src/HOL/Library/Float.thy
author wenzelm
Wed Jun 17 11:03:05 2015 +0200 (2015-06-17)
changeset 60500 903bb1495239
parent 60376 528a48f4ad87
child 60679 ade12ef2773c
permissions -rw-r--r--
isabelle update_cartouches;
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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 Complex_Main 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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instantiation float :: real_of
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begin
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definition real_float :: "float \<Rightarrow> real" where
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  real_of_float_def[code_unfold]: "real \<equiv> real_of_float"
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instance ..
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end
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lemma type_definition_float': "type_definition real float_of float"
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  using type_definition_float unfolding real_of_float_def .
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setup_lifting type_definition_float'
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lemmas float_of_inject[simp]
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declare [[coercion "real :: float \<Rightarrow> real"]]
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lemma real_of_float_eq:
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  fixes f1 f2 :: float shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"
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  unfolding real_of_float_def real_of_float_inject ..
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lemma float_of_real[simp]: "float_of (real x) = x"
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  unfolding real_of_float_def by (rule real_of_float_inverse)
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lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
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  unfolding real_of_float_def by (rule float_of_inverse)
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subsection \<open>Real operations preserving the representation as floating point number\<close>
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lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"
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  by (auto simp: float_def)
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lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)
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lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp
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lemma numeral_float[simp]: "numeral i \<in> float" by (intro floatI[of "numeral i" 0]) simp
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lemma neg_numeral_float[simp]: "- numeral i \<in> float" by (intro floatI[of "- numeral i" 0]) simp
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lemma real_of_int_float[simp]: "real (x :: int) \<in> float" by (intro floatI[of x 0]) simp
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lemma real_of_nat_float[simp]: "real (x :: nat) \<in> float" by (intro floatI[of x 0]) simp
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lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float" by (intro floatI[of 1 i]) simp
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lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) \<in> float" by (intro floatI[of 1 i]) simp
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lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float" by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float" by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" 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" by (intro floatI[of 1 "- numeral i"]) simp
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lemma two_pow_float[simp]: "2 ^ n \<in> float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
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lemma real_of_float_float[simp]: "real (f::float) \<in> float" by (cases f) simp
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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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  fix e1 m1 e2 m2 :: int
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  { fix e1 m1 e2 m2 :: int assume "e1 \<le> e2"
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    then 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_divide2[symmetric] field_simps)
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    then have "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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      by blast }
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  note * = this
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  show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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  proof (cases e1 e2 rule: linorder_le_cases)
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    assume "e2 \<le> e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)
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  qed (rule *)
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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> abs x \<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: "x / numeral n \<in> float" 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 x 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: "x / numeral n \<in> float" 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" using x 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]: assumes "a \<in> float" 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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  thus ?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" by simp
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declare Float.rep_eq[simp]
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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: real_of_float_def[symmetric] powr_int)
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code_datatype Float
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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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  proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+
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end
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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]: fixes f::float shows "real (f^n) = real f^n"
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  by (induct n) simp_all
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lemma fixes x y::float
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  shows real_of_float_min: "real (min x y) = min (real x) (real y)"
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    and real_of_float_max: "real (max x y) = max (real x) (real y)"
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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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  assume "a < b"
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  then show "\<exists>c. a < c \<and> c < b"
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    apply (intro exI[of _ "(a + b) * Float 1 (- 1)"])
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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 default
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     (transfer, simp_all 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 (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 real_float
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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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  unfolding rel_fun_def float.pcr_cr_eq  cr_float_def by simp
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lemma float_neg_numeral[simp]: "real (- 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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  unfolding rel_fun_def float.pcr_cr_eq cr_float_def by simp
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lemma
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  shows 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 (%x. Quickcheck_Exhaustive.exhaustive (%y. f (Float x y)) d) d"
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instance ..
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end
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definition (in term_syntax) [code_unfold]:
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  "valtermify_float x y = Code_Evaluation.valtermify Float {\<cdot>} x {\<cdot>} y"
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instantiation float :: full_exhaustive
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begin
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definition full_exhaustive_float where
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  "full_exhaustive_float f d =
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    Quickcheck_Exhaustive.full_exhaustive
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      (\<lambda>x. Quickcheck_Exhaustive.full_exhaustive (\<lambda>y. f (valtermify_float x y)) d) d"
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instance ..
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end
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instantiation float :: random
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begin
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definition "Quickcheck_Random.random i =
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  scomp (Quickcheck_Random.random (2 ^ nat_of_natural i))
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    (\<lambda>man. scomp (Quickcheck_Random.random i) (\<lambda>exp. Pair (valtermify_float man exp)))"
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instance ..
immler@58987
   303
immler@58987
   304
end
immler@58987
   305
immler@58987
   306
wenzelm@60500
   307
subsection \<open>Represent floats as unique mantissa and exponent\<close>
huffman@47108
   308
hoelzl@47599
   309
lemma int_induct_abs[case_names less]:
hoelzl@47599
   310
  fixes j :: int
hoelzl@47599
   311
  assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
hoelzl@47599
   312
  shows "P j"
hoelzl@47599
   313
proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)
hoelzl@47599
   314
  case less show ?case by (rule H[OF less]) simp
hoelzl@47599
   315
qed
hoelzl@47599
   316
hoelzl@47599
   317
lemma int_cancel_factors:
hoelzl@47599
   318
  fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
hoelzl@47599
   319
proof (induct n rule: int_induct_abs)
hoelzl@47599
   320
  case (less n)
hoelzl@47599
   321
  { fix m assume n: "n \<noteq> 0" "n = m * r"
hoelzl@47599
   322
    then have "\<bar>m \<bar> < \<bar>n\<bar>"
wenzelm@60500
   323
      using \<open>1 < r\<close> by (simp add: abs_mult)
hoelzl@47599
   324
    from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
hoelzl@47599
   325
  then show ?case
haftmann@59554
   326
    by (metis dvd_def monoid_mult_class.mult.right_neutral mult.commute power_0)
hoelzl@47599
   327
qed
hoelzl@47599
   328
hoelzl@47599
   329
lemma mult_powr_eq_mult_powr_iff_asym:
hoelzl@47599
   330
  fixes m1 m2 e1 e2 :: int
hoelzl@47599
   331
  assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"
hoelzl@47599
   332
  shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
hoelzl@47599
   333
proof
hoelzl@47599
   334
  have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
hoelzl@47599
   335
  assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
wenzelm@60500
   336
  with \<open>e1 \<le> e2\<close> have "m1 = m2 * 2 powr nat (e2 - e1)"
hoelzl@47599
   337
    by (simp add: powr_divide2[symmetric] field_simps)
hoelzl@47599
   338
  also have "\<dots> = m2 * 2^nat (e2 - e1)"
hoelzl@47599
   339
    by (simp add: powr_realpow)
hoelzl@47599
   340
  finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
hoelzl@47599
   341
    unfolding real_of_int_inject .
hoelzl@47599
   342
  with m1 have "m1 = m2"
hoelzl@47599
   343
    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
hoelzl@47599
   344
  then show "m1 = m2 \<and> e1 = e2"
wenzelm@60500
   345
    using eq \<open>m1 \<noteq> 0\<close> by (simp add: powr_inj)
hoelzl@47599
   346
qed simp
hoelzl@47599
   347
hoelzl@47599
   348
lemma mult_powr_eq_mult_powr_iff:
hoelzl@47599
   349
  fixes m1 m2 e1 e2 :: int
hoelzl@47599
   350
  shows "\<not> 2 dvd m1 \<Longrightarrow> \<not> 2 dvd m2 \<Longrightarrow> m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
hoelzl@47599
   351
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
hoelzl@47599
   352
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
hoelzl@47599
   353
  by (cases e1 e2 rule: linorder_le_cases) auto
hoelzl@47599
   354
hoelzl@47599
   355
lemma floatE_normed:
hoelzl@47599
   356
  assumes x: "x \<in> float"
hoelzl@47599
   357
  obtains (zero) "x = 0"
hoelzl@47599
   358
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
hoelzl@47599
   359
proof atomize_elim
hoelzl@47599
   360
  { assume "x \<noteq> 0"
hoelzl@47599
   361
    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
wenzelm@60500
   362
    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
   363
      by auto
wenzelm@60500
   364
    with \<open>\<not> 2 dvd k\<close> x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
hoelzl@47599
   365
      by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
hoelzl@47599
   366
         (simp add: powr_add powr_realpow) }
hoelzl@47599
   367
  then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)"
hoelzl@47599
   368
    by blast
hoelzl@47599
   369
qed
hoelzl@47599
   370
hoelzl@47599
   371
lemma float_normed_cases:
hoelzl@47599
   372
  fixes f :: float
hoelzl@47599
   373
  obtains (zero) "f = 0"
hoelzl@47599
   374
   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
hoelzl@47599
   375
proof (atomize_elim, induct f)
hoelzl@47599
   376
  case (float_of y) then show ?case
hoelzl@47600
   377
    by (cases rule: floatE_normed) (auto simp: zero_float_def)
hoelzl@47599
   378
qed
hoelzl@47599
   379
hoelzl@47599
   380
definition mantissa :: "float \<Rightarrow> int" where
hoelzl@47599
   381
  "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   382
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   383
hoelzl@47599
   384
definition exponent :: "float \<Rightarrow> int" where
hoelzl@47599
   385
  "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   386
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   387
wenzelm@53381
   388
lemma
hoelzl@47599
   389
  shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
hoelzl@47599
   390
    and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
hoelzl@47599
   391
proof -
hoelzl@47599
   392
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
hoelzl@47599
   393
  then show ?E ?M
hoelzl@47600
   394
    by (auto simp add: mantissa_def exponent_def zero_float_def)
hoelzl@29804
   395
qed
hoelzl@29804
   396
hoelzl@47599
   397
lemma
hoelzl@47599
   398
  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
hoelzl@47599
   399
    and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
hoelzl@47599
   400
proof cases
hoelzl@47599
   401
  assume [simp]: "f \<noteq> (float_of 0)"
hoelzl@47599
   402
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
hoelzl@47599
   403
  proof (cases f rule: float_normed_cases)
hoelzl@47599
   404
    case (powr m e)
hoelzl@47599
   405
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   406
     \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
hoelzl@47599
   407
      by auto
hoelzl@47599
   408
    then show ?thesis
hoelzl@47599
   409
      unfolding exponent_def mantissa_def
hoelzl@47600
   410
      by (rule someI2_ex) (simp add: zero_float_def)
hoelzl@47600
   411
  qed (simp add: zero_float_def)
hoelzl@47599
   412
  then show ?E ?D by auto
hoelzl@47599
   413
qed simp
hoelzl@47599
   414
hoelzl@47599
   415
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
hoelzl@47599
   416
  using mantissa_not_dvd[of f] by auto
hoelzl@47599
   417
wenzelm@53381
   418
lemma
hoelzl@47599
   419
  fixes m e :: int
hoelzl@47599
   420
  defines "f \<equiv> float_of (m * 2 powr e)"
hoelzl@47599
   421
  assumes dvd: "\<not> 2 dvd m"
hoelzl@47599
   422
  shows mantissa_float: "mantissa f = m" (is "?M")
hoelzl@47599
   423
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
hoelzl@47599
   424
proof cases
hoelzl@47599
   425
  assume "m = 0" with dvd show "mantissa f = m" by auto
hoelzl@47599
   426
next
hoelzl@47599
   427
  assume "m \<noteq> 0"
hoelzl@47599
   428
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
hoelzl@47599
   429
  from mantissa_exponent[of f]
hoelzl@47599
   430
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   431
    by (auto simp add: f_def)
hoelzl@47599
   432
  then show "?M" "?E"
hoelzl@47599
   433
    using mantissa_not_dvd[OF f_not_0] dvd
hoelzl@47599
   434
    by (auto simp: mult_powr_eq_mult_powr_iff)
hoelzl@47599
   435
qed
hoelzl@47599
   436
wenzelm@60500
   437
subsection \<open>Compute arithmetic operations\<close>
hoelzl@47600
   438
hoelzl@47600
   439
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
hoelzl@47600
   440
  unfolding real_of_float_eq mantissa_exponent[of f] by simp
hoelzl@47600
   441
hoelzl@47600
   442
lemma Float_cases[case_names Float, cases type: float]:
hoelzl@47600
   443
  fixes f :: float
hoelzl@47600
   444
  obtains (Float) m e :: int where "f = Float m e"
hoelzl@47600
   445
  using Float_mantissa_exponent[symmetric]
hoelzl@47600
   446
  by (atomize_elim) auto
hoelzl@47600
   447
hoelzl@47599
   448
lemma denormalize_shift:
hoelzl@47599
   449
  assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
hoelzl@47599
   450
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
hoelzl@47599
   451
proof
hoelzl@47599
   452
  from mantissa_exponent[of f] f_def
hoelzl@47599
   453
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   454
    by simp
hoelzl@47599
   455
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
hoelzl@47599
   456
    by (simp add: powr_divide2[symmetric] field_simps)
hoelzl@47599
   457
  moreover
hoelzl@47599
   458
  have "e \<le> exponent f"
hoelzl@47599
   459
  proof (rule ccontr)
hoelzl@47599
   460
    assume "\<not> e \<le> exponent f"
hoelzl@47599
   461
    then have pos: "exponent f < e" by simp
hoelzl@47599
   462
    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
hoelzl@47599
   463
      by simp
hoelzl@47599
   464
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
hoelzl@47599
   465
      using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47599
   466
    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
hoelzl@47599
   467
      using eq by simp
hoelzl@47599
   468
    then have "mantissa f = m * 2^nat (e - exponent f)"
hoelzl@47599
   469
      unfolding real_of_int_inject by simp
wenzelm@60500
   470
    with \<open>exponent f < e\<close> have "2 dvd mantissa f"
hoelzl@47599
   471
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
hoelzl@47599
   472
      apply (cases "nat (e - exponent f)")
hoelzl@47599
   473
      apply auto
hoelzl@47599
   474
      done
hoelzl@47599
   475
    then show False using mantissa_not_dvd[OF not_0] by simp
hoelzl@47599
   476
  qed
hoelzl@47599
   477
  ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
hoelzl@47599
   478
    by (simp add: powr_realpow[symmetric])
wenzelm@60500
   479
  with \<open>e \<le> exponent f\<close>
hoelzl@47599
   480
  show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
hoelzl@47599
   481
    unfolding real_of_int_inject by auto
hoelzl@29804
   482
qed
hoelzl@29804
   483
hoelzl@47621
   484
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
hoelzl@47600
   485
  by transfer simp
hoelzl@47621
   486
hide_fact (open) compute_float_zero
hoelzl@47600
   487
hoelzl@47621
   488
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
hoelzl@47600
   489
  by transfer simp
hoelzl@47621
   490
hide_fact (open) compute_float_one
hoelzl@47600
   491
immler@58982
   492
lift_definition normfloat :: "float \<Rightarrow> float" is "\<lambda>x. x" .
immler@58982
   493
lemma normloat_id[simp]: "normfloat x = x" by transfer rule
hoelzl@47600
   494
hoelzl@47600
   495
lemma compute_normfloat[code]: "normfloat (Float m e) =
hoelzl@47600
   496
  (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
hoelzl@47600
   497
                           else if m = 0 then 0 else Float m e)"
hoelzl@47600
   498
  by transfer (auto simp add: powr_add zmod_eq_0_iff)
hoelzl@47621
   499
hide_fact (open) compute_normfloat
hoelzl@47599
   500
hoelzl@47599
   501
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
hoelzl@47600
   502
  by transfer simp
hoelzl@47621
   503
hide_fact (open) compute_float_numeral
hoelzl@47599
   504
haftmann@54489
   505
lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
hoelzl@47600
   506
  by transfer simp
hoelzl@47621
   507
hide_fact (open) compute_float_neg_numeral
hoelzl@47599
   508
hoelzl@47599
   509
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
hoelzl@47600
   510
  by transfer simp
hoelzl@47621
   511
hide_fact (open) compute_float_uminus
hoelzl@47599
   512
hoelzl@47599
   513
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
hoelzl@47600
   514
  by transfer (simp add: field_simps powr_add)
hoelzl@47621
   515
hide_fact (open) compute_float_times
hoelzl@47599
   516
hoelzl@47599
   517
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
immler@54783
   518
  (if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else
immler@54783
   519
  if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
hoelzl@47599
   520
              else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
hoelzl@47600
   521
  by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47621
   522
hide_fact (open) compute_float_plus
hoelzl@47599
   523
hoelzl@47600
   524
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
hoelzl@47600
   525
  by simp
hoelzl@47621
   526
hide_fact (open) compute_float_minus
hoelzl@47599
   527
hoelzl@47599
   528
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
hoelzl@47600
   529
  by transfer (simp add: sgn_times)
hoelzl@47621
   530
hide_fact (open) compute_float_sgn
hoelzl@47599
   531
kuncar@55565
   532
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   533
hoelzl@47599
   534
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
hoelzl@47600
   535
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
hoelzl@47621
   536
hide_fact (open) compute_is_float_pos
hoelzl@47599
   537
hoelzl@47599
   538
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
hoelzl@47600
   539
  by transfer (simp add: field_simps)
hoelzl@47621
   540
hide_fact (open) compute_float_less
hoelzl@47599
   541
kuncar@55565
   542
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   543
hoelzl@47599
   544
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
hoelzl@47600
   545
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
hoelzl@47621
   546
hide_fact (open) compute_is_float_nonneg
hoelzl@47599
   547
hoelzl@47599
   548
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
hoelzl@47600
   549
  by transfer (simp add: field_simps)
hoelzl@47621
   550
hide_fact (open) compute_float_le
hoelzl@47599
   551
kuncar@55565
   552
lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   553
hoelzl@47599
   554
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
hoelzl@47600
   555
  by transfer (auto simp add: is_float_zero_def)
hoelzl@47621
   556
hide_fact (open) compute_is_float_zero
hoelzl@47599
   557
hoelzl@47600
   558
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
hoelzl@47600
   559
  by transfer (simp add: abs_mult)
hoelzl@47621
   560
hide_fact (open) compute_float_abs
hoelzl@47599
   561
hoelzl@47600
   562
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
hoelzl@47600
   563
  by transfer simp
hoelzl@47621
   564
hide_fact (open) compute_float_eq
hoelzl@47599
   565
immler@58982
   566
wenzelm@60500
   567
subsection \<open>Lemmas for types @{typ real}, @{typ nat}, @{typ int}\<close>
immler@58982
   568
immler@58982
   569
lemmas real_of_ints =
immler@58982
   570
  real_of_int_zero
immler@58982
   571
  real_of_one
immler@58982
   572
  real_of_int_add
immler@58982
   573
  real_of_int_minus
immler@58982
   574
  real_of_int_diff
immler@58982
   575
  real_of_int_mult
immler@58982
   576
  real_of_int_power
immler@58982
   577
  real_numeral
immler@58982
   578
lemmas real_of_nats =
immler@58982
   579
  real_of_nat_zero
immler@58982
   580
  real_of_nat_one
immler@58982
   581
  real_of_nat_1
immler@58982
   582
  real_of_nat_add
immler@58982
   583
  real_of_nat_mult
immler@58982
   584
  real_of_nat_power
immler@58989
   585
  real_of_nat_numeral
immler@58982
   586
immler@58982
   587
lemmas int_of_reals = real_of_ints[symmetric]
immler@58982
   588
lemmas nat_of_reals = real_of_nats[symmetric]
immler@58982
   589
immler@58982
   590
wenzelm@60500
   591
subsection \<open>Rounding Real Numbers\<close>
hoelzl@47599
   592
hoelzl@47599
   593
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   594
  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   595
hoelzl@47599
   596
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   597
  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   598
hoelzl@47599
   599
lemma round_down_float[simp]: "round_down prec x \<in> float"
hoelzl@47599
   600
  unfolding round_down_def
hoelzl@47599
   601
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   602
hoelzl@47599
   603
lemma round_up_float[simp]: "round_up prec x \<in> float"
hoelzl@47599
   604
  unfolding round_up_def
hoelzl@47599
   605
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   606
hoelzl@47599
   607
lemma round_up: "x \<le> round_up prec x"
hoelzl@47599
   608
  by (simp add: powr_minus_divide le_divide_eq round_up_def)
hoelzl@47599
   609
hoelzl@47599
   610
lemma round_down: "round_down prec x \<le> x"
hoelzl@47599
   611
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
hoelzl@47599
   612
hoelzl@47599
   613
lemma round_up_0[simp]: "round_up p 0 = 0"
hoelzl@47599
   614
  unfolding round_up_def by simp
hoelzl@47599
   615
hoelzl@47599
   616
lemma round_down_0[simp]: "round_down p 0 = 0"
hoelzl@47599
   617
  unfolding round_down_def by simp
hoelzl@47599
   618
hoelzl@47599
   619
lemma round_up_diff_round_down:
hoelzl@47599
   620
  "round_up prec x - round_down prec x \<le> 2 powr -prec"
hoelzl@47599
   621
proof -
hoelzl@47599
   622
  have "round_up prec x - round_down prec x =
hoelzl@47599
   623
    (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
hoelzl@47599
   624
    by (simp add: round_up_def round_down_def field_simps)
hoelzl@47599
   625
  also have "\<dots> \<le> 1 * 2 powr -prec"
hoelzl@47599
   626
    by (rule mult_mono)
hoelzl@47599
   627
       (auto simp del: real_of_int_diff
hoelzl@47599
   628
             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
hoelzl@47599
   629
  finally show ?thesis by simp
hoelzl@29804
   630
qed
hoelzl@29804
   631
hoelzl@47599
   632
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
hoelzl@47599
   633
  unfolding round_down_def
hoelzl@47599
   634
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   635
    (simp add: powr_add[symmetric])
hoelzl@29804
   636
hoelzl@47599
   637
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
hoelzl@47599
   638
  unfolding round_up_def
hoelzl@47599
   639
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   640
    (simp add: powr_add[symmetric])
hoelzl@47599
   641
immler@58982
   642
lemma round_up_uminus_eq: "round_up p (-x) = - round_down p x"
immler@58982
   643
  and round_down_uminus_eq: "round_down p (-x) = - round_up p x"
immler@58982
   644
  by (auto simp: round_up_def round_down_def ceiling_def)
immler@58982
   645
immler@58982
   646
lemma round_up_mono: "x \<le> y \<Longrightarrow> round_up p x \<le> round_up p y"
immler@58982
   647
  by (auto intro!: ceiling_mono simp: round_up_def)
immler@58982
   648
immler@58982
   649
lemma round_up_le1:
immler@58982
   650
  assumes "x \<le> 1" "prec \<ge> 0"
immler@58982
   651
  shows "round_up prec x \<le> 1"
immler@58982
   652
proof -
immler@58982
   653
  have "real \<lceil>x * 2 powr prec\<rceil> \<le> real \<lceil>2 powr real prec\<rceil>"
immler@58982
   654
    using assms by (auto intro!: ceiling_mono)
immler@58982
   655
  also have "\<dots> = 2 powr prec" using assms by (auto simp: powr_int intro!: exI[where x="2^nat prec"])
immler@58982
   656
  finally show ?thesis
immler@58982
   657
    by (simp add: round_up_def) (simp add: powr_minus inverse_eq_divide)
immler@58982
   658
qed
immler@58982
   659
immler@58982
   660
lemma round_up_less1:
immler@58982
   661
  assumes "x < 1 / 2" "p > 0"
immler@58982
   662
  shows "round_up p x < 1"
immler@58982
   663
proof -
immler@58982
   664
  have "x * 2 powr p < 1 / 2 * 2 powr p"
immler@58982
   665
    using assms by simp
wenzelm@60500
   666
  also have "\<dots> \<le> 2 powr p - 1" using \<open>p > 0\<close>
immler@58989
   667
    by (auto simp: powr_divide2[symmetric] powr_int field_simps self_le_power)
wenzelm@60500
   668
  finally show ?thesis using \<open>p > 0\<close>
immler@58989
   669
    by (simp add: round_up_def field_simps powr_minus powr_int ceiling_less_eq)
immler@58982
   670
qed
immler@58982
   671
immler@58982
   672
lemma round_down_ge1:
immler@58982
   673
  assumes x: "x \<ge> 1"
immler@58982
   674
  assumes prec: "p \<ge> - log 2 x"
immler@58982
   675
  shows "1 \<le> round_down p x"
immler@58982
   676
proof cases
immler@58982
   677
  assume nonneg: "0 \<le> p"
immler@58985
   678
  have "2 powr p = real \<lfloor>2 powr real p\<rfloor>"
immler@58985
   679
    using nonneg by (auto simp: powr_int)
immler@58985
   680
  also have "\<dots> \<le> real \<lfloor>x * 2 powr p\<rfloor>"
immler@58985
   681
    using assms by (auto intro!: floor_mono)
immler@58985
   682
  finally show ?thesis
immler@58985
   683
    by (simp add: round_down_def) (simp add: powr_minus inverse_eq_divide)
immler@58982
   684
next
immler@58982
   685
  assume neg: "\<not> 0 \<le> p"
immler@58982
   686
  have "x = 2 powr (log 2 x)"
immler@58982
   687
    using x by simp
immler@58982
   688
  also have "2 powr (log 2 x) \<ge> 2 powr - p"
immler@58982
   689
    using prec by auto
immler@58982
   690
  finally have x_le: "x \<ge> 2 powr -p" .
immler@58982
   691
immler@58982
   692
  from neg have "2 powr real p \<le> 2 powr 0"
immler@58982
   693
    by (intro powr_mono) auto
lp15@60017
   694
  also have "\<dots> \<le> \<lfloor>2 powr 0::real\<rfloor>" by simp
lp15@60017
   695
  also have "... \<le> \<lfloor>x * 2 powr (real p)\<rfloor>" 
lp15@60017
   696
    unfolding real_of_int_le_iff
immler@58982
   697
    using x x_le by (intro floor_mono) (simp add: powr_minus_divide field_simps)
immler@58982
   698
  finally show ?thesis
immler@58982
   699
    using prec x
immler@58982
   700
    by (simp add: round_down_def powr_minus_divide pos_le_divide_eq)
immler@58982
   701
qed
immler@58982
   702
immler@58982
   703
lemma round_up_le0: "x \<le> 0 \<Longrightarrow> round_up p x \<le> 0"
immler@58982
   704
  unfolding round_up_def
immler@58982
   705
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
immler@58982
   706
immler@58982
   707
wenzelm@60500
   708
subsection \<open>Rounding Floats\<close>
hoelzl@29804
   709
immler@58985
   710
definition div_twopow::"int \<Rightarrow> nat \<Rightarrow> int" where [simp]: "div_twopow x n = x div (2 ^ n)"
immler@58985
   711
immler@58985
   712
definition mod_twopow::"int \<Rightarrow> nat \<Rightarrow> int" where [simp]: "mod_twopow x n = x mod (2 ^ n)"
immler@58985
   713
immler@58985
   714
lemma compute_div_twopow[code]:
immler@58985
   715
  "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
   716
  by (cases n) (auto simp: zdiv_zmult2_eq div_eq_minus1)
immler@58985
   717
immler@58985
   718
lemma compute_mod_twopow[code]:
immler@58985
   719
  "mod_twopow x n = (if n = 0 then 0 else x mod 2 + 2 * mod_twopow (x div 2) (n - 1))"
immler@58985
   720
  by (cases n) (auto simp: zmod_zmult2_eq)
immler@58985
   721
hoelzl@47600
   722
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
hoelzl@47601
   723
declare float_up.rep_eq[simp]
hoelzl@29804
   724
immler@54782
   725
lemma round_up_correct:
immler@54782
   726
  shows "round_up e f - f \<in> {0..2 powr -e}"
hoelzl@47599
   727
unfolding atLeastAtMost_iff
hoelzl@47599
   728
proof
hoelzl@47599
   729
  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
hoelzl@47599
   730
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
lp15@60017
   731
  finally show "round_up e f - f \<le> 2 powr - (real e)"
hoelzl@47600
   732
    by simp
hoelzl@47600
   733
qed (simp add: algebra_simps round_up)
hoelzl@29804
   734
immler@54782
   735
lemma float_up_correct:
immler@54782
   736
  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
immler@54782
   737
  by transfer (rule round_up_correct)
immler@54782
   738
hoelzl@47600
   739
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
hoelzl@47601
   740
declare float_down.rep_eq[simp]
obua@16782
   741
immler@54782
   742
lemma round_down_correct:
immler@54782
   743
  shows "f - (round_down e f) \<in> {0..2 powr -e}"
hoelzl@47599
   744
unfolding atLeastAtMost_iff
hoelzl@47599
   745
proof
hoelzl@47599
   746
  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
hoelzl@47599
   747
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
lp15@60017
   748
  finally show "f - round_down e f \<le> 2 powr - (real e)"
hoelzl@47600
   749
    by simp
hoelzl@47600
   750
qed (simp add: algebra_simps round_down)
obua@24301
   751
immler@54782
   752
lemma float_down_correct:
immler@54782
   753
  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
immler@54782
   754
  by transfer (rule round_down_correct)
immler@54782
   755
hoelzl@47599
   756
lemma compute_float_down[code]:
hoelzl@47599
   757
  "float_down p (Float m e) =
immler@58985
   758
    (if p + e < 0 then Float (div_twopow m (nat (-(p + e)))) (-p) else Float m e)"
hoelzl@47599
   759
proof cases
hoelzl@47599
   760
  assume "p + e < 0"
hoelzl@47599
   761
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   762
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   763
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47600
   764
    unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   765
  finally show ?thesis
wenzelm@60500
   766
    using \<open>p + e < 0\<close>
hoelzl@47600
   767
    by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
hoelzl@47599
   768
next
hoelzl@47600
   769
  assume "\<not> p + e < 0"
hoelzl@47600
   770
  then have r: "real e + real p = real (nat (e + p))" by simp
hoelzl@47600
   771
  have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
hoelzl@47600
   772
    by (auto intro: exI[where x="m*2^nat (e+p)"]
hoelzl@47600
   773
             simp add: ac_simps powr_add[symmetric] r powr_realpow)
wenzelm@60500
   774
  with \<open>\<not> p + e < 0\<close> show ?thesis
wenzelm@57862
   775
    by transfer (auto simp add: round_down_def field_simps powr_add powr_minus)
hoelzl@47599
   776
qed
hoelzl@47621
   777
hide_fact (open) compute_float_down
obua@24301
   778
immler@54782
   779
lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"
immler@54782
   780
  using round_down_correct[of f e] by simp
immler@54782
   781
immler@54782
   782
lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e"
immler@54782
   783
  using round_up_correct[of e f] by simp
immler@54782
   784
immler@54782
   785
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s"
nipkow@56536
   786
  by (auto simp: round_down_def)
immler@54782
   787
hoelzl@47599
   788
lemma ceil_divide_floor_conv:
hoelzl@47599
   789
assumes "b \<noteq> 0"
hoelzl@47599
   790
shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)"
hoelzl@47599
   791
proof cases
hoelzl@47599
   792
  assume "\<not> b dvd a"
hoelzl@47599
   793
  hence "a mod b \<noteq> 0" by auto
wenzelm@60500
   794
  hence ne: "real (a mod b) / real b \<noteq> 0" using \<open>b \<noteq> 0\<close> by auto
hoelzl@47599
   795
  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
hoelzl@47599
   796
  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
hoelzl@47599
   797
  proof -
hoelzl@47599
   798
    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
hoelzl@47599
   799
    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
wenzelm@60500
   800
    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne \<open>b \<noteq> 0\<close> by auto
hoelzl@47599
   801
    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
hoelzl@47599
   802
  qed
wenzelm@60500
   803
  thus ?thesis using \<open>\<not> b dvd a\<close> by simp
hoelzl@47599
   804
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
hoelzl@56479
   805
  floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
wenzelm@19765
   806
hoelzl@47599
   807
lemma compute_float_up[code]:
immler@58982
   808
  "float_up p x = - float_down p (-x)"
immler@58982
   809
  by transfer (simp add: round_down_uminus_eq)
hoelzl@47621
   810
hide_fact (open) compute_float_up
hoelzl@29804
   811
hoelzl@47599
   812
wenzelm@60500
   813
subsection \<open>Compute bitlen of integers\<close>
hoelzl@47599
   814
hoelzl@47600
   815
definition bitlen :: "int \<Rightarrow> int" where
hoelzl@47600
   816
  "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
hoelzl@47599
   817
hoelzl@47599
   818
lemma bitlen_nonneg: "0 \<le> bitlen x"
hoelzl@29804
   819
proof -
hoelzl@47599
   820
  {
hoelzl@47599
   821
    assume "0 > x"
hoelzl@47599
   822
    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
wenzelm@60500
   823
    also have "... < log 2 (-x)" using \<open>0 > x\<close> by auto
hoelzl@47599
   824
    finally have "-1 < log 2 (-x)" .
hoelzl@47599
   825
  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
hoelzl@47599
   826
qed
hoelzl@47599
   827
hoelzl@47599
   828
lemma bitlen_bounds:
hoelzl@47599
   829
  assumes "x > 0"
hoelzl@47599
   830
  shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
hoelzl@47599
   831
proof
hoelzl@47599
   832
  have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
wenzelm@60500
   833
    using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] \<open>x > 0\<close>
hoelzl@47599
   834
    using real_nat_eq_real[of "floor (log 2 (real x))"]
hoelzl@47599
   835
    by simp
hoelzl@47599
   836
  also have "... \<le> 2 powr log 2 (real x)"
hoelzl@47599
   837
    by simp
hoelzl@47599
   838
  also have "... = real x"
wenzelm@60500
   839
    using \<open>0 < x\<close> by simp
hoelzl@47599
   840
  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
wenzelm@60500
   841
  thus "2 ^ nat (bitlen x - 1) \<le> x" using \<open>x > 0\<close>
hoelzl@47599
   842
    by (simp add: bitlen_def)
hoelzl@47599
   843
next
wenzelm@60500
   844
  have "x \<le> 2 powr (log 2 x)" using \<open>x > 0\<close> by simp
hoelzl@47599
   845
  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
hoelzl@47599
   846
    apply (simp add: powr_realpow[symmetric])
wenzelm@60500
   847
    using \<open>x > 0\<close> by simp
wenzelm@60500
   848
  finally show "x < 2 ^ nat (bitlen x)" using \<open>x > 0\<close>
immler@58989
   849
    by (simp add: bitlen_def ac_simps)
hoelzl@47599
   850
qed
hoelzl@47599
   851
hoelzl@47599
   852
lemma bitlen_pow2[simp]:
hoelzl@47599
   853
  assumes "b > 0"
hoelzl@47599
   854
  shows "bitlen (b * 2 ^ c) = bitlen b + c"
hoelzl@47599
   855
proof -
nipkow@56544
   856
  from assms have "b * 2 ^ c > 0" by auto
hoelzl@47599
   857
  thus ?thesis
hoelzl@47599
   858
    using floor_add[of "log 2 b" c] assms
hoelzl@47599
   859
    by (auto simp add: log_mult log_nat_power bitlen_def)
hoelzl@29804
   860
qed
hoelzl@29804
   861
hoelzl@47599
   862
lemma bitlen_Float:
wenzelm@53381
   863
  fixes m e
wenzelm@53381
   864
  defines "f \<equiv> Float m e"
wenzelm@53381
   865
  shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
wenzelm@53381
   866
proof (cases "m = 0")
wenzelm@53381
   867
  case True
wenzelm@53381
   868
  then show ?thesis by (simp add: f_def bitlen_def Float_def)
wenzelm@53381
   869
next
wenzelm@53381
   870
  case False
hoelzl@47600
   871
  hence "f \<noteq> float_of 0"
hoelzl@47600
   872
    unfolding real_of_float_eq by (simp add: f_def)
hoelzl@47600
   873
  hence "mantissa f \<noteq> 0"
hoelzl@47599
   874
    by (simp add: mantissa_noteq_0)
hoelzl@47599
   875
  moreover
wenzelm@53381
   876
  obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
wenzelm@60500
   877
    by (rule f_def[THEN denormalize_shift, OF \<open>f \<noteq> float_of 0\<close>])
hoelzl@47599
   878
  ultimately show ?thesis by (simp add: abs_mult)
wenzelm@53381
   879
qed
hoelzl@29804
   880
hoelzl@47599
   881
lemma compute_bitlen[code]:
hoelzl@47599
   882
  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
hoelzl@47599
   883
proof -
hoelzl@47599
   884
  { assume "2 \<le> x"
hoelzl@47599
   885
    then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
hoelzl@47599
   886
      by (simp add: log_mult zmod_zdiv_equality')
hoelzl@47599
   887
    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   888
    proof cases
hoelzl@47599
   889
      assume "x mod 2 = 0" then show ?thesis by simp
hoelzl@47599
   890
    next
hoelzl@47599
   891
      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   892
      then have "0 \<le> n"
wenzelm@60500
   893
        using \<open>2 \<le> x\<close> by simp
hoelzl@47599
   894
      assume "x mod 2 \<noteq> 0"
wenzelm@60500
   895
      with \<open>2 \<le> x\<close> have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
wenzelm@60500
   896
      with \<open>2 \<le> x\<close> have "x \<noteq> 2^nat n" by (cases "nat n") auto
hoelzl@47599
   897
      moreover
hoelzl@47599
   898
      { have "real (2^nat n :: int) = 2 powr (nat n)"
hoelzl@47599
   899
          by (simp add: powr_realpow)
hoelzl@47599
   900
        also have "\<dots> \<le> 2 powr (log 2 x)"
wenzelm@60500
   901
          using \<open>2 \<le> x\<close> by (simp add: n_def del: powr_log_cancel)
wenzelm@60500
   902
        finally have "2^nat n \<le> x" using \<open>2 \<le> x\<close> by simp }
hoelzl@47599
   903
      ultimately have "2^nat n \<le> x - 1" by simp
hoelzl@47599
   904
      then have "2^nat n \<le> real (x - 1)"
hoelzl@47599
   905
        unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47599
   906
      { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
wenzelm@60500
   907
          using \<open>0 \<le> n\<close> by (simp add: log_nat_power)
hoelzl@47599
   908
        also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
wenzelm@60500
   909
          using \<open>2^nat n \<le> real (x - 1)\<close> \<open>0 \<le> n\<close> \<open>2 \<le> x\<close> by (auto intro: floor_mono)
hoelzl@47599
   910
        finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
hoelzl@47599
   911
      moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
wenzelm@60500
   912
        using \<open>2 \<le> x\<close> by (auto simp add: n_def intro!: floor_mono)
hoelzl@47599
   913
      ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
wenzelm@60500
   914
        unfolding n_def \<open>x mod 2 = 1\<close> by auto
hoelzl@47599
   915
    qed
hoelzl@47599
   916
    finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
hoelzl@47599
   917
  moreover
hoelzl@47599
   918
  { assume "x < 2" "0 < x"
hoelzl@47599
   919
    then have "x = 1" by simp
hoelzl@47599
   920
    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
hoelzl@47599
   921
  ultimately show ?thesis
hoelzl@47599
   922
    unfolding bitlen_def
hoelzl@47599
   923
    by (auto simp: pos_imp_zdiv_pos_iff not_le)
hoelzl@47599
   924
qed
hoelzl@47621
   925
hide_fact (open) compute_bitlen
hoelzl@29804
   926
hoelzl@47599
   927
lemma float_gt1_scale: assumes "1 \<le> Float m e"
hoelzl@47599
   928
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@47599
   929
proof -
hoelzl@47599
   930
  have "0 < Float m e" using assms by auto
lp15@60017
   931
  hence "0 < m" using powr_gt_zero[of 2 e]  
lp15@60017
   932
    apply (auto simp: zero_less_mult_iff)
lp15@60017
   933
    using not_le powr_ge_pzero by blast
hoelzl@47599
   934
  hence "m \<noteq> 0" by auto
hoelzl@47599
   935
  show ?thesis
hoelzl@47599
   936
  proof (cases "0 \<le> e")
wenzelm@60500
   937
    case True thus ?thesis using \<open>0 < m\<close>  by (simp add: bitlen_def)
hoelzl@29804
   938
  next
hoelzl@47599
   939
    have "(1::int) < 2" by simp
hoelzl@47599
   940
    case False let ?S = "2^(nat (-e))"
hoelzl@47599
   941
    have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
wenzelm@57862
   942
      by (auto simp: powr_minus field_simps)
hoelzl@47599
   943
    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   944
      by (auto simp: powr_minus)
hoelzl@47599
   945
    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
haftmann@57512
   946
    hence "?S \<le> real m" unfolding mult.assoc by auto
hoelzl@47599
   947
    hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
wenzelm@60500
   948
    from this bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
wenzelm@60500
   949
    have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF \<open>1 < 2\<close>, symmetric]
immler@58985
   950
      by (rule order_le_less_trans)
hoelzl@47599
   951
    hence "-e < bitlen m" using False by auto
hoelzl@47599
   952
    thus ?thesis by auto
hoelzl@29804
   953
  qed
hoelzl@47599
   954
qed
hoelzl@29804
   955
immler@58985
   956
lemma bitlen_div:
immler@58985
   957
  assumes "0 < m"
immler@58985
   958
  shows "1 \<le> real m / 2^nat (bitlen m - 1)" and "real m / 2^nat (bitlen m - 1) < 2"
hoelzl@29804
   959
proof -
hoelzl@29804
   960
  let ?B = "2^nat(bitlen m - 1)"
hoelzl@29804
   961
wenzelm@60500
   962
  have "?B \<le> m" using bitlen_bounds[OF \<open>0 <m\<close>] ..
hoelzl@29804
   963
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   964
  thus "1 \<le> real m / ?B" by auto
hoelzl@29804
   965
hoelzl@29804
   966
  have "m \<noteq> 0" using assms by auto
wenzelm@60500
   967
  have "0 \<le> bitlen m - 1" using \<open>0 < m\<close> by (auto simp: bitlen_def)
obua@16782
   968
wenzelm@60500
   969
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF \<open>0 <m\<close>] ..
wenzelm@60500
   970
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using \<open>0 < m\<close> by (auto simp: bitlen_def)
wenzelm@60500
   971
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF \<open>0 \<le> bitlen m - 1\<close> zero_le_one] by auto
hoelzl@29804
   972
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   973
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
hoelzl@29804
   974
  thus "real m / ?B < 2" by auto
hoelzl@29804
   975
qed
hoelzl@29804
   976
wenzelm@60500
   977
subsection \<open>Truncating Real Numbers\<close>
immler@58985
   978
immler@58985
   979
definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where
immler@58985
   980
  "truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
immler@58985
   981
immler@58985
   982
lemma truncate_down: "truncate_down prec x \<le> x"
immler@58985
   983
  using round_down by (simp add: truncate_down_def)
immler@58985
   984
immler@58985
   985
lemma truncate_down_le: "x \<le> y \<Longrightarrow> truncate_down prec x \<le> y"
immler@58985
   986
  by (rule order_trans[OF truncate_down])
immler@58985
   987
immler@58985
   988
lemma truncate_down_zero[simp]: "truncate_down prec 0 = 0"
immler@58985
   989
  by (simp add: truncate_down_def)
immler@58985
   990
immler@58985
   991
lemma truncate_down_float[simp]: "truncate_down p x \<in> float"
immler@58985
   992
  by (auto simp: truncate_down_def)
immler@58985
   993
immler@58985
   994
definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real" where
immler@58985
   995
  "truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
immler@58985
   996
immler@58985
   997
lemma truncate_up: "x \<le> truncate_up prec x"
immler@58985
   998
  using round_up by (simp add: truncate_up_def)
immler@58985
   999
immler@58985
  1000
lemma truncate_up_le: "x \<le> y \<Longrightarrow> x \<le> truncate_up prec y"
immler@58985
  1001
  by (rule order_trans[OF _ truncate_up])
immler@58985
  1002
immler@58985
  1003
lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0"
immler@58985
  1004
  by (simp add: truncate_up_def)
immler@58985
  1005
immler@58985
  1006
lemma truncate_up_uminus_eq: "truncate_up prec (-x) = - truncate_down prec x"
immler@58985
  1007
  and truncate_down_uminus_eq: "truncate_down prec (-x) = - truncate_up prec x"
immler@58985
  1008
  by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
immler@58985
  1009
immler@58985
  1010
lemma truncate_up_float[simp]: "truncate_up p x \<in> float"
immler@58985
  1011
  by (auto simp: truncate_up_def)
immler@58985
  1012
immler@58985
  1013
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
  1014
  by (simp_all add: powr_add)
immler@58985
  1015
immler@58985
  1016
lemma truncate_down_pos:
immler@58985
  1017
  assumes "x > 0" "p > 0"
immler@58985
  1018
  shows "truncate_down p x > 0"
immler@58985
  1019
proof -
immler@58985
  1020
  have "0 \<le> log 2 x - real \<lfloor>log 2 x\<rfloor>"
immler@58985
  1021
    by (simp add: algebra_simps)
immler@58985
  1022
  from this assms
immler@58985
  1023
  show ?thesis
immler@58985
  1024
    by (auto simp: truncate_down_def round_down_def mult_powr_eq
immler@58985
  1025
      intro!: ge_one_powr_ge_zero mult_pos_pos)
immler@58985
  1026
qed
immler@58985
  1027
immler@58985
  1028
lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y"
immler@58985
  1029
  by (auto simp: truncate_down_def round_down_def)
immler@58985
  1030
immler@58985
  1031
lemma truncate_down_ge1: "1 \<le> x \<Longrightarrow> 1 \<le> p \<Longrightarrow> 1 \<le> truncate_down p x"
immler@58985
  1032
  by (auto simp: truncate_down_def algebra_simps intro!: round_down_ge1 add_mono)
immler@58985
  1033
immler@58985
  1034
lemma truncate_up_nonpos: "x \<le> 0 \<Longrightarrow> truncate_up prec x \<le> 0"
immler@58985
  1035
  by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)
hoelzl@47599
  1036
immler@58985
  1037
lemma truncate_up_le1:
immler@58985
  1038
  assumes "x \<le> 1" "1 \<le> p" shows "truncate_up p x \<le> 1"
immler@58985
  1039
proof -
immler@58985
  1040
  {
immler@58985
  1041
    assume "x \<le> 0"
immler@58985
  1042
    with truncate_up_nonpos[OF this, of p] have ?thesis by simp
immler@58985
  1043
  } moreover {
immler@58985
  1044
    assume "x > 0"
immler@58985
  1045
    hence le: "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<le> 0"
immler@58985
  1046
      using assms by (auto simp: log_less_iff)
immler@58985
  1047
    from assms have "1 \<le> int p" by simp
immler@58985
  1048
    from add_mono[OF this le]
immler@58985
  1049
    have ?thesis using assms
immler@58985
  1050
      by (simp add: truncate_up_def round_up_le1 add_mono)
immler@58985
  1051
  } ultimately show ?thesis by arith
immler@58985
  1052
qed
immler@58985
  1053
wenzelm@60500
  1054
subsection \<open>Truncating Floats\<close>
immler@58985
  1055
immler@58985
  1056
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
immler@58985
  1057
  by (simp add: truncate_up_def)
immler@58985
  1058
immler@58985
  1059
lemma float_round_up: "real x \<le> real (float_round_up prec x)"
immler@58985
  1060
  using truncate_up by transfer simp
immler@58985
  1061
immler@58985
  1062
lemma float_round_up_zero[simp]: "float_round_up prec 0 = 0"
immler@58985
  1063
  by transfer simp
immler@58985
  1064
immler@58985
  1065
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down
immler@58985
  1066
  by (simp add: truncate_down_def)
immler@58985
  1067
immler@58985
  1068
lemma float_round_down: "real (float_round_down prec x) \<le> real x"
immler@58985
  1069
  using truncate_down by transfer simp
immler@58985
  1070
immler@58985
  1071
lemma float_round_down_zero[simp]: "float_round_down prec 0 = 0"
immler@58985
  1072
  by transfer simp
immler@58985
  1073
immler@58985
  1074
lemmas float_round_up_le = order_trans[OF _ float_round_up]
immler@58985
  1075
  and float_round_down_le = order_trans[OF float_round_down]
immler@58985
  1076
immler@58985
  1077
lemma minus_float_round_up_eq: "- float_round_up prec x = float_round_down prec (- x)"
immler@58985
  1078
  and minus_float_round_down_eq: "- float_round_down prec x = float_round_up prec (- x)"
immler@58985
  1079
  by (transfer, simp add: truncate_down_uminus_eq truncate_up_uminus_eq)+
immler@58985
  1080
immler@58985
  1081
lemma compute_float_round_down[code]:
immler@58985
  1082
  "float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
immler@58985
  1083
    if 0 < d then Float (div_twopow m (nat d)) (e + d)
immler@58985
  1084
             else Float m e)"
immler@58985
  1085
  using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
immler@58985
  1086
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_down_def
immler@58985
  1087
    cong del: if_weak_cong)
immler@58985
  1088
hide_fact (open) compute_float_round_down
immler@58985
  1089
immler@58985
  1090
lemma compute_float_round_up[code]:
immler@58985
  1091
  "float_round_up prec x = - float_round_down prec (-x)"
immler@58985
  1092
  by transfer (simp add: truncate_down_uminus_eq)
immler@58985
  1093
hide_fact (open) compute_float_round_up
immler@58985
  1094
immler@58985
  1095
wenzelm@60500
  1096
subsection \<open>Approximation of positive rationals\<close>
hoelzl@29804
  1097
hoelzl@47599
  1098
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
hoelzl@47599
  1099
  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
hoelzl@29804
  1100
hoelzl@47599
  1101
lemma real_div_nat_eq_floor_of_divide:
haftmann@59984
  1102
  fixes a b :: nat
haftmann@59984
  1103
  shows "a div b = real \<lfloor>a / b\<rfloor>"
haftmann@59984
  1104
  by (simp add: floor_divide_of_nat_eq [of a b] real_eq_of_nat)
hoelzl@29804
  1105
hoelzl@47599
  1106
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
hoelzl@29804
  1107
hoelzl@47600
  1108
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
  1109
  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp
obua@16782
  1110
hoelzl@47599
  1111
lemma compute_lapprox_posrat[code]:
wenzelm@53381
  1112
  fixes prec x y
wenzelm@53381
  1113
  shows "lapprox_posrat prec x y =
wenzelm@53381
  1114
   (let
hoelzl@47599
  1115
       l = rat_precision prec x y;
hoelzl@47599
  1116
       d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
hoelzl@47599
  1117
    in normfloat (Float d (- l)))"
immler@58982
  1118
    unfolding div_mult_twopow_eq
hoelzl@47600
  1119
    by transfer
hoelzl@47615
  1120
       (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
hoelzl@47599
  1121
             del: two_powr_minus_int_float)
hoelzl@47621
  1122
hide_fact (open) compute_lapprox_posrat
hoelzl@29804
  1123
hoelzl@47600
  1124
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
  1125
  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
hoelzl@29804
  1126
wenzelm@60376
  1127
context
wenzelm@60376
  1128
  notes divmod_int_mod_div[simp]
wenzelm@60376
  1129
begin
wenzelm@60376
  1130
wenzelm@60376
  1131
qualified lemma compute_rapprox_posrat[code]:
hoelzl@47599
  1132
  fixes prec x y
hoelzl@47599
  1133
  defines "l \<equiv> rat_precision prec x y"
hoelzl@47599
  1134
  shows "rapprox_posrat prec x y = (let
hoelzl@47599
  1135
     l = l ;
hoelzl@47599
  1136
     X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
immler@58982
  1137
     (d, m) = divmod_int (fst X) (snd X)
hoelzl@47599
  1138
   in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
hoelzl@47599
  1139
proof (cases "y = 0")
immler@58982
  1140
  assume "y = 0" thus ?thesis by transfer simp
hoelzl@47599
  1141
next
hoelzl@47599
  1142
  assume "y \<noteq> 0"
hoelzl@29804
  1143
  show ?thesis
hoelzl@47599
  1144
  proof (cases "0 \<le> l")
hoelzl@47599
  1145
    assume "0 \<le> l"
wenzelm@56777
  1146
    def x' \<equiv> "x * 2 ^ nat l"
hoelzl@47599
  1147
    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
hoelzl@47599
  1148
    moreover have "real x * 2 powr real l = real x'"
wenzelm@60500
  1149
      by (simp add: powr_realpow[symmetric] \<open>0 \<le> l\<close> x'_def)
hoelzl@47599
  1150
    ultimately show ?thesis
wenzelm@60500
  1151
      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
  1152
        l_def[symmetric, THEN meta_eq_to_obj_eq]
haftmann@58834
  1153
      by transfer (auto simp add: floor_divide_eq_div [symmetric] round_up_def)
hoelzl@47599
  1154
   next
hoelzl@47599
  1155
    assume "\<not> 0 \<le> l"
wenzelm@56777
  1156
    def y' \<equiv> "y * 2 ^ nat (- l)"
wenzelm@60500
  1157
    from \<open>y \<noteq> 0\<close> have "y' \<noteq> 0" by (simp add: y'_def)
hoelzl@47599
  1158
    have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
hoelzl@47599
  1159
    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
wenzelm@60500
  1160
      using \<open>\<not> 0 \<le> l\<close>
wenzelm@57862
  1161
      by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps)
hoelzl@47599
  1162
    ultimately show ?thesis
wenzelm@60500
  1163
      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
  1164
        l_def[symmetric, THEN meta_eq_to_obj_eq]
hoelzl@47600
  1165
      by transfer
haftmann@58834
  1166
         (auto simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div [symmetric])
hoelzl@29804
  1167
  qed
hoelzl@29804
  1168
qed
wenzelm@60376
  1169
wenzelm@60376
  1170
end
hoelzl@29804
  1171
hoelzl@47599
  1172
lemma rat_precision_pos:
hoelzl@47599
  1173
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@47599
  1174
  shows "rat_precision n (int x) (int y) > 0"
hoelzl@29804
  1175
proof -
hoelzl@47599
  1176
  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
hoelzl@47599
  1177
  hence "bitlen (int x) < bitlen (int y)" using assms
hoelzl@47599
  1178
    by (simp add: bitlen_def del: floor_add_one)
hoelzl@47599
  1179
      (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
hoelzl@47599
  1180
  thus ?thesis
hoelzl@47599
  1181
    using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
hoelzl@29804
  1182
qed
obua@16782
  1183
hoelzl@47601
  1184
lemma rapprox_posrat_less1:
immler@58982
  1185
  shows "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 2 * x < y \<Longrightarrow> 0 < n \<Longrightarrow> real (rapprox_posrat n x y) < 1"
immler@58982
  1186
  by transfer (simp add: rat_precision_pos round_up_less1)
hoelzl@29804
  1187
hoelzl@47600
  1188
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
  1189
  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
obua@16782
  1190
hoelzl@29804
  1191
lemma compute_lapprox_rat[code]:
hoelzl@47599
  1192
  "lapprox_rat prec x y =
hoelzl@47599
  1193
    (if y = 0 then 0
hoelzl@47599
  1194
    else if 0 \<le> x then
hoelzl@47599
  1195
      (if 0 < y then lapprox_posrat prec (nat x) (nat y)
wenzelm@53381
  1196
      else - (rapprox_posrat prec (nat x) (nat (-y))))
hoelzl@47599
  1197
      else (if 0 < y
hoelzl@47599
  1198
        then - (rapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
  1199
        else lapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@56479
  1200
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hoelzl@47621
  1201
hide_fact (open) compute_lapprox_rat
hoelzl@47599
  1202
hoelzl@47600
  1203
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
  1204
  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
hoelzl@47599
  1205
immler@58982
  1206
lemma "rapprox_rat = rapprox_posrat"
immler@58982
  1207
  by transfer auto
immler@58982
  1208
immler@58982
  1209
lemma "lapprox_rat = lapprox_posrat"
immler@58982
  1210
  by transfer auto
immler@58982
  1211
hoelzl@47599
  1212
lemma compute_rapprox_rat[code]:
immler@58982
  1213
  "rapprox_rat prec x y = - lapprox_rat prec (-x) y"
immler@58982
  1214
  by transfer (simp add: round_down_uminus_eq)
hoelzl@47621
  1215
hide_fact (open) compute_rapprox_rat
hoelzl@47599
  1216
wenzelm@60500
  1217
subsection \<open>Division\<close>
hoelzl@47599
  1218
immler@54782
  1219
definition "real_divl prec a b = round_down (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)"
immler@54782
  1220
immler@54782
  1221
definition "real_divr prec a b = round_up (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)"
immler@54782
  1222
immler@54782
  1223
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl
immler@54782
  1224
  by (simp add: real_divl_def)
hoelzl@47599
  1225
hoelzl@47599
  1226
lemma compute_float_divl[code]:
hoelzl@47600
  1227
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1228
proof cases
hoelzl@47601
  1229
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
hoelzl@47601
  1230
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
hoelzl@47601
  1231
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
hoelzl@47601
  1232
  then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"
hoelzl@47601
  1233
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
hoelzl@47601
  1234
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
hoelzl@47601
  1235
    by (simp add: field_simps powr_divide2[symmetric])
hoelzl@47599
  1236
hoelzl@47601
  1237
  show ?thesis
wenzelm@53381
  1238
    using not_0
immler@54782
  1239
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,
immler@54782
  1240
      simp add: field_simps)
immler@54782
  1241
qed (transfer, auto simp: real_divl_def)
hoelzl@47621
  1242
hide_fact (open) compute_float_divl
hoelzl@47600
  1243
immler@54782
  1244
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr
immler@54782
  1245
  by (simp add: real_divr_def)
hoelzl@47599
  1246
hoelzl@47599
  1247
lemma compute_float_divr[code]:
immler@58982
  1248
  "float_divr prec x y = - float_divl prec (-x) y"
immler@58982
  1249
  by transfer (simp add: real_divr_def real_divl_def round_down_uminus_eq)
immler@58982
  1250
hide_fact (open) compute_float_divr
hoelzl@47600
  1251
obua@16782
  1252
wenzelm@60500
  1253
subsection \<open>Approximate Power\<close>
immler@58985
  1254
immler@58985
  1255
lemma div2_less_self[termination_simp]: fixes n::nat shows "odd n \<Longrightarrow> n div 2 < n"
immler@58985
  1256
  by (simp add: odd_pos)
immler@58985
  1257
immler@58985
  1258
fun power_down :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real" where
immler@58985
  1259
  "power_down p x 0 = 1"
immler@58985
  1260
| "power_down p x (Suc n) =
immler@58985
  1261
    (if odd n then truncate_down (Suc p) ((power_down p x (Suc n div 2))\<^sup>2) else truncate_down (Suc p) (x * power_down p x n))"
immler@58985
  1262
immler@58985
  1263
fun power_up :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real" where
immler@58985
  1264
  "power_up p x 0 = 1"
immler@58985
  1265
| "power_up p x (Suc n) =
immler@58985
  1266
    (if odd n then truncate_up p ((power_up p x (Suc n div 2))\<^sup>2) else truncate_up p (x * power_up p x n))"
immler@58985
  1267
immler@58985
  1268
lift_definition power_up_fl :: "nat \<Rightarrow> float \<Rightarrow> nat \<Rightarrow> float" is power_up
immler@58985
  1269
  by (induct_tac rule: power_up.induct) simp_all
immler@58985
  1270
immler@58985
  1271
lift_definition power_down_fl :: "nat \<Rightarrow> float \<Rightarrow> nat \<Rightarrow> float" is power_down
immler@58985
  1272
  by (induct_tac rule: power_down.induct) simp_all
immler@58985
  1273
immler@58985
  1274
lemma power_float_transfer[transfer_rule]:
immler@58985
  1275
  "(rel_fun pcr_float (rel_fun op = pcr_float)) op ^ op ^"
immler@58985
  1276
  unfolding power_def
immler@58985
  1277
  by transfer_prover
immler@58985
  1278
immler@58985
  1279
lemma compute_power_up_fl[code]:
immler@58985
  1280
  "power_up_fl p x 0 = 1"
immler@58985
  1281
  "power_up_fl p x (Suc n) =
immler@58985
  1282
    (if odd n then float_round_up p ((power_up_fl p x (Suc n div 2))\<^sup>2) else float_round_up p (x * power_up_fl p x n))"
immler@58985
  1283
  and compute_power_down_fl[code]:
immler@58985
  1284
  "power_down_fl p x 0 = 1"
immler@58985
  1285
  "power_down_fl p x (Suc n) =
immler@58985
  1286
    (if odd n then float_round_down (Suc p) ((power_down_fl p x (Suc n div 2))\<^sup>2) else float_round_down (Suc p) (x * power_down_fl p x n))"
immler@58985
  1287
  unfolding atomize_conj
immler@58985
  1288
  by transfer simp
immler@58985
  1289
immler@58985
  1290
lemma power_down_pos: "0 < x \<Longrightarrow> 0 < power_down p x n"
immler@58985
  1291
  by (induct p x n rule: power_down.induct)
immler@58985
  1292
    (auto simp del: odd_Suc_div_two intro!: truncate_down_pos)
immler@58985
  1293
immler@58985
  1294
lemma power_down_nonneg: "0 \<le> x \<Longrightarrow> 0 \<le> power_down p x n"
immler@58985
  1295
  by (induct p x n rule: power_down.induct)
immler@58985
  1296
    (auto simp del: odd_Suc_div_two intro!: truncate_down_nonneg mult_nonneg_nonneg)
immler@58985
  1297
immler@58985
  1298
lemma power_down: "0 \<le> x \<Longrightarrow> power_down p x n \<le> x ^ n"
immler@58985
  1299
proof (induct p x n rule: power_down.induct)
immler@58985
  1300
  case (2 p x n)
immler@58985
  1301
  {
immler@58985
  1302
    assume "odd n"
immler@58985
  1303
    hence "(power_down p x (Suc n div 2)) ^ 2 \<le> (x ^ (Suc n div 2)) ^ 2"
immler@58985
  1304
      using 2
immler@58985
  1305
      by (auto intro: power_mono power_down_nonneg simp del: odd_Suc_div_two)
immler@58985
  1306
    also have "\<dots> = x ^ (Suc n div 2 * 2)"
immler@58985
  1307
      by (simp add: power_mult[symmetric])
immler@58985
  1308
    also have "Suc n div 2 * 2 = Suc n"
wenzelm@60500
  1309
      using \<open>odd n\<close> by presburger
immler@58985
  1310
    finally have ?case
wenzelm@60500
  1311
      using \<open>odd n\<close>
immler@58985
  1312
      by (auto intro!: truncate_down_le simp del: odd_Suc_div_two)
immler@58985
  1313
  } thus ?case
immler@58985
  1314
    by (auto intro!: truncate_down_le mult_left_mono 2 mult_nonneg_nonneg power_down_nonneg)
immler@58985
  1315
qed simp
immler@58985
  1316
immler@58985
  1317
lemma power_up: "0 \<le> x \<Longrightarrow> x ^ n \<le> power_up p x n"
immler@58985
  1318
proof (induct p x n rule: power_up.induct)
immler@58985
  1319
  case (2 p x n)
immler@58985
  1320
  {
immler@58985
  1321
    assume "odd n"
immler@58985
  1322
    hence "Suc n = Suc n div 2 * 2"
wenzelm@60500
  1323
      using \<open>odd n\<close> even_Suc by presburger
immler@58985
  1324
    hence "x ^ Suc n \<le> (x ^ (Suc n div 2))\<^sup>2"
immler@58985
  1325
      by (simp add: power_mult[symmetric])
immler@58985
  1326
    also have "\<dots> \<le> (power_up p x (Suc n div 2))\<^sup>2"
wenzelm@60500
  1327
      using 2 \<open>odd n\<close>
immler@58985
  1328
      by (auto intro: power_mono simp del: odd_Suc_div_two )
immler@58985
  1329
    finally have ?case
wenzelm@60500
  1330
      using \<open>odd n\<close>
immler@58985
  1331
      by (auto intro!: truncate_up_le simp del: odd_Suc_div_two )
immler@58985
  1332
  } thus ?case
immler@58985
  1333
    by (auto intro!: truncate_up_le mult_left_mono 2)
immler@58985
  1334
qed simp
immler@58985
  1335
immler@58985
  1336
lemmas power_up_le = order_trans[OF _ power_up]
immler@58985
  1337
  and power_up_less = less_le_trans[OF _ power_up]
immler@58985
  1338
  and power_down_le = order_trans[OF power_down]
immler@58985
  1339
immler@58985
  1340
lemma power_down_fl: "0 \<le> x \<Longrightarrow> power_down_fl p x n \<le> x ^ n"
immler@58985
  1341
  by transfer (rule power_down)
immler@58985
  1342
immler@58985
  1343
lemma power_up_fl: "0 \<le> x \<Longrightarrow> x ^ n \<le> power_up_fl p x n"
immler@58985
  1344
  by transfer (rule power_up)
immler@58985
  1345
immler@58985
  1346
lemma real_power_up_fl: "real (power_up_fl p x n) = power_up p x n"
immler@58985
  1347
  by transfer simp
immler@58985
  1348
immler@58985
  1349
lemma real_power_down_fl: "real (power_down_fl p x n) = power_down p x n"
immler@58985
  1350
  by transfer simp
immler@58985
  1351
immler@58985
  1352
wenzelm@60500
  1353
subsection \<open>Approximate Addition\<close>
immler@58985
  1354
immler@58985
  1355
definition "plus_down prec x y = truncate_down prec (x + y)"
immler@58985
  1356
immler@58985
  1357
definition "plus_up prec x y = truncate_up prec (x + y)"
immler@58985
  1358
immler@58985
  1359
lemma float_plus_down_float[intro, simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> plus_down p x y \<in> float"
immler@58985
  1360
  by (simp add: plus_down_def)
immler@58985
  1361
immler@58985
  1362
lemma float_plus_up_float[intro, simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> plus_up p x y \<in> float"
immler@58985
  1363
  by (simp add: plus_up_def)
immler@58985
  1364
immler@58985
  1365
lift_definition float_plus_down::"nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is plus_down ..
immler@58985
  1366
immler@58985
  1367
lift_definition float_plus_up::"nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is plus_up ..
immler@58985
  1368
immler@58985
  1369
lemma plus_down: "plus_down prec x y \<le> x + y"
immler@58985
  1370
  and plus_up: "x + y \<le> plus_up prec x y"
immler@58985
  1371
  by (auto simp: plus_down_def truncate_down plus_up_def truncate_up)
immler@58985
  1372
immler@58985
  1373
lemma float_plus_down: "real (float_plus_down prec x y) \<le> x + y"
immler@58985
  1374
  and float_plus_up: "x + y \<le> real (float_plus_up prec x y)"
immler@58985
  1375
  by (transfer, rule plus_down plus_up)+
immler@58985
  1376
immler@58985
  1377
lemmas plus_down_le = order_trans[OF plus_down]
immler@58985
  1378
  and plus_up_le = order_trans[OF _ plus_up]
immler@58985
  1379
  and float_plus_down_le = order_trans[OF float_plus_down]
immler@58985
  1380
  and float_plus_up_le = order_trans[OF _ float_plus_up]
immler@58985
  1381
immler@58985
  1382
lemma compute_plus_up[code]: "plus_up p x y = - plus_down p (-x) (-y)"
immler@58985
  1383
  using truncate_down_uminus_eq[of p "x + y"]
immler@58985
  1384
  by (auto simp: plus_down_def plus_up_def)
immler@58985
  1385
immler@58985
  1386
lemma
immler@58985
  1387
  truncate_down_log2_eqI:
immler@58985
  1388
  assumes "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@58985
  1389
  assumes "\<lfloor>x * 2 powr (p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)\<rfloor> = \<lfloor>y * 2 powr (p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)\<rfloor>"
immler@58985
  1390
  shows "truncate_down p x = truncate_down p y"
immler@58985
  1391
  using assms by (auto simp: truncate_down_def round_down_def)
immler@58985
  1392
immler@58985
  1393
lemma bitlen_eq_zero_iff: "bitlen x = 0 \<longleftrightarrow> x \<le> 0"
immler@58985
  1394
  by (clarsimp simp add: bitlen_def)
immler@58985
  1395
    (metis Float.compute_bitlen add.commute bitlen_def bitlen_nonneg less_add_same_cancel2 not_less
immler@58985
  1396
      zero_less_one)
immler@58985
  1397
immler@58985
  1398
lemma
immler@58985
  1399
  sum_neq_zeroI:
immler@58985
  1400
  fixes a k::real
immler@58985
  1401
  shows "abs a \<ge> k \<Longrightarrow> abs b < k \<Longrightarrow> a + b \<noteq> 0"
immler@58985
  1402
    and "abs a > k \<Longrightarrow> abs b \<le> k \<Longrightarrow> a + b \<noteq> 0"
immler@58985
  1403
  by auto
immler@58985
  1404
immler@58985
  1405
lemma
immler@58985
  1406
  abs_real_le_2_powr_bitlen[simp]:
immler@58985
  1407
  "\<bar>real m2\<bar> < 2 powr real (bitlen \<bar>m2\<bar>)"
immler@58985
  1408
proof cases
immler@58985
  1409
  assume "m2 \<noteq> 0"
immler@58985
  1410
  hence "\<bar>m2\<bar> < 2 ^ nat (bitlen \<bar>m2\<bar>)"
immler@58985
  1411
    using bitlen_bounds[of "\<bar>m2\<bar>"]
immler@58985
  1412
    by (auto simp: powr_add bitlen_nonneg)
immler@58985
  1413
  thus ?thesis
immler@58985
  1414
    by (simp add: powr_int bitlen_nonneg real_of_int_less_iff[symmetric])
immler@58985
  1415
qed simp
immler@58985
  1416
immler@58985
  1417
lemma floor_sum_times_2_powr_sgn_eq:
immler@58985
  1418
  fixes ai p q::int
immler@58985
  1419
  and a b::real
immler@58985
  1420
  assumes "a * 2 powr p = ai"
immler@58985
  1421
  assumes b_le_1: "abs (b * 2 powr (p + 1)) \<le> 1"
immler@58985
  1422
  assumes leqp: "q \<le> p"
immler@58985
  1423
  shows "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(2 * ai + sgn b) * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1424
proof -
immler@58985
  1425
  {
immler@58985
  1426
    assume "b = 0"
immler@58985
  1427
    hence ?thesis
immler@58985
  1428
      by (simp add: assms(1)[symmetric] powr_add[symmetric] algebra_simps powr_mult_base)
immler@58985
  1429
  } moreover {
immler@58985
  1430
    assume "b > 0"
immler@58985
  1431
    hence "b * 2 powr p < abs (b * 2 powr (p + 1))" by simp
immler@58985
  1432
    also note b_le_1
immler@58985
  1433
    finally have b_less_1: "b * 2 powr real p < 1" .
immler@58985
  1434
wenzelm@60500
  1435
    from b_less_1 \<open>b > 0\<close> have floor_eq: "\<lfloor>b * 2 powr real p\<rfloor> = 0" "\<lfloor>sgn b / 2\<rfloor> = 0"
immler@58985
  1436
      by (simp_all add: floor_eq_iff)
immler@58985
  1437
immler@58985
  1438
    have "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(a + b) * 2 powr p * 2 powr (q - p)\<rfloor>"
immler@58985
  1439
      by (simp add: algebra_simps powr_realpow[symmetric] powr_add[symmetric])
immler@58985
  1440
    also have "\<dots> = \<lfloor>(ai + b * 2 powr p) * 2 powr (q - p)\<rfloor>"
immler@58985
  1441
      by (simp add: assms algebra_simps)
immler@58985
  1442
    also have "\<dots> = \<lfloor>(ai + b * 2 powr p) / real ((2::int) ^ nat (p - q))\<rfloor>"
immler@58985
  1443
      using assms
immler@58985
  1444
      by (simp add: algebra_simps powr_realpow[symmetric] divide_powr_uminus powr_add[symmetric])
immler@58985
  1445
    also have "\<dots> = \<lfloor>ai / real ((2::int) ^ nat (p - q))\<rfloor>"
immler@58985
  1446
      by (simp del: real_of_int_power add: floor_divide_real_eq_div floor_eq)
immler@58985
  1447
    finally have "\<lfloor>(a + b) * 2 powr real q\<rfloor> = \<lfloor>real ai / real ((2::int) ^ nat (p - q))\<rfloor>" .
immler@58985
  1448
    moreover
immler@58985
  1449
    {
immler@58985
  1450
      have "\<lfloor>(2 * ai + sgn b) * 2 powr (real (q - p) - 1)\<rfloor> = \<lfloor>(ai + sgn b / 2) * 2 powr (q - p)\<rfloor>"
immler@58985
  1451
        by (subst powr_divide2[symmetric]) (simp add: field_simps)
immler@58985
  1452
      also have "\<dots> = \<lfloor>(ai + sgn b / 2) / real ((2::int) ^ nat (p - q))\<rfloor>"
immler@58985
  1453
        using leqp by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
immler@58985
  1454
      also have "\<dots> = \<lfloor>ai / real ((2::int) ^ nat (p - q))\<rfloor>"
immler@58985
  1455
        by (simp del: real_of_int_power add: floor_divide_real_eq_div floor_eq)
immler@58985
  1456
      finally
immler@58985
  1457
      have "\<lfloor>(2 * ai + (sgn b)) * 2 powr (real (q - p) - 1)\<rfloor> =
immler@58985
  1458
          \<lfloor>real ai / real ((2::int) ^ nat (p - q))\<rfloor>"
immler@58985
  1459
        .
immler@58985
  1460
    } ultimately have ?thesis by simp
immler@58985
  1461
  } moreover {
immler@58985
  1462
    assume "\<not> 0 \<le> b"
immler@58985
  1463
    hence "0 > b" by simp
immler@58985
  1464
    hence floor_eq: "\<lfloor>b * 2 powr (real p + 1)\<rfloor> = -1"
immler@58985
  1465
      using b_le_1
immler@58985
  1466
      by (auto simp: floor_eq_iff algebra_simps pos_divide_le_eq[symmetric] abs_if divide_powr_uminus
immler@58985
  1467
        intro!: mult_neg_pos split: split_if_asm)
immler@58985
  1468
    have "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(2*a + 2*b) * 2 powr p * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1469
      by (simp add: algebra_simps powr_realpow[symmetric] powr_add[symmetric] powr_mult_base)
immler@58985
  1470
    also have "\<dots> = \<lfloor>(2 * (a * 2 powr p) + 2 * b * 2 powr p) * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1471
      by (simp add: algebra_simps)
immler@58985
  1472
    also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / 2 powr (1 - q + p)\<rfloor>"
immler@58985
  1473
      using assms by (simp add: algebra_simps powr_mult_base divide_powr_uminus)
immler@58985
  1474
    also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / real ((2::int) ^ nat (p - q + 1))\<rfloor>"
immler@58985
  1475
      using assms by (simp add: algebra_simps powr_realpow[symmetric])
immler@58985
  1476
    also have "\<dots> = \<lfloor>(2 * ai - 1) / real ((2::int) ^ nat (p - q + 1))\<rfloor>"
wenzelm@60500
  1477
      using \<open>b < 0\<close> assms
immler@58985
  1478
      by (simp add: floor_divide_eq_div floor_eq floor_divide_real_eq_div
immler@58985
  1479
        del: real_of_int_mult real_of_int_power real_of_int_diff)
immler@58985
  1480
    also have "\<dots> = \<lfloor>(2 * ai - 1) * 2 powr (q - p - 1)\<rfloor>"
immler@58985
  1481
      using assms by (simp add: algebra_simps divide_powr_uminus powr_realpow[symmetric])
wenzelm@60500
  1482
    finally have ?thesis using \<open>b < 0\<close> by simp
immler@58985
  1483
  } ultimately show ?thesis by arith
immler@58985
  1484
qed
immler@58985
  1485
immler@58985
  1486
lemma
immler@58985
  1487
  log2_abs_int_add_less_half_sgn_eq:
immler@58985
  1488
  fixes ai::int and b::real
immler@58985
  1489
  assumes "abs b \<le> 1/2" "ai \<noteq> 0"
immler@58985
  1490
  shows "\<lfloor>log 2 \<bar>real ai + b\<bar>\<rfloor> = \<lfloor>log 2 \<bar>ai + sgn b / 2\<bar>\<rfloor>"
immler@58985
  1491
proof cases
immler@58985
  1492
  assume "b = 0" thus ?thesis by simp
immler@58985
  1493
next
immler@58985
  1494
  assume "b \<noteq> 0"
immler@58985
  1495
  def k \<equiv> "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor>"
immler@58985
  1496
  hence "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor> = k" by simp
immler@58985
  1497
  hence k: "2 powr k \<le> \<bar>ai\<bar>" "\<bar>ai\<bar> < 2 powr (k + 1)"
wenzelm@60500
  1498
    by (simp_all add: floor_log_eq_powr_iff \<open>ai \<noteq> 0\<close>)
immler@58985
  1499
  have "k \<ge> 0"
immler@58985
  1500
    using assms by (auto simp: k_def)
immler@58985
  1501
  def r \<equiv> "\<bar>ai\<bar> - 2 ^ nat k"
immler@58985
  1502
  have r: "0 \<le> r" "r < 2 powr k"
wenzelm@60500
  1503
    using \<open>k \<ge> 0\<close> k
immler@58985
  1504
    by (auto simp: r_def k_def algebra_simps powr_add abs_if powr_int)
immler@58985
  1505
  hence "r \<le> (2::int) ^ nat k - 1"
wenzelm@60500
  1506
    using \<open>k \<ge> 0\<close> by (auto simp: powr_int)
wenzelm@60500
  1507
  from this[simplified real_of_int_le_iff[symmetric]] \<open>0 \<le> k\<close>
immler@58985
  1508
  have r_le: "r \<le> 2 powr k - 1"
immler@58985
  1509
    by (auto simp: algebra_simps powr_int simp del: real_of_int_le_iff)
immler@58985
  1510
immler@58985
  1511
  have "\<bar>ai\<bar> = 2 powr k + r"
wenzelm@60500
  1512
    using \<open>k \<ge> 0\<close> by (auto simp: k_def r_def powr_realpow[symmetric])
immler@58985
  1513
immler@58985
  1514
  have pos: "\<And>b::real. abs b < 1 \<Longrightarrow> 0 < 2 powr k + (r + b)"
wenzelm@60500
  1515
    using \<open>0 \<le> k\<close> \<open>ai \<noteq> 0\<close>
immler@58985
  1516
    by (auto simp add: r_def powr_realpow[symmetric] abs_if sgn_if algebra_simps
immler@58985
  1517
      split: split_if_asm)
immler@58985
  1518
  have less: "\<bar>sgn ai * b\<bar> < 1"
immler@58985
  1519
    and less': "\<bar>sgn (sgn ai * b) / 2\<bar> < 1"
wenzelm@60500
  1520
    using \<open>abs b \<le> _\<close> by (auto simp: abs_if sgn_if split: split_if_asm)
immler@58985
  1521
immler@58985
  1522
  have floor_eq: "\<And>b::real. abs b \<le> 1 / 2 \<Longrightarrow>
immler@58985
  1523
      \<lfloor>log 2 (1 + (r + b) / 2 powr k)\<rfloor> = (if r = 0 \<and> b < 0 then -1 else 0)"
wenzelm@60500
  1524
    using \<open>k \<ge> 0\<close> r r_le
immler@58985
  1525
    by (auto simp: floor_log_eq_powr_iff powr_minus_divide field_simps sgn_if)
immler@58985
  1526
wenzelm@60500
  1527
  from \<open>real \<bar>ai\<bar> = _\<close> have "\<bar>ai + b\<bar> = 2 powr k + (r + sgn ai * b)"
wenzelm@60500
  1528
    using \<open>abs b <= _\<close> \<open>0 \<le> k\<close> r
immler@58985
  1529
    by (auto simp add: sgn_if abs_if)
immler@58985
  1530
  also have "\<lfloor>log 2 \<dots>\<rfloor> = \<lfloor>log 2 (2 powr k + r + sgn (sgn ai * b) / 2)\<rfloor>"
immler@58985
  1531
  proof -
immler@58985
  1532
    have "2 powr k + (r + (sgn ai) * b) = 2 powr k * (1 + (r + sgn ai * b) / 2 powr k)"
immler@58985
  1533
      by (simp add: field_simps)
immler@58985
  1534
    also have "\<lfloor>log 2 \<dots>\<rfloor> = k + \<lfloor>log 2 (1 + (r + sgn ai * b) / 2 powr k)\<rfloor>"
immler@58985
  1535
      using pos[OF less]
immler@58985
  1536
      by (subst log_mult) (simp_all add: log_mult powr_mult field_simps)
immler@58985
  1537
    also
immler@58985
  1538
    let ?if = "if r = 0 \<and> sgn ai * b < 0 then -1 else 0"
immler@58985
  1539
    have "\<lfloor>log 2 (1 + (r + sgn ai * b) / 2 powr k)\<rfloor> = ?if"
wenzelm@60500
  1540
      using \<open>abs b <= _\<close>
immler@58985
  1541
      by (intro floor_eq) (auto simp: abs_mult sgn_if)
immler@58985
  1542
    also
immler@58985
  1543
    have "\<dots> = \<lfloor>log 2 (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k)\<rfloor>"
immler@58985
  1544
      by (subst floor_eq) (auto simp: sgn_if)
immler@58985
  1545
    also have "k + \<dots> = \<lfloor>log 2 (2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k))\<rfloor>"
immler@58985
  1546
      unfolding floor_add2[symmetric]
wenzelm@60500
  1547
      using pos[OF less'] \<open>abs b \<le> _\<close>
immler@58985
  1548
      by (simp add: field_simps add_log_eq_powr)
immler@58985
  1549
    also have "2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k) =
immler@58985
  1550
        2 powr k + r + sgn (sgn ai * b) / 2"
immler@58985
  1551
      by (simp add: sgn_if field_simps)
immler@58985
  1552
    finally show ?thesis .
immler@58985
  1553
  qed
immler@58985
  1554
  also have "2 powr k + r + sgn (sgn ai * b) / 2 = \<bar>ai + sgn b / 2\<bar>"
wenzelm@60500
  1555
    unfolding \<open>real \<bar>ai\<bar> = _\<close>[symmetric] using \<open>ai \<noteq> 0\<close>
immler@58985
  1556
    by (auto simp: abs_if sgn_if algebra_simps)
immler@58985
  1557
  finally show ?thesis .
immler@58985
  1558
qed
immler@58985
  1559
immler@58985
  1560
lemma compute_far_float_plus_down:
immler@58985
  1561
  fixes m1 e1 m2 e2::int and p::nat
immler@58985
  1562
  defines "k1 \<equiv> p - nat (bitlen \<bar>m1\<bar>)"
immler@58985
  1563
  assumes H: "bitlen \<bar>m2\<bar> \<le> e1 - e2 - k1 - 2" "m1 \<noteq> 0" "m2 \<noteq> 0" "e1 \<ge> e2"
immler@58985
  1564
  shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
immler@58985
  1565
    float_round_down p (Float (m1 * 2 ^ (Suc (Suc k1)) + sgn m2) (e1 - int k1 - 2))"
immler@58985
  1566
proof -
immler@58985
  1567
  let ?a = "real (Float m1 e1)"
immler@58985
  1568
  let ?b = "real (Float m2 e2)"
immler@58985
  1569
  let ?sum = "?a + ?b"
immler@58985
  1570
  let ?shift = "real e2 - real e1 + real k1 + 1"
immler@58985
  1571
  let ?m1 = "m1 * 2 ^ Suc k1"
immler@58985
  1572
  let ?m2 = "m2 * 2 powr ?shift"
immler@58985
  1573
  let ?m2' = "sgn m2 / 2"
immler@58985
  1574
  let ?e = "e1 - int k1 - 1"
immler@58985
  1575
immler@58985
  1576
  have sum_eq: "?sum = (?m1 + ?m2) * 2 powr ?e"
immler@58985
  1577
    by (auto simp: powr_add[symmetric] powr_mult[symmetric] algebra_simps
immler@58985
  1578
      powr_realpow[symmetric] powr_mult_base)
immler@58985
  1579
immler@58985
  1580
  have "\<bar>?m2\<bar> * 2 < 2 powr (bitlen \<bar>m2\<bar> + ?shift + 1)"
immler@58985
  1581
    by (auto simp: field_simps powr_add powr_mult_base powr_numeral powr_divide2[symmetric] abs_mult)
immler@58985
  1582
  also have "\<dots> \<le> 2 powr 0"
immler@58985
  1583
    using H by (intro powr_mono) auto
immler@58985
  1584
  finally have abs_m2_less_half: "\<bar>?m2\<bar> < 1 / 2"
immler@58985
  1585
    by simp
immler@58985
  1586
immler@58985
  1587
  hence "\<bar>real m2\<bar> < 2 powr -(?shift + 1)"
immler@58985
  1588
    unfolding powr_minus_divide by (auto simp: bitlen_def field_simps powr_mult_base abs_mult)
immler@58985
  1589
  also have "\<dots> \<le> 2 powr real (e1 - e2 - 2)"
immler@58985
  1590
    by simp
immler@58985
  1591
  finally have b_less_quarter: "\<bar>?b\<bar> < 1/4 * 2 powr real e1"
immler@58985
  1592
    by (simp add: powr_add field_simps powr_divide2[symmetric] powr_numeral abs_mult)
wenzelm@60500
  1593
  also have "1/4 < \<bar>real m1\<bar> / 2" using \<open>m1 \<noteq> 0\<close> by simp
immler@58985
  1594
  finally have b_less_half_a: "\<bar>?b\<bar> < 1/2 * \<bar>?a\<bar>"
immler@58985
  1595
    by (simp add: algebra_simps powr_mult_base abs_mult)
immler@58985
  1596
  hence a_half_less_sum: "\<bar>?a\<bar> / 2 < \<bar>?sum\<bar>"
immler@58985
  1597
    by (auto simp: field_simps abs_if split: split_if_asm)
immler@58985
  1598
immler@58985
  1599
  from b_less_half_a have "\<bar>?b\<bar> < \<bar>?a\<bar>" "\<bar>?b\<bar> \<le> \<bar>?a\<bar>"
immler@58985
  1600
    by simp_all
immler@58985
  1601
immler@58985
  1602
  have "\<bar>real (Float m1 e1)\<bar> \<ge> 1/4 * 2 powr real e1"
wenzelm@60500
  1603
    using \<open>m1 \<noteq> 0\<close>
immler@58985
  1604
    by (auto simp: powr_add powr_int bitlen_nonneg divide_right_mono abs_mult)
immler@58985
  1605
  hence "?sum \<noteq> 0" using b_less_quarter
immler@58985
  1606
    by (rule sum_neq_zeroI)
immler@58985
  1607
  hence "?m1 + ?m2 \<noteq> 0"
immler@58985
  1608
    unfolding sum_eq by (simp add: abs_mult zero_less_mult_iff)
immler@58985
  1609
immler@58985
  1610
  have "\<bar>real ?m1\<bar> \<ge> 2 ^ Suc k1" "\<bar>?m2'\<bar> < 2 ^ Suc k1"
wenzelm@60500
  1611
    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)
immler@58985
  1612
  hence sum'_nz: "?m1 + ?m2' \<noteq> 0"
immler@58985
  1613
    by (intro sum_neq_zeroI)
immler@58985
  1614
immler@58985
  1615
  have "\<lfloor>log 2 \<bar>real (Float m1 e1) + real (Float m2 e2)\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> + ?e"
wenzelm@60500
  1616
    using \<open>?m1 + ?m2 \<noteq> 0\<close>
immler@58985
  1617
    unfolding floor_add[symmetric] sum_eq
immler@58985
  1618
    by (simp add: abs_mult log_mult)
immler@58985
  1619
  also have "\<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + sgn (real m2 * 2 powr ?shift) / 2\<bar>\<rfloor>"
wenzelm@60500
  1620
    using abs_m2_less_half \<open>m1 \<noteq> 0\<close>
immler@58985
  1621
    by (intro log2_abs_int_add_less_half_sgn_eq) (auto simp: abs_mult)
immler@58985
  1622
  also have "sgn (real m2 * 2 powr ?shift) = sgn m2"
immler@58985
  1623
    by (auto simp: sgn_if zero_less_mult_iff less_not_sym)
immler@58985
  1624
  also
immler@58985
  1625
  have "\<bar>?m1 + ?m2'\<bar> * 2 powr ?e = \<bar>?m1 * 2 + sgn m2\<bar> * 2 powr (?e - 1)"
immler@58985
  1626
    by (auto simp: field_simps powr_minus[symmetric] powr_divide2[symmetric] powr_mult_base)
immler@58985
  1627
  hence "\<lfloor>log 2 \<bar>?m1 + ?m2'\<bar>\<rfloor> + ?e = \<lfloor>log 2 \<bar>real (Float (?m1 * 2 + sgn m2) (?e - 1))\<bar>\<rfloor>"
wenzelm@60500
  1628
    using \<open>?m1 + ?m2' \<noteq> 0\<close>
immler@58985
  1629
    unfolding floor_add[symmetric]
immler@58985
  1630
    by (simp add: log_add_eq_powr abs_mult_pos)
immler@58985
  1631
  finally
immler@58985
  1632
  have "\<lfloor>log 2 \<bar>?sum\<bar>\<rfloor> = \<lfloor>log 2 \<bar>real (Float (?m1*2 + sgn m2) (?e - 1))\<bar>\<rfloor>" .
immler@58985
  1633
  hence "plus_down p (Float m1 e1) (Float m2 e2) =
immler@58985
  1634
      truncate_down p (Float (?m1*2 + sgn m2) (?e - 1))"
immler@58985
  1635
    unfolding plus_down_def
immler@58985
  1636
  proof (rule truncate_down_log2_eqI)
immler@58985
  1637
    let ?f = "(int p - \<lfloor>log 2 \<bar>real (Float m1 e1) + real (Float m2 e2)\<bar>\<rfloor> - 1)"
immler@58985
  1638
    let ?ai = "m1 * 2 ^ (Suc k1)"
immler@58985
  1639
    have "\<lfloor>(?a + ?b) * 2 powr real ?f\<rfloor> = \<lfloor>(real (2 * ?ai) + sgn ?b) * 2 powr real (?f - - ?e - 1)\<rfloor>"
immler@58985
  1640
    proof (rule floor_sum_times_2_powr_sgn_eq)
immler@58985
  1641
      show "?a * 2 powr real (-?e) = real ?ai"
immler@58985
  1642
        by (simp add: powr_add powr_realpow[symmetric] powr_divide2[symmetric])
immler@58985
  1643
      show "\<bar>?b * 2 powr real (-?e + 1)\<bar> \<le> 1"
immler@58985
  1644
        using abs_m2_less_half
immler@58985
  1645
        by (simp add: abs_mult powr_add[symmetric] algebra_simps powr_mult_base)
immler@58985
  1646
    next
immler@58985
  1647
      have "e1 + \<lfloor>log 2 \<bar>real m1\<bar>\<rfloor> - 1 = \<lfloor>log 2 \<bar>?a\<bar>\<rfloor> - 1"
wenzelm@60500
  1648
        using \<open>m1 \<noteq> 0\<close>
immler@58985
  1649
        by (simp add: floor_add2[symmetric] algebra_simps log_mult abs_mult del: floor_add2)
immler@58985
  1650
      also have "\<dots> \<le> \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor>"
wenzelm@60500
  1651
        using a_half_less_sum \<open>m1 \<noteq> 0\<close> \<open>?sum \<noteq> 0\<close>
immler@58985
  1652
        unfolding floor_subtract[symmetric]
immler@58985
  1653
        by (auto simp add: log_minus_eq_powr powr_minus_divide
immler@58985
  1654
          intro!: floor_mono)
immler@58985
  1655
      finally
immler@58985
  1656
      have "int p - \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor> \<le> p - (bitlen \<bar>m1\<bar>) - e1 + 2"
wenzelm@60500
  1657
        by (auto simp: algebra_simps bitlen_def \<open>m1 \<noteq> 0\<close>)
immler@58985
  1658
      also have "\<dots> \<le> 1 - ?e"
immler@58985
  1659
        using bitlen_nonneg[of "\<bar>m1\<bar>"] by (simp add: k1_def)
immler@58985
  1660
      finally show "?f \<le> - ?e" by simp
immler@58985
  1661
    qed
immler@58985
  1662
    also have "sgn ?b = sgn m2"
immler@58985
  1663
      using powr_gt_zero[of 2 e2]
immler@58985
  1664
      by (auto simp add: sgn_if zero_less_mult_iff simp del: powr_gt_zero)
immler@58985
  1665
    also have "\<lfloor>(real (2 * ?m1) + real (sgn m2)) * 2 powr real (?f - - ?e - 1)\<rfloor> =
immler@58985
  1666
        \<lfloor>Float (?m1 * 2 + sgn m2) (?e - 1) * 2 powr ?f\<rfloor>"
immler@58985
  1667
      by (simp add: powr_add[symmetric] algebra_simps powr_realpow[symmetric])
immler@58985
  1668
    finally
immler@58985
  1669
    show "\<lfloor>(?a + ?b) * 2 powr ?f\<rfloor> = \<lfloor>real (Float (?m1 * 2 + sgn m2) (?e - 1)) * 2 powr ?f\<rfloor>" .
immler@58985
  1670
  qed
immler@58985
  1671
  thus ?thesis
immler@58985
  1672
    by transfer (simp add: plus_down_def ac_simps Let_def)
immler@58985
  1673
qed
immler@58985
  1674
immler@58985
  1675
lemma compute_float_plus_down_naive[code]: "float_plus_down p x y = float_round_down p (x + y)"
immler@58985
  1676
  by transfer (auto simp: plus_down_def)
immler@58985
  1677
immler@58985
  1678
lemma compute_float_plus_down[code]:
immler@58985
  1679
  fixes p::nat and m1 e1 m2 e2::int
immler@58985
  1680
  shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
immler@58985
  1681
    (if m1 = 0 then float_round_down p (Float m2 e2)
immler@58985
  1682
    else if m2 = 0 then float_round_down p (Float m1 e1)
immler@58985
  1683
    else (if e1 \<ge> e2 then
immler@58985
  1684
      (let
immler@58985
  1685
        k1 = p - nat (bitlen \<bar>m1\<bar>)
immler@58985
  1686
      in
immler@58985
  1687
        if bitlen \<bar>m2\<bar> > e1 - e2 - k1 - 2 then float_round_down p ((Float m1 e1) + (Float m2 e2))
immler@58985
  1688
        else float_round_down p (Float (m1 * 2 ^ (Suc (Suc k1)) + sgn m2) (e1 - int k1 - 2)))
immler@58985
  1689
    else float_plus_down p (Float m2 e2) (Float m1 e1)))"
immler@58985
  1690
proof -
immler@58985
  1691
  {
immler@58985
  1692
    assume H: "bitlen \<bar>m2\<bar> \<le> e1 - e2 - (p - nat (bitlen \<bar>m1\<bar>)) - 2" "m1 \<noteq> 0" "m2 \<noteq> 0" "e1 \<ge> e2"
immler@58985
  1693
    note compute_far_float_plus_down[OF H]
immler@58985
  1694
  }
immler@58985
  1695
  thus ?thesis
immler@58985
  1696
    by transfer (simp add: Let_def plus_down_def ac_simps)
immler@58985
  1697
qed
immler@58985
  1698
hide_fact (open) compute_far_float_plus_down
immler@58985
  1699
hide_fact (open) compute_float_plus_down
immler@58985
  1700
immler@58985
  1701
lemma compute_float_plus_up[code]: "float_plus_up p x y = - float_plus_down p (-x) (-y)"
immler@58985
  1702
  using truncate_down_uminus_eq[of p "x + y"]
immler@58985
  1703
  by transfer (simp add: plus_down_def plus_up_def ac_simps)
immler@58985
  1704
hide_fact (open) compute_float_plus_up
immler@58985
  1705
immler@58985
  1706
lemma mantissa_zero[simp]: "mantissa 0 = 0"
immler@58985
  1707
by (metis mantissa_0 zero_float.abs_eq)
immler@58985
  1708
immler@58985
  1709
wenzelm@60500
  1710
subsection \<open>Lemmas needed by Approximate\<close>
hoelzl@47599
  1711
hoelzl@47599
  1712
lemma Float_num[simp]: shows
hoelzl@47599
  1713
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
haftmann@58410
  1714
   "real (Float 1 (- 1)) = 1/2" and "real (Float 1 (- 2)) = 1/4" and "real (Float 1 (- 3)) = 1/8" and
haftmann@58410
  1715
   "real (Float (- 1) 0) = -1" and "real (Float (number_of n) 0) = number_of n"
hoelzl@47599
  1716
using two_powr_int_float[of 2] two_powr_int_float[of "-1"] two_powr_int_float[of "-2"] two_powr_int_float[of "-3"]
hoelzl@47599
  1717
using powr_realpow[of 2 2] powr_realpow[of 2 3]
hoelzl@47599
  1718
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
hoelzl@47599
  1719
by auto
hoelzl@47599
  1720
hoelzl@47599
  1721
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp
hoelzl@47599
  1722
hoelzl@47599
  1723
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
hoelzl@47599
  1724
hoelzl@47599
  1725
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
hoelzl@47599
  1726
by arith
hoelzl@29804
  1727
hoelzl@47599
  1728
lemma lapprox_rat:
hoelzl@47599
  1729
  shows "real (lapprox_rat prec x y) \<le> real x / real y"
hoelzl@47599
  1730
  using round_down by (simp add: lapprox_rat_def)
obua@16782
  1731
hoelzl@47599
  1732
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
hoelzl@47599
  1733
proof -
hoelzl@47599
  1734
  from zmod_zdiv_equality'[of a b]
hoelzl@47599
  1735
  have "a = b * (a div b) + a mod b" by simp
hoelzl@47599
  1736
  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
hoelzl@47599
  1737
  using assms by simp
hoelzl@47599
  1738
  finally show ?thesis by simp
hoelzl@47599
  1739
qed
hoelzl@47599
  1740
hoelzl@47599
  1741
lemma lapprox_rat_nonneg:
hoelzl@47599
  1742
  fixes n x y
immler@58982
  1743
  assumes "0 \<le> x" and "0 \<le> y"
hoelzl@47599
  1744
  shows "0 \<le> real (lapprox_rat n x y)"
immler@58982
  1745
  using assms by (auto simp: lapprox_rat_def simp: round_down_nonneg)
obua@16782
  1746
hoelzl@31098
  1747
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
hoelzl@47599
  1748
  using round_up by (simp add: rapprox_rat_def)
hoelzl@47599
  1749
hoelzl@47599
  1750
lemma rapprox_rat_le1:
hoelzl@47599
  1751
  fixes n x y
hoelzl@47599
  1752
  assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
hoelzl@47599
  1753
  shows "real (rapprox_rat n x y) \<le> 1"
hoelzl@47599
  1754
proof -
hoelzl@47599
  1755
  have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
hoelzl@47599
  1756
    using xy unfolding bitlen_def by (auto intro!: floor_mono)
immler@58982
  1757
  from this assms show ?thesis
immler@58982
  1758
    by transfer (auto intro!: round_up_le1 simp: rat_precision_def)
hoelzl@29804
  1759
qed
obua@16782
  1760
immler@58982
  1761
lemma rapprox_rat_nonneg_nonpos:
immler@58982
  1762
  "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
immler@58982
  1763
  by transfer (simp add: round_up_le0 divide_nonneg_nonpos)
obua@16782
  1764
immler@58982
  1765
lemma rapprox_rat_nonpos_nonneg:
immler@58982
  1766
  "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
immler@58982
  1767
  by transfer (simp add: round_up_le0 divide_nonpos_nonneg)
obua@16782
  1768
immler@54782
  1769
lemma real_divl: "real_divl prec x y \<le> x / y"
immler@54782
  1770
  by (simp add: real_divl_def round_down)
immler@54782
  1771
immler@54782
  1772
lemma real_divr: "x / y \<le> real_divr prec x y"
immler@54782
  1773
  using round_up by (simp add: real_divr_def)
immler@54782
  1774
hoelzl@31098
  1775
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
immler@54782
  1776
  by transfer (rule real_divl)
immler@54782
  1777
immler@54782
  1778
lemma real_divl_lower_bound:
immler@54782
  1779
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y"
immler@58982
  1780
  by (simp add: real_divl_def round_down_nonneg)
hoelzl@47599
  1781
hoelzl@47599
  1782
lemma float_divl_lower_bound:
immler@54782
  1783
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
immler@54782
  1784
  by transfer (rule real_divl_lower_bound)
hoelzl@47599
  1785
hoelzl@47599
  1786
lemma exponent_1: "exponent 1 = 0"
hoelzl@47599
  1787
  using exponent_float[of 1 0] by (simp add: one_float_def)
hoelzl@47599
  1788
hoelzl@47599
  1789
lemma mantissa_1: "mantissa 1 = 1"
hoelzl@47599
  1790
  using mantissa_float[of 1 0] by (simp add: one_float_def)
obua@16782
  1791
hoelzl@47599
  1792
lemma bitlen_1: "bitlen 1 = 1"
hoelzl@47599
  1793
  by (simp add: bitlen_def)
hoelzl@47599
  1794
hoelzl@47599
  1795
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
hoelzl@47599
  1796
proof
hoelzl@47599
  1797
  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
hoelzl@47599
  1798
  show "x = 0" by (simp add: zero_float_def z)
hoelzl@47599
  1799
qed (simp add: zero_float_def)
obua@16782
  1800
hoelzl@47599
  1801
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
hoelzl@47599
  1802
proof (cases "x = 0", simp)
hoelzl@47599
  1803
  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
hoelzl@47599
  1804
  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
hoelzl@47599
  1805
  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
hoelzl@47599
  1806
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
wenzelm@60500
  1807
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg \<open>mantissa x \<noteq> 0\<close>
immler@58989
  1808
    by (auto simp del: real_of_int_abs simp add: powr_int)
hoelzl@47599
  1809
  finally show ?thesis by (simp add: powr_add)
hoelzl@29804
  1810
qed
hoelzl@29804
  1811
immler@54782
  1812
lemma real_divl_pos_less1_bound:
immler@58982
  1813
  assumes "0 < x" "x \<le> 1" "prec \<ge> 1"
immler@58982
  1814
  shows "1 \<le> real_divl prec 1 x"
immler@58982
  1815
proof -
wenzelm@60500
  1816
  have "log 2 x \<le> real prec + real \<lfloor>log 2 x\<rfloor>" using \<open>prec \<ge> 1\<close> by arith
immler@58982
  1817
  from this assms show ?thesis
immler@58982
  1818
    by (simp add: real_divl_def log_divide round_down_ge1)
hoelzl@29804
  1819
qed
obua@16782
  1820
immler@54782
  1821
lemma float_divl_pos_less1_bound:
immler@58982
  1822
  "0 < real x \<Longrightarrow> real x \<le> 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
immler@54782
  1823
  by (transfer, rule real_divl_pos_less1_bound)
obua@16782
  1824
immler@54782
  1825
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
immler@54782
  1826
  by transfer (rule real_divr)
immler@54782
  1827
immler@58982
  1828
lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x \<le> 1" shows "1 \<le> real_divr prec 1 x"
hoelzl@29804
  1829
proof -
wenzelm@60500
  1830
  have "1 \<le> 1 / x" using \<open>0 < x\<close> and \<open>x <= 1\<close> by auto
immler@54782
  1831
  also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto
hoelzl@47600
  1832
  finally show ?thesis by auto
hoelzl@29804
  1833
qed
hoelzl@29804
  1834
immler@58982
  1835
lemma float_divr_pos_less1_lower_bound: "0 < x \<Longrightarrow> x \<le> 1 \<Longrightarrow> 1 \<le> float_divr prec 1 x"
immler@54782
  1836
  by transfer (rule real_divr_pos_less1_lower_bound)
immler@54782
  1837
immler@54782
  1838
lemma real_divr_nonpos_pos_upper_bound:
immler@58982
  1839
  "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real_divr prec x y \<le> 0"
immler@58982
  1840
  by (simp add: real_divr_def round_up_le0 divide_le_0_iff)
immler@54782
  1841
hoelzl@47599
  1842
lemma float_divr_nonpos_pos_upper_bound:
immler@58982
  1843
  "real x \<le> 0 \<Longrightarrow> 0 \<le> real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"
immler@54782
  1844
  by transfer (rule real_divr_nonpos_pos_upper_bound)
immler@54782
  1845
immler@54782
  1846
lemma real_divr_nonneg_neg_upper_bound:
immler@58982
  1847
  "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real_divr prec x y \<le> 0"
immler@58982
  1848
  by (simp add: real_divr_def round_up_le0 divide_le_0_iff)
obua@16782
  1849
hoelzl@47599
  1850
lemma float_divr_nonneg_neg_upper_bound:
immler@58982
  1851
  "0 \<le> real x \<Longrightarrow> real y \<le> 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"
immler@54782
  1852
  by transfer (rule real_divr_nonneg_neg_upper_bound)
immler@54782
  1853
immler@54784
  1854
lemma truncate_up_nonneg_mono:
immler@54784
  1855
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  1856
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  1857
proof -
immler@54784
  1858
  {
immler@54784
  1859
    assume "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>"
immler@54784
  1860
    hence ?thesis
immler@54784
  1861
      using assms
immler@54784
  1862
      by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
immler@54784
  1863
  } moreover {
immler@54784
  1864
    assume "0 < x"
immler@54784
  1865
    hence "log 2 x \<le> log 2 y" using assms by auto
immler@54784
  1866
    moreover
immler@54784
  1867
    assume "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>"
immler@54784
  1868
    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
immler@54784
  1869
      unfolding atomize_conj
immler@54784
  1870
      by (metis floor_less_cancel linorder_cases not_le)
immler@54784
  1871
    have "truncate_up prec x =
immler@54784
  1872
      real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> * 2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1)"
immler@54784
  1873
      using assms by (simp add: truncate_up_def round_up_def)
immler@54784
  1874
    also have "\<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)"
immler@54784
  1875
    proof (unfold ceiling_le_eq)
immler@54784
  1876
      have "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> x * (2 powr real prec / (2 powr log 2 x))"
immler@54784
  1877
        using real_of_int_floor_add_one_ge[of "log 2 x"] assms
immler@54784
  1878
        by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono)
immler@54784
  1879
      thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
wenzelm@60500
  1880
        using \<open>0 < x\<close> by (simp add: powr_realpow)
immler@54784
  1881
    qed
immler@54784
  1882
    hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
immler@54784
  1883
      by (auto simp: powr_realpow)
immler@54784
  1884
    also
immler@54784
  1885
    have "2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
immler@54784
  1886
      using logless flogless by (auto intro!: floor_mono)
immler@54784
  1887
    also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))"
wenzelm@60500
  1888
      using assms \<open>0 < x\<close>
immler@54784
  1889
      by (auto simp: algebra_simps)
immler@54784
  1890
    finally have "truncate_up prec x \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>)) * 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
immler@54784
  1891
      by simp
immler@54784
  1892
    also have "\<dots> = 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>) - real (int prec - \<lfloor>log 2 y\<rfloor>))"
immler@54784
  1893
      by (subst powr_add[symmetric]) simp
immler@54784
  1894
    also have "\<dots> = y"
wenzelm@60500
  1895
      using \<open>0 < x\<close> assms
immler@54784
  1896
      by (simp add: powr_add)
immler@54784
  1897
    also have "\<dots> \<le> truncate_up prec y"
immler@54784
  1898
      by (rule truncate_up)
immler@54784
  1899
    finally have ?thesis .
immler@54784
  1900
  } moreover {
immler@54784
  1901
    assume "~ 0 < x"
immler@54784
  1902
    hence ?thesis
immler@54784
  1903
      using assms
immler@54784
  1904
      by (auto intro!: truncate_up_le)
immler@54784
  1905
  } ultimately show ?thesis
immler@54784
  1906
    by blast
immler@54784
  1907
qed
immler@54784
  1908
immler@54784
  1909
lemma truncate_up_switch_sign_mono:
immler@54784
  1910
  assumes "x \<le> 0" "0 \<le> y"
immler@54784
  1911
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  1912
proof -
wenzelm@60500
  1913
  note truncate_up_nonpos[OF \<open>x \<le> 0\<close>]
wenzelm@60500
  1914
  also note truncate_up_le[OF \<open>0 \<le> y\<close>]
immler@54784
  1915
  finally show ?thesis .
immler@54784
  1916
qed
immler@54784
  1917
immler@54784
  1918
lemma truncate_down_zeroprec_mono:
immler@54784
  1919
  assumes "0 < x" "x \<le> y"
immler@54784
  1920
  shows "truncate_down 0 x \<le> truncate_down 0 y"
immler@54784
  1921
proof -
immler@54784
  1922
  have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))"
immler@54784
  1923
    by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide)
immler@54784
  1924
  also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)"
wenzelm@60500
  1925
    using \<open>0 < x\<close>
wenzelm@57862
  1926
    by (auto simp: field_simps powr_add powr_divide2[symmetric])
immler@54784
  1927
  also have "\<dots> < 2 powr 0"
immler@54784
  1928
    using real_of_int_floor_add_one_gt
immler@54784
  1929
    unfolding neg_less_iff_less
immler@54784
  1930
    by (intro powr_less_mono) (auto simp: algebra_simps)
immler@54784
  1931
  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1"
immler@54784
  1932
    unfolding less_ceiling_eq real_of_int_minus real_of_one
immler@54784
  1933
    by simp
immler@54784
  1934
  moreover
immler@54784
  1935
  have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"
wenzelm@60500
  1936
    using \<open>x > 0\<close> by auto
immler@54784
  1937
  ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"
immler@54784
  1938
    by simp
immler@54784
  1939
  also have "\<dots> \<subseteq> {0}" by auto
immler@54784
  1940
  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0" by simp
immler@54784
  1941
  with assms show ?thesis
nipkow@56536
  1942
    by (auto simp: truncate_down_def round_down_def)
immler@54784
  1943
qed
immler@54784
  1944
immler@54784
  1945
lemma truncate_down_switch_sign_mono:
immler@54784
  1946
  assumes "x \<le> 0" "0 \<le> y"
immler@54784
  1947
  assumes "x \<le> y"
immler@54784
  1948
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  1949
proof -
wenzelm@60500
  1950
  note truncate_down_le[OF \<open>x \<le> 0\<close>]
wenzelm@60500
  1951
  also note truncate_down_nonneg[OF \<open>0 \<le> y\<close>]
immler@54784
  1952
  finally show ?thesis .
immler@54784
  1953
qed
immler@54784
  1954
immler@54784
  1955
lemma truncate_down_nonneg_mono:
immler@54784
  1956
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  1957
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  1958
proof -
immler@54784
  1959
  {
immler@54784
  1960
    assume "0 < x" "prec = 0"
immler@54784
  1961
    with assms have ?thesis
immler@54784
  1962
      by (simp add: truncate_down_zeroprec_mono)
immler@54784
  1963
  } moreover {
immler@54784
  1964
    assume "~ 0 < x"
immler@54784
  1965
    with assms have "x = 0" "0 \<le> y" by simp_all
immler@54784
  1966
    hence ?thesis
immler@58985
  1967
      by (auto intro!: truncate_down_nonneg)
immler@54784
  1968
  } moreover {
immler@54784
  1969
    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@54784
  1970
    hence ?thesis
immler@54784
  1971
      using assms
immler@54784
  1972
      by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
immler@54784
  1973
  } moreover {
immler@54784
  1974
    assume "0 < x"
immler@54784
  1975
    hence "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y" using assms by auto
immler@54784
  1976
    moreover
immler@54784
  1977
    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@54784
  1978
    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
wenzelm@60500
  1979
      unfolding atomize_conj abs_of_pos[OF \<open>0 < x\<close>] abs_of_pos[OF \<open>0 < y\<close>]
immler@54784
  1980
      by (metis floor_less_cancel linorder_cases not_le)
immler@54784
  1981
    assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto
immler@54784
  1982
    have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)"
wenzelm@60500
  1983
      using \<open>0 < y\<close>
immler@54784
  1984
      by simp
immler@54784
  1985
    also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))"
wenzelm@60500
  1986
      using \<open>0 \<le> y\<close> \<open>0 \<le> x\<close> assms(2)
nipkow@56544
  1987
      by (auto intro!: powr_mono divide_left_mono
immler@54784
  1988
        simp: real_of_nat_diff powr_add
immler@54784
  1989
        powr_divide2[symmetric])
immler@54784
  1990
    also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)"
immler@54784
  1991
      by (auto simp: powr_add)
immler@54784
  1992
    finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
wenzelm@60500
  1993
      using \<open>0 \<le> y\<close>
immler@54784
  1994
      by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow)
immler@54784
  1995
    hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
immler@54784
  1996
      by (auto simp: truncate_down_def round_down_def)
immler@54784
  1997
    moreover
immler@54784
  1998
    {
wenzelm@60500
  1999
      have "x = 2 powr (log 2 \<bar>x\<bar>)" using \<open>0 < x\<close> by simp
immler@54784
  2000
      also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"
immler@54784
  2001
        using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"]
immler@54784
  2002
        by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps)
immler@54784
  2003
      also
immler@54784
  2004
      have "2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)"
wenzelm@60500
  2005
        using logless flogless \<open>x > 0\<close> \<open>y > 0\<close>
immler@54784
  2006
        by (auto intro!: floor_mono)
immler@54784
  2007
      finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"
immler@54784
  2008
        by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff)
immler@54784
  2009
    } ultimately have ?thesis
immler@54784
  2010
      by (metis dual_order.trans truncate_down)
immler@54784
  2011
  } ultimately show ?thesis by blast
immler@54784
  2012
qed
immler@54784
  2013
immler@58982
  2014
lemma truncate_down_eq_truncate_up: "truncate_down p x = - truncate_up p (-x)"
immler@58982
  2015
  and truncate_up_eq_truncate_down: "truncate_up p x = - truncate_down p (-x)"
immler@58982
  2016
  by (auto simp: truncate_up_uminus_eq truncate_down_uminus_eq)
immler@58982
  2017
immler@54784
  2018
lemma truncate_down_mono: "x \<le> y \<Longrightarrow> truncate_down p x \<le> truncate_down p y"
immler@54784
  2019
  apply (cases "0 \<le> x")
immler@54784
  2020
  apply (rule truncate_down_nonneg_mono, assumption+)
immler@58982
  2021
  apply (simp add: truncate_down_eq_truncate_up)
immler@54784
  2022
  apply (cases "0 \<le> y")
immler@54784
  2023
  apply (auto intro: truncate_up_nonneg_mono truncate_up_switch_sign_mono)
immler@54784
  2024
  done
immler@54784
  2025
immler@54784
  2026
lemma truncate_up_mono: "x \<le> y \<Longrightarrow> truncate_up p x \<le> truncate_up p y"
immler@58982
  2027
  by (simp add: truncate_up_eq_truncate_down truncate_down_mono)
immler@54784
  2028
hoelzl@47599
  2029
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
lp15@60017
  2030
 by (auto simp: zero_float_def mult_le_0_iff) (simp add: not_less [symmetric])
hoelzl@47599
  2031
hoelzl@47621
  2032
lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
hoelzl@47600
  2033
  unfolding pprt_def sup_float_def max_def sup_real_def by auto
hoelzl@47599
  2034
hoelzl@47621
  2035
lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
hoelzl@47600
  2036
  unfolding nprt_def inf_float_def min_def inf_real_def by auto
hoelzl@47599
  2037
kuncar@55565
  2038
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor .
obua@16782
  2039
hoelzl@47599
  2040
lemma compute_int_floor_fl[code]:
hoelzl@47601
  2041
  "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
hoelzl@47600
  2042
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  2043
hide_fact (open) compute_int_floor_fl
hoelzl@47599
  2044
hoelzl@47600
  2045
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp
hoelzl@47599
  2046
hoelzl@47599
  2047
lemma compute_floor_fl[code]:
hoelzl@47601
  2048
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
hoelzl@47600
  2049
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  2050
hide_fact (open) compute_floor_fl
obua@16782
  2051
hoelzl@47600
  2052
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp
hoelzl@47600
  2053
hoelzl@47600
  2054
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp
hoelzl@29804
  2055
hoelzl@47599
  2056
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
wenzelm@53381
  2057
proof (cases "floor_fl x = float_of 0")
wenzelm@53381
  2058
  case True
wenzelm@53381
  2059
  then show ?thesis by (simp add: floor_fl_def)
wenzelm@53381
  2060
next
wenzelm@53381
  2061
  case False
wenzelm@53381
  2062
  have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp
wenzelm@53381
  2063
  obtain i where "\<lfloor>real x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
wenzelm@53381
  2064
    by (rule denormalize_shift[OF eq[THEN eq_reflection] False])
wenzelm@53381
  2065
  then show ?thesis by simp
wenzelm@53381
  2066
qed
obua@16782
  2067
immler@58985
  2068
lemma compute_mantissa[code]:
immler@58985
  2069
  "mantissa (Float m e) = (if m = 0 then 0 else if 2 dvd m then mantissa (normfloat (Float m e)) else m)"
immler@58985
  2070
  by (auto simp: mantissa_float Float.abs_eq)
immler@58985
  2071
immler@58985
  2072
lemma compute_exponent[code]:
immler@58985
  2073
  "exponent (Float m e) = (if m = 0 then 0 else if 2 dvd m then exponent (normfloat (Float m e)) else e)"
immler@58985
  2074
  by (auto simp: exponent_float Float.abs_eq)
immler@58985
  2075
obua@16782
  2076
end
hoelzl@47599
  2077