symmetric]
lp15@61609
  1721
        by (auto simp add: log_minus_eq_powr powr_minus_divide intro!: floor_mono)
immler@58985
  1722
      finally
immler@58985
  1723
      have "int p - \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor> \<le> p - (bitlen \<bar>m1\<bar>) - e1 + 2"
immler@63248
  1724
        by (auto simp: algebra_simps bitlen_alt_def \<open>m1 \<noteq> 0\<close>)
immler@62420
  1725
      also have "\<dots> \<le> - ?e"
immler@58985
  1726
        using bitlen_nonneg[of "\<bar>m1\<bar>"] by (simp add: k1_def)
immler@58985
  1727
      finally show "?f \<le> - ?e" by simp
immler@58985
  1728
    qed
immler@58985
  1729
    also have "sgn ?b = sgn m2"
immler@58985
  1730
      using powr_gt_zero[of 2 e2]
immler@58985
  1731
      by (auto simp add: sgn_if zero_less_mult_iff simp del: powr_gt_zero)
lp15@61609
  1732
    also have "\<lfloor>(real_of_int (2 * ?m1) + real_of_int (sgn m2)) * 2 powr real_of_int (?f - - ?e - 1)\<rfloor> =
immler@58985
  1733
        \<lfloor>Float (?m1 * 2 + sgn m2) (?e - 1) * 2 powr ?f\<rfloor>"
immler@58985
  1734
      by (simp add: powr_add[symmetric] algebra_simps powr_realpow[symmetric])
immler@58985
  1735
    finally
lp15@61609
  1736
    show "\<lfloor>(?a + ?b) * 2 powr ?f\<rfloor> = \<lfloor>real_of_float (Float (?m1 * 2 + sgn m2) (?e - 1)) * 2 powr ?f\<rfloor>" .
immler@58985
  1737
  qed
wenzelm@60698
  1738
  then show ?thesis
immler@58985
  1739
    by transfer (simp add: plus_down_def ac_simps Let_def)
immler@58985
  1740
qed
immler@58985
  1741
immler@58985
  1742
lemma compute_float_plus_down_naive[code]: "float_plus_down p x y = float_round_down p (x + y)"
immler@58985
  1743
  by transfer (auto simp: plus_down_def)
immler@58985
  1744
wenzelm@60698
  1745
qualified lemma compute_float_plus_down[code]:
immler@58985
  1746
  fixes p::nat and m1 e1 m2 e2::int
immler@58985
  1747
  shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
immler@58985
  1748
    (if m1 = 0 then float_round_down p (Float m2 e2)
immler@58985
  1749
    else if m2 = 0 then float_round_down p (Float m1 e1)
wenzelm@63356
  1750
    else
wenzelm@63356
  1751
      (if e1 \<ge> e2 then
wenzelm@63356
  1752
        (let k1 = Suc p - nat (bitlen \<bar>m1\<bar>) in
wenzelm@63356
  1753
          if bitlen \<bar>m2\<bar> > e1 - e2 - k1 - 2
wenzelm@63356
  1754
          then float_round_down p ((Float m1 e1) + (Float m2 e2))
wenzelm@63356
  1755
          else float_round_down p (Float (m1 * 2 ^ (Suc (Suc k1)) + sgn m2) (e1 - int k1 - 2)))
immler@58985
  1756
    else float_plus_down p (Float m2 e2) (Float m1 e1)))"
immler@58985
  1757
proof -
immler@58985
  1758
  {
immler@62420
  1759
    assume "bitlen \<bar>m2\<bar> \<le> e1 - e2 - (Suc p - nat (bitlen \<bar>m1\<bar>)) - 2" "m1 \<noteq> 0" "m2 \<noteq> 0" "e1 \<ge> e2"
wenzelm@60698
  1760
    note compute_far_float_plus_down[OF this]
immler@58985
  1761
  }
wenzelm@60698
  1762
  then show ?thesis
immler@58985
  1763
    by transfer (simp add: Let_def plus_down_def ac_simps)
immler@58985
  1764
qed
immler@58985
  1765
wenzelm@60698
  1766
qualified lemma compute_float_plus_up[code]: "float_plus_up p x y = - float_plus_down p (-x) (-y)"
immler@58985
  1767
  using truncate_down_uminus_eq[of p "x + y"]
immler@58985
  1768
  by transfer (simp add: plus_down_def plus_up_def ac_simps)
immler@58985
  1769
immler@58985
  1770
lemma mantissa_zero[simp]: "mantissa 0 = 0"
wenzelm@60698
  1771
  by (metis mantissa_0 zero_float.abs_eq)
wenzelm@60698
  1772
immler@62421
  1773
qualified lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (float_plus_down 0 b (- a))"
immler@62421
  1774
  using truncate_down[of 0 "b - a"] truncate_down_pos[of "b - a" 0]
immler@62421
  1775
  by transfer (auto simp: plus_down_def)
immler@62421
  1776
immler@62421
  1777
qualified lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (float_plus_down 0 b (- a))"
immler@62421
  1778
  using truncate_down[of 0 "b - a"] truncate_down_nonneg[of "b - a" 0]
immler@62421
  1779
  by transfer (auto simp: plus_down_def)
immler@62421
  1780
wenzelm@60698
  1781
end
immler@58985
  1782
immler@58985
  1783
wenzelm@60500
  1784
subsection \<open>Lemmas needed by Approximate\<close>
hoelzl@47599
  1785
wenzelm@60698
  1786
lemma Float_num[simp]:
lp15@61609
  1787
   "real_of_float (Float 1 0) = 1"
lp15@61609
  1788
   "real_of_float (Float 1 1) = 2"
lp15@61609
  1789
   "real_of_float (Float 1 2) = 4"
lp15@61609
  1790
   "real_of_float (Float 1 (- 1)) = 1/2"
lp15@61609
  1791
   "real_of_float (Float 1 (- 2)) = 1/4"
lp15@61609
  1792
   "real_of_float (Float 1 (- 3)) = 1/8"
lp15@61609
  1793
   "real_of_float (Float (- 1) 0) = -1"
immler@62420
  1794
   "real_of_float (Float (numeral n) 0) = numeral n"
immler@62420
  1795
   "real_of_float (Float (- numeral n) 0) = - numeral n"
wenzelm@60698
  1796
  using two_powr_int_float[of 2] two_powr_int_float[of "-1"] two_powr_int_float[of "-2"]
wenzelm@60698
  1797
    two_powr_int_float[of "-3"]
wenzelm@60698
  1798
  using powr_realpow[of 2 2] powr_realpow[of 2 3]
lp15@65583
  1799
  using powr_minus[of "2::real" 1] powr_minus[of "2::real" 2] powr_minus[of "2::real" 3]
wenzelm@60698
  1800
  by auto
hoelzl@47599
  1801
lp15@61609
  1802
lemma real_of_Float_int[simp]: "real_of_float (Float n 0) = real n"
wenzelm@60698
  1803
  by simp
hoelzl@47599
  1804
lp15@61609
  1805
lemma float_zero[simp]: "real_of_float (Float 0 e) = 0"
wenzelm@60698
  1806
  by simp
hoelzl@47599
  1807
wenzelm@61945
  1808
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> \<bar>(a::int) div 2\<bar> < \<bar>a\<bar>"
wenzelm@60698
  1809
  by arith
hoelzl@29804
  1810
lp15@61609
  1811
lemma lapprox_rat: "real_of_float (lapprox_rat prec x y) \<le> real_of_int x / real_of_int y"
immler@62420
  1812
  by (simp add: lapprox_rat.rep_eq truncate_down)
obua@16782
  1813
wenzelm@60698
  1814
lemma mult_div_le:
wenzelm@60698
  1815
  fixes a b :: int
wenzelm@60698
  1816
  assumes "b > 0"
wenzelm@60698
  1817
  shows "a \<ge> b * (a div b)"
hoelzl@47599
  1818
proof -
haftmann@64246
  1819
  from minus_div_mult_eq_mod [symmetric, of a b] have "a = b * (a div b) + a mod b"
wenzelm@60698
  1820
    by simp
wenzelm@60698
  1821
  also have "\<dots> \<ge> b * (a div b) + 0"
wenzelm@60698
  1822
    apply (rule add_left_mono)
wenzelm@60698
  1823
    apply (rule pos_mod_sign)
wenzelm@63356
  1824
    using assms
wenzelm@63356
  1825
    apply simp
wenzelm@60698
  1826
    done
wenzelm@60698
  1827
  finally show ?thesis
wenzelm@60698
  1828
    by simp
hoelzl@47599
  1829
qed
hoelzl@47599
  1830
hoelzl@47599
  1831
lemma lapprox_rat_nonneg:
immler@58982
  1832
  assumes "0 \<le> x" and "0 \<le> y"
lp15@61609
  1833
  shows "0 \<le> real_of_float (lapprox_rat n x y)"
immler@62420
  1834
  using assms
immler@62420
  1835
  by transfer (simp add: truncate_down_nonneg)
obua@16782
  1836
lp15@61609
  1837
lemma rapprox_rat: "real_of_int x / real_of_int y \<le> real_of_float (rapprox_rat prec x y)"
immler@62420
  1838
  by transfer (simp add: truncate_up)
hoelzl@47599
  1839
hoelzl@47599
  1840
lemma rapprox_rat_le1:
wenzelm@63356
  1841
  assumes "0 \<le> x" "0 < y" "x \<le> y"
lp15@61609
  1842
  shows "real_of_float (rapprox_rat n x y) \<le> 1"
immler@62420
  1843
  using assms
immler@62420
  1844
  by transfer (simp add: truncate_up_le1)
obua@16782
  1845
lp15@61609
  1846
lemma rapprox_rat_nonneg_nonpos: "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real_of_float (rapprox_rat n x y) \<le> 0"
immler@62420
  1847
  by transfer (simp add: truncate_up_nonpos divide_nonneg_nonpos)
obua@16782
  1848
lp15@61609
  1849
lemma rapprox_rat_nonpos_nonneg: "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real_of_float (rapprox_rat n x y) \<le> 0"
immler@62420
  1850
  by transfer (simp add: truncate_up_nonpos divide_nonpos_nonneg)
obua@16782
  1851
immler@54782
  1852
lemma real_divl: "real_divl prec x y \<le> x / y"
immler@62420
  1853
  by (simp add: real_divl_def truncate_down)
immler@54782
  1854
immler@54782
  1855
lemma real_divr: "x / y \<le> real_divr prec x y"
immler@62420
  1856
  by (simp add: real_divr_def truncate_up)
immler@54782
  1857
lp15@61609
  1858
lemma float_divl: "real_of_float (float_divl prec x y) \<le> x / y"
immler@54782
  1859
  by transfer (rule real_divl)
immler@54782
  1860
wenzelm@63356
  1861
lemma real_divl_lower_bound: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y"
immler@62420
  1862
  by (simp add: real_divl_def truncate_down_nonneg)
hoelzl@47599
  1863
wenzelm@63356
  1864
lemma float_divl_lower_bound: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_of_float (float_divl prec x y)"
immler@54782
  1865
  by transfer (rule real_divl_lower_bound)
hoelzl@47599
  1866
hoelzl@47599
  1867
lemma exponent_1: "exponent 1 = 0"
hoelzl@47599
  1868
  using exponent_float[of 1 0] by (simp add: one_float_def)
hoelzl@47599
  1869
hoelzl@47599
  1870
lemma mantissa_1: "mantissa 1 = 1"
hoelzl@47599
  1871
  using mantissa_float[of 1 0] by (simp add: one_float_def)
obua@16782
  1872
hoelzl@47599
  1873
lemma bitlen_1: "bitlen 1 = 1"
immler@63248
  1874
  by (simp add: bitlen_alt_def)
hoelzl@47599
  1875
hoelzl@47599
  1876
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
wenzelm@60698
  1877
  (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@47599
  1878
proof
wenzelm@60698
  1879
  show ?rhs if ?lhs
wenzelm@60698
  1880
  proof -
lp15@61609
  1881
    from that have z: "0 = real_of_float x"
wenzelm@60698
  1882
      using mantissa_exponent by simp
wenzelm@60698
  1883
    show ?thesis
wenzelm@60698
  1884
      by (simp add: zero_float_def z)
wenzelm@60698
  1885
  qed
wenzelm@60698
  1886
  show ?lhs if ?rhs
wenzelm@60698
  1887
    using that by (simp add: zero_float_def)
wenzelm@60698
  1888
qed
obua@16782
  1889
hoelzl@47599
  1890
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
wenzelm@60698
  1891
proof (cases "x = 0")
wenzelm@60698
  1892
  case True
wenzelm@60698
  1893
  then show ?thesis by simp
wenzelm@60698
  1894
next
wenzelm@60698
  1895
  case False
wenzelm@60698
  1896
  then have "mantissa x \<noteq> 0"
wenzelm@60698
  1897
    using mantissa_eq_zero_iff by auto
wenzelm@60698
  1898
  have "x = mantissa x * 2 powr (exponent x)"
wenzelm@60698
  1899
    by (rule mantissa_exponent)
wenzelm@60698
  1900
  also have "mantissa x \<le> \<bar>mantissa x\<bar>"
wenzelm@60698
  1901
    by simp
wenzelm@60698
  1902
  also have "\<dots> \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
wenzelm@60500
  1903
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg \<open>mantissa x \<noteq> 0\<close>
lp15@61649
  1904
    by (auto simp del: of_int_abs simp add: powr_int)
hoelzl@47599
  1905
  finally show ?thesis by (simp add: powr_add)
hoelzl@29804
  1906
qed
hoelzl@29804
  1907
immler@54782
  1908
lemma real_divl_pos_less1_bound:
immler@62420
  1909
  assumes "0 < x" "x \<le> 1"
immler@58982
  1910
  shows "1 \<le> real_divl prec 1 x"
immler@62420
  1911
  using assms
immler@62420
  1912
  by (auto intro!: truncate_down_ge1 simp: real_divl_def)
obua@16782
  1913
immler@54782
  1914
lemma float_divl_pos_less1_bound:
wenzelm@63356
  1915
  "0 < real_of_float x \<Longrightarrow> real_of_float x \<le> 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow>
wenzelm@63356
  1916
    1 \<le> real_of_float (float_divl prec 1 x)"
wenzelm@60698
  1917
  by transfer (rule real_divl_pos_less1_bound)
obua@16782
  1918
lp15@61609
  1919
lemma float_divr: "real_of_float x / real_of_float y \<le> real_of_float (float_divr prec x y)"
immler@54782
  1920
  by transfer (rule real_divr)
immler@54782
  1921
wenzelm@60698
  1922
lemma real_divr_pos_less1_lower_bound:
wenzelm@60698
  1923
  assumes "0 < x"
wenzelm@60698
  1924
    and "x \<le> 1"
wenzelm@60698
  1925
  shows "1 \<le> real_divr prec 1 x"
hoelzl@29804
  1926
proof -
wenzelm@60698
  1927
  have "1 \<le> 1 / x"
wenzelm@63356
  1928
    using \<open>0 < x\<close> and \<open>x \<le> 1\<close> by auto
wenzelm@60698
  1929
  also have "\<dots> \<le> real_divr prec 1 x"
wenzelm@63356
  1930
    using real_divr[where x = 1 and y = x] by auto
hoelzl@47600
  1931
  finally show ?thesis by auto
hoelzl@29804
  1932
qed
hoelzl@29804
  1933
immler@58982
  1934
lemma float_divr_pos_less1_lower_bound: "0 < x \<Longrightarrow> x \<le> 1 \<Longrightarrow> 1 \<le> float_divr prec 1 x"
immler@54782
  1935
  by transfer (rule real_divr_pos_less1_lower_bound)
immler@54782
  1936
wenzelm@63356
  1937
lemma real_divr_nonpos_pos_upper_bound: "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real_divr prec x y \<le> 0"
immler@62420
  1938
  by (simp add: real_divr_def truncate_up_nonpos divide_le_0_iff)
immler@54782
  1939
hoelzl@47599
  1940
lemma float_divr_nonpos_pos_upper_bound:
lp15@61609
  1941
  "real_of_float x \<le> 0 \<Longrightarrow> 0 \<le> real_of_float y \<Longrightarrow> real_of_float (float_divr prec x y) \<le> 0"
immler@54782
  1942
  by transfer (rule real_divr_nonpos_pos_upper_bound)
immler@54782
  1943
wenzelm@63356
  1944
lemma real_divr_nonneg_neg_upper_bound: "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real_divr prec x y \<le> 0"
immler@62420
  1945
  by (simp add: real_divr_def truncate_up_nonpos divide_le_0_iff)
obua@16782
  1946
hoelzl@47599
  1947
lemma float_divr_nonneg_neg_upper_bound:
lp15@61609
  1948
  "0 \<le> real_of_float x \<Longrightarrow> real_of_float y \<le> 0 \<Longrightarrow> real_of_float (float_divr prec x y) \<le> 0"
immler@54782
  1949
  by transfer (rule real_divr_nonneg_neg_upper_bound)
immler@54782
  1950
immler@54784
  1951
lemma truncate_up_nonneg_mono:
immler@54784
  1952
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  1953
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  1954
proof -
wenzelm@60698
  1955
  consider "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>" | "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>" "0 < x" | "x \<le> 0"
wenzelm@60698
  1956
    by arith
wenzelm@60698
  1957
  then show ?thesis
wenzelm@60698
  1958
  proof cases
wenzelm@60698
  1959
    case 1
wenzelm@60698
  1960
    then show ?thesis
immler@54784
  1961
      using assms
immler@54784
  1962
      by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
wenzelm@60698
  1963
  next
wenzelm@60698
  1964
    case 2
wenzelm@60698
  1965
    from assms \<open>0 < x\<close> have "log 2 x \<le> log 2 y"
wenzelm@60698
  1966
      by auto
wenzelm@60698
  1967
    with \<open>\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>\<close>
wenzelm@60698
  1968
    have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
wenzelm@60698
  1969
      by (metis floor_less_cancel linorder_cases not_le)+
immler@54784
  1970
    have "truncate_up prec x =
immler@62420
  1971
      real_of_int \<lceil>x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor> )\<rceil> * 2 powr - real_of_int (int prec - \<lfloor>log 2 x\<rfloor>)"
immler@54784
  1972
      using assms by (simp add: truncate_up_def round_up_def)
immler@62420
  1973
    also have "\<lceil>x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor>)\<rceil> \<le> (2 ^ (Suc prec))"
wenzelm@63356
  1974
    proof (simp only: ceiling_le_iff)
wenzelm@63356
  1975
      have "x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor>) \<le>
wenzelm@63356
  1976
        x * (2 powr real (Suc prec) / (2 powr log 2 x))"
immler@54784
  1977
        using real_of_int_floor_add_one_ge[of "log 2 x"] assms
lp15@65583
  1978
        by (auto simp add: algebra_simps powr_diff [symmetric]  intro!: mult_left_mono)
immler@62420
  1979
      then show "x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor>) \<le> real_of_int ((2::int) ^ (Suc prec))"
immler@62420
  1980
        using \<open>0 < x\<close> by (simp add: powr_realpow powr_add)
immler@54784
  1981
    qed
immler@62420
  1982
    then have "real_of_int \<lceil>x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor>)\<rceil> \<le> 2 powr int (Suc prec)"
wenzelm@63356
  1983
      by (auto simp: powr_realpow powr_add)
wenzelm@63356
  1984
        (metis power_Suc real_of_int_le_numeral_power_cancel_iff)
immler@54784
  1985
    also
immler@62420
  1986
    have "2 powr - real_of_int (int prec - \<lfloor>log 2 x\<rfloor>) \<le> 2 powr - real_of_int (int prec - \<lfloor>log 2 y\<rfloor> + 1)"
immler@54784
  1987
      using logless flogless by (auto intro!: floor_mono)
wenzelm@63356
  1988
    also have "2 powr real_of_int (int (Suc prec)) \<le>
wenzelm@63356
  1989
        2 powr (log 2 y + real_of_int (int prec - \<lfloor>log 2 y\<rfloor> + 1))"
wenzelm@60500
  1990
      using assms \<open>0 < x\<close>
immler@54784
  1991
      by (auto simp: algebra_simps)
wenzelm@63356
  1992
    finally have "truncate_up prec x \<le>
wenzelm@63356
  1993
        2 powr (log 2 y + real_of_int (int prec - \<lfloor>log 2 y\<rfloor> + 1)) * 2 powr - real_of_int (int prec - \<lfloor>log 2 y\<rfloor> + 1)"
immler@54784
  1994
      by simp
lp15@61609
  1995
    also have "\<dots> = 2 powr (log 2 y + real_of_int (int prec - \<lfloor>log 2 y\<rfloor>) - real_of_int (int prec - \<lfloor>log 2 y\<rfloor>))"
immler@54784
  1996
      by (subst powr_add[symmetric]) simp
immler@54784
  1997
    also have "\<dots> = y"
wenzelm@60500
  1998
      using \<open>0 < x\<close> assms
immler@54784
  1999
      by (simp add: powr_add)
immler@54784
  2000
    also have "\<dots> \<le> truncate_up prec y"
immler@54784
  2001
      by (rule truncate_up)
wenzelm@60698
  2002
    finally show ?thesis .
wenzelm@60698
  2003
  next
wenzelm@60698
  2004
    case 3
wenzelm@60698
  2005
    then show ?thesis
immler@54784
  2006
      using assms
immler@54784
  2007
      by (auto intro!: truncate_up_le)
wenzelm@60698
  2008
  qed
immler@54784
  2009
qed
immler@54784
  2010
immler@54784
  2011
lemma truncate_up_switch_sign_mono:
immler@54784
  2012
  assumes "x \<le> 0" "0 \<le> y"
immler@54784
  2013
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  2014
proof -
wenzelm@60500
  2015
  note truncate_up_nonpos[OF \<open>x \<le> 0\<close>]
wenzelm@60500
  2016
  also note truncate_up_le[OF \<open>0 \<le> y\<close>]
immler@54784
  2017
  finally show ?thesis .
immler@54784
  2018
qed
immler@54784
  2019
immler@54784
  2020
lemma truncate_down_switch_sign_mono:
wenzelm@60698
  2021
  assumes "x \<le> 0"
wenzelm@60698
  2022
    and "0 \<le> y"
wenzelm@60698
  2023
    and "x \<le> y"
immler@54784
  2024
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  2025
proof -
wenzelm@60500
  2026
  note truncate_down_le[OF \<open>x \<le> 0\<close>]
wenzelm@60500
  2027
  also note truncate_down_nonneg[OF \<open>0 \<le> y\<close>]
immler@54784
  2028
  finally show ?thesis .
immler@54784
  2029
qed
immler@54784
  2030
immler@54784
  2031
lemma truncate_down_nonneg_mono:
immler@54784
  2032
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  2033
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  2034
proof -
immler@62420
  2035
  consider "x \<le> 0" | "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>" |
immler@62420
  2036
    "0 < x" "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
wenzelm@60698
  2037
    by arith
wenzelm@60698
  2038
  then show ?thesis
wenzelm@60698
  2039
  proof cases
wenzelm@60698
  2040
    case 1
immler@54784
  2041
    with assms have "x = 0" "0 \<le> y" by simp_all
wenzelm@60698
  2042
    then show ?thesis
immler@58985
  2043
      by (auto intro!: truncate_down_nonneg)
wenzelm@60698
  2044
  next
immler@62420
  2045
    case 2
wenzelm@60698
  2046
    then show ?thesis
immler@54784
  2047
      using assms
immler@54784
  2048
      by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
wenzelm@60698
  2049
  next
immler@62420
  2050
    case 3
wenzelm@60698
  2051
    from \<open>0 < x\<close> have "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y"
wenzelm@60698
  2052
      using assms by auto
wenzelm@60698
  2053
    with \<open>\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>\<close>
wenzelm@60698
  2054
    have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
wenzelm@60500
  2055
      unfolding atomize_conj abs_of_pos[OF \<open>0 < x\<close>] abs_of_pos[OF \<open>0 < y\<close>]
immler@54784
  2056
      by (metis floor_less_cancel linorder_cases not_le)
immler@62420
  2057
    have "2 powr prec \<le> y * 2 powr real prec / (2 powr log 2 y)"
wenzelm@60698
  2058
      using \<open>0 < y\<close> by simp
immler@62420
  2059
    also have "\<dots> \<le> y * 2 powr real (Suc prec) / (2 powr (real_of_int \<lfloor>log 2 y\<rfloor> + 1))"
wenzelm@60500
  2060
      using \<open>0 \<le> y\<close> \<open>0 \<le> x\<close> assms(2)
nipkow@56544
  2061
      by (auto intro!: powr_mono divide_left_mono
lp15@65583
  2062
          simp: of_nat_diff powr_add powr_diff)
immler@62420
  2063
    also have "\<dots> = y * 2 powr real (Suc prec) / (2 powr real_of_int \<lfloor>log 2 y\<rfloor> * 2)"
immler@54784
  2064
      by (auto simp: powr_add)
immler@62420
  2065
    finally have "(2 ^ prec) \<le> \<lfloor>y * 2 powr real_of_int (int (Suc prec) - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
wenzelm@60500
  2066
      using \<open>0 \<le> y\<close>
lp15@65583
  2067
      by (auto simp: powr_diff le_floor_iff powr_realpow powr_add)
immler@62420
  2068
    then have "(2 ^ (prec)) * 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>) \<le> truncate_down prec y"
immler@54784
  2069
      by (auto simp: truncate_down_def round_down_def)
wenzelm@63356
  2070
    moreover have "x \<le> (2 ^ prec) * 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)"
wenzelm@63356
  2071
    proof -
wenzelm@60500
  2072
      have "x = 2 powr (log 2 \<bar>x\<bar>)" using \<open>0 < x\<close> by simp
immler@62420
  2073
      also have "\<dots> \<le> (2 ^ (Suc prec )) * 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>)"
immler@62420
  2074
        using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"] \<open>0 < x\<close>
immler@62420
  2075
        by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps
immler@62420
  2076
          powr_mult_base le_powr_iff)
immler@54784
  2077
      also
immler@62420
  2078
      have "2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>) \<le> 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> + 1)"
wenzelm@60500
  2079
        using logless flogless \<open>x > 0\<close> \<open>y > 0\<close>
immler@54784
  2080
        by (auto intro!: floor_mono)
wenzelm@63356
  2081
      finally show ?thesis
lp15@65583
  2082
        by (auto simp: powr_realpow[symmetric] powr_diff assms of_nat_diff)
wenzelm@63356
  2083
    qed
wenzelm@60698
  2084
    ultimately show ?thesis
immler@54784
  2085
      by (metis dual_order.trans truncate_down)
wenzelm@60698
  2086
  qed
immler@54784
  2087
qed
immler@54784
  2088
immler@58982
  2089
lemma truncate_down_eq_truncate_up: "truncate_down p x = - truncate_up p (-x)"
immler@58982
  2090
  and truncate_up_eq_truncate_down: "truncate_up p x = - truncate_down p (-x)"
immler@58982
  2091
  by (auto simp: truncate_up_uminus_eq truncate_down_uminus_eq)
immler@58982
  2092
immler@54784
  2093
lemma truncate_down_mono: "x \<le> y \<Longrightarrow> truncate_down p x \<le> truncate_down p y"
immler@54784
  2094
  apply (cases "0 \<le> x")
immler@54784
  2095
  apply (rule truncate_down_nonneg_mono, assumption+)
immler@58982
  2096
  apply (simp add: truncate_down_eq_truncate_up)
immler@54784
  2097
  apply (cases "0 \<le> y")
immler@54784
  2098
  apply (auto intro: truncate_up_nonneg_mono truncate_up_switch_sign_mono)
immler@54784
  2099
  done
immler@54784
  2100
immler@54784
  2101
lemma truncate_up_mono: "x \<le> y \<Longrightarrow> truncate_up p x \<le> truncate_up p y"
immler@58982
  2102
  by (simp add: truncate_up_eq_truncate_down truncate_down_mono)
immler@54784
  2103
hoelzl@47599
  2104
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
lp15@60017
  2105
 by (auto simp: zero_float_def mult_le_0_iff) (simp add: not_less [symmetric])
hoelzl@47599
  2106
wenzelm@60698
  2107
lemma real_of_float_pprt[simp]:
wenzelm@60698
  2108
  fixes a :: float
lp15@61609
  2109
  shows "real_of_float (pprt a) = pprt (real_of_float a)"
hoelzl@47600
  2110
  unfolding pprt_def sup_float_def max_def sup_real_def by auto
hoelzl@47599
  2111
wenzelm@60698
  2112
lemma real_of_float_nprt[simp]:
wenzelm@60698
  2113
  fixes a :: float
lp15@61609
  2114
  shows "real_of_float (nprt a) = nprt (real_of_float a)"
hoelzl@47600
  2115
  unfolding nprt_def inf_float_def min_def inf_real_def by auto
hoelzl@47599
  2116
wenzelm@60698
  2117
context
wenzelm@60698
  2118
begin
wenzelm@60698
  2119
kuncar@55565
  2120
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor .
obua@16782
  2121
wenzelm@60698
  2122
qualified lemma compute_int_floor_fl[code]:
hoelzl@47601
  2123
  "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
lp15@61609
  2124
  apply transfer
lp15@61609
  2125
  apply (simp add: powr_int floor_divide_of_int_eq)
wenzelm@61942
  2126
  apply (metis (no_types, hide_lams)floor_divide_of_int_eq of_int_numeral of_int_power floor_of_int of_int_mult)
wenzelm@61942
  2127
  done
hoelzl@47599
  2128
wenzelm@61942
  2129
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real_of_int \<lfloor>x\<rfloor>"
wenzelm@61942
  2130
  by simp
hoelzl@47599
  2131
wenzelm@60698
  2132
qualified lemma compute_floor_fl[code]:
hoelzl@47601
  2133
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
lp15@61609
  2134
  apply transfer
lp15@61609
  2135
  apply (simp add: powr_int floor_divide_of_int_eq)
wenzelm@61942
  2136
  apply (metis (no_types, hide_lams)floor_divide_of_int_eq of_int_numeral of_int_power of_int_mult)
wenzelm@61942
  2137
  done
wenzelm@60698
  2138
wenzelm@60698
  2139
end
obua@16782
  2140
lp15@61609
  2141
lemma floor_fl: "real_of_float (floor_fl x) \<le> real_of_float x"
wenzelm@60698
  2142
  by transfer simp
hoelzl@47600
  2143
lp15@61609
  2144
lemma int_floor_fl: "real_of_int (int_floor_fl x) \<le> real_of_float x"
wenzelm@60698
  2145
  by transfer simp
hoelzl@29804
  2146
hoelzl@47599
  2147
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
wenzelm@53381
  2148
proof (cases "floor_fl x = float_of 0")
wenzelm@53381
  2149
  case True
wenzelm@60698
  2150
  then show ?thesis
wenzelm@60698
  2151
    by (simp add: floor_fl_def)
wenzelm@53381
  2152
next
wenzelm@53381
  2153
  case False
lp15@61609
  2154
  have eq: "floor_fl x = Float \<lfloor>real_of_float x\<rfloor> 0"
wenzelm@60698
  2155
    by transfer simp
lp15@61609
  2156
  obtain i where "\<lfloor>real_of_float x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
wenzelm@53381
  2157
    by (rule denormalize_shift[OF eq[THEN eq_reflection] False])
wenzelm@60698
  2158
  then show ?thesis
wenzelm@60698
  2159
    by simp
wenzelm@53381
  2160
qed
obua@16782
  2161
immler@58985
  2162
lemma compute_mantissa[code]:
wenzelm@60698
  2163
  "mantissa (Float m e) =
wenzelm@60698
  2164
    (if m = 0 then 0 else if 2 dvd m then mantissa (normfloat (Float m e)) else m)"
immler@58985
  2165
  by (auto simp: mantissa_float Float.abs_eq)
immler@58985
  2166
immler@58985
  2167
lemma compute_exponent[code]:
wenzelm@60698
  2168
  "exponent (Float m e) =
wenzelm@60698
  2169
    (if m = 0 then 0 else if 2 dvd m then exponent (normfloat (Float m e)) else e)"
immler@58985
  2170
  by (auto simp: exponent_float Float.abs_eq)
immler@58985
  2171
obua@16782
  2172
end