src/HOL/Library/Float.thy
author wenzelm
Tue Aug 05 12:56:15 2014 +0200 (2014-08-05)
changeset 57862 8f074e6e22fc
parent 57512 cc97b347b301
child 58042 ffa9e39763e3
permissions -rw-r--r--
tuned proofs;
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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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header {* Floating-Point Numbers *}
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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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defs (overloaded)
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  real_of_float_def[code_unfold]: "real \<equiv> real_of_float"
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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 (no_code) 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 {* Real operations preserving the representation as floating point number *}
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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 (rule_tac x="-x" in exI)
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  apply (rule_tac x="xa" 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 (rule_tac x="x * xa" in exI)
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  apply (rule_tac x="xb + xc" 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 (rule_tac x="x" in exI)
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  apply (rule_tac x="xa - 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 (rule_tac x="x" in exI)
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  apply (rule_tac x="xa - 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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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 {* Arithmetic operations on floating point numbers *}
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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 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 {* Represent floats as unique mantissa and exponent *}
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lemma int_induct_abs[case_names less]:
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  fixes j :: int
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  assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
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  shows "P j"
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proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)
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  case less show ?case by (rule H[OF less]) simp
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qed
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lemma int_cancel_factors:
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  fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
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proof (induct n rule: int_induct_abs)
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  case (less n)
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  { fix m assume n: "n \<noteq> 0" "n = m * r"
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    then have "\<bar>m \<bar> < \<bar>n\<bar>"
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      by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7)
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                dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le
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                mult_eq_0_iff zdvd_mult_cancel1)
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    from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
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  then show ?case
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    by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0)
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qed
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lemma mult_powr_eq_mult_powr_iff_asym:
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  fixes m1 m2 e1 e2 :: int
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  assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"
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  shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
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proof
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  have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
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  assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
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  with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)"
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    by (simp add: powr_divide2[symmetric] field_simps)
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  also have "\<dots> = m2 * 2^nat (e2 - e1)"
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    by (simp add: powr_realpow)
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  finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
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    unfolding real_of_int_inject .
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  with m1 have "m1 = m2"
hoelzl@47599
   283
    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
hoelzl@47599
   284
  then show "m1 = m2 \<and> e1 = e2"
hoelzl@47599
   285
    using eq `m1 \<noteq> 0` by (simp add: powr_inj)
hoelzl@47599
   286
qed simp
hoelzl@47599
   287
hoelzl@47599
   288
lemma mult_powr_eq_mult_powr_iff:
hoelzl@47599
   289
  fixes m1 m2 e1 e2 :: int
hoelzl@47599
   290
  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
   291
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
hoelzl@47599
   292
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
hoelzl@47599
   293
  by (cases e1 e2 rule: linorder_le_cases) auto
hoelzl@47599
   294
hoelzl@47599
   295
lemma floatE_normed:
hoelzl@47599
   296
  assumes x: "x \<in> float"
hoelzl@47599
   297
  obtains (zero) "x = 0"
hoelzl@47599
   298
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
hoelzl@47599
   299
proof atomize_elim
hoelzl@47599
   300
  { assume "x \<noteq> 0"
hoelzl@47599
   301
    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
hoelzl@47599
   302
    with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
hoelzl@47599
   303
      by auto
hoelzl@47599
   304
    with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
hoelzl@47599
   305
      by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
hoelzl@47599
   306
         (simp add: powr_add powr_realpow) }
hoelzl@47599
   307
  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
   308
    by blast
hoelzl@47599
   309
qed
hoelzl@47599
   310
hoelzl@47599
   311
lemma float_normed_cases:
hoelzl@47599
   312
  fixes f :: float
hoelzl@47599
   313
  obtains (zero) "f = 0"
hoelzl@47599
   314
   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
hoelzl@47599
   315
proof (atomize_elim, induct f)
hoelzl@47599
   316
  case (float_of y) then show ?case
hoelzl@47600
   317
    by (cases rule: floatE_normed) (auto simp: zero_float_def)
hoelzl@47599
   318
qed
hoelzl@47599
   319
hoelzl@47599
   320
definition mantissa :: "float \<Rightarrow> int" where
hoelzl@47599
   321
  "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   322
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   323
hoelzl@47599
   324
definition exponent :: "float \<Rightarrow> int" where
hoelzl@47599
   325
  "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   326
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   327
wenzelm@53381
   328
lemma
hoelzl@47599
   329
  shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
hoelzl@47599
   330
    and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
hoelzl@47599
   331
proof -
hoelzl@47599
   332
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
hoelzl@47599
   333
  then show ?E ?M
hoelzl@47600
   334
    by (auto simp add: mantissa_def exponent_def zero_float_def)
hoelzl@29804
   335
qed
hoelzl@29804
   336
hoelzl@47599
   337
lemma
hoelzl@47599
   338
  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
hoelzl@47599
   339
    and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
hoelzl@47599
   340
proof cases
hoelzl@47599
   341
  assume [simp]: "f \<noteq> (float_of 0)"
hoelzl@47599
   342
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
hoelzl@47599
   343
  proof (cases f rule: float_normed_cases)
hoelzl@47599
   344
    case (powr m e)
hoelzl@47599
   345
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   346
     \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
hoelzl@47599
   347
      by auto
hoelzl@47599
   348
    then show ?thesis
hoelzl@47599
   349
      unfolding exponent_def mantissa_def
hoelzl@47600
   350
      by (rule someI2_ex) (simp add: zero_float_def)
hoelzl@47600
   351
  qed (simp add: zero_float_def)
hoelzl@47599
   352
  then show ?E ?D by auto
hoelzl@47599
   353
qed simp
hoelzl@47599
   354
hoelzl@47599
   355
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
hoelzl@47599
   356
  using mantissa_not_dvd[of f] by auto
hoelzl@47599
   357
wenzelm@53381
   358
lemma
hoelzl@47599
   359
  fixes m e :: int
hoelzl@47599
   360
  defines "f \<equiv> float_of (m * 2 powr e)"
hoelzl@47599
   361
  assumes dvd: "\<not> 2 dvd m"
hoelzl@47599
   362
  shows mantissa_float: "mantissa f = m" (is "?M")
hoelzl@47599
   363
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
hoelzl@47599
   364
proof cases
hoelzl@47599
   365
  assume "m = 0" with dvd show "mantissa f = m" by auto
hoelzl@47599
   366
next
hoelzl@47599
   367
  assume "m \<noteq> 0"
hoelzl@47599
   368
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
hoelzl@47599
   369
  from mantissa_exponent[of f]
hoelzl@47599
   370
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   371
    by (auto simp add: f_def)
hoelzl@47599
   372
  then show "?M" "?E"
hoelzl@47599
   373
    using mantissa_not_dvd[OF f_not_0] dvd
hoelzl@47599
   374
    by (auto simp: mult_powr_eq_mult_powr_iff)
hoelzl@47599
   375
qed
hoelzl@47599
   376
hoelzl@47600
   377
subsection {* Compute arithmetic operations *}
hoelzl@47600
   378
hoelzl@47600
   379
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
hoelzl@47600
   380
  unfolding real_of_float_eq mantissa_exponent[of f] by simp
hoelzl@47600
   381
hoelzl@47600
   382
lemma Float_cases[case_names Float, cases type: float]:
hoelzl@47600
   383
  fixes f :: float
hoelzl@47600
   384
  obtains (Float) m e :: int where "f = Float m e"
hoelzl@47600
   385
  using Float_mantissa_exponent[symmetric]
hoelzl@47600
   386
  by (atomize_elim) auto
hoelzl@47600
   387
hoelzl@47599
   388
lemma denormalize_shift:
hoelzl@47599
   389
  assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
hoelzl@47599
   390
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
hoelzl@47599
   391
proof
hoelzl@47599
   392
  from mantissa_exponent[of f] f_def
hoelzl@47599
   393
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   394
    by simp
hoelzl@47599
   395
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
hoelzl@47599
   396
    by (simp add: powr_divide2[symmetric] field_simps)
hoelzl@47599
   397
  moreover
hoelzl@47599
   398
  have "e \<le> exponent f"
hoelzl@47599
   399
  proof (rule ccontr)
hoelzl@47599
   400
    assume "\<not> e \<le> exponent f"
hoelzl@47599
   401
    then have pos: "exponent f < e" by simp
hoelzl@47599
   402
    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
hoelzl@47599
   403
      by simp
hoelzl@47599
   404
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
hoelzl@47599
   405
      using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47599
   406
    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
hoelzl@47599
   407
      using eq by simp
hoelzl@47599
   408
    then have "mantissa f = m * 2^nat (e - exponent f)"
hoelzl@47599
   409
      unfolding real_of_int_inject by simp
hoelzl@47599
   410
    with `exponent f < e` have "2 dvd mantissa f"
hoelzl@47599
   411
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
hoelzl@47599
   412
      apply (cases "nat (e - exponent f)")
hoelzl@47599
   413
      apply auto
hoelzl@47599
   414
      done
hoelzl@47599
   415
    then show False using mantissa_not_dvd[OF not_0] by simp
hoelzl@47599
   416
  qed
hoelzl@47599
   417
  ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
hoelzl@47599
   418
    by (simp add: powr_realpow[symmetric])
hoelzl@47599
   419
  with `e \<le> exponent f`
hoelzl@47599
   420
  show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
hoelzl@47599
   421
    unfolding real_of_int_inject by auto
hoelzl@29804
   422
qed
hoelzl@29804
   423
hoelzl@47621
   424
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
hoelzl@47600
   425
  by transfer simp
hoelzl@47621
   426
hide_fact (open) compute_float_zero
hoelzl@47600
   427
hoelzl@47621
   428
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
hoelzl@47600
   429
  by transfer simp
hoelzl@47621
   430
hide_fact (open) compute_float_one
hoelzl@47600
   431
hoelzl@47600
   432
definition normfloat :: "float \<Rightarrow> float" where
hoelzl@47600
   433
  [simp]: "normfloat x = x"
hoelzl@47600
   434
hoelzl@47600
   435
lemma compute_normfloat[code]: "normfloat (Float m e) =
hoelzl@47600
   436
  (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
hoelzl@47600
   437
                           else if m = 0 then 0 else Float m e)"
hoelzl@47600
   438
  unfolding normfloat_def
hoelzl@47600
   439
  by transfer (auto simp add: powr_add zmod_eq_0_iff)
hoelzl@47621
   440
hide_fact (open) compute_normfloat
hoelzl@47599
   441
hoelzl@47599
   442
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
hoelzl@47600
   443
  by transfer simp
hoelzl@47621
   444
hide_fact (open) compute_float_numeral
hoelzl@47599
   445
haftmann@54489
   446
lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
hoelzl@47600
   447
  by transfer simp
hoelzl@47621
   448
hide_fact (open) compute_float_neg_numeral
hoelzl@47599
   449
hoelzl@47599
   450
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
hoelzl@47600
   451
  by transfer simp
hoelzl@47621
   452
hide_fact (open) compute_float_uminus
hoelzl@47599
   453
hoelzl@47599
   454
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
hoelzl@47600
   455
  by transfer (simp add: field_simps powr_add)
hoelzl@47621
   456
hide_fact (open) compute_float_times
hoelzl@47599
   457
hoelzl@47599
   458
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
immler@54783
   459
  (if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else
immler@54783
   460
  if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
hoelzl@47599
   461
              else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
hoelzl@47600
   462
  by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47621
   463
hide_fact (open) compute_float_plus
hoelzl@47599
   464
hoelzl@47600
   465
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
hoelzl@47600
   466
  by simp
hoelzl@47621
   467
hide_fact (open) compute_float_minus
hoelzl@47599
   468
hoelzl@47599
   469
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
hoelzl@47600
   470
  by transfer (simp add: sgn_times)
hoelzl@47621
   471
hide_fact (open) compute_float_sgn
hoelzl@47599
   472
kuncar@55565
   473
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   474
hoelzl@47599
   475
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
hoelzl@47600
   476
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
hoelzl@47621
   477
hide_fact (open) compute_is_float_pos
hoelzl@47599
   478
hoelzl@47599
   479
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
hoelzl@47600
   480
  by transfer (simp add: field_simps)
hoelzl@47621
   481
hide_fact (open) compute_float_less
hoelzl@47599
   482
kuncar@55565
   483
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   484
hoelzl@47599
   485
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
hoelzl@47600
   486
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
hoelzl@47621
   487
hide_fact (open) compute_is_float_nonneg
hoelzl@47599
   488
hoelzl@47599
   489
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
hoelzl@47600
   490
  by transfer (simp add: field_simps)
hoelzl@47621
   491
hide_fact (open) compute_float_le
hoelzl@47599
   492
kuncar@55565
   493
lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   494
hoelzl@47599
   495
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
hoelzl@47600
   496
  by transfer (auto simp add: is_float_zero_def)
hoelzl@47621
   497
hide_fact (open) compute_is_float_zero
hoelzl@47599
   498
hoelzl@47600
   499
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
hoelzl@47600
   500
  by transfer (simp add: abs_mult)
hoelzl@47621
   501
hide_fact (open) compute_float_abs
hoelzl@47599
   502
hoelzl@47600
   503
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
hoelzl@47600
   504
  by transfer simp
hoelzl@47621
   505
hide_fact (open) compute_float_eq
hoelzl@47599
   506
hoelzl@47599
   507
subsection {* Rounding Real numbers *}
hoelzl@47599
   508
hoelzl@47599
   509
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   510
  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   511
hoelzl@47599
   512
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   513
  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   514
hoelzl@47599
   515
lemma round_down_float[simp]: "round_down prec x \<in> float"
hoelzl@47599
   516
  unfolding round_down_def
hoelzl@47599
   517
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   518
hoelzl@47599
   519
lemma round_up_float[simp]: "round_up prec x \<in> float"
hoelzl@47599
   520
  unfolding round_up_def
hoelzl@47599
   521
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   522
hoelzl@47599
   523
lemma round_up: "x \<le> round_up prec x"
hoelzl@47599
   524
  by (simp add: powr_minus_divide le_divide_eq round_up_def)
hoelzl@47599
   525
hoelzl@47599
   526
lemma round_down: "round_down prec x \<le> x"
hoelzl@47599
   527
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
hoelzl@47599
   528
hoelzl@47599
   529
lemma round_up_0[simp]: "round_up p 0 = 0"
hoelzl@47599
   530
  unfolding round_up_def by simp
hoelzl@47599
   531
hoelzl@47599
   532
lemma round_down_0[simp]: "round_down p 0 = 0"
hoelzl@47599
   533
  unfolding round_down_def by simp
hoelzl@47599
   534
hoelzl@47599
   535
lemma round_up_diff_round_down:
hoelzl@47599
   536
  "round_up prec x - round_down prec x \<le> 2 powr -prec"
hoelzl@47599
   537
proof -
hoelzl@47599
   538
  have "round_up prec x - round_down prec x =
hoelzl@47599
   539
    (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
hoelzl@47599
   540
    by (simp add: round_up_def round_down_def field_simps)
hoelzl@47599
   541
  also have "\<dots> \<le> 1 * 2 powr -prec"
hoelzl@47599
   542
    by (rule mult_mono)
hoelzl@47599
   543
       (auto simp del: real_of_int_diff
hoelzl@47599
   544
             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
hoelzl@47599
   545
  finally show ?thesis by simp
hoelzl@29804
   546
qed
hoelzl@29804
   547
hoelzl@47599
   548
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
hoelzl@47599
   549
  unfolding round_down_def
hoelzl@47599
   550
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   551
    (simp add: powr_add[symmetric])
hoelzl@29804
   552
hoelzl@47599
   553
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
hoelzl@47599
   554
  unfolding round_up_def
hoelzl@47599
   555
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   556
    (simp add: powr_add[symmetric])
hoelzl@47599
   557
hoelzl@47599
   558
subsection {* Rounding Floats *}
hoelzl@29804
   559
hoelzl@47600
   560
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
hoelzl@47601
   561
declare float_up.rep_eq[simp]
hoelzl@29804
   562
immler@54782
   563
lemma round_up_correct:
immler@54782
   564
  shows "round_up e f - f \<in> {0..2 powr -e}"
hoelzl@47599
   565
unfolding atLeastAtMost_iff
hoelzl@47599
   566
proof
hoelzl@47599
   567
  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
hoelzl@47599
   568
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
immler@54782
   569
  finally show "round_up e f - f \<le> 2 powr real (- e)"
hoelzl@47600
   570
    by simp
hoelzl@47600
   571
qed (simp add: algebra_simps round_up)
hoelzl@29804
   572
immler@54782
   573
lemma float_up_correct:
immler@54782
   574
  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
immler@54782
   575
  by transfer (rule round_up_correct)
immler@54782
   576
hoelzl@47600
   577
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
hoelzl@47601
   578
declare float_down.rep_eq[simp]
obua@16782
   579
immler@54782
   580
lemma round_down_correct:
immler@54782
   581
  shows "f - (round_down e f) \<in> {0..2 powr -e}"
hoelzl@47599
   582
unfolding atLeastAtMost_iff
hoelzl@47599
   583
proof
hoelzl@47599
   584
  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
hoelzl@47599
   585
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
immler@54782
   586
  finally show "f - round_down e f \<le> 2 powr real (- e)"
hoelzl@47600
   587
    by simp
hoelzl@47600
   588
qed (simp add: algebra_simps round_down)
obua@24301
   589
immler@54782
   590
lemma float_down_correct:
immler@54782
   591
  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
immler@54782
   592
  by transfer (rule round_down_correct)
immler@54782
   593
hoelzl@47599
   594
lemma compute_float_down[code]:
hoelzl@47599
   595
  "float_down p (Float m e) =
hoelzl@47599
   596
    (if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"
hoelzl@47599
   597
proof cases
hoelzl@47599
   598
  assume "p + e < 0"
hoelzl@47599
   599
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   600
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   601
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47600
   602
    unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   603
  finally show ?thesis
hoelzl@47600
   604
    using `p + e < 0`
hoelzl@47600
   605
    by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
hoelzl@47599
   606
next
hoelzl@47600
   607
  assume "\<not> p + e < 0"
hoelzl@47600
   608
  then have r: "real e + real p = real (nat (e + p))" by simp
hoelzl@47600
   609
  have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
hoelzl@47600
   610
    by (auto intro: exI[where x="m*2^nat (e+p)"]
hoelzl@47600
   611
             simp add: ac_simps powr_add[symmetric] r powr_realpow)
hoelzl@47600
   612
  with `\<not> p + e < 0` show ?thesis
wenzelm@57862
   613
    by transfer (auto simp add: round_down_def field_simps powr_add powr_minus)
hoelzl@47599
   614
qed
hoelzl@47621
   615
hide_fact (open) compute_float_down
obua@24301
   616
immler@54782
   617
lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"
immler@54782
   618
  using round_down_correct[of f e] by simp
immler@54782
   619
immler@54782
   620
lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e"
immler@54782
   621
  using round_up_correct[of e f] by simp
immler@54782
   622
immler@54782
   623
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s"
nipkow@56536
   624
  by (auto simp: round_down_def)
immler@54782
   625
hoelzl@47599
   626
lemma ceil_divide_floor_conv:
hoelzl@47599
   627
assumes "b \<noteq> 0"
hoelzl@47599
   628
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
   629
proof cases
hoelzl@47599
   630
  assume "\<not> b dvd a"
hoelzl@47599
   631
  hence "a mod b \<noteq> 0" by auto
hoelzl@47599
   632
  hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
hoelzl@47599
   633
  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
hoelzl@47599
   634
  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
hoelzl@47599
   635
  proof -
hoelzl@47599
   636
    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
hoelzl@47599
   637
    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
hoelzl@47599
   638
    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
hoelzl@47599
   639
    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
hoelzl@47599
   640
  qed
hoelzl@47599
   641
  thus ?thesis using `\<not> b dvd a` by simp
hoelzl@47599
   642
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
hoelzl@56479
   643
  floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
wenzelm@19765
   644
hoelzl@47599
   645
lemma compute_float_up[code]:
hoelzl@47599
   646
  "float_up p (Float m e) =
hoelzl@47599
   647
    (let P = 2^nat (-(p + e)); r = m mod P in
hoelzl@47599
   648
      if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (-p) else Float m e)"
hoelzl@47599
   649
proof cases
hoelzl@47599
   650
  assume "p + e < 0"
hoelzl@47599
   651
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   652
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   653
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47599
   654
  unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   655
  finally have twopow_rewrite:
hoelzl@47599
   656
    "real ((2::int) ^ nat (- (p + e))) = 1 / 2 powr real p / 2 powr real e" .
hoelzl@47599
   657
  with `p + e < 0` have powr_rewrite:
hoelzl@47599
   658
    "2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat (- (p + e)))"
hoelzl@47599
   659
    unfolding powr_divide2 by simp
hoelzl@47599
   660
  show ?thesis
hoelzl@47599
   661
  proof cases
hoelzl@47599
   662
    assume "2^nat (-(p + e)) dvd m"
hoelzl@47615
   663
    with `p + e < 0` twopow_rewrite show ?thesis
hoelzl@47600
   664
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div dvd_eq_mod_eq_0)
hoelzl@47599
   665
  next
hoelzl@47599
   666
    assume ndvd: "\<not> 2 ^ nat (- (p + e)) dvd m"
hoelzl@47599
   667
    have one_div: "real m * (1 / real ((2::int) ^ nat (- (p + e)))) =
hoelzl@47599
   668
      real m / real ((2::int) ^ nat (- (p + e)))"
hoelzl@47599
   669
      by (simp add: field_simps)
hoelzl@47599
   670
    have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> =
hoelzl@47599
   671
      real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1"
hoelzl@47599
   672
      using ndvd unfolding powr_rewrite one_div
hoelzl@47599
   673
      by (subst ceil_divide_floor_conv) (auto simp: field_simps)
hoelzl@47599
   674
    thus ?thesis using `p + e < 0` twopow_rewrite
hoelzl@47600
   675
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div[symmetric])
hoelzl@29804
   676
  qed
hoelzl@47599
   677
next
hoelzl@47600
   678
  assume "\<not> p + e < 0"
hoelzl@47600
   679
  then have r1: "real e + real p = real (nat (e + p))" by simp
hoelzl@47600
   680
  have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p"
hoelzl@47600
   681
    by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow
hoelzl@47600
   682
      intro: exI[where x="m*2^nat (e+p)"])
hoelzl@47600
   683
  then show ?thesis using `\<not> p + e < 0`
wenzelm@57862
   684
    by transfer (simp add: round_up_def floor_divide_eq_div field_simps powr_add powr_minus)
hoelzl@29804
   685
qed
hoelzl@47621
   686
hide_fact (open) compute_float_up
hoelzl@29804
   687
hoelzl@47599
   688
lemmas real_of_ints =
hoelzl@47599
   689
  real_of_int_zero
hoelzl@47599
   690
  real_of_one
hoelzl@47599
   691
  real_of_int_add
hoelzl@47599
   692
  real_of_int_minus
hoelzl@47599
   693
  real_of_int_diff
hoelzl@47599
   694
  real_of_int_mult
hoelzl@47599
   695
  real_of_int_power
hoelzl@47599
   696
  real_numeral
hoelzl@47599
   697
lemmas real_of_nats =
hoelzl@47599
   698
  real_of_nat_zero
hoelzl@47599
   699
  real_of_nat_one
hoelzl@47599
   700
  real_of_nat_1
hoelzl@47599
   701
  real_of_nat_add
hoelzl@47599
   702
  real_of_nat_mult
hoelzl@47599
   703
  real_of_nat_power
hoelzl@47599
   704
hoelzl@47599
   705
lemmas int_of_reals = real_of_ints[symmetric]
hoelzl@47599
   706
lemmas nat_of_reals = real_of_nats[symmetric]
hoelzl@47599
   707
hoelzl@47599
   708
lemma two_real_int: "(2::real) = real (2::int)" by simp
hoelzl@47599
   709
lemma two_real_nat: "(2::real) = real (2::nat)" by simp
hoelzl@47599
   710
hoelzl@47599
   711
lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp
hoelzl@47599
   712
hoelzl@47599
   713
subsection {* Compute bitlen of integers *}
hoelzl@47599
   714
hoelzl@47600
   715
definition bitlen :: "int \<Rightarrow> int" where
hoelzl@47600
   716
  "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
hoelzl@47599
   717
hoelzl@47599
   718
lemma bitlen_nonneg: "0 \<le> bitlen x"
hoelzl@29804
   719
proof -
hoelzl@47599
   720
  {
hoelzl@47599
   721
    assume "0 > x"
hoelzl@47599
   722
    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
hoelzl@47599
   723
    also have "... < log 2 (-x)" using `0 > x` by auto
hoelzl@47599
   724
    finally have "-1 < log 2 (-x)" .
hoelzl@47599
   725
  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
hoelzl@47599
   726
qed
hoelzl@47599
   727
hoelzl@47599
   728
lemma bitlen_bounds:
hoelzl@47599
   729
  assumes "x > 0"
hoelzl@47599
   730
  shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
hoelzl@47599
   731
proof
hoelzl@47599
   732
  have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
hoelzl@47599
   733
    using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
hoelzl@47599
   734
    using real_nat_eq_real[of "floor (log 2 (real x))"]
hoelzl@47599
   735
    by simp
hoelzl@47599
   736
  also have "... \<le> 2 powr log 2 (real x)"
hoelzl@47599
   737
    by simp
hoelzl@47599
   738
  also have "... = real x"
hoelzl@47599
   739
    using `0 < x` by simp
hoelzl@47599
   740
  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
hoelzl@47599
   741
  thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
hoelzl@47599
   742
    by (simp add: bitlen_def)
hoelzl@47599
   743
next
hoelzl@47599
   744
  have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
hoelzl@47599
   745
  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
hoelzl@47599
   746
    apply (simp add: powr_realpow[symmetric])
hoelzl@47599
   747
    using `x > 0` by simp
hoelzl@47599
   748
  finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
hoelzl@47599
   749
    by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)
hoelzl@47599
   750
qed
hoelzl@47599
   751
hoelzl@47599
   752
lemma bitlen_pow2[simp]:
hoelzl@47599
   753
  assumes "b > 0"
hoelzl@47599
   754
  shows "bitlen (b * 2 ^ c) = bitlen b + c"
hoelzl@47599
   755
proof -
nipkow@56544
   756
  from assms have "b * 2 ^ c > 0" by auto
hoelzl@47599
   757
  thus ?thesis
hoelzl@47599
   758
    using floor_add[of "log 2 b" c] assms
hoelzl@47599
   759
    by (auto simp add: log_mult log_nat_power bitlen_def)
hoelzl@29804
   760
qed
hoelzl@29804
   761
hoelzl@47599
   762
lemma bitlen_Float:
wenzelm@53381
   763
  fixes m e
wenzelm@53381
   764
  defines "f \<equiv> Float m e"
wenzelm@53381
   765
  shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
wenzelm@53381
   766
proof (cases "m = 0")
wenzelm@53381
   767
  case True
wenzelm@53381
   768
  then show ?thesis by (simp add: f_def bitlen_def Float_def)
wenzelm@53381
   769
next
wenzelm@53381
   770
  case False
hoelzl@47600
   771
  hence "f \<noteq> float_of 0"
hoelzl@47600
   772
    unfolding real_of_float_eq by (simp add: f_def)
hoelzl@47600
   773
  hence "mantissa f \<noteq> 0"
hoelzl@47599
   774
    by (simp add: mantissa_noteq_0)
hoelzl@47599
   775
  moreover
wenzelm@53381
   776
  obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
wenzelm@53381
   777
    by (rule f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`])
hoelzl@47599
   778
  ultimately show ?thesis by (simp add: abs_mult)
wenzelm@53381
   779
qed
hoelzl@29804
   780
hoelzl@47599
   781
lemma compute_bitlen[code]:
hoelzl@47599
   782
  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
hoelzl@47599
   783
proof -
hoelzl@47599
   784
  { assume "2 \<le> x"
hoelzl@47599
   785
    then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
hoelzl@47599
   786
      by (simp add: log_mult zmod_zdiv_equality')
hoelzl@47599
   787
    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   788
    proof cases
hoelzl@47599
   789
      assume "x mod 2 = 0" then show ?thesis by simp
hoelzl@47599
   790
    next
hoelzl@47599
   791
      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   792
      then have "0 \<le> n"
hoelzl@47599
   793
        using `2 \<le> x` by simp
hoelzl@47599
   794
      assume "x mod 2 \<noteq> 0"
hoelzl@47599
   795
      with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
hoelzl@47599
   796
      with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
hoelzl@47599
   797
      moreover
hoelzl@47599
   798
      { have "real (2^nat n :: int) = 2 powr (nat n)"
hoelzl@47599
   799
          by (simp add: powr_realpow)
hoelzl@47599
   800
        also have "\<dots> \<le> 2 powr (log 2 x)"
hoelzl@47599
   801
          using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
hoelzl@47599
   802
        finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
hoelzl@47599
   803
      ultimately have "2^nat n \<le> x - 1" by simp
hoelzl@47599
   804
      then have "2^nat n \<le> real (x - 1)"
hoelzl@47599
   805
        unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47599
   806
      { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
hoelzl@47599
   807
          using `0 \<le> n` by (simp add: log_nat_power)
hoelzl@47599
   808
        also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
hoelzl@47599
   809
          using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
hoelzl@47599
   810
        finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
hoelzl@47599
   811
      moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
hoelzl@47599
   812
        using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
hoelzl@47599
   813
      ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
hoelzl@47599
   814
        unfolding n_def `x mod 2 = 1` by auto
hoelzl@47599
   815
    qed
hoelzl@47599
   816
    finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
hoelzl@47599
   817
  moreover
hoelzl@47599
   818
  { assume "x < 2" "0 < x"
hoelzl@47599
   819
    then have "x = 1" by simp
hoelzl@47599
   820
    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
hoelzl@47599
   821
  ultimately show ?thesis
hoelzl@47599
   822
    unfolding bitlen_def
hoelzl@47599
   823
    by (auto simp: pos_imp_zdiv_pos_iff not_le)
hoelzl@47599
   824
qed
hoelzl@47621
   825
hide_fact (open) compute_bitlen
hoelzl@29804
   826
hoelzl@47599
   827
lemma float_gt1_scale: assumes "1 \<le> Float m e"
hoelzl@47599
   828
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@47599
   829
proof -
hoelzl@47599
   830
  have "0 < Float m e" using assms by auto
hoelzl@47599
   831
  hence "0 < m" using powr_gt_zero[of 2 e]
hoelzl@47600
   832
    by (auto simp: zero_less_mult_iff)
hoelzl@47599
   833
  hence "m \<noteq> 0" by auto
hoelzl@47599
   834
  show ?thesis
hoelzl@47599
   835
  proof (cases "0 \<le> e")
hoelzl@47599
   836
    case True thus ?thesis using `0 < m`  by (simp add: bitlen_def)
hoelzl@29804
   837
  next
hoelzl@47599
   838
    have "(1::int) < 2" by simp
hoelzl@47599
   839
    case False let ?S = "2^(nat (-e))"
hoelzl@47599
   840
    have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
wenzelm@57862
   841
      by (auto simp: powr_minus field_simps)
hoelzl@47599
   842
    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   843
      by (auto simp: powr_minus)
hoelzl@47599
   844
    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
haftmann@57512
   845
    hence "?S \<le> real m" unfolding mult.assoc by auto
hoelzl@47599
   846
    hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@47599
   847
    from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
hoelzl@47599
   848
    have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
hoelzl@47599
   849
    hence "-e < bitlen m" using False by auto
hoelzl@47599
   850
    thus ?thesis by auto
hoelzl@29804
   851
  qed
hoelzl@47599
   852
qed
hoelzl@29804
   853
hoelzl@29804
   854
lemma bitlen_div: assumes "0 < m" shows "1 \<le> real m / 2^nat (bitlen m - 1)" and "real m / 2^nat (bitlen m - 1) < 2"
hoelzl@29804
   855
proof -
hoelzl@29804
   856
  let ?B = "2^nat(bitlen m - 1)"
hoelzl@29804
   857
hoelzl@29804
   858
  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   859
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   860
  thus "1 \<le> real m / ?B" by auto
hoelzl@29804
   861
hoelzl@29804
   862
  have "m \<noteq> 0" using assms by auto
hoelzl@47599
   863
  have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)
obua@16782
   864
hoelzl@29804
   865
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
hoelzl@47599
   866
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
hoelzl@29804
   867
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
hoelzl@29804
   868
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   869
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
hoelzl@29804
   870
  thus "real m / ?B < 2" by auto
hoelzl@29804
   871
qed
hoelzl@29804
   872
hoelzl@47599
   873
subsection {* Approximation of positive rationals *}
hoelzl@47599
   874
hoelzl@47599
   875
lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"
hoelzl@47599
   876
by (simp add: zdiv_zmult2_eq)
hoelzl@29804
   877
hoelzl@47599
   878
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
hoelzl@47599
   879
  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
hoelzl@29804
   880
hoelzl@47599
   881
lemma real_div_nat_eq_floor_of_divide:
hoelzl@47599
   882
  fixes a b::nat
hoelzl@47599
   883
  shows "a div b = real (floor (a/b))"
hoelzl@47599
   884
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)
hoelzl@29804
   885
hoelzl@47599
   886
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
hoelzl@29804
   887
hoelzl@47600
   888
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
   889
  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp
obua@16782
   890
hoelzl@47599
   891
lemma compute_lapprox_posrat[code]:
wenzelm@53381
   892
  fixes prec x y
wenzelm@53381
   893
  shows "lapprox_posrat prec x y =
wenzelm@53381
   894
   (let
hoelzl@47599
   895
       l = rat_precision prec x y;
hoelzl@47599
   896
       d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
hoelzl@47599
   897
    in normfloat (Float d (- l)))"
hoelzl@47615
   898
    unfolding div_mult_twopow_eq normfloat_def
hoelzl@47600
   899
    by transfer
hoelzl@47615
   900
       (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
hoelzl@47599
   901
             del: two_powr_minus_int_float)
hoelzl@47621
   902
hide_fact (open) compute_lapprox_posrat
hoelzl@29804
   903
hoelzl@47600
   904
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
   905
  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
hoelzl@29804
   906
hoelzl@47599
   907
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *)
hoelzl@47599
   908
lemma compute_rapprox_posrat[code]:
hoelzl@47599
   909
  fixes prec x y
hoelzl@47599
   910
  defines "l \<equiv> rat_precision prec x y"
hoelzl@47599
   911
  shows "rapprox_posrat prec x y = (let
hoelzl@47599
   912
     l = l ;
hoelzl@47599
   913
     X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
hoelzl@47599
   914
     d = fst X div snd X ;
hoelzl@47599
   915
     m = fst X mod snd X
hoelzl@47599
   916
   in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
hoelzl@47599
   917
proof (cases "y = 0")
hoelzl@47615
   918
  assume "y = 0" thus ?thesis unfolding normfloat_def by transfer simp
hoelzl@47599
   919
next
hoelzl@47599
   920
  assume "y \<noteq> 0"
hoelzl@29804
   921
  show ?thesis
hoelzl@47599
   922
  proof (cases "0 \<le> l")
hoelzl@47599
   923
    assume "0 \<le> l"
wenzelm@56777
   924
    def x' \<equiv> "x * 2 ^ nat l"
hoelzl@47599
   925
    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
hoelzl@47599
   926
    moreover have "real x * 2 powr real l = real x'"
hoelzl@47599
   927
      by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
hoelzl@47599
   928
    ultimately show ?thesis
hoelzl@47615
   929
      unfolding normfloat_def
hoelzl@47599
   930
      using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
hoelzl@47600
   931
        l_def[symmetric, THEN meta_eq_to_obj_eq]
hoelzl@47600
   932
      by transfer
hoelzl@47600
   933
         (simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def)
hoelzl@47599
   934
   next
hoelzl@47599
   935
    assume "\<not> 0 \<le> l"
wenzelm@56777
   936
    def y' \<equiv> "y * 2 ^ nat (- l)"
hoelzl@47599
   937
    from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
hoelzl@47599
   938
    have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
hoelzl@47599
   939
    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
hoelzl@47599
   940
      using `\<not> 0 \<le> l`
wenzelm@57862
   941
      by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps)
hoelzl@47599
   942
    ultimately show ?thesis
hoelzl@47615
   943
      unfolding normfloat_def
hoelzl@47599
   944
      using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
hoelzl@47600
   945
        l_def[symmetric, THEN meta_eq_to_obj_eq]
hoelzl@47600
   946
      by transfer
hoelzl@47600
   947
         (simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0)
hoelzl@29804
   948
  qed
hoelzl@29804
   949
qed
hoelzl@47621
   950
hide_fact (open) compute_rapprox_posrat
hoelzl@29804
   951
hoelzl@47599
   952
lemma rat_precision_pos:
hoelzl@47599
   953
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@47599
   954
  shows "rat_precision n (int x) (int y) > 0"
hoelzl@29804
   955
proof -
hoelzl@47599
   956
  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
hoelzl@47599
   957
  hence "bitlen (int x) < bitlen (int y)" using assms
hoelzl@47599
   958
    by (simp add: bitlen_def del: floor_add_one)
hoelzl@47599
   959
      (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
hoelzl@47599
   960
  thus ?thesis
hoelzl@47599
   961
    using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
hoelzl@29804
   962
qed
obua@16782
   963
wenzelm@56777
   964
lemma power_aux:
wenzelm@56777
   965
  assumes "x > 0"
wenzelm@56777
   966
  shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1"
hoelzl@47599
   967
proof -
wenzelm@56777
   968
  def y \<equiv> "nat (x - 1)"
wenzelm@56777
   969
  moreover
hoelzl@47599
   970
  have "(2::int) ^ y \<le> (2 ^ (y + 1)) - 1" by simp
hoelzl@47599
   971
  ultimately show ?thesis using assms by simp
hoelzl@29804
   972
qed
hoelzl@29804
   973
hoelzl@47601
   974
lemma rapprox_posrat_less1:
hoelzl@47601
   975
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@31098
   976
  shows "real (rapprox_posrat n x y) < 1"
hoelzl@47599
   977
proof -
wenzelm@53381
   978
  have powr1: "2 powr real (rat_precision n (int x) (int y)) =
hoelzl@47599
   979
    2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms
hoelzl@47599
   980
    by (simp add: powr_realpow[symmetric])
hoelzl@47599
   981
  have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) *
hoelzl@47599
   982
     2 powr real (rat_precision n (int x) (int y))" by simp
hoelzl@47599
   983
  also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"
hoelzl@47599
   984
    apply (rule mult_strict_right_mono) by (insert assms) auto
hoelzl@47599
   985
  also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"
haftmann@54489
   986
    using powr_add [of 2 _ "- 1", simplified add_uminus_conv_diff] by (simp add: powr_minus)
hoelzl@47599
   987
  also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"
hoelzl@47599
   988
    using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])
hoelzl@47599
   989
  also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"
hoelzl@47599
   990
    unfolding int_of_reals real_of_int_le_iff
hoelzl@47599
   991
    using rat_precision_pos[OF assms] by (rule power_aux)
hoelzl@47600
   992
  finally show ?thesis
hoelzl@47601
   993
    apply (transfer fixing: n x y)
wenzelm@57862
   994
    apply (simp add: round_up_def field_simps powr_minus powr1)
hoelzl@47599
   995
    unfolding int_of_reals real_of_int_less_iff
hoelzl@47601
   996
    apply (simp add: ceiling_less_eq)
hoelzl@47600
   997
    done
hoelzl@29804
   998
qed
hoelzl@29804
   999
hoelzl@47600
  1000
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
  1001
  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
obua@16782
  1002
hoelzl@29804
  1003
lemma compute_lapprox_rat[code]:
hoelzl@47599
  1004
  "lapprox_rat prec x y =
hoelzl@47599
  1005
    (if y = 0 then 0
hoelzl@47599
  1006
    else if 0 \<le> x then
hoelzl@47599
  1007
      (if 0 < y then lapprox_posrat prec (nat x) (nat y)
wenzelm@53381
  1008
      else - (rapprox_posrat prec (nat x) (nat (-y))))
hoelzl@47599
  1009
      else (if 0 < y
hoelzl@47599
  1010
        then - (rapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
  1011
        else lapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@56479
  1012
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hoelzl@47621
  1013
hide_fact (open) compute_lapprox_rat
hoelzl@47599
  1014
hoelzl@47600
  1015
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
  1016
  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
hoelzl@47599
  1017
hoelzl@47599
  1018
lemma compute_rapprox_rat[code]:
hoelzl@47599
  1019
  "rapprox_rat prec x y =
hoelzl@47599
  1020
    (if y = 0 then 0
hoelzl@47599
  1021
    else if 0 \<le> x then
hoelzl@47599
  1022
      (if 0 < y then rapprox_posrat prec (nat x) (nat y)
wenzelm@53381
  1023
      else - (lapprox_posrat prec (nat x) (nat (-y))))
hoelzl@47599
  1024
      else (if 0 < y
hoelzl@47599
  1025
        then - (lapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
  1026
        else rapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@56479
  1027
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hoelzl@47621
  1028
hide_fact (open) compute_rapprox_rat
hoelzl@47599
  1029
hoelzl@47599
  1030
subsection {* Division *}
hoelzl@47599
  1031
immler@54782
  1032
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
  1033
immler@54782
  1034
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
  1035
immler@54782
  1036
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl
immler@54782
  1037
  by (simp add: real_divl_def)
hoelzl@47599
  1038
hoelzl@47599
  1039
lemma compute_float_divl[code]:
hoelzl@47600
  1040
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1041
proof cases
hoelzl@47601
  1042
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
hoelzl@47601
  1043
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
hoelzl@47601
  1044
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
hoelzl@47601
  1045
  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
  1046
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
hoelzl@47601
  1047
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
hoelzl@47601
  1048
    by (simp add: field_simps powr_divide2[symmetric])
hoelzl@47599
  1049
hoelzl@47601
  1050
  show ?thesis
wenzelm@53381
  1051
    using not_0
immler@54782
  1052
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,
immler@54782
  1053
      simp add: field_simps)
immler@54782
  1054
qed (transfer, auto simp: real_divl_def)
hoelzl@47621
  1055
hide_fact (open) compute_float_divl
hoelzl@47600
  1056
immler@54782
  1057
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr
immler@54782
  1058
  by (simp add: real_divr_def)
hoelzl@47599
  1059
hoelzl@47599
  1060
lemma compute_float_divr[code]:
hoelzl@47600
  1061
  "float_divr prec (Float m1 s1) (Float m2 s2) = rapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1062
proof cases
hoelzl@47601
  1063
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
hoelzl@47601
  1064
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
hoelzl@47601
  1065
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
hoelzl@47601
  1066
  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
  1067
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
hoelzl@47601
  1068
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
hoelzl@47601
  1069
    by (simp add: field_simps powr_divide2[symmetric])
hoelzl@47600
  1070
hoelzl@47601
  1071
  show ?thesis
wenzelm@53381
  1072
    using not_0
immler@54782
  1073
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift real_divr_def,
immler@54782
  1074
      simp add: field_simps)
immler@54782
  1075
qed (transfer, auto simp: real_divr_def)
hoelzl@47621
  1076
hide_fact (open) compute_float_divr
obua@16782
  1077
hoelzl@47599
  1078
subsection {* Lemmas needed by Approximate *}
hoelzl@47599
  1079
hoelzl@47599
  1080
lemma Float_num[simp]: shows
hoelzl@47599
  1081
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
hoelzl@47599
  1082
   "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
hoelzl@47599
  1083
   "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"
hoelzl@47599
  1084
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
  1085
using powr_realpow[of 2 2] powr_realpow[of 2 3]
hoelzl@47599
  1086
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
hoelzl@47599
  1087
by auto
hoelzl@47599
  1088
hoelzl@47599
  1089
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp
hoelzl@47599
  1090
hoelzl@47599
  1091
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
hoelzl@47599
  1092
hoelzl@47599
  1093
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
hoelzl@47599
  1094
by arith
hoelzl@29804
  1095
hoelzl@47599
  1096
lemma lapprox_rat:
hoelzl@47599
  1097
  shows "real (lapprox_rat prec x y) \<le> real x / real y"
hoelzl@47599
  1098
  using round_down by (simp add: lapprox_rat_def)
obua@16782
  1099
hoelzl@47599
  1100
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
hoelzl@47599
  1101
proof -
hoelzl@47599
  1102
  from zmod_zdiv_equality'[of a b]
hoelzl@47599
  1103
  have "a = b * (a div b) + a mod b" by simp
hoelzl@47599
  1104
  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
hoelzl@47599
  1105
  using assms by simp
hoelzl@47599
  1106
  finally show ?thesis by simp
hoelzl@47599
  1107
qed
hoelzl@47599
  1108
hoelzl@47599
  1109
lemma lapprox_rat_nonneg:
hoelzl@47599
  1110
  fixes n x y
wenzelm@56777
  1111
  defines "p \<equiv> int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))"
wenzelm@56777
  1112
  assumes "0 \<le> x" and "0 < y"
hoelzl@47599
  1113
  shows "0 \<le> real (lapprox_rat n x y)"
hoelzl@47599
  1114
using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric]
hoelzl@47599
  1115
   powr_int[of 2, simplified]
hoelzl@56571
  1116
  by auto
obua@16782
  1117
hoelzl@31098
  1118
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
hoelzl@47599
  1119
  using round_up by (simp add: rapprox_rat_def)
hoelzl@47599
  1120
hoelzl@47599
  1121
lemma rapprox_rat_le1:
hoelzl@47599
  1122
  fixes n x y
hoelzl@47599
  1123
  assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
hoelzl@47599
  1124
  shows "real (rapprox_rat n x y) \<le> 1"
hoelzl@47599
  1125
proof -
hoelzl@47599
  1126
  have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
hoelzl@47599
  1127
    using xy unfolding bitlen_def by (auto intro!: floor_mono)
hoelzl@47599
  1128
  then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def)
hoelzl@47599
  1129
  have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>
hoelzl@47599
  1130
      \<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>"
hoelzl@47599
  1131
    using xy by (auto intro!: ceiling_mono simp: field_simps)
hoelzl@47599
  1132
  also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"
hoelzl@47599
  1133
    using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>`
hoelzl@47599
  1134
    by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int)
hoelzl@47599
  1135
  finally show ?thesis
hoelzl@47599
  1136
    by (simp add: rapprox_rat_def round_up_def)
hoelzl@47599
  1137
       (simp add: powr_minus inverse_eq_divide)
hoelzl@29804
  1138
qed
obua@16782
  1139
wenzelm@53381
  1140
lemma rapprox_rat_nonneg_neg:
hoelzl@47599
  1141
  "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1142
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1143
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
obua@16782
  1144
hoelzl@47599
  1145
lemma rapprox_rat_neg:
hoelzl@47599
  1146
  "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1147
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1148
  by (auto simp: field_simps mult_le_0_iff)
hoelzl@29804
  1149
hoelzl@47599
  1150
lemma rapprox_rat_nonpos_pos:
hoelzl@47599
  1151
  "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1152
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1153
  by (auto simp: field_simps mult_le_0_iff)
obua@16782
  1154
immler@54782
  1155
lemma real_divl: "real_divl prec x y \<le> x / y"
immler@54782
  1156
  by (simp add: real_divl_def round_down)
immler@54782
  1157
immler@54782
  1158
lemma real_divr: "x / y \<le> real_divr prec x y"
immler@54782
  1159
  using round_up by (simp add: real_divr_def)
immler@54782
  1160
hoelzl@31098
  1161
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
immler@54782
  1162
  by transfer (rule real_divl)
immler@54782
  1163
immler@54782
  1164
lemma real_divl_lower_bound:
immler@54782
  1165
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y"
immler@54782
  1166
  by (simp add: real_divl_def round_down_def zero_le_mult_iff zero_le_divide_iff)
hoelzl@47599
  1167
hoelzl@47599
  1168
lemma float_divl_lower_bound:
immler@54782
  1169
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
immler@54782
  1170
  by transfer (rule real_divl_lower_bound)
hoelzl@47599
  1171
hoelzl@47599
  1172
lemma exponent_1: "exponent 1 = 0"
hoelzl@47599
  1173
  using exponent_float[of 1 0] by (simp add: one_float_def)
hoelzl@47599
  1174
hoelzl@47599
  1175
lemma mantissa_1: "mantissa 1 = 1"
hoelzl@47599
  1176
  using mantissa_float[of 1 0] by (simp add: one_float_def)
obua@16782
  1177
hoelzl@47599
  1178
lemma bitlen_1: "bitlen 1 = 1"
hoelzl@47599
  1179
  by (simp add: bitlen_def)
hoelzl@47599
  1180
hoelzl@47599
  1181
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
hoelzl@47599
  1182
proof
hoelzl@47599
  1183
  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
hoelzl@47599
  1184
  show "x = 0" by (simp add: zero_float_def z)
hoelzl@47599
  1185
qed (simp add: zero_float_def)
obua@16782
  1186
hoelzl@47599
  1187
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
hoelzl@47599
  1188
proof (cases "x = 0", simp)
hoelzl@47599
  1189
  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
hoelzl@47599
  1190
  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
hoelzl@47599
  1191
  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
hoelzl@47599
  1192
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
hoelzl@47599
  1193
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
hoelzl@47599
  1194
    by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]
hoelzl@47599
  1195
      real_of_int_le_iff less_imp_le)
hoelzl@47599
  1196
  finally show ?thesis by (simp add: powr_add)
hoelzl@29804
  1197
qed
hoelzl@29804
  1198
immler@54782
  1199
lemma real_divl_pos_less1_bound:
immler@54782
  1200
  "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real_divl prec 1 x"
immler@54782
  1201
proof (unfold real_divl_def)
immler@54782
  1202
  fix prec :: nat and x :: real assume x: "0 < x" "x < 1" and prec: "1 \<le> prec"
wenzelm@53381
  1203
  def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>"
hoelzl@47600
  1204
  show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) "
hoelzl@47600
  1205
  proof cases
hoelzl@47600
  1206
    assume nonneg: "0 \<le> p"
hoelzl@47600
  1207
    hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)"
hoelzl@47600
  1208
      by (simp add: powr_int del: real_of_int_power) simp
hoelzl@47600
  1209
    also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp
hoelzl@47600
  1210
    also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le>
hoelzl@47600
  1211
      floor (real ((2::int) ^ nat p) * (1 / x))"
hoelzl@47600
  1212
      by (rule le_mult_floor) (auto simp: x prec less_imp_le)
hoelzl@47600
  1213
    finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow)
hoelzl@47600
  1214
    thus ?thesis unfolding p_def[symmetric]
hoelzl@47600
  1215
      using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def)
hoelzl@47600
  1216
  next
hoelzl@47600
  1217
    assume neg: "\<not> 0 \<le> p"
hoelzl@47600
  1218
hoelzl@47600
  1219
    have "x = 2 powr (log 2 x)"
hoelzl@47600
  1220
      using x by simp
hoelzl@47600
  1221
    also have "2 powr (log 2 x) \<le> 2 powr p"
hoelzl@47600
  1222
    proof (rule powr_mono)
hoelzl@47600
  1223
      have "log 2 x \<le> \<lceil>log 2 x\<rceil>"
hoelzl@47600
  1224
        by simp
hoelzl@47600
  1225
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1"
hoelzl@47600
  1226
        using ceiling_diff_floor_le_1[of "log 2 x"] by simp
hoelzl@47600
  1227
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec"
hoelzl@47600
  1228
        using prec by simp
hoelzl@47600
  1229
      finally show "log 2 x \<le> real p"
hoelzl@47600
  1230
        using x by (simp add: p_def)
hoelzl@47600
  1231
    qed simp
hoelzl@47600
  1232
    finally have x_le: "x \<le> 2 powr p" .
hoelzl@47600
  1233
hoelzl@47600
  1234
    from neg have "2 powr real p \<le> 2 powr 0"
hoelzl@47600
  1235
      by (intro powr_mono) auto
hoelzl@47600
  1236
    also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp
hoelzl@47600
  1237
    also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff
nipkow@56544
  1238
      using x x_le by (intro floor_mono) (simp add:  pos_le_divide_eq)
hoelzl@47600
  1239
    finally show ?thesis
hoelzl@47600
  1240
      using prec x unfolding p_def[symmetric]
nipkow@56544
  1241
      by (simp add: round_down_def powr_minus_divide pos_le_divide_eq)
hoelzl@47600
  1242
  qed
hoelzl@29804
  1243
qed
obua@16782
  1244
immler@54782
  1245
lemma float_divl_pos_less1_bound:
immler@54782
  1246
  "0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
immler@54782
  1247
  by (transfer, rule real_divl_pos_less1_bound)
obua@16782
  1248
immler@54782
  1249
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
immler@54782
  1250
  by transfer (rule real_divr)
immler@54782
  1251
immler@54782
  1252
lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> real_divr prec 1 x"
hoelzl@29804
  1253
proof -
immler@54782
  1254
  have "1 \<le> 1 / x" using `0 < x` and `x < 1` by auto
immler@54782
  1255
  also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto
hoelzl@47600
  1256
  finally show ?thesis by auto
hoelzl@29804
  1257
qed
hoelzl@29804
  1258
immler@54782
  1259
lemma float_divr_pos_less1_lower_bound: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> 1 \<le> float_divr prec 1 x"
immler@54782
  1260
  by transfer (rule real_divr_pos_less1_lower_bound)
immler@54782
  1261
immler@54782
  1262
lemma real_divr_nonpos_pos_upper_bound:
immler@54782
  1263
  "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real_divr prec x y \<le> 0"
immler@54782
  1264
  by (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def real_divr_def)
immler@54782
  1265
hoelzl@47599
  1266
lemma float_divr_nonpos_pos_upper_bound:
hoelzl@47600
  1267
  "real x \<le> 0 \<Longrightarrow> 0 < real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"
immler@54782
  1268
  by transfer (rule real_divr_nonpos_pos_upper_bound)
immler@54782
  1269
immler@54782
  1270
lemma real_divr_nonneg_neg_upper_bound:
immler@54782
  1271
  "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real_divr prec x y \<le> 0"
immler@54782
  1272
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def real_divr_def)
obua@16782
  1273
hoelzl@47599
  1274
lemma float_divr_nonneg_neg_upper_bound:
hoelzl@47600
  1275
  "0 \<le> real x \<Longrightarrow> real y < 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"
immler@54782
  1276
  by transfer (rule real_divr_nonneg_neg_upper_bound)
immler@54782
  1277
immler@54782
  1278
definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where
immler@54782
  1279
  "truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
immler@54782
  1280
immler@54782
  1281
lemma truncate_down: "truncate_down prec x \<le> x"
immler@54782
  1282
  using round_down by (simp add: truncate_down_def)
immler@54782
  1283
immler@54782
  1284
lemma truncate_down_le: "x \<le> y \<Longrightarrow> truncate_down prec x \<le> y"
immler@54782
  1285
  by (rule order_trans[OF truncate_down])
hoelzl@47600
  1286
immler@54782
  1287
definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real" where
immler@54782
  1288
  "truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
immler@54782
  1289
immler@54782
  1290
lemma truncate_up: "x \<le> truncate_up prec x"
immler@54782
  1291
  using round_up by (simp add: truncate_up_def)
immler@54782
  1292
immler@54782
  1293
lemma truncate_up_le: "x \<le> y \<Longrightarrow> x \<le> truncate_up prec y"
immler@54782
  1294
  by (rule order_trans[OF _ truncate_up])
immler@54782
  1295
immler@54782
  1296
lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0"
immler@54782
  1297
  by (simp add: truncate_up_def)
immler@54782
  1298
immler@54782
  1299
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
immler@54782
  1300
  by (simp add: truncate_up_def)
hoelzl@47600
  1301
hoelzl@47600
  1302
lemma float_round_up: "real x \<le> real (float_round_up prec x)"
immler@54782
  1303
  using truncate_up by transfer simp
hoelzl@47599
  1304
immler@54782
  1305
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down
immler@54782
  1306
  by (simp add: truncate_down_def)
hoelzl@47599
  1307
hoelzl@47600
  1308
lemma float_round_down: "real (float_round_down prec x) \<le> real x"
immler@54782
  1309
  using truncate_down by transfer simp
hoelzl@47599
  1310
hoelzl@47600
  1311
lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>"
hoelzl@47600
  1312
  using floor_add[of x i] by (simp del: floor_add add: ac_simps)
obua@16782
  1313
hoelzl@47599
  1314
lemma compute_float_round_down[code]:
hoelzl@47600
  1315
  "float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
hoelzl@47600
  1316
    if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
hoelzl@47600
  1317
             else Float m e)"
hoelzl@47621
  1318
  using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
immler@54782
  1319
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_down_def
immler@54782
  1320
    cong del: if_weak_cong)
hoelzl@47621
  1321
hide_fact (open) compute_float_round_down
hoelzl@47599
  1322
hoelzl@47600
  1323
lemma compute_float_round_up[code]:
hoelzl@47600
  1324
  "float_round_up prec (Float m e) = (let d = (bitlen (abs m) - int prec) in
hoelzl@47600
  1325
     if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P
hoelzl@47600
  1326
                   in Float (n + (if r = 0 then 0 else 1)) (e + d)
hoelzl@47600
  1327
              else Float m e)"
hoelzl@47621
  1328
  using Float.compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
hoelzl@47600
  1329
  unfolding Let_def
immler@54782
  1330
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_up_def
immler@54782
  1331
    cong del: if_weak_cong)
hoelzl@47621
  1332
hide_fact (open) compute_float_round_up
obua@16782
  1333
immler@54784
  1334
lemma round_up_mono: "x \<le> y \<Longrightarrow> round_up p x \<le> round_up p y"
immler@54784
  1335
  by (auto intro!: ceiling_mono simp: round_up_def)
immler@54784
  1336
immler@54784
  1337
lemma truncate_up_nonneg_mono:
immler@54784
  1338
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  1339
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  1340
proof -
immler@54784
  1341
  {
immler@54784
  1342
    assume "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>"
immler@54784
  1343
    hence ?thesis
immler@54784
  1344
      using assms
immler@54784
  1345
      by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
immler@54784
  1346
  } moreover {
immler@54784
  1347
    assume "0 < x"
immler@54784
  1348
    hence "log 2 x \<le> log 2 y" using assms by auto
immler@54784
  1349
    moreover
immler@54784
  1350
    assume "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>"
immler@54784
  1351
    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
immler@54784
  1352
      unfolding atomize_conj
immler@54784
  1353
      by (metis floor_less_cancel linorder_cases not_le)
immler@54784
  1354
    have "truncate_up prec x =
immler@54784
  1355
      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
  1356
      using assms by (simp add: truncate_up_def round_up_def)
immler@54784
  1357
    also have "\<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)"
immler@54784
  1358
    proof (unfold ceiling_le_eq)
immler@54784
  1359
      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
  1360
        using real_of_int_floor_add_one_ge[of "log 2 x"] assms
immler@54784
  1361
        by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono)
immler@54784
  1362
      thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
immler@54784
  1363
        using `0 < x` by (simp add: powr_realpow)
immler@54784
  1364
    qed
immler@54784
  1365
    hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
immler@54784
  1366
      by (auto simp: powr_realpow)
immler@54784
  1367
    also
immler@54784
  1368
    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
  1369
      using logless flogless by (auto intro!: floor_mono)
immler@54784
  1370
    also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))"
immler@54784
  1371
      using assms `0 < x`
immler@54784
  1372
      by (auto simp: algebra_simps)
immler@54784
  1373
    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
  1374
      by simp
immler@54784
  1375
    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
  1376
      by (subst powr_add[symmetric]) simp
immler@54784
  1377
    also have "\<dots> = y"
immler@54784
  1378
      using `0 < x` assms
immler@54784
  1379
      by (simp add: powr_add)
immler@54784
  1380
    also have "\<dots> \<le> truncate_up prec y"
immler@54784
  1381
      by (rule truncate_up)
immler@54784
  1382
    finally have ?thesis .
immler@54784
  1383
  } moreover {
immler@54784
  1384
    assume "~ 0 < x"
immler@54784
  1385
    hence ?thesis
immler@54784
  1386
      using assms
immler@54784
  1387
      by (auto intro!: truncate_up_le)
immler@54784
  1388
  } ultimately show ?thesis
immler@54784
  1389
    by blast
immler@54784
  1390
qed
immler@54784
  1391
immler@54784
  1392
lemma truncate_up_nonpos: "x \<le> 0 \<Longrightarrow> truncate_up prec x \<le> 0"
immler@54784
  1393
  by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)
immler@54784
  1394
immler@54784
  1395
lemma truncate_down_nonpos: "x \<le> 0 \<Longrightarrow> truncate_down prec x \<le> 0"
immler@54784
  1396
  by (auto simp: truncate_down_def round_down_def intro!: mult_nonpos_nonneg
immler@54784
  1397
    order_le_less_trans[of _ 0, OF mult_nonpos_nonneg])
immler@54784
  1398
immler@54784
  1399
lemma truncate_up_switch_sign_mono:
immler@54784
  1400
  assumes "x \<le> 0" "0 \<le> y"
immler@54784
  1401
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  1402
proof -
immler@54784
  1403
  note truncate_up_nonpos[OF `x \<le> 0`]
immler@54784
  1404
  also note truncate_up_le[OF `0 \<le> y`]
immler@54784
  1405
  finally show ?thesis .
immler@54784
  1406
qed
immler@54784
  1407
immler@54784
  1408
lemma truncate_down_zeroprec_mono:
immler@54784
  1409
  assumes "0 < x" "x \<le> y"
immler@54784
  1410
  shows "truncate_down 0 x \<le> truncate_down 0 y"
immler@54784
  1411
proof -
immler@54784
  1412
  have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))"
immler@54784
  1413
    by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide)
immler@54784
  1414
  also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)"
immler@54784
  1415
    using `0 < x`
wenzelm@57862
  1416
    by (auto simp: field_simps powr_add powr_divide2[symmetric])
immler@54784
  1417
  also have "\<dots> < 2 powr 0"
immler@54784
  1418
    using real_of_int_floor_add_one_gt
immler@54784
  1419
    unfolding neg_less_iff_less
immler@54784
  1420
    by (intro powr_less_mono) (auto simp: algebra_simps)
immler@54784
  1421
  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1"
immler@54784
  1422
    unfolding less_ceiling_eq real_of_int_minus real_of_one
immler@54784
  1423
    by simp
immler@54784
  1424
  moreover
immler@54784
  1425
  have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"
nipkow@56536
  1426
    using `x > 0` by auto
immler@54784
  1427
  ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"
immler@54784
  1428
    by simp
immler@54784
  1429
  also have "\<dots> \<subseteq> {0}" by auto
immler@54784
  1430
  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0" by simp
immler@54784
  1431
  with assms show ?thesis
nipkow@56536
  1432
    by (auto simp: truncate_down_def round_down_def)
immler@54784
  1433
qed
immler@54784
  1434
immler@54784
  1435
lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y"
nipkow@56536
  1436
  by (auto simp: truncate_down_def round_down_def)
immler@54784
  1437
immler@54784
  1438
lemma truncate_down_zero: "truncate_down prec 0 = 0"
nipkow@56536
  1439
  by (auto simp: truncate_down_def round_down_def)
immler@54784
  1440
immler@54784
  1441
lemma truncate_down_switch_sign_mono:
immler@54784
  1442
  assumes "x \<le> 0" "0 \<le> y"
immler@54784
  1443
  assumes "x \<le> y"
immler@54784
  1444
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  1445
proof -
immler@54784
  1446
  note truncate_down_nonpos[OF `x \<le> 0`]
immler@54784
  1447
  also note truncate_down_nonneg[OF `0 \<le> y`]
immler@54784
  1448
  finally show ?thesis .
immler@54784
  1449
qed
immler@54784
  1450
immler@54784
  1451
lemma truncate_up_uminus_truncate_down:
immler@54784
  1452
  "truncate_up prec x = - truncate_down prec (- x)"
immler@54784
  1453
  "truncate_up prec (-x) = - truncate_down prec x"
immler@54784
  1454
  by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
immler@54784
  1455
immler@54784
  1456
lemma truncate_down_uminus_truncate_up:
immler@54784
  1457
  "truncate_down prec x = - truncate_up prec (- x)"
immler@54784
  1458
  "truncate_down prec (-x) = - truncate_up prec x"
immler@54784
  1459
  by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
immler@54784
  1460
immler@54784
  1461
lemma truncate_down_nonneg_mono:
immler@54784
  1462
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  1463
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  1464
proof -
immler@54784
  1465
  {
immler@54784
  1466
    assume "0 < x" "prec = 0"
immler@54784
  1467
    with assms have ?thesis
immler@54784
  1468
      by (simp add: truncate_down_zeroprec_mono)
immler@54784
  1469
  } moreover {
immler@54784
  1470
    assume "~ 0 < x"
immler@54784
  1471
    with assms have "x = 0" "0 \<le> y" by simp_all
immler@54784
  1472
    hence ?thesis
immler@54784
  1473
      by (auto simp add: truncate_down_zero intro!: truncate_down_nonneg)
immler@54784
  1474
  } moreover {
immler@54784
  1475
    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@54784
  1476
    hence ?thesis
immler@54784
  1477
      using assms
immler@54784
  1478
      by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
immler@54784
  1479
  } moreover {
immler@54784
  1480
    assume "0 < x"
immler@54784
  1481
    hence "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y" using assms by auto
immler@54784
  1482
    moreover
immler@54784
  1483
    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@54784
  1484
    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
immler@54784
  1485
      unfolding atomize_conj abs_of_pos[OF `0 < x`] abs_of_pos[OF `0 < y`]
immler@54784
  1486
      by (metis floor_less_cancel linorder_cases not_le)
immler@54784
  1487
    assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto
immler@54784
  1488
    have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)"
immler@54784
  1489
      using `0 < y`
immler@54784
  1490
      by simp
immler@54784
  1491
    also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))"
immler@54784
  1492
      using `0 \<le> y` `0 \<le> x` assms(2)
nipkow@56544
  1493
      by (auto intro!: powr_mono divide_left_mono
immler@54784
  1494
        simp: real_of_nat_diff powr_add
immler@54784
  1495
        powr_divide2[symmetric])
immler@54784
  1496
    also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)"
immler@54784
  1497
      by (auto simp: powr_add)
immler@54784
  1498
    finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
immler@54784
  1499
      using `0 \<le> y`
immler@54784
  1500
      by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow)
immler@54784
  1501
    hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
immler@54784
  1502
      by (auto simp: truncate_down_def round_down_def)
immler@54784
  1503
    moreover
immler@54784
  1504
    {
immler@54784
  1505
      have "x = 2 powr (log 2 \<bar>x\<bar>)" using `0 < x` by simp
immler@54784
  1506
      also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"
immler@54784
  1507
        using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"]
immler@54784
  1508
        by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps)
immler@54784
  1509
      also
immler@54784
  1510
      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>)"
immler@54784
  1511
        using logless flogless `x > 0` `y > 0`
immler@54784
  1512
        by (auto intro!: floor_mono)
immler@54784
  1513
      finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"
immler@54784
  1514
        by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff)
immler@54784
  1515
    } ultimately have ?thesis
immler@54784
  1516
      by (metis dual_order.trans truncate_down)
immler@54784
  1517
  } ultimately show ?thesis by blast
immler@54784
  1518
qed
immler@54784
  1519
immler@54784
  1520
lemma truncate_down_mono: "x \<le> y \<Longrightarrow> truncate_down p x \<le> truncate_down p y"
immler@54784
  1521
  apply (cases "0 \<le> x")
immler@54784
  1522
  apply (rule truncate_down_nonneg_mono, assumption+)
immler@54784
  1523
  apply (simp add: truncate_down_uminus_truncate_up)
immler@54784
  1524
  apply (cases "0 \<le> y")
immler@54784
  1525
  apply (auto intro: truncate_up_nonneg_mono truncate_up_switch_sign_mono)
immler@54784
  1526
  done
immler@54784
  1527
immler@54784
  1528
lemma truncate_up_mono: "x \<le> y \<Longrightarrow> truncate_up p x \<le> truncate_up p y"
immler@54784
  1529
  by (simp add: truncate_up_uminus_truncate_down truncate_down_mono)
immler@54784
  1530
hoelzl@47599
  1531
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
hoelzl@47599
  1532
 apply (auto simp: zero_float_def mult_le_0_iff)
hoelzl@47599
  1533
 using powr_gt_zero[of 2 b] by simp
hoelzl@47599
  1534
hoelzl@47621
  1535
lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
hoelzl@47600
  1536
  unfolding pprt_def sup_float_def max_def sup_real_def by auto
hoelzl@47599
  1537
hoelzl@47621
  1538
lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
hoelzl@47600
  1539
  unfolding nprt_def inf_float_def min_def inf_real_def by auto
hoelzl@47599
  1540
kuncar@55565
  1541
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor .
obua@16782
  1542
hoelzl@47599
  1543
lemma compute_int_floor_fl[code]:
hoelzl@47601
  1544
  "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
hoelzl@47600
  1545
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  1546
hide_fact (open) compute_int_floor_fl
hoelzl@47599
  1547
hoelzl@47600
  1548
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp
hoelzl@47599
  1549
hoelzl@47599
  1550
lemma compute_floor_fl[code]:
hoelzl@47601
  1551
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
hoelzl@47600
  1552
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  1553
hide_fact (open) compute_floor_fl
obua@16782
  1554
hoelzl@47600
  1555
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp
hoelzl@47600
  1556
hoelzl@47600
  1557
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp
hoelzl@29804
  1558
hoelzl@47599
  1559
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
wenzelm@53381
  1560
proof (cases "floor_fl x = float_of 0")
wenzelm@53381
  1561
  case True
wenzelm@53381
  1562
  then show ?thesis by (simp add: floor_fl_def)
wenzelm@53381
  1563
next
wenzelm@53381
  1564
  case False
wenzelm@53381
  1565
  have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp
wenzelm@53381
  1566
  obtain i where "\<lfloor>real x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
wenzelm@53381
  1567
    by (rule denormalize_shift[OF eq[THEN eq_reflection] False])
wenzelm@53381
  1568
  then show ?thesis by simp
wenzelm@53381
  1569
qed
obua@16782
  1570
obua@16782
  1571
end
hoelzl@47599
  1572