src/HOL/Library/Float.thy
author hoelzl
Fri Apr 20 11:14:39 2012 +0200 (2012-04-20)
changeset 47621 4cf6011fb884
parent 47615 341fd902ef1c
child 47780 3357688660ff
permissions -rw-r--r--
hide code generation facts in the Float theory, they are only exported for Approximation
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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 "~~/src/HOL/Library/Lattice_Algebras"
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begin
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typedef float = "{m * 2 powr e | (m :: int) (e :: int). True }"
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  morphisms real_of_float float_of
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  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_abs_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]: "neg_numeral i \<in> float" by (intro floatI[of "neg_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 neg_numeral i \<in> float" by (intro floatI[of 1 "neg_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 (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 (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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  unfolding ab_diff_minus by (intro uminus_float plus_float)
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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 (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 (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 / (neg_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 / neg_numeral (Num.Bit0 n)"
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    unfolding neg_numeral_def by (simp del: minus_numeral)
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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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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 :: 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_neg_numeral) 
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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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  "fun_rel (op =) cr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
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  unfolding fun_rel_def cr_float_def by simp
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lemma float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x"
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  by (simp add: minus_numeral[symmetric] del: minus_numeral)
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lemma transfer_neg_numeral [transfer_rule]: 
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  "fun_rel (op =) cr_float (neg_numeral :: _ \<Rightarrow> real) (neg_numeral :: _ \<Rightarrow> float)"
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  unfolding fun_rel_def 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]: "neg_numeral k = float_of (neg_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"
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    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
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  then show "m1 = m2 \<and> e1 = e2"
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    using eq `m1 \<noteq> 0` by (simp add: powr_inj)
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qed simp
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   278
lemma mult_powr_eq_mult_powr_iff:
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   279
  fixes m1 m2 e1 e2 :: int
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   280
  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"
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   281
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
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   282
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
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   283
  by (cases e1 e2 rule: linorder_le_cases) auto
hoelzl@47599
   284
hoelzl@47599
   285
lemma floatE_normed:
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   286
  assumes x: "x \<in> float"
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   287
  obtains (zero) "x = 0"
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   288
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
hoelzl@47599
   289
proof atomize_elim
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   290
  { assume "x \<noteq> 0"
hoelzl@47599
   291
    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
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   292
    with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
hoelzl@47599
   293
      by auto
hoelzl@47599
   294
    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
   295
      by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
hoelzl@47599
   296
         (simp add: powr_add powr_realpow) }
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   297
  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)"
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   298
    by blast
hoelzl@47599
   299
qed
hoelzl@47599
   300
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   301
lemma float_normed_cases:
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   302
  fixes f :: float
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   303
  obtains (zero) "f = 0"
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   304
   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
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   305
proof (atomize_elim, induct f)
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   306
  case (float_of y) then show ?case
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   307
    by (cases rule: floatE_normed) (auto simp: zero_float_def)
hoelzl@47599
   308
qed
hoelzl@47599
   309
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   310
definition mantissa :: "float \<Rightarrow> int" where
hoelzl@47599
   311
  "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
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   312
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   313
hoelzl@47599
   314
definition exponent :: "float \<Rightarrow> int" where
hoelzl@47599
   315
  "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   316
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   317
hoelzl@47599
   318
lemma 
hoelzl@47599
   319
  shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
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   320
    and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
hoelzl@47599
   321
proof -
hoelzl@47599
   322
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
hoelzl@47599
   323
  then show ?E ?M
hoelzl@47600
   324
    by (auto simp add: mantissa_def exponent_def zero_float_def)
hoelzl@29804
   325
qed
hoelzl@29804
   326
hoelzl@47599
   327
lemma
hoelzl@47599
   328
  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
hoelzl@47599
   329
    and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
hoelzl@47599
   330
proof cases
hoelzl@47599
   331
  assume [simp]: "f \<noteq> (float_of 0)"
hoelzl@47599
   332
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
hoelzl@47599
   333
  proof (cases f rule: float_normed_cases)
hoelzl@47599
   334
    case (powr m e)
hoelzl@47599
   335
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   336
     \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
hoelzl@47599
   337
      by auto
hoelzl@47599
   338
    then show ?thesis
hoelzl@47599
   339
      unfolding exponent_def mantissa_def
hoelzl@47600
   340
      by (rule someI2_ex) (simp add: zero_float_def)
hoelzl@47600
   341
  qed (simp add: zero_float_def)
hoelzl@47599
   342
  then show ?E ?D by auto
hoelzl@47599
   343
qed simp
hoelzl@47599
   344
hoelzl@47599
   345
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
hoelzl@47599
   346
  using mantissa_not_dvd[of f] by auto
hoelzl@47599
   347
hoelzl@47599
   348
lemma 
hoelzl@47599
   349
  fixes m e :: int
hoelzl@47599
   350
  defines "f \<equiv> float_of (m * 2 powr e)"
hoelzl@47599
   351
  assumes dvd: "\<not> 2 dvd m"
hoelzl@47599
   352
  shows mantissa_float: "mantissa f = m" (is "?M")
hoelzl@47599
   353
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
hoelzl@47599
   354
proof cases
hoelzl@47599
   355
  assume "m = 0" with dvd show "mantissa f = m" by auto
hoelzl@47599
   356
next
hoelzl@47599
   357
  assume "m \<noteq> 0"
hoelzl@47599
   358
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
hoelzl@47599
   359
  from mantissa_exponent[of f]
hoelzl@47599
   360
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   361
    by (auto simp add: f_def)
hoelzl@47599
   362
  then show "?M" "?E"
hoelzl@47599
   363
    using mantissa_not_dvd[OF f_not_0] dvd
hoelzl@47599
   364
    by (auto simp: mult_powr_eq_mult_powr_iff)
hoelzl@47599
   365
qed
hoelzl@47599
   366
hoelzl@47600
   367
subsection {* Compute arithmetic operations *}
hoelzl@47600
   368
hoelzl@47600
   369
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
hoelzl@47600
   370
  unfolding real_of_float_eq mantissa_exponent[of f] by simp
hoelzl@47600
   371
hoelzl@47600
   372
lemma Float_cases[case_names Float, cases type: float]:
hoelzl@47600
   373
  fixes f :: float
hoelzl@47600
   374
  obtains (Float) m e :: int where "f = Float m e"
hoelzl@47600
   375
  using Float_mantissa_exponent[symmetric]
hoelzl@47600
   376
  by (atomize_elim) auto
hoelzl@47600
   377
hoelzl@47599
   378
lemma denormalize_shift:
hoelzl@47599
   379
  assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
hoelzl@47599
   380
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
hoelzl@47599
   381
proof
hoelzl@47599
   382
  from mantissa_exponent[of f] f_def
hoelzl@47599
   383
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   384
    by simp
hoelzl@47599
   385
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
hoelzl@47599
   386
    by (simp add: powr_divide2[symmetric] field_simps)
hoelzl@47599
   387
  moreover
hoelzl@47599
   388
  have "e \<le> exponent f"
hoelzl@47599
   389
  proof (rule ccontr)
hoelzl@47599
   390
    assume "\<not> e \<le> exponent f"
hoelzl@47599
   391
    then have pos: "exponent f < e" by simp
hoelzl@47599
   392
    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
hoelzl@47599
   393
      by simp
hoelzl@47599
   394
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
hoelzl@47599
   395
      using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47599
   396
    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
hoelzl@47599
   397
      using eq by simp
hoelzl@47599
   398
    then have "mantissa f = m * 2^nat (e - exponent f)"
hoelzl@47599
   399
      unfolding real_of_int_inject by simp
hoelzl@47599
   400
    with `exponent f < e` have "2 dvd mantissa f"
hoelzl@47599
   401
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
hoelzl@47599
   402
      apply (cases "nat (e - exponent f)")
hoelzl@47599
   403
      apply auto
hoelzl@47599
   404
      done
hoelzl@47599
   405
    then show False using mantissa_not_dvd[OF not_0] by simp
hoelzl@47599
   406
  qed
hoelzl@47599
   407
  ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
hoelzl@47599
   408
    by (simp add: powr_realpow[symmetric])
hoelzl@47599
   409
  with `e \<le> exponent f`
hoelzl@47599
   410
  show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
hoelzl@47599
   411
    unfolding real_of_int_inject by auto
hoelzl@29804
   412
qed
hoelzl@29804
   413
hoelzl@47621
   414
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
hoelzl@47600
   415
  by transfer simp
hoelzl@47621
   416
hide_fact (open) compute_float_zero
hoelzl@47600
   417
hoelzl@47621
   418
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
hoelzl@47600
   419
  by transfer simp
hoelzl@47621
   420
hide_fact (open) compute_float_one
hoelzl@47600
   421
hoelzl@47600
   422
definition normfloat :: "float \<Rightarrow> float" where
hoelzl@47600
   423
  [simp]: "normfloat x = x"
hoelzl@47600
   424
hoelzl@47600
   425
lemma compute_normfloat[code]: "normfloat (Float m e) =
hoelzl@47600
   426
  (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
hoelzl@47600
   427
                           else if m = 0 then 0 else Float m e)"
hoelzl@47600
   428
  unfolding normfloat_def
hoelzl@47600
   429
  by transfer (auto simp add: powr_add zmod_eq_0_iff)
hoelzl@47621
   430
hide_fact (open) compute_normfloat
hoelzl@47599
   431
hoelzl@47599
   432
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
hoelzl@47600
   433
  by transfer simp
hoelzl@47621
   434
hide_fact (open) compute_float_numeral
hoelzl@47599
   435
hoelzl@47599
   436
lemma compute_float_neg_numeral[code_abbrev]: "Float (neg_numeral k) 0 = neg_numeral k"
hoelzl@47600
   437
  by transfer simp
hoelzl@47621
   438
hide_fact (open) compute_float_neg_numeral
hoelzl@47599
   439
hoelzl@47599
   440
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
hoelzl@47600
   441
  by transfer simp
hoelzl@47621
   442
hide_fact (open) compute_float_uminus
hoelzl@47599
   443
hoelzl@47599
   444
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
hoelzl@47600
   445
  by transfer (simp add: field_simps powr_add)
hoelzl@47621
   446
hide_fact (open) compute_float_times
hoelzl@47599
   447
hoelzl@47599
   448
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
hoelzl@47599
   449
  (if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
hoelzl@47599
   450
              else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
hoelzl@47600
   451
  by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47621
   452
hide_fact (open) compute_float_plus
hoelzl@47599
   453
hoelzl@47600
   454
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
hoelzl@47600
   455
  by simp
hoelzl@47621
   456
hide_fact (open) compute_float_minus
hoelzl@47599
   457
hoelzl@47599
   458
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
hoelzl@47600
   459
  by transfer (simp add: sgn_times)
hoelzl@47621
   460
hide_fact (open) compute_float_sgn
hoelzl@47599
   461
hoelzl@47600
   462
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" ..
hoelzl@47599
   463
hoelzl@47599
   464
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
hoelzl@47600
   465
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
hoelzl@47621
   466
hide_fact (open) compute_is_float_pos
hoelzl@47599
   467
hoelzl@47599
   468
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
hoelzl@47600
   469
  by transfer (simp add: field_simps)
hoelzl@47621
   470
hide_fact (open) compute_float_less
hoelzl@47599
   471
hoelzl@47600
   472
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" ..
hoelzl@47599
   473
hoelzl@47599
   474
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
hoelzl@47600
   475
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
hoelzl@47621
   476
hide_fact (open) compute_is_float_nonneg
hoelzl@47599
   477
hoelzl@47599
   478
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
hoelzl@47600
   479
  by transfer (simp add: field_simps)
hoelzl@47621
   480
hide_fact (open) compute_float_le
hoelzl@47599
   481
hoelzl@47600
   482
lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" by simp
hoelzl@47599
   483
hoelzl@47599
   484
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
hoelzl@47600
   485
  by transfer (auto simp add: is_float_zero_def)
hoelzl@47621
   486
hide_fact (open) compute_is_float_zero
hoelzl@47599
   487
hoelzl@47600
   488
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
hoelzl@47600
   489
  by transfer (simp add: abs_mult)
hoelzl@47621
   490
hide_fact (open) compute_float_abs
hoelzl@47599
   491
hoelzl@47600
   492
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
hoelzl@47600
   493
  by transfer simp
hoelzl@47621
   494
hide_fact (open) compute_float_eq
hoelzl@47599
   495
hoelzl@47599
   496
subsection {* Rounding Real numbers *}
hoelzl@47599
   497
hoelzl@47599
   498
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   499
  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   500
hoelzl@47599
   501
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   502
  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   503
hoelzl@47599
   504
lemma round_down_float[simp]: "round_down prec x \<in> float"
hoelzl@47599
   505
  unfolding round_down_def
hoelzl@47599
   506
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   507
hoelzl@47599
   508
lemma round_up_float[simp]: "round_up prec x \<in> float"
hoelzl@47599
   509
  unfolding round_up_def
hoelzl@47599
   510
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   511
hoelzl@47599
   512
lemma round_up: "x \<le> round_up prec x"
hoelzl@47599
   513
  by (simp add: powr_minus_divide le_divide_eq round_up_def)
hoelzl@47599
   514
hoelzl@47599
   515
lemma round_down: "round_down prec x \<le> x"
hoelzl@47599
   516
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
hoelzl@47599
   517
hoelzl@47599
   518
lemma round_up_0[simp]: "round_up p 0 = 0"
hoelzl@47599
   519
  unfolding round_up_def by simp
hoelzl@47599
   520
hoelzl@47599
   521
lemma round_down_0[simp]: "round_down p 0 = 0"
hoelzl@47599
   522
  unfolding round_down_def by simp
hoelzl@47599
   523
hoelzl@47599
   524
lemma round_up_diff_round_down:
hoelzl@47599
   525
  "round_up prec x - round_down prec x \<le> 2 powr -prec"
hoelzl@47599
   526
proof -
hoelzl@47599
   527
  have "round_up prec x - round_down prec x =
hoelzl@47599
   528
    (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
hoelzl@47599
   529
    by (simp add: round_up_def round_down_def field_simps)
hoelzl@47599
   530
  also have "\<dots> \<le> 1 * 2 powr -prec"
hoelzl@47599
   531
    by (rule mult_mono)
hoelzl@47599
   532
       (auto simp del: real_of_int_diff
hoelzl@47599
   533
             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
hoelzl@47599
   534
  finally show ?thesis by simp
hoelzl@29804
   535
qed
hoelzl@29804
   536
hoelzl@47599
   537
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
hoelzl@47599
   538
  unfolding round_down_def
hoelzl@47599
   539
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   540
    (simp add: powr_add[symmetric])
hoelzl@29804
   541
hoelzl@47599
   542
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
hoelzl@47599
   543
  unfolding round_up_def
hoelzl@47599
   544
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   545
    (simp add: powr_add[symmetric])
hoelzl@47599
   546
hoelzl@47599
   547
subsection {* Rounding Floats *}
hoelzl@29804
   548
hoelzl@47600
   549
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
hoelzl@47601
   550
declare float_up.rep_eq[simp]
hoelzl@29804
   551
hoelzl@47599
   552
lemma float_up_correct:
hoelzl@47599
   553
  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
hoelzl@47599
   554
unfolding atLeastAtMost_iff
hoelzl@47599
   555
proof
hoelzl@47599
   556
  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
hoelzl@47599
   557
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
hoelzl@47599
   558
  finally show "real (float_up e f) - real f \<le> 2 powr real (- e)"
hoelzl@47600
   559
    by simp
hoelzl@47600
   560
qed (simp add: algebra_simps round_up)
hoelzl@29804
   561
hoelzl@47600
   562
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
hoelzl@47601
   563
declare float_down.rep_eq[simp]
obua@16782
   564
hoelzl@47599
   565
lemma float_down_correct:
hoelzl@47599
   566
  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
hoelzl@47599
   567
unfolding atLeastAtMost_iff
hoelzl@47599
   568
proof
hoelzl@47599
   569
  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
hoelzl@47599
   570
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
hoelzl@47599
   571
  finally show "real f - real (float_down e f) \<le> 2 powr real (- e)"
hoelzl@47600
   572
    by simp
hoelzl@47600
   573
qed (simp add: algebra_simps round_down)
obua@24301
   574
hoelzl@47599
   575
lemma compute_float_down[code]:
hoelzl@47599
   576
  "float_down p (Float m e) =
hoelzl@47599
   577
    (if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"
hoelzl@47599
   578
proof cases
hoelzl@47599
   579
  assume "p + e < 0"
hoelzl@47599
   580
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   581
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   582
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47600
   583
    unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   584
  finally show ?thesis
hoelzl@47600
   585
    using `p + e < 0`
hoelzl@47600
   586
    by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
hoelzl@47599
   587
next
hoelzl@47600
   588
  assume "\<not> p + e < 0"
hoelzl@47600
   589
  then have r: "real e + real p = real (nat (e + p))" by simp
hoelzl@47600
   590
  have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
hoelzl@47600
   591
    by (auto intro: exI[where x="m*2^nat (e+p)"]
hoelzl@47600
   592
             simp add: ac_simps powr_add[symmetric] r powr_realpow)
hoelzl@47600
   593
  with `\<not> p + e < 0` show ?thesis
hoelzl@47600
   594
    by transfer
hoelzl@47600
   595
       (auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide)
hoelzl@47599
   596
qed
hoelzl@47621
   597
hide_fact (open) compute_float_down
obua@24301
   598
hoelzl@47599
   599
lemma ceil_divide_floor_conv:
hoelzl@47599
   600
assumes "b \<noteq> 0"
hoelzl@47599
   601
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
   602
proof cases
hoelzl@47599
   603
  assume "\<not> b dvd a"
hoelzl@47599
   604
  hence "a mod b \<noteq> 0" by auto
hoelzl@47599
   605
  hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
hoelzl@47599
   606
  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
hoelzl@47599
   607
  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
hoelzl@47599
   608
  proof -
hoelzl@47599
   609
    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
hoelzl@47599
   610
    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
hoelzl@47599
   611
    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
hoelzl@47599
   612
    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
hoelzl@47599
   613
  qed
hoelzl@47599
   614
  thus ?thesis using `\<not> b dvd a` by simp
hoelzl@47599
   615
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
hoelzl@47599
   616
  floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
wenzelm@19765
   617
hoelzl@47599
   618
lemma compute_float_up[code]:
hoelzl@47599
   619
  "float_up p (Float m e) =
hoelzl@47599
   620
    (let P = 2^nat (-(p + e)); r = m mod P in
hoelzl@47599
   621
      if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (-p) else Float m e)"
hoelzl@47599
   622
proof cases
hoelzl@47599
   623
  assume "p + e < 0"
hoelzl@47599
   624
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   625
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   626
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47599
   627
  unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   628
  finally have twopow_rewrite:
hoelzl@47599
   629
    "real ((2::int) ^ nat (- (p + e))) = 1 / 2 powr real p / 2 powr real e" .
hoelzl@47599
   630
  with `p + e < 0` have powr_rewrite:
hoelzl@47599
   631
    "2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat (- (p + e)))"
hoelzl@47599
   632
    unfolding powr_divide2 by simp
hoelzl@47599
   633
  show ?thesis
hoelzl@47599
   634
  proof cases
hoelzl@47599
   635
    assume "2^nat (-(p + e)) dvd m"
hoelzl@47615
   636
    with `p + e < 0` twopow_rewrite show ?thesis
hoelzl@47600
   637
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div dvd_eq_mod_eq_0)
hoelzl@47599
   638
  next
hoelzl@47599
   639
    assume ndvd: "\<not> 2 ^ nat (- (p + e)) dvd m"
hoelzl@47599
   640
    have one_div: "real m * (1 / real ((2::int) ^ nat (- (p + e)))) =
hoelzl@47599
   641
      real m / real ((2::int) ^ nat (- (p + e)))"
hoelzl@47599
   642
      by (simp add: field_simps)
hoelzl@47599
   643
    have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> =
hoelzl@47599
   644
      real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1"
hoelzl@47599
   645
      using ndvd unfolding powr_rewrite one_div
hoelzl@47599
   646
      by (subst ceil_divide_floor_conv) (auto simp: field_simps)
hoelzl@47599
   647
    thus ?thesis using `p + e < 0` twopow_rewrite
hoelzl@47600
   648
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div[symmetric])
hoelzl@29804
   649
  qed
hoelzl@47599
   650
next
hoelzl@47600
   651
  assume "\<not> p + e < 0"
hoelzl@47600
   652
  then have r1: "real e + real p = real (nat (e + p))" by simp
hoelzl@47600
   653
  have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p"
hoelzl@47600
   654
    by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow
hoelzl@47600
   655
      intro: exI[where x="m*2^nat (e+p)"])
hoelzl@47600
   656
  then show ?thesis using `\<not> p + e < 0`
hoelzl@47600
   657
    by transfer
hoelzl@47600
   658
       (simp add: round_up_def floor_divide_eq_div field_simps powr_add powr_minus inverse_eq_divide)
hoelzl@29804
   659
qed
hoelzl@47621
   660
hide_fact (open) compute_float_up
hoelzl@29804
   661
hoelzl@47599
   662
lemmas real_of_ints =
hoelzl@47599
   663
  real_of_int_zero
hoelzl@47599
   664
  real_of_one
hoelzl@47599
   665
  real_of_int_add
hoelzl@47599
   666
  real_of_int_minus
hoelzl@47599
   667
  real_of_int_diff
hoelzl@47599
   668
  real_of_int_mult
hoelzl@47599
   669
  real_of_int_power
hoelzl@47599
   670
  real_numeral
hoelzl@47599
   671
lemmas real_of_nats =
hoelzl@47599
   672
  real_of_nat_zero
hoelzl@47599
   673
  real_of_nat_one
hoelzl@47599
   674
  real_of_nat_1
hoelzl@47599
   675
  real_of_nat_add
hoelzl@47599
   676
  real_of_nat_mult
hoelzl@47599
   677
  real_of_nat_power
hoelzl@47599
   678
hoelzl@47599
   679
lemmas int_of_reals = real_of_ints[symmetric]
hoelzl@47599
   680
lemmas nat_of_reals = real_of_nats[symmetric]
hoelzl@47599
   681
hoelzl@47599
   682
lemma two_real_int: "(2::real) = real (2::int)" by simp
hoelzl@47599
   683
lemma two_real_nat: "(2::real) = real (2::nat)" by simp
hoelzl@47599
   684
hoelzl@47599
   685
lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp
hoelzl@47599
   686
hoelzl@47599
   687
subsection {* Compute bitlen of integers *}
hoelzl@47599
   688
hoelzl@47600
   689
definition bitlen :: "int \<Rightarrow> int" where
hoelzl@47600
   690
  "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
hoelzl@47599
   691
hoelzl@47599
   692
lemma bitlen_nonneg: "0 \<le> bitlen x"
hoelzl@29804
   693
proof -
hoelzl@47599
   694
  {
hoelzl@47599
   695
    assume "0 > x"
hoelzl@47599
   696
    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
hoelzl@47599
   697
    also have "... < log 2 (-x)" using `0 > x` by auto
hoelzl@47599
   698
    finally have "-1 < log 2 (-x)" .
hoelzl@47599
   699
  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
hoelzl@47599
   700
qed
hoelzl@47599
   701
hoelzl@47599
   702
lemma bitlen_bounds:
hoelzl@47599
   703
  assumes "x > 0"
hoelzl@47599
   704
  shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
hoelzl@47599
   705
proof
hoelzl@47599
   706
  have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
hoelzl@47599
   707
    using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
hoelzl@47599
   708
    using real_nat_eq_real[of "floor (log 2 (real x))"]
hoelzl@47599
   709
    by simp
hoelzl@47599
   710
  also have "... \<le> 2 powr log 2 (real x)"
hoelzl@47599
   711
    by simp
hoelzl@47599
   712
  also have "... = real x"
hoelzl@47599
   713
    using `0 < x` by simp
hoelzl@47599
   714
  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
hoelzl@47599
   715
  thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
hoelzl@47599
   716
    by (simp add: bitlen_def)
hoelzl@47599
   717
next
hoelzl@47599
   718
  have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
hoelzl@47599
   719
  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
hoelzl@47599
   720
    apply (simp add: powr_realpow[symmetric])
hoelzl@47599
   721
    using `x > 0` by simp
hoelzl@47599
   722
  finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
hoelzl@47599
   723
    by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)
hoelzl@47599
   724
qed
hoelzl@47599
   725
hoelzl@47599
   726
lemma bitlen_pow2[simp]:
hoelzl@47599
   727
  assumes "b > 0"
hoelzl@47599
   728
  shows "bitlen (b * 2 ^ c) = bitlen b + c"
hoelzl@47599
   729
proof -
hoelzl@47599
   730
  from assms have "b * 2 ^ c > 0" by (auto intro: mult_pos_pos)
hoelzl@47599
   731
  thus ?thesis
hoelzl@47599
   732
    using floor_add[of "log 2 b" c] assms
hoelzl@47599
   733
    by (auto simp add: log_mult log_nat_power bitlen_def)
hoelzl@29804
   734
qed
hoelzl@29804
   735
hoelzl@47599
   736
lemma bitlen_Float:
hoelzl@47599
   737
fixes m e
hoelzl@47599
   738
defines "f \<equiv> Float m e"
hoelzl@47599
   739
shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
hoelzl@47599
   740
proof cases
hoelzl@47600
   741
  assume "m \<noteq> 0"
hoelzl@47600
   742
  hence "f \<noteq> float_of 0"
hoelzl@47600
   743
    unfolding real_of_float_eq by (simp add: f_def)
hoelzl@47600
   744
  hence "mantissa f \<noteq> 0"
hoelzl@47599
   745
    by (simp add: mantissa_noteq_0)
hoelzl@47599
   746
  moreover
hoelzl@47599
   747
  from f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`] guess i .
hoelzl@47599
   748
  ultimately show ?thesis by (simp add: abs_mult)
hoelzl@47600
   749
qed (simp add: f_def bitlen_def Float_def)
hoelzl@29804
   750
hoelzl@47599
   751
lemma compute_bitlen[code]:
hoelzl@47599
   752
  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
hoelzl@47599
   753
proof -
hoelzl@47599
   754
  { assume "2 \<le> x"
hoelzl@47599
   755
    then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
hoelzl@47599
   756
      by (simp add: log_mult zmod_zdiv_equality')
hoelzl@47599
   757
    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   758
    proof cases
hoelzl@47599
   759
      assume "x mod 2 = 0" then show ?thesis by simp
hoelzl@47599
   760
    next
hoelzl@47599
   761
      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   762
      then have "0 \<le> n"
hoelzl@47599
   763
        using `2 \<le> x` by simp
hoelzl@47599
   764
      assume "x mod 2 \<noteq> 0"
hoelzl@47599
   765
      with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
hoelzl@47599
   766
      with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
hoelzl@47599
   767
      moreover
hoelzl@47599
   768
      { have "real (2^nat n :: int) = 2 powr (nat n)"
hoelzl@47599
   769
          by (simp add: powr_realpow)
hoelzl@47599
   770
        also have "\<dots> \<le> 2 powr (log 2 x)"
hoelzl@47599
   771
          using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
hoelzl@47599
   772
        finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
hoelzl@47599
   773
      ultimately have "2^nat n \<le> x - 1" by simp
hoelzl@47599
   774
      then have "2^nat n \<le> real (x - 1)"
hoelzl@47599
   775
        unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47599
   776
      { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
hoelzl@47599
   777
          using `0 \<le> n` by (simp add: log_nat_power)
hoelzl@47599
   778
        also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
hoelzl@47599
   779
          using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
hoelzl@47599
   780
        finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
hoelzl@47599
   781
      moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
hoelzl@47599
   782
        using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
hoelzl@47599
   783
      ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
hoelzl@47599
   784
        unfolding n_def `x mod 2 = 1` by auto
hoelzl@47599
   785
    qed
hoelzl@47599
   786
    finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
hoelzl@47599
   787
  moreover
hoelzl@47599
   788
  { assume "x < 2" "0 < x"
hoelzl@47599
   789
    then have "x = 1" by simp
hoelzl@47599
   790
    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
hoelzl@47599
   791
  ultimately show ?thesis
hoelzl@47599
   792
    unfolding bitlen_def
hoelzl@47599
   793
    by (auto simp: pos_imp_zdiv_pos_iff not_le)
hoelzl@47599
   794
qed
hoelzl@47621
   795
hide_fact (open) compute_bitlen
hoelzl@29804
   796
hoelzl@47599
   797
lemma float_gt1_scale: assumes "1 \<le> Float m e"
hoelzl@47599
   798
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@47599
   799
proof -
hoelzl@47599
   800
  have "0 < Float m e" using assms by auto
hoelzl@47599
   801
  hence "0 < m" using powr_gt_zero[of 2 e]
hoelzl@47600
   802
    by (auto simp: zero_less_mult_iff)
hoelzl@47599
   803
  hence "m \<noteq> 0" by auto
hoelzl@47599
   804
  show ?thesis
hoelzl@47599
   805
  proof (cases "0 \<le> e")
hoelzl@47599
   806
    case True thus ?thesis using `0 < m`  by (simp add: bitlen_def)
hoelzl@29804
   807
  next
hoelzl@47599
   808
    have "(1::int) < 2" by simp
hoelzl@47599
   809
    case False let ?S = "2^(nat (-e))"
hoelzl@47599
   810
    have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   811
      by (auto simp: powr_minus field_simps inverse_eq_divide)
hoelzl@47599
   812
    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   813
      by (auto simp: powr_minus)
hoelzl@47599
   814
    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hoelzl@47599
   815
    hence "?S \<le> real m" unfolding mult_assoc by auto
hoelzl@47599
   816
    hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@47599
   817
    from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
hoelzl@47599
   818
    have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
hoelzl@47599
   819
    hence "-e < bitlen m" using False by auto
hoelzl@47599
   820
    thus ?thesis by auto
hoelzl@29804
   821
  qed
hoelzl@47599
   822
qed
hoelzl@29804
   823
hoelzl@29804
   824
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
   825
proof -
hoelzl@29804
   826
  let ?B = "2^nat(bitlen m - 1)"
hoelzl@29804
   827
hoelzl@29804
   828
  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   829
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   830
  thus "1 \<le> real m / ?B" by auto
hoelzl@29804
   831
hoelzl@29804
   832
  have "m \<noteq> 0" using assms by auto
hoelzl@47599
   833
  have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)
obua@16782
   834
hoelzl@29804
   835
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
hoelzl@47599
   836
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
hoelzl@29804
   837
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
hoelzl@29804
   838
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   839
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
hoelzl@29804
   840
  thus "real m / ?B < 2" by auto
hoelzl@29804
   841
qed
hoelzl@29804
   842
hoelzl@47599
   843
subsection {* Approximation of positive rationals *}
hoelzl@47599
   844
hoelzl@47599
   845
lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"
hoelzl@47599
   846
by (simp add: zdiv_zmult2_eq)
hoelzl@29804
   847
hoelzl@47599
   848
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
hoelzl@47599
   849
  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
hoelzl@29804
   850
hoelzl@47599
   851
lemma real_div_nat_eq_floor_of_divide:
hoelzl@47599
   852
  fixes a b::nat
hoelzl@47599
   853
  shows "a div b = real (floor (a/b))"
hoelzl@47599
   854
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)
hoelzl@29804
   855
hoelzl@47599
   856
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
hoelzl@29804
   857
hoelzl@47600
   858
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
   859
  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp
obua@16782
   860
hoelzl@47599
   861
lemma compute_lapprox_posrat[code]:
hoelzl@47599
   862
  fixes prec x y 
hoelzl@47599
   863
  shows "lapprox_posrat prec x y = 
hoelzl@47599
   864
   (let 
hoelzl@47599
   865
       l = rat_precision prec x y;
hoelzl@47599
   866
       d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
hoelzl@47599
   867
    in normfloat (Float d (- l)))"
hoelzl@47615
   868
    unfolding div_mult_twopow_eq normfloat_def
hoelzl@47600
   869
    by transfer
hoelzl@47615
   870
       (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
hoelzl@47599
   871
             del: two_powr_minus_int_float)
hoelzl@47621
   872
hide_fact (open) compute_lapprox_posrat
hoelzl@29804
   873
hoelzl@47600
   874
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
   875
  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
hoelzl@29804
   876
hoelzl@47599
   877
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *)
hoelzl@47599
   878
lemma compute_rapprox_posrat[code]:
hoelzl@47599
   879
  fixes prec x y
hoelzl@47599
   880
  defines "l \<equiv> rat_precision prec x y"
hoelzl@47599
   881
  shows "rapprox_posrat prec x y = (let
hoelzl@47599
   882
     l = l ;
hoelzl@47599
   883
     X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
hoelzl@47599
   884
     d = fst X div snd X ;
hoelzl@47599
   885
     m = fst X mod snd X
hoelzl@47599
   886
   in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
hoelzl@47599
   887
proof (cases "y = 0")
hoelzl@47615
   888
  assume "y = 0" thus ?thesis unfolding normfloat_def by transfer simp
hoelzl@47599
   889
next
hoelzl@47599
   890
  assume "y \<noteq> 0"
hoelzl@29804
   891
  show ?thesis
hoelzl@47599
   892
  proof (cases "0 \<le> l")
hoelzl@47599
   893
    assume "0 \<le> l"
hoelzl@47599
   894
    def x' == "x * 2 ^ nat l"
hoelzl@47599
   895
    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
hoelzl@47599
   896
    moreover have "real x * 2 powr real l = real x'"
hoelzl@47599
   897
      by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
hoelzl@47599
   898
    ultimately show ?thesis
hoelzl@47615
   899
      unfolding normfloat_def
hoelzl@47599
   900
      using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
hoelzl@47600
   901
        l_def[symmetric, THEN meta_eq_to_obj_eq]
hoelzl@47600
   902
      by transfer
hoelzl@47600
   903
         (simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def)
hoelzl@47599
   904
   next
hoelzl@47599
   905
    assume "\<not> 0 \<le> l"
hoelzl@47599
   906
    def y' == "y * 2 ^ nat (- l)"
hoelzl@47599
   907
    from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
hoelzl@47599
   908
    have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
hoelzl@47599
   909
    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
hoelzl@47599
   910
      using `\<not> 0 \<le> l`
hoelzl@47599
   911
      by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps inverse_eq_divide)
hoelzl@47599
   912
    ultimately show ?thesis
hoelzl@47615
   913
      unfolding normfloat_def
hoelzl@47599
   914
      using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
hoelzl@47600
   915
        l_def[symmetric, THEN meta_eq_to_obj_eq]
hoelzl@47600
   916
      by transfer
hoelzl@47600
   917
         (simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0)
hoelzl@29804
   918
  qed
hoelzl@29804
   919
qed
hoelzl@47621
   920
hide_fact (open) compute_rapprox_posrat
hoelzl@29804
   921
hoelzl@47599
   922
lemma rat_precision_pos:
hoelzl@47599
   923
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@47599
   924
  shows "rat_precision n (int x) (int y) > 0"
hoelzl@29804
   925
proof -
hoelzl@47599
   926
  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
hoelzl@47599
   927
  hence "bitlen (int x) < bitlen (int y)" using assms
hoelzl@47599
   928
    by (simp add: bitlen_def del: floor_add_one)
hoelzl@47599
   929
      (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
hoelzl@47599
   930
  thus ?thesis
hoelzl@47599
   931
    using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
hoelzl@29804
   932
qed
obua@16782
   933
hoelzl@47599
   934
lemma power_aux: assumes "x > 0" shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1"
hoelzl@47599
   935
proof -
hoelzl@47599
   936
  def y \<equiv> "nat (x - 1)" moreover
hoelzl@47599
   937
  have "(2::int) ^ y \<le> (2 ^ (y + 1)) - 1" by simp
hoelzl@47599
   938
  ultimately show ?thesis using assms by simp
hoelzl@29804
   939
qed
hoelzl@29804
   940
hoelzl@47601
   941
lemma rapprox_posrat_less1:
hoelzl@47601
   942
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@31098
   943
  shows "real (rapprox_posrat n x y) < 1"
hoelzl@47599
   944
proof -
hoelzl@47599
   945
  have powr1: "2 powr real (rat_precision n (int x) (int y)) = 
hoelzl@47599
   946
    2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms
hoelzl@47599
   947
    by (simp add: powr_realpow[symmetric])
hoelzl@47599
   948
  have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) *
hoelzl@47599
   949
     2 powr real (rat_precision n (int x) (int y))" by simp
hoelzl@47599
   950
  also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"
hoelzl@47599
   951
    apply (rule mult_strict_right_mono) by (insert assms) auto
hoelzl@47599
   952
  also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"
hoelzl@47599
   953
    by (simp add: powr_add diff_def powr_neg_numeral)
hoelzl@47599
   954
  also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"
hoelzl@47599
   955
    using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])
hoelzl@47599
   956
  also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"
hoelzl@47599
   957
    unfolding int_of_reals real_of_int_le_iff
hoelzl@47599
   958
    using rat_precision_pos[OF assms] by (rule power_aux)
hoelzl@47600
   959
  finally show ?thesis
hoelzl@47601
   960
    apply (transfer fixing: n x y)
hoelzl@47601
   961
    apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide powr1)
hoelzl@47599
   962
    unfolding int_of_reals real_of_int_less_iff
hoelzl@47601
   963
    apply (simp add: ceiling_less_eq)
hoelzl@47600
   964
    done
hoelzl@29804
   965
qed
hoelzl@29804
   966
hoelzl@47600
   967
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
   968
  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
obua@16782
   969
hoelzl@29804
   970
lemma compute_lapprox_rat[code]:
hoelzl@47599
   971
  "lapprox_rat prec x y =
hoelzl@47599
   972
    (if y = 0 then 0
hoelzl@47599
   973
    else if 0 \<le> x then
hoelzl@47599
   974
      (if 0 < y then lapprox_posrat prec (nat x) (nat y)
hoelzl@47599
   975
      else - (rapprox_posrat prec (nat x) (nat (-y)))) 
hoelzl@47599
   976
      else (if 0 < y
hoelzl@47599
   977
        then - (rapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
   978
        else lapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@47601
   979
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hoelzl@47621
   980
hide_fact (open) compute_lapprox_rat
hoelzl@47599
   981
hoelzl@47600
   982
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
   983
  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
hoelzl@47599
   984
hoelzl@47599
   985
lemma compute_rapprox_rat[code]:
hoelzl@47599
   986
  "rapprox_rat prec x y =
hoelzl@47599
   987
    (if y = 0 then 0
hoelzl@47599
   988
    else if 0 \<le> x then
hoelzl@47599
   989
      (if 0 < y then rapprox_posrat prec (nat x) (nat y)
hoelzl@47599
   990
      else - (lapprox_posrat prec (nat x) (nat (-y)))) 
hoelzl@47599
   991
      else (if 0 < y
hoelzl@47599
   992
        then - (lapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
   993
        else rapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@47601
   994
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hoelzl@47621
   995
hide_fact (open) compute_rapprox_rat
hoelzl@47599
   996
hoelzl@47599
   997
subsection {* Division *}
hoelzl@47599
   998
hoelzl@47600
   999
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is
hoelzl@47600
  1000
  "\<lambda>(prec::nat) a b. round_down (prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" by simp
hoelzl@47599
  1001
hoelzl@47599
  1002
lemma compute_float_divl[code]:
hoelzl@47600
  1003
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1004
proof cases
hoelzl@47601
  1005
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
hoelzl@47601
  1006
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
hoelzl@47601
  1007
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
hoelzl@47601
  1008
  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
  1009
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
hoelzl@47601
  1010
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
hoelzl@47601
  1011
    by (simp add: field_simps powr_divide2[symmetric])
hoelzl@47599
  1012
hoelzl@47601
  1013
  show ?thesis
hoelzl@47601
  1014
    using not_0 
hoelzl@47601
  1015
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift, simp add: field_simps)
hoelzl@47600
  1016
qed (transfer, auto)
hoelzl@47621
  1017
hide_fact (open) compute_float_divl
hoelzl@47600
  1018
hoelzl@47600
  1019
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is
hoelzl@47600
  1020
  "\<lambda>(prec::nat) a b. round_up (prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" by simp
hoelzl@47599
  1021
hoelzl@47599
  1022
lemma compute_float_divr[code]:
hoelzl@47600
  1023
  "float_divr prec (Float m1 s1) (Float m2 s2) = rapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1024
proof cases
hoelzl@47601
  1025
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
hoelzl@47601
  1026
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
hoelzl@47601
  1027
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
hoelzl@47601
  1028
  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
  1029
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
hoelzl@47601
  1030
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
hoelzl@47601
  1031
    by (simp add: field_simps powr_divide2[symmetric])
hoelzl@47600
  1032
hoelzl@47601
  1033
  show ?thesis
hoelzl@47601
  1034
    using not_0 
hoelzl@47601
  1035
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift, simp add: field_simps)
hoelzl@47600
  1036
qed (transfer, auto)
hoelzl@47621
  1037
hide_fact (open) compute_float_divr
obua@16782
  1038
hoelzl@47599
  1039
subsection {* Lemmas needed by Approximate *}
hoelzl@47599
  1040
hoelzl@47599
  1041
lemma Float_num[simp]: shows
hoelzl@47599
  1042
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
hoelzl@47599
  1043
   "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
hoelzl@47599
  1044
   "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"
hoelzl@47599
  1045
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
  1046
using powr_realpow[of 2 2] powr_realpow[of 2 3]
hoelzl@47599
  1047
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
hoelzl@47599
  1048
by auto
hoelzl@47599
  1049
hoelzl@47599
  1050
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp
hoelzl@47599
  1051
hoelzl@47599
  1052
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
hoelzl@47599
  1053
hoelzl@47599
  1054
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
hoelzl@47599
  1055
by arith
hoelzl@29804
  1056
hoelzl@47599
  1057
lemma lapprox_rat:
hoelzl@47599
  1058
  shows "real (lapprox_rat prec x y) \<le> real x / real y"
hoelzl@47599
  1059
  using round_down by (simp add: lapprox_rat_def)
obua@16782
  1060
hoelzl@47599
  1061
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
hoelzl@47599
  1062
proof -
hoelzl@47599
  1063
  from zmod_zdiv_equality'[of a b]
hoelzl@47599
  1064
  have "a = b * (a div b) + a mod b" by simp
hoelzl@47599
  1065
  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
hoelzl@47599
  1066
  using assms by simp
hoelzl@47599
  1067
  finally show ?thesis by simp
hoelzl@47599
  1068
qed
hoelzl@47599
  1069
hoelzl@47599
  1070
lemma lapprox_rat_nonneg:
hoelzl@47599
  1071
  fixes n x y
hoelzl@47599
  1072
  defines "p == int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))"
hoelzl@47599
  1073
  assumes "0 \<le> x" "0 < y"
hoelzl@47599
  1074
  shows "0 \<le> real (lapprox_rat n x y)"
hoelzl@47599
  1075
using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric]
hoelzl@47599
  1076
   powr_int[of 2, simplified]
hoelzl@47599
  1077
  by (auto simp add: inverse_eq_divide intro!: mult_nonneg_nonneg divide_nonneg_pos mult_pos_pos)
obua@16782
  1078
hoelzl@31098
  1079
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
hoelzl@47599
  1080
  using round_up by (simp add: rapprox_rat_def)
hoelzl@47599
  1081
hoelzl@47599
  1082
lemma rapprox_rat_le1:
hoelzl@47599
  1083
  fixes n x y
hoelzl@47599
  1084
  assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
hoelzl@47599
  1085
  shows "real (rapprox_rat n x y) \<le> 1"
hoelzl@47599
  1086
proof -
hoelzl@47599
  1087
  have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
hoelzl@47599
  1088
    using xy unfolding bitlen_def by (auto intro!: floor_mono)
hoelzl@47599
  1089
  then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def)
hoelzl@47599
  1090
  have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>
hoelzl@47599
  1091
      \<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>"
hoelzl@47599
  1092
    using xy by (auto intro!: ceiling_mono simp: field_simps)
hoelzl@47599
  1093
  also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"
hoelzl@47599
  1094
    using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>`
hoelzl@47599
  1095
    by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int)
hoelzl@47599
  1096
  finally show ?thesis
hoelzl@47599
  1097
    by (simp add: rapprox_rat_def round_up_def)
hoelzl@47599
  1098
       (simp add: powr_minus inverse_eq_divide)
hoelzl@29804
  1099
qed
obua@16782
  1100
hoelzl@47599
  1101
lemma rapprox_rat_nonneg_neg: 
hoelzl@47599
  1102
  "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1103
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1104
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
obua@16782
  1105
hoelzl@47599
  1106
lemma rapprox_rat_neg:
hoelzl@47599
  1107
  "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1108
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1109
  by (auto simp: field_simps mult_le_0_iff)
hoelzl@29804
  1110
hoelzl@47599
  1111
lemma rapprox_rat_nonpos_pos:
hoelzl@47599
  1112
  "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
hoelzl@47599
  1113
  unfolding rapprox_rat_def round_up_def
hoelzl@47599
  1114
  by (auto simp: field_simps mult_le_0_iff)
obua@16782
  1115
hoelzl@31098
  1116
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
hoelzl@47600
  1117
  by transfer (simp add: round_down)
hoelzl@47599
  1118
hoelzl@47599
  1119
lemma float_divl_lower_bound:
hoelzl@47600
  1120
  "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
hoelzl@47600
  1121
  by transfer (simp add: round_down_def zero_le_mult_iff zero_le_divide_iff)
hoelzl@47599
  1122
hoelzl@47599
  1123
lemma exponent_1: "exponent 1 = 0"
hoelzl@47599
  1124
  using exponent_float[of 1 0] by (simp add: one_float_def)
hoelzl@47599
  1125
hoelzl@47599
  1126
lemma mantissa_1: "mantissa 1 = 1"
hoelzl@47599
  1127
  using mantissa_float[of 1 0] by (simp add: one_float_def)
obua@16782
  1128
hoelzl@47599
  1129
lemma bitlen_1: "bitlen 1 = 1"
hoelzl@47599
  1130
  by (simp add: bitlen_def)
hoelzl@47599
  1131
hoelzl@47599
  1132
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
hoelzl@47599
  1133
proof
hoelzl@47599
  1134
  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
hoelzl@47599
  1135
  show "x = 0" by (simp add: zero_float_def z)
hoelzl@47599
  1136
qed (simp add: zero_float_def)
obua@16782
  1137
hoelzl@47599
  1138
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
hoelzl@47599
  1139
proof (cases "x = 0", simp)
hoelzl@47599
  1140
  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
hoelzl@47599
  1141
  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
hoelzl@47599
  1142
  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
hoelzl@47599
  1143
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
hoelzl@47599
  1144
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
hoelzl@47599
  1145
    by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]
hoelzl@47599
  1146
      real_of_int_le_iff less_imp_le)
hoelzl@47599
  1147
  finally show ?thesis by (simp add: powr_add)
hoelzl@29804
  1148
qed
hoelzl@29804
  1149
wenzelm@41528
  1150
lemma float_divl_pos_less1_bound:
hoelzl@47600
  1151
  "0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
hoelzl@47600
  1152
proof transfer
hoelzl@47600
  1153
  fix prec :: nat and x :: real assume x: "0 < x" "x < 1" "x \<in> float" and prec: "1 \<le> prec"
hoelzl@47600
  1154
  def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>" 
hoelzl@47600
  1155
  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
  1156
  proof cases
hoelzl@47600
  1157
    assume nonneg: "0 \<le> p"
hoelzl@47600
  1158
    hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)"
hoelzl@47600
  1159
      by (simp add: powr_int del: real_of_int_power) simp
hoelzl@47600
  1160
    also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp
hoelzl@47600
  1161
    also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le>
hoelzl@47600
  1162
      floor (real ((2::int) ^ nat p) * (1 / x))"
hoelzl@47600
  1163
      by (rule le_mult_floor) (auto simp: x prec less_imp_le)
hoelzl@47600
  1164
    finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow)
hoelzl@47600
  1165
    thus ?thesis unfolding p_def[symmetric]
hoelzl@47600
  1166
      using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def)
hoelzl@47600
  1167
  next
hoelzl@47600
  1168
    assume neg: "\<not> 0 \<le> p"
hoelzl@47600
  1169
hoelzl@47600
  1170
    have "x = 2 powr (log 2 x)"
hoelzl@47600
  1171
      using x by simp
hoelzl@47600
  1172
    also have "2 powr (log 2 x) \<le> 2 powr p"
hoelzl@47600
  1173
    proof (rule powr_mono)
hoelzl@47600
  1174
      have "log 2 x \<le> \<lceil>log 2 x\<rceil>"
hoelzl@47600
  1175
        by simp
hoelzl@47600
  1176
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1"
hoelzl@47600
  1177
        using ceiling_diff_floor_le_1[of "log 2 x"] by simp
hoelzl@47600
  1178
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec"
hoelzl@47600
  1179
        using prec by simp
hoelzl@47600
  1180
      finally show "log 2 x \<le> real p"
hoelzl@47600
  1181
        using x by (simp add: p_def)
hoelzl@47600
  1182
    qed simp
hoelzl@47600
  1183
    finally have x_le: "x \<le> 2 powr p" .
hoelzl@47600
  1184
hoelzl@47600
  1185
    from neg have "2 powr real p \<le> 2 powr 0"
hoelzl@47600
  1186
      by (intro powr_mono) auto
hoelzl@47600
  1187
    also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp
hoelzl@47600
  1188
    also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff
hoelzl@47600
  1189
      using x x_le by (intro floor_mono) (simp add:  pos_le_divide_eq mult_pos_pos)
hoelzl@47600
  1190
    finally show ?thesis
hoelzl@47600
  1191
      using prec x unfolding p_def[symmetric]
hoelzl@47600
  1192
      by (simp add: round_down_def powr_minus_divide pos_le_divide_eq mult_pos_pos)
hoelzl@47600
  1193
  qed
hoelzl@29804
  1194
qed
obua@16782
  1195
hoelzl@31098
  1196
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
hoelzl@47600
  1197
  using round_up by transfer simp
obua@16782
  1198
hoelzl@29804
  1199
lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
hoelzl@29804
  1200
proof -
hoelzl@47600
  1201
  have "1 \<le> 1 / real x" using `0 < x` and `x < 1` by auto
hoelzl@31098
  1202
  also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
hoelzl@47600
  1203
  finally show ?thesis by auto
hoelzl@29804
  1204
qed
hoelzl@29804
  1205
hoelzl@47599
  1206
lemma float_divr_nonpos_pos_upper_bound:
hoelzl@47600
  1207
  "real x \<le> 0 \<Longrightarrow> 0 < real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"
hoelzl@47600
  1208
  by transfer (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def)
obua@16782
  1209
hoelzl@47599
  1210
lemma float_divr_nonneg_neg_upper_bound:
hoelzl@47600
  1211
  "0 \<le> real x \<Longrightarrow> real y < 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"
hoelzl@47600
  1212
  by transfer (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def)
hoelzl@47600
  1213
hoelzl@47600
  1214
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is
hoelzl@47600
  1215
  "\<lambda>(prec::nat) x. round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x" by simp
hoelzl@47600
  1216
hoelzl@47600
  1217
lemma float_round_up: "real x \<le> real (float_round_up prec x)"
hoelzl@47600
  1218
  using round_up by transfer simp
hoelzl@47599
  1219
hoelzl@47600
  1220
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is
hoelzl@47600
  1221
  "\<lambda>(prec::nat) x. round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x" by simp
hoelzl@47599
  1222
hoelzl@47600
  1223
lemma float_round_down: "real (float_round_down prec x) \<le> real x"
hoelzl@47600
  1224
  using round_down by transfer simp
hoelzl@47599
  1225
hoelzl@47600
  1226
lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>"
hoelzl@47600
  1227
  using floor_add[of x i] by (simp del: floor_add add: ac_simps)
obua@16782
  1228
hoelzl@47599
  1229
lemma compute_float_round_down[code]:
hoelzl@47600
  1230
  "float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
hoelzl@47600
  1231
    if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
hoelzl@47600
  1232
             else Float m e)"
hoelzl@47621
  1233
  using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
hoelzl@47601
  1234
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong)
hoelzl@47621
  1235
hide_fact (open) compute_float_round_down
hoelzl@47599
  1236
hoelzl@47600
  1237
lemma compute_float_round_up[code]:
hoelzl@47600
  1238
  "float_round_up prec (Float m e) = (let d = (bitlen (abs m) - int prec) in
hoelzl@47600
  1239
     if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P
hoelzl@47600
  1240
                   in Float (n + (if r = 0 then 0 else 1)) (e + d)
hoelzl@47600
  1241
              else Float m e)"
hoelzl@47621
  1242
  using Float.compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
hoelzl@47600
  1243
  unfolding Let_def
hoelzl@47601
  1244
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong)
hoelzl@47621
  1245
hide_fact (open) compute_float_round_up
obua@16782
  1246
hoelzl@47599
  1247
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
hoelzl@47599
  1248
 apply (auto simp: zero_float_def mult_le_0_iff)
hoelzl@47599
  1249
 using powr_gt_zero[of 2 b] by simp
hoelzl@47599
  1250
hoelzl@47621
  1251
lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
hoelzl@47600
  1252
  unfolding pprt_def sup_float_def max_def sup_real_def by auto
hoelzl@47599
  1253
hoelzl@47621
  1254
lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
hoelzl@47600
  1255
  unfolding nprt_def inf_float_def min_def inf_real_def by auto
hoelzl@47599
  1256
hoelzl@47600
  1257
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor by simp
obua@16782
  1258
hoelzl@47599
  1259
lemma compute_int_floor_fl[code]:
hoelzl@47601
  1260
  "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
hoelzl@47600
  1261
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  1262
hide_fact (open) compute_int_floor_fl
hoelzl@47599
  1263
hoelzl@47600
  1264
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp
hoelzl@47599
  1265
hoelzl@47599
  1266
lemma compute_floor_fl[code]:
hoelzl@47601
  1267
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
hoelzl@47600
  1268
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  1269
hide_fact (open) compute_floor_fl
obua@16782
  1270
hoelzl@47600
  1271
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp
hoelzl@47600
  1272
hoelzl@47600
  1273
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp
hoelzl@29804
  1274
hoelzl@47599
  1275
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
hoelzl@47599
  1276
proof cases
hoelzl@47599
  1277
  assume nzero: "floor_fl x \<noteq> float_of 0"
hoelzl@47600
  1278
  have "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp
hoelzl@47600
  1279
  from denormalize_shift[OF this[THEN eq_reflection] nzero] guess i . note i = this
hoelzl@47599
  1280
  thus ?thesis by simp
hoelzl@47599
  1281
qed (simp add: floor_fl_def)
obua@16782
  1282
obua@16782
  1283
end
hoelzl@47599
  1284