src/HOL/Library/Float.thy
author immler
Wed Nov 12 17:36:29 2014 +0100 (2014-11-12)
changeset 58982 27e7e3f9e665
parent 58881 b9556a055632
child 58985 bf498e0af9e3
permissions -rw-r--r--
simplified computations based on round_up by reducing to round_down;
more general round_up_le1, round_up_less1, round_down_ge1, round_up_le0
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(*  Title:      HOL/Library/Float.thy
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    Author:     Johannes Hölzl, Fabian Immler
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    Copyright   2012  TU München
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*)
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section {* 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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instantiation float :: real_of
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begin
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definition real_float :: "float \<Rightarrow> real" where
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  real_of_float_def[code_unfold]: "real \<equiv> real_of_float"
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instance ..
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end
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lemma type_definition_float': "type_definition real float_of float"
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  using type_definition_float unfolding real_of_float_def .
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setup_lifting (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)
hoelzl@47599
   284
  also have "\<dots> = m2 * 2^nat (e2 - e1)"
hoelzl@47599
   285
    by (simp add: powr_realpow)
hoelzl@47599
   286
  finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
hoelzl@47599
   287
    unfolding real_of_int_inject .
hoelzl@47599
   288
  with m1 have "m1 = m2"
hoelzl@47599
   289
    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
hoelzl@47599
   290
  then show "m1 = m2 \<and> e1 = e2"
hoelzl@47599
   291
    using eq `m1 \<noteq> 0` by (simp add: powr_inj)
hoelzl@47599
   292
qed simp
hoelzl@47599
   293
hoelzl@47599
   294
lemma mult_powr_eq_mult_powr_iff:
hoelzl@47599
   295
  fixes m1 m2 e1 e2 :: int
hoelzl@47599
   296
  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
   297
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
hoelzl@47599
   298
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
hoelzl@47599
   299
  by (cases e1 e2 rule: linorder_le_cases) auto
hoelzl@47599
   300
hoelzl@47599
   301
lemma floatE_normed:
hoelzl@47599
   302
  assumes x: "x \<in> float"
hoelzl@47599
   303
  obtains (zero) "x = 0"
hoelzl@47599
   304
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
hoelzl@47599
   305
proof atomize_elim
hoelzl@47599
   306
  { assume "x \<noteq> 0"
hoelzl@47599
   307
    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
hoelzl@47599
   308
    with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
hoelzl@47599
   309
      by auto
hoelzl@47599
   310
    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
   311
      by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
hoelzl@47599
   312
         (simp add: powr_add powr_realpow) }
hoelzl@47599
   313
  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
   314
    by blast
hoelzl@47599
   315
qed
hoelzl@47599
   316
hoelzl@47599
   317
lemma float_normed_cases:
hoelzl@47599
   318
  fixes f :: float
hoelzl@47599
   319
  obtains (zero) "f = 0"
hoelzl@47599
   320
   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
hoelzl@47599
   321
proof (atomize_elim, induct f)
hoelzl@47599
   322
  case (float_of y) then show ?case
hoelzl@47600
   323
    by (cases rule: floatE_normed) (auto simp: zero_float_def)
hoelzl@47599
   324
qed
hoelzl@47599
   325
hoelzl@47599
   326
definition mantissa :: "float \<Rightarrow> int" where
hoelzl@47599
   327
  "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   328
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   329
hoelzl@47599
   330
definition exponent :: "float \<Rightarrow> int" where
hoelzl@47599
   331
  "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   332
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
hoelzl@47599
   333
wenzelm@53381
   334
lemma
hoelzl@47599
   335
  shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
hoelzl@47599
   336
    and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
hoelzl@47599
   337
proof -
hoelzl@47599
   338
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
hoelzl@47599
   339
  then show ?E ?M
hoelzl@47600
   340
    by (auto simp add: mantissa_def exponent_def zero_float_def)
hoelzl@29804
   341
qed
hoelzl@29804
   342
hoelzl@47599
   343
lemma
hoelzl@47599
   344
  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
hoelzl@47599
   345
    and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
hoelzl@47599
   346
proof cases
hoelzl@47599
   347
  assume [simp]: "f \<noteq> (float_of 0)"
hoelzl@47599
   348
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
hoelzl@47599
   349
  proof (cases f rule: float_normed_cases)
hoelzl@47599
   350
    case (powr m e)
hoelzl@47599
   351
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
hoelzl@47599
   352
     \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
hoelzl@47599
   353
      by auto
hoelzl@47599
   354
    then show ?thesis
hoelzl@47599
   355
      unfolding exponent_def mantissa_def
hoelzl@47600
   356
      by (rule someI2_ex) (simp add: zero_float_def)
hoelzl@47600
   357
  qed (simp add: zero_float_def)
hoelzl@47599
   358
  then show ?E ?D by auto
hoelzl@47599
   359
qed simp
hoelzl@47599
   360
hoelzl@47599
   361
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
hoelzl@47599
   362
  using mantissa_not_dvd[of f] by auto
hoelzl@47599
   363
wenzelm@53381
   364
lemma
hoelzl@47599
   365
  fixes m e :: int
hoelzl@47599
   366
  defines "f \<equiv> float_of (m * 2 powr e)"
hoelzl@47599
   367
  assumes dvd: "\<not> 2 dvd m"
hoelzl@47599
   368
  shows mantissa_float: "mantissa f = m" (is "?M")
hoelzl@47599
   369
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
hoelzl@47599
   370
proof cases
hoelzl@47599
   371
  assume "m = 0" with dvd show "mantissa f = m" by auto
hoelzl@47599
   372
next
hoelzl@47599
   373
  assume "m \<noteq> 0"
hoelzl@47599
   374
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
hoelzl@47599
   375
  from mantissa_exponent[of f]
hoelzl@47599
   376
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   377
    by (auto simp add: f_def)
hoelzl@47599
   378
  then show "?M" "?E"
hoelzl@47599
   379
    using mantissa_not_dvd[OF f_not_0] dvd
hoelzl@47599
   380
    by (auto simp: mult_powr_eq_mult_powr_iff)
hoelzl@47599
   381
qed
hoelzl@47599
   382
hoelzl@47600
   383
subsection {* Compute arithmetic operations *}
hoelzl@47600
   384
hoelzl@47600
   385
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
hoelzl@47600
   386
  unfolding real_of_float_eq mantissa_exponent[of f] by simp
hoelzl@47600
   387
hoelzl@47600
   388
lemma Float_cases[case_names Float, cases type: float]:
hoelzl@47600
   389
  fixes f :: float
hoelzl@47600
   390
  obtains (Float) m e :: int where "f = Float m e"
hoelzl@47600
   391
  using Float_mantissa_exponent[symmetric]
hoelzl@47600
   392
  by (atomize_elim) auto
hoelzl@47600
   393
hoelzl@47599
   394
lemma denormalize_shift:
hoelzl@47599
   395
  assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
hoelzl@47599
   396
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
hoelzl@47599
   397
proof
hoelzl@47599
   398
  from mantissa_exponent[of f] f_def
hoelzl@47599
   399
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
hoelzl@47599
   400
    by simp
hoelzl@47599
   401
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
hoelzl@47599
   402
    by (simp add: powr_divide2[symmetric] field_simps)
hoelzl@47599
   403
  moreover
hoelzl@47599
   404
  have "e \<le> exponent f"
hoelzl@47599
   405
  proof (rule ccontr)
hoelzl@47599
   406
    assume "\<not> e \<le> exponent f"
hoelzl@47599
   407
    then have pos: "exponent f < e" by simp
hoelzl@47599
   408
    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
hoelzl@47599
   409
      by simp
hoelzl@47599
   410
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
hoelzl@47599
   411
      using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47599
   412
    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
hoelzl@47599
   413
      using eq by simp
hoelzl@47599
   414
    then have "mantissa f = m * 2^nat (e - exponent f)"
hoelzl@47599
   415
      unfolding real_of_int_inject by simp
hoelzl@47599
   416
    with `exponent f < e` have "2 dvd mantissa f"
hoelzl@47599
   417
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
hoelzl@47599
   418
      apply (cases "nat (e - exponent f)")
hoelzl@47599
   419
      apply auto
hoelzl@47599
   420
      done
hoelzl@47599
   421
    then show False using mantissa_not_dvd[OF not_0] by simp
hoelzl@47599
   422
  qed
hoelzl@47599
   423
  ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
hoelzl@47599
   424
    by (simp add: powr_realpow[symmetric])
hoelzl@47599
   425
  with `e \<le> exponent f`
hoelzl@47599
   426
  show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
hoelzl@47599
   427
    unfolding real_of_int_inject by auto
hoelzl@29804
   428
qed
hoelzl@29804
   429
hoelzl@47621
   430
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
hoelzl@47600
   431
  by transfer simp
hoelzl@47621
   432
hide_fact (open) compute_float_zero
hoelzl@47600
   433
hoelzl@47621
   434
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
hoelzl@47600
   435
  by transfer simp
hoelzl@47621
   436
hide_fact (open) compute_float_one
hoelzl@47600
   437
immler@58982
   438
lift_definition normfloat :: "float \<Rightarrow> float" is "\<lambda>x. x" .
immler@58982
   439
lemma normloat_id[simp]: "normfloat x = x" by transfer rule
hoelzl@47600
   440
hoelzl@47600
   441
lemma compute_normfloat[code]: "normfloat (Float m e) =
hoelzl@47600
   442
  (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
hoelzl@47600
   443
                           else if m = 0 then 0 else Float m e)"
hoelzl@47600
   444
  by transfer (auto simp add: powr_add zmod_eq_0_iff)
hoelzl@47621
   445
hide_fact (open) compute_normfloat
hoelzl@47599
   446
hoelzl@47599
   447
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
hoelzl@47600
   448
  by transfer simp
hoelzl@47621
   449
hide_fact (open) compute_float_numeral
hoelzl@47599
   450
haftmann@54489
   451
lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
hoelzl@47600
   452
  by transfer simp
hoelzl@47621
   453
hide_fact (open) compute_float_neg_numeral
hoelzl@47599
   454
hoelzl@47599
   455
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
hoelzl@47600
   456
  by transfer simp
hoelzl@47621
   457
hide_fact (open) compute_float_uminus
hoelzl@47599
   458
hoelzl@47599
   459
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
hoelzl@47600
   460
  by transfer (simp add: field_simps powr_add)
hoelzl@47621
   461
hide_fact (open) compute_float_times
hoelzl@47599
   462
hoelzl@47599
   463
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
immler@54783
   464
  (if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else
immler@54783
   465
  if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
hoelzl@47599
   466
              else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
hoelzl@47600
   467
  by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
hoelzl@47621
   468
hide_fact (open) compute_float_plus
hoelzl@47599
   469
hoelzl@47600
   470
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
hoelzl@47600
   471
  by simp
hoelzl@47621
   472
hide_fact (open) compute_float_minus
hoelzl@47599
   473
hoelzl@47599
   474
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
hoelzl@47600
   475
  by transfer (simp add: sgn_times)
hoelzl@47621
   476
hide_fact (open) compute_float_sgn
hoelzl@47599
   477
kuncar@55565
   478
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   479
hoelzl@47599
   480
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
hoelzl@47600
   481
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
hoelzl@47621
   482
hide_fact (open) compute_is_float_pos
hoelzl@47599
   483
hoelzl@47599
   484
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
hoelzl@47600
   485
  by transfer (simp add: field_simps)
hoelzl@47621
   486
hide_fact (open) compute_float_less
hoelzl@47599
   487
kuncar@55565
   488
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   489
hoelzl@47599
   490
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
hoelzl@47600
   491
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
hoelzl@47621
   492
hide_fact (open) compute_is_float_nonneg
hoelzl@47599
   493
hoelzl@47599
   494
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
hoelzl@47600
   495
  by transfer (simp add: field_simps)
hoelzl@47621
   496
hide_fact (open) compute_float_le
hoelzl@47599
   497
kuncar@55565
   498
lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" .
hoelzl@47599
   499
hoelzl@47599
   500
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
hoelzl@47600
   501
  by transfer (auto simp add: is_float_zero_def)
hoelzl@47621
   502
hide_fact (open) compute_is_float_zero
hoelzl@47599
   503
hoelzl@47600
   504
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
hoelzl@47600
   505
  by transfer (simp add: abs_mult)
hoelzl@47621
   506
hide_fact (open) compute_float_abs
hoelzl@47599
   507
hoelzl@47600
   508
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
hoelzl@47600
   509
  by transfer simp
hoelzl@47621
   510
hide_fact (open) compute_float_eq
hoelzl@47599
   511
immler@58982
   512
immler@58982
   513
subsection {* Lemmas for types @{typ real}, @{typ nat}, @{typ int}*}
immler@58982
   514
immler@58982
   515
lemmas real_of_ints =
immler@58982
   516
  real_of_int_zero
immler@58982
   517
  real_of_one
immler@58982
   518
  real_of_int_add
immler@58982
   519
  real_of_int_minus
immler@58982
   520
  real_of_int_diff
immler@58982
   521
  real_of_int_mult
immler@58982
   522
  real_of_int_power
immler@58982
   523
  real_numeral
immler@58982
   524
lemmas real_of_nats =
immler@58982
   525
  real_of_nat_zero
immler@58982
   526
  real_of_nat_one
immler@58982
   527
  real_of_nat_1
immler@58982
   528
  real_of_nat_add
immler@58982
   529
  real_of_nat_mult
immler@58982
   530
  real_of_nat_power
immler@58982
   531
immler@58982
   532
lemmas int_of_reals = real_of_ints[symmetric]
immler@58982
   533
lemmas nat_of_reals = real_of_nats[symmetric]
immler@58982
   534
immler@58982
   535
lemma two_real_int: "(2::real) = real (2::int)" by simp
immler@58982
   536
lemma two_real_nat: "(2::real) = real (2::nat)" by simp
immler@58982
   537
immler@58982
   538
hoelzl@47599
   539
subsection {* Rounding Real numbers *}
hoelzl@47599
   540
hoelzl@47599
   541
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   542
  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   543
hoelzl@47599
   544
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
hoelzl@47599
   545
  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
hoelzl@47599
   546
hoelzl@47599
   547
lemma round_down_float[simp]: "round_down prec x \<in> float"
hoelzl@47599
   548
  unfolding round_down_def
hoelzl@47599
   549
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   550
hoelzl@47599
   551
lemma round_up_float[simp]: "round_up prec x \<in> float"
hoelzl@47599
   552
  unfolding round_up_def
hoelzl@47599
   553
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
hoelzl@47599
   554
hoelzl@47599
   555
lemma round_up: "x \<le> round_up prec x"
hoelzl@47599
   556
  by (simp add: powr_minus_divide le_divide_eq round_up_def)
hoelzl@47599
   557
hoelzl@47599
   558
lemma round_down: "round_down prec x \<le> x"
hoelzl@47599
   559
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
hoelzl@47599
   560
hoelzl@47599
   561
lemma round_up_0[simp]: "round_up p 0 = 0"
hoelzl@47599
   562
  unfolding round_up_def by simp
hoelzl@47599
   563
hoelzl@47599
   564
lemma round_down_0[simp]: "round_down p 0 = 0"
hoelzl@47599
   565
  unfolding round_down_def by simp
hoelzl@47599
   566
hoelzl@47599
   567
lemma round_up_diff_round_down:
hoelzl@47599
   568
  "round_up prec x - round_down prec x \<le> 2 powr -prec"
hoelzl@47599
   569
proof -
hoelzl@47599
   570
  have "round_up prec x - round_down prec x =
hoelzl@47599
   571
    (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
hoelzl@47599
   572
    by (simp add: round_up_def round_down_def field_simps)
hoelzl@47599
   573
  also have "\<dots> \<le> 1 * 2 powr -prec"
hoelzl@47599
   574
    by (rule mult_mono)
hoelzl@47599
   575
       (auto simp del: real_of_int_diff
hoelzl@47599
   576
             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
hoelzl@47599
   577
  finally show ?thesis by simp
hoelzl@29804
   578
qed
hoelzl@29804
   579
hoelzl@47599
   580
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
hoelzl@47599
   581
  unfolding round_down_def
hoelzl@47599
   582
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   583
    (simp add: powr_add[symmetric])
hoelzl@29804
   584
hoelzl@47599
   585
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
hoelzl@47599
   586
  unfolding round_up_def
hoelzl@47599
   587
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
hoelzl@47599
   588
    (simp add: powr_add[symmetric])
hoelzl@47599
   589
immler@58982
   590
lemma round_up_uminus_eq: "round_up p (-x) = - round_down p x"
immler@58982
   591
  and round_down_uminus_eq: "round_down p (-x) = - round_up p x"
immler@58982
   592
  by (auto simp: round_up_def round_down_def ceiling_def)
immler@58982
   593
immler@58982
   594
lemma round_up_mono: "x \<le> y \<Longrightarrow> round_up p x \<le> round_up p y"
immler@58982
   595
  by (auto intro!: ceiling_mono simp: round_up_def)
immler@58982
   596
immler@58982
   597
lemma round_up_le1:
immler@58982
   598
  assumes "x \<le> 1" "prec \<ge> 0"
immler@58982
   599
  shows "round_up prec x \<le> 1"
immler@58982
   600
proof -
immler@58982
   601
  have "real \<lceil>x * 2 powr prec\<rceil> \<le> real \<lceil>2 powr real prec\<rceil>"
immler@58982
   602
    using assms by (auto intro!: ceiling_mono)
immler@58982
   603
  also have "\<dots> = 2 powr prec" using assms by (auto simp: powr_int intro!: exI[where x="2^nat prec"])
immler@58982
   604
  finally show ?thesis
immler@58982
   605
    by (simp add: round_up_def) (simp add: powr_minus inverse_eq_divide)
immler@58982
   606
qed
immler@58982
   607
immler@58982
   608
lemma round_up_less1:
immler@58982
   609
  assumes "x < 1 / 2" "p > 0"
immler@58982
   610
  shows "round_up p x < 1"
immler@58982
   611
proof -
immler@58982
   612
  have powr1: "2 powr p = 2 ^ nat p"
immler@58982
   613
    using `p > 0` by (simp add: powr_realpow[symmetric])
immler@58982
   614
  have "x * 2 powr p < 1 / 2 * 2 powr p"
immler@58982
   615
    using assms by simp
immler@58982
   616
  also have "\<dots> = 2 powr (p - 1)"
immler@58982
   617
    by (simp add: algebra_simps powr_mult_base)
immler@58982
   618
  also have "\<dots> = 2 ^ nat (p - 1)"
immler@58982
   619
    using `p > 0` by (simp add: powr_realpow[symmetric])
immler@58982
   620
  also have "\<dots> \<le> 2 ^ nat p - 1"
immler@58982
   621
    using `p > 0`
immler@58982
   622
    unfolding int_of_reals real_of_int_le_iff
immler@58982
   623
    by simp
immler@58982
   624
  finally show ?thesis
immler@58982
   625
    apply (simp add: round_up_def field_simps powr_minus powr1)
immler@58982
   626
    unfolding int_of_reals real_of_int_less_iff
immler@58982
   627
    apply (simp add: ceiling_less_eq)
immler@58982
   628
    done
immler@58982
   629
qed
immler@58982
   630
immler@58982
   631
lemma round_down_ge1:
immler@58982
   632
  assumes x: "x \<ge> 1"
immler@58982
   633
  assumes prec: "p \<ge> - log 2 x"
immler@58982
   634
  shows "1 \<le> round_down p x"
immler@58982
   635
proof cases
immler@58982
   636
  assume nonneg: "0 \<le> p"
immler@58982
   637
  hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)"
immler@58982
   638
    by (simp add: powr_int del: real_of_int_power) simp
immler@58982
   639
  also have "floor (1::real) \<le> floor x" using x by simp
immler@58982
   640
  also have "floor (real ((2::int) ^ nat p)) * floor x \<le>
immler@58982
   641
    floor (real ((2::int) ^ nat p) * x)"
immler@58982
   642
    using x
immler@58982
   643
    by (intro le_mult_floor) (auto simp: less_imp_le)
immler@58982
   644
  finally have "2 powr real p \<le> floor (2 powr nat p * x)" by (simp add: powr_realpow)
immler@58982
   645
  thus ?thesis
immler@58982
   646
    using x nonneg by (simp add: powr_minus inverse_eq_divide round_down_def mult.commute)
immler@58982
   647
next
immler@58982
   648
  assume neg: "\<not> 0 \<le> p"
immler@58982
   649
  have "x = 2 powr (log 2 x)"
immler@58982
   650
    using x by simp
immler@58982
   651
  also have "2 powr (log 2 x) \<ge> 2 powr - p"
immler@58982
   652
    using prec by auto
immler@58982
   653
  finally have x_le: "x \<ge> 2 powr -p" .
immler@58982
   654
immler@58982
   655
  from neg have "2 powr real p \<le> 2 powr 0"
immler@58982
   656
    by (intro powr_mono) auto
immler@58982
   657
  also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp
immler@58982
   658
  also have "\<dots> \<le> \<lfloor>x * 2 powr real p\<rfloor>" unfolding real_of_int_le_iff
immler@58982
   659
    using x x_le by (intro floor_mono) (simp add: powr_minus_divide field_simps)
immler@58982
   660
  finally show ?thesis
immler@58982
   661
    using prec x
immler@58982
   662
    by (simp add: round_down_def powr_minus_divide pos_le_divide_eq)
immler@58982
   663
qed
immler@58982
   664
immler@58982
   665
lemma round_up_le0: "x \<le> 0 \<Longrightarrow> round_up p x \<le> 0"
immler@58982
   666
  unfolding round_up_def
immler@58982
   667
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
immler@58982
   668
immler@58982
   669
hoelzl@47599
   670
subsection {* Rounding Floats *}
hoelzl@29804
   671
hoelzl@47600
   672
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
hoelzl@47601
   673
declare float_up.rep_eq[simp]
hoelzl@29804
   674
immler@54782
   675
lemma round_up_correct:
immler@54782
   676
  shows "round_up e f - f \<in> {0..2 powr -e}"
hoelzl@47599
   677
unfolding atLeastAtMost_iff
hoelzl@47599
   678
proof
hoelzl@47599
   679
  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
hoelzl@47599
   680
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
immler@54782
   681
  finally show "round_up e f - f \<le> 2 powr real (- e)"
hoelzl@47600
   682
    by simp
hoelzl@47600
   683
qed (simp add: algebra_simps round_up)
hoelzl@29804
   684
immler@54782
   685
lemma float_up_correct:
immler@54782
   686
  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
immler@54782
   687
  by transfer (rule round_up_correct)
immler@54782
   688
hoelzl@47600
   689
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
hoelzl@47601
   690
declare float_down.rep_eq[simp]
obua@16782
   691
immler@54782
   692
lemma round_down_correct:
immler@54782
   693
  shows "f - (round_down e f) \<in> {0..2 powr -e}"
hoelzl@47599
   694
unfolding atLeastAtMost_iff
hoelzl@47599
   695
proof
hoelzl@47599
   696
  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
hoelzl@47599
   697
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
immler@54782
   698
  finally show "f - round_down e f \<le> 2 powr real (- e)"
hoelzl@47600
   699
    by simp
hoelzl@47600
   700
qed (simp add: algebra_simps round_down)
obua@24301
   701
immler@54782
   702
lemma float_down_correct:
immler@54782
   703
  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
immler@54782
   704
  by transfer (rule round_down_correct)
immler@54782
   705
hoelzl@47599
   706
lemma compute_float_down[code]:
hoelzl@47599
   707
  "float_down p (Float m e) =
hoelzl@47599
   708
    (if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"
hoelzl@47599
   709
proof cases
hoelzl@47599
   710
  assume "p + e < 0"
hoelzl@47599
   711
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
hoelzl@47599
   712
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
hoelzl@47599
   713
  also have "... = 1 / 2 powr p / 2 powr e"
hoelzl@47600
   714
    unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
hoelzl@47599
   715
  finally show ?thesis
hoelzl@47600
   716
    using `p + e < 0`
hoelzl@47600
   717
    by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
hoelzl@47599
   718
next
hoelzl@47600
   719
  assume "\<not> p + e < 0"
hoelzl@47600
   720
  then have r: "real e + real p = real (nat (e + p))" by simp
hoelzl@47600
   721
  have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
hoelzl@47600
   722
    by (auto intro: exI[where x="m*2^nat (e+p)"]
hoelzl@47600
   723
             simp add: ac_simps powr_add[symmetric] r powr_realpow)
hoelzl@47600
   724
  with `\<not> p + e < 0` show ?thesis
wenzelm@57862
   725
    by transfer (auto simp add: round_down_def field_simps powr_add powr_minus)
hoelzl@47599
   726
qed
hoelzl@47621
   727
hide_fact (open) compute_float_down
obua@24301
   728
immler@54782
   729
lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"
immler@54782
   730
  using round_down_correct[of f e] by simp
immler@54782
   731
immler@54782
   732
lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e"
immler@54782
   733
  using round_up_correct[of e f] by simp
immler@54782
   734
immler@54782
   735
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s"
nipkow@56536
   736
  by (auto simp: round_down_def)
immler@54782
   737
hoelzl@47599
   738
lemma ceil_divide_floor_conv:
hoelzl@47599
   739
assumes "b \<noteq> 0"
hoelzl@47599
   740
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
   741
proof cases
hoelzl@47599
   742
  assume "\<not> b dvd a"
hoelzl@47599
   743
  hence "a mod b \<noteq> 0" by auto
hoelzl@47599
   744
  hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
hoelzl@47599
   745
  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
hoelzl@47599
   746
  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
hoelzl@47599
   747
  proof -
hoelzl@47599
   748
    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
hoelzl@47599
   749
    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
hoelzl@47599
   750
    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
hoelzl@47599
   751
    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
hoelzl@47599
   752
  qed
hoelzl@47599
   753
  thus ?thesis using `\<not> b dvd a` by simp
hoelzl@47599
   754
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
hoelzl@56479
   755
  floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
wenzelm@19765
   756
hoelzl@47599
   757
lemma compute_float_up[code]:
immler@58982
   758
  "float_up p x = - float_down p (-x)"
immler@58982
   759
  by transfer (simp add: round_down_uminus_eq)
hoelzl@47621
   760
hide_fact (open) compute_float_up
hoelzl@29804
   761
hoelzl@47599
   762
hoelzl@47599
   763
subsection {* Compute bitlen of integers *}
hoelzl@47599
   764
hoelzl@47600
   765
definition bitlen :: "int \<Rightarrow> int" where
hoelzl@47600
   766
  "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
hoelzl@47599
   767
hoelzl@47599
   768
lemma bitlen_nonneg: "0 \<le> bitlen x"
hoelzl@29804
   769
proof -
hoelzl@47599
   770
  {
hoelzl@47599
   771
    assume "0 > x"
hoelzl@47599
   772
    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
hoelzl@47599
   773
    also have "... < log 2 (-x)" using `0 > x` by auto
hoelzl@47599
   774
    finally have "-1 < log 2 (-x)" .
hoelzl@47599
   775
  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
hoelzl@47599
   776
qed
hoelzl@47599
   777
hoelzl@47599
   778
lemma bitlen_bounds:
hoelzl@47599
   779
  assumes "x > 0"
hoelzl@47599
   780
  shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
hoelzl@47599
   781
proof
hoelzl@47599
   782
  have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
hoelzl@47599
   783
    using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
hoelzl@47599
   784
    using real_nat_eq_real[of "floor (log 2 (real x))"]
hoelzl@47599
   785
    by simp
hoelzl@47599
   786
  also have "... \<le> 2 powr log 2 (real x)"
hoelzl@47599
   787
    by simp
hoelzl@47599
   788
  also have "... = real x"
hoelzl@47599
   789
    using `0 < x` by simp
hoelzl@47599
   790
  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
hoelzl@47599
   791
  thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
hoelzl@47599
   792
    by (simp add: bitlen_def)
hoelzl@47599
   793
next
hoelzl@47599
   794
  have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
hoelzl@47599
   795
  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
hoelzl@47599
   796
    apply (simp add: powr_realpow[symmetric])
hoelzl@47599
   797
    using `x > 0` by simp
hoelzl@47599
   798
  finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
hoelzl@47599
   799
    by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)
hoelzl@47599
   800
qed
hoelzl@47599
   801
hoelzl@47599
   802
lemma bitlen_pow2[simp]:
hoelzl@47599
   803
  assumes "b > 0"
hoelzl@47599
   804
  shows "bitlen (b * 2 ^ c) = bitlen b + c"
hoelzl@47599
   805
proof -
nipkow@56544
   806
  from assms have "b * 2 ^ c > 0" by auto
hoelzl@47599
   807
  thus ?thesis
hoelzl@47599
   808
    using floor_add[of "log 2 b" c] assms
hoelzl@47599
   809
    by (auto simp add: log_mult log_nat_power bitlen_def)
hoelzl@29804
   810
qed
hoelzl@29804
   811
hoelzl@47599
   812
lemma bitlen_Float:
wenzelm@53381
   813
  fixes m e
wenzelm@53381
   814
  defines "f \<equiv> Float m e"
wenzelm@53381
   815
  shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
wenzelm@53381
   816
proof (cases "m = 0")
wenzelm@53381
   817
  case True
wenzelm@53381
   818
  then show ?thesis by (simp add: f_def bitlen_def Float_def)
wenzelm@53381
   819
next
wenzelm@53381
   820
  case False
hoelzl@47600
   821
  hence "f \<noteq> float_of 0"
hoelzl@47600
   822
    unfolding real_of_float_eq by (simp add: f_def)
hoelzl@47600
   823
  hence "mantissa f \<noteq> 0"
hoelzl@47599
   824
    by (simp add: mantissa_noteq_0)
hoelzl@47599
   825
  moreover
wenzelm@53381
   826
  obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
wenzelm@53381
   827
    by (rule f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`])
hoelzl@47599
   828
  ultimately show ?thesis by (simp add: abs_mult)
wenzelm@53381
   829
qed
hoelzl@29804
   830
hoelzl@47599
   831
lemma compute_bitlen[code]:
hoelzl@47599
   832
  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
hoelzl@47599
   833
proof -
hoelzl@47599
   834
  { assume "2 \<le> x"
hoelzl@47599
   835
    then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
hoelzl@47599
   836
      by (simp add: log_mult zmod_zdiv_equality')
hoelzl@47599
   837
    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   838
    proof cases
hoelzl@47599
   839
      assume "x mod 2 = 0" then show ?thesis by simp
hoelzl@47599
   840
    next
hoelzl@47599
   841
      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
hoelzl@47599
   842
      then have "0 \<le> n"
hoelzl@47599
   843
        using `2 \<le> x` by simp
hoelzl@47599
   844
      assume "x mod 2 \<noteq> 0"
hoelzl@47599
   845
      with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
hoelzl@47599
   846
      with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
hoelzl@47599
   847
      moreover
hoelzl@47599
   848
      { have "real (2^nat n :: int) = 2 powr (nat n)"
hoelzl@47599
   849
          by (simp add: powr_realpow)
hoelzl@47599
   850
        also have "\<dots> \<le> 2 powr (log 2 x)"
hoelzl@47599
   851
          using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
hoelzl@47599
   852
        finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
hoelzl@47599
   853
      ultimately have "2^nat n \<le> x - 1" by simp
hoelzl@47599
   854
      then have "2^nat n \<le> real (x - 1)"
hoelzl@47599
   855
        unfolding real_of_int_le_iff[symmetric] by simp
hoelzl@47599
   856
      { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
hoelzl@47599
   857
          using `0 \<le> n` by (simp add: log_nat_power)
hoelzl@47599
   858
        also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
hoelzl@47599
   859
          using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
hoelzl@47599
   860
        finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
hoelzl@47599
   861
      moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
hoelzl@47599
   862
        using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
hoelzl@47599
   863
      ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
hoelzl@47599
   864
        unfolding n_def `x mod 2 = 1` by auto
hoelzl@47599
   865
    qed
hoelzl@47599
   866
    finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
hoelzl@47599
   867
  moreover
hoelzl@47599
   868
  { assume "x < 2" "0 < x"
hoelzl@47599
   869
    then have "x = 1" by simp
hoelzl@47599
   870
    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
hoelzl@47599
   871
  ultimately show ?thesis
hoelzl@47599
   872
    unfolding bitlen_def
hoelzl@47599
   873
    by (auto simp: pos_imp_zdiv_pos_iff not_le)
hoelzl@47599
   874
qed
hoelzl@47621
   875
hide_fact (open) compute_bitlen
hoelzl@29804
   876
hoelzl@47599
   877
lemma float_gt1_scale: assumes "1 \<le> Float m e"
hoelzl@47599
   878
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@47599
   879
proof -
hoelzl@47599
   880
  have "0 < Float m e" using assms by auto
hoelzl@47599
   881
  hence "0 < m" using powr_gt_zero[of 2 e]
hoelzl@47600
   882
    by (auto simp: zero_less_mult_iff)
hoelzl@47599
   883
  hence "m \<noteq> 0" by auto
hoelzl@47599
   884
  show ?thesis
hoelzl@47599
   885
  proof (cases "0 \<le> e")
hoelzl@47599
   886
    case True thus ?thesis using `0 < m`  by (simp add: bitlen_def)
hoelzl@29804
   887
  next
hoelzl@47599
   888
    have "(1::int) < 2" by simp
hoelzl@47599
   889
    case False let ?S = "2^(nat (-e))"
hoelzl@47599
   890
    have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
wenzelm@57862
   891
      by (auto simp: powr_minus field_simps)
hoelzl@47599
   892
    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
hoelzl@47599
   893
      by (auto simp: powr_minus)
hoelzl@47599
   894
    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
haftmann@57512
   895
    hence "?S \<le> real m" unfolding mult.assoc by auto
hoelzl@47599
   896
    hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@47599
   897
    from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
hoelzl@47599
   898
    have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
hoelzl@47599
   899
    hence "-e < bitlen m" using False by auto
hoelzl@47599
   900
    thus ?thesis by auto
hoelzl@29804
   901
  qed
hoelzl@47599
   902
qed
hoelzl@29804
   903
hoelzl@29804
   904
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
   905
proof -
hoelzl@29804
   906
  let ?B = "2^nat(bitlen m - 1)"
hoelzl@29804
   907
hoelzl@29804
   908
  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   909
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   910
  thus "1 \<le> real m / ?B" by auto
hoelzl@29804
   911
hoelzl@29804
   912
  have "m \<noteq> 0" using assms by auto
hoelzl@47599
   913
  have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)
obua@16782
   914
hoelzl@29804
   915
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
hoelzl@47599
   916
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
hoelzl@29804
   917
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
hoelzl@29804
   918
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   919
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
hoelzl@29804
   920
  thus "real m / ?B < 2" by auto
hoelzl@29804
   921
qed
hoelzl@29804
   922
hoelzl@47599
   923
subsection {* Approximation of positive rationals *}
hoelzl@47599
   924
hoelzl@47599
   925
lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"
hoelzl@47599
   926
by (simp add: zdiv_zmult2_eq)
hoelzl@29804
   927
hoelzl@47599
   928
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
hoelzl@47599
   929
  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
hoelzl@29804
   930
hoelzl@47599
   931
lemma real_div_nat_eq_floor_of_divide:
hoelzl@47599
   932
  fixes a b::nat
hoelzl@47599
   933
  shows "a div b = real (floor (a/b))"
hoelzl@47599
   934
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)
hoelzl@29804
   935
hoelzl@47599
   936
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
hoelzl@29804
   937
hoelzl@47600
   938
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
   939
  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp
obua@16782
   940
hoelzl@47599
   941
lemma compute_lapprox_posrat[code]:
wenzelm@53381
   942
  fixes prec x y
wenzelm@53381
   943
  shows "lapprox_posrat prec x y =
wenzelm@53381
   944
   (let
hoelzl@47599
   945
       l = rat_precision prec x y;
hoelzl@47599
   946
       d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
hoelzl@47599
   947
    in normfloat (Float d (- l)))"
immler@58982
   948
    unfolding div_mult_twopow_eq
hoelzl@47600
   949
    by transfer
hoelzl@47615
   950
       (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
hoelzl@47599
   951
             del: two_powr_minus_int_float)
hoelzl@47621
   952
hide_fact (open) compute_lapprox_posrat
hoelzl@29804
   953
hoelzl@47600
   954
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
hoelzl@47600
   955
  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
hoelzl@29804
   956
hoelzl@47599
   957
lemma compute_rapprox_posrat[code]:
hoelzl@47599
   958
  fixes prec x y
immler@58982
   959
  notes divmod_int_mod_div[simp]
hoelzl@47599
   960
  defines "l \<equiv> rat_precision prec x y"
hoelzl@47599
   961
  shows "rapprox_posrat prec x y = (let
hoelzl@47599
   962
     l = l ;
hoelzl@47599
   963
     X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
immler@58982
   964
     (d, m) = divmod_int (fst X) (snd X)
hoelzl@47599
   965
   in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
hoelzl@47599
   966
proof (cases "y = 0")
immler@58982
   967
  assume "y = 0" thus ?thesis by transfer simp
hoelzl@47599
   968
next
hoelzl@47599
   969
  assume "y \<noteq> 0"
hoelzl@29804
   970
  show ?thesis
hoelzl@47599
   971
  proof (cases "0 \<le> l")
hoelzl@47599
   972
    assume "0 \<le> l"
wenzelm@56777
   973
    def x' \<equiv> "x * 2 ^ nat l"
hoelzl@47599
   974
    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
hoelzl@47599
   975
    moreover have "real x * 2 powr real l = real x'"
hoelzl@47599
   976
      by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
hoelzl@47599
   977
    ultimately show ?thesis
hoelzl@47599
   978
      using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
hoelzl@47600
   979
        l_def[symmetric, THEN meta_eq_to_obj_eq]
haftmann@58834
   980
      by transfer (auto simp add: floor_divide_eq_div [symmetric] round_up_def)
hoelzl@47599
   981
   next
hoelzl@47599
   982
    assume "\<not> 0 \<le> l"
wenzelm@56777
   983
    def y' \<equiv> "y * 2 ^ nat (- l)"
hoelzl@47599
   984
    from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
hoelzl@47599
   985
    have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
hoelzl@47599
   986
    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
hoelzl@47599
   987
      using `\<not> 0 \<le> l`
wenzelm@57862
   988
      by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps)
hoelzl@47599
   989
    ultimately show ?thesis
hoelzl@47599
   990
      using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
hoelzl@47600
   991
        l_def[symmetric, THEN meta_eq_to_obj_eq]
hoelzl@47600
   992
      by transfer
haftmann@58834
   993
         (auto simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div [symmetric])
hoelzl@29804
   994
  qed
hoelzl@29804
   995
qed
hoelzl@47621
   996
hide_fact (open) compute_rapprox_posrat
hoelzl@29804
   997
hoelzl@47599
   998
lemma rat_precision_pos:
hoelzl@47599
   999
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@47599
  1000
  shows "rat_precision n (int x) (int y) > 0"
hoelzl@29804
  1001
proof -
hoelzl@47599
  1002
  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
hoelzl@47599
  1003
  hence "bitlen (int x) < bitlen (int y)" using assms
hoelzl@47599
  1004
    by (simp add: bitlen_def del: floor_add_one)
hoelzl@47599
  1005
      (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
hoelzl@47599
  1006
  thus ?thesis
hoelzl@47599
  1007
    using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
hoelzl@29804
  1008
qed
obua@16782
  1009
hoelzl@47601
  1010
lemma rapprox_posrat_less1:
immler@58982
  1011
  shows "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 2 * x < y \<Longrightarrow> 0 < n \<Longrightarrow> real (rapprox_posrat n x y) < 1"
immler@58982
  1012
  by transfer (simp add: rat_precision_pos round_up_less1)
hoelzl@29804
  1013
hoelzl@47600
  1014
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
  1015
  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
obua@16782
  1016
hoelzl@29804
  1017
lemma compute_lapprox_rat[code]:
hoelzl@47599
  1018
  "lapprox_rat prec x y =
hoelzl@47599
  1019
    (if y = 0 then 0
hoelzl@47599
  1020
    else if 0 \<le> x then
hoelzl@47599
  1021
      (if 0 < y then lapprox_posrat prec (nat x) (nat y)
wenzelm@53381
  1022
      else - (rapprox_posrat prec (nat x) (nat (-y))))
hoelzl@47599
  1023
      else (if 0 < y
hoelzl@47599
  1024
        then - (rapprox_posrat prec (nat (-x)) (nat y))
hoelzl@47599
  1025
        else lapprox_posrat prec (nat (-x)) (nat (-y))))"
hoelzl@56479
  1026
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hoelzl@47621
  1027
hide_fact (open) compute_lapprox_rat
hoelzl@47599
  1028
hoelzl@47600
  1029
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
hoelzl@47600
  1030
  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
hoelzl@47599
  1031
immler@58982
  1032
lemma "rapprox_rat = rapprox_posrat"
immler@58982
  1033
  by transfer auto
immler@58982
  1034
immler@58982
  1035
lemma "lapprox_rat = lapprox_posrat"
immler@58982
  1036
  by transfer auto
immler@58982
  1037
hoelzl@47599
  1038
lemma compute_rapprox_rat[code]:
immler@58982
  1039
  "rapprox_rat prec x y = - lapprox_rat prec (-x) y"
immler@58982
  1040
  by transfer (simp add: round_down_uminus_eq)
hoelzl@47621
  1041
hide_fact (open) compute_rapprox_rat
hoelzl@47599
  1042
hoelzl@47599
  1043
subsection {* Division *}
hoelzl@47599
  1044
immler@54782
  1045
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
  1046
immler@54782
  1047
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
  1048
immler@54782
  1049
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl
immler@54782
  1050
  by (simp add: real_divl_def)
hoelzl@47599
  1051
hoelzl@47599
  1052
lemma compute_float_divl[code]:
hoelzl@47600
  1053
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
hoelzl@47599
  1054
proof cases
hoelzl@47601
  1055
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
hoelzl@47601
  1056
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
hoelzl@47601
  1057
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
hoelzl@47601
  1058
  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
  1059
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
hoelzl@47601
  1060
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
hoelzl@47601
  1061
    by (simp add: field_simps powr_divide2[symmetric])
hoelzl@47599
  1062
hoelzl@47601
  1063
  show ?thesis
wenzelm@53381
  1064
    using not_0
immler@54782
  1065
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,
immler@54782
  1066
      simp add: field_simps)
immler@54782
  1067
qed (transfer, auto simp: real_divl_def)
hoelzl@47621
  1068
hide_fact (open) compute_float_divl
hoelzl@47600
  1069
immler@54782
  1070
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr
immler@54782
  1071
  by (simp add: real_divr_def)
hoelzl@47599
  1072
hoelzl@47599
  1073
lemma compute_float_divr[code]:
immler@58982
  1074
  "float_divr prec x y = - float_divl prec (-x) y"
immler@58982
  1075
  by transfer (simp add: real_divr_def real_divl_def round_down_uminus_eq)
immler@58982
  1076
hide_fact (open) compute_float_divr
hoelzl@47600
  1077
obua@16782
  1078
hoelzl@47599
  1079
subsection {* Lemmas needed by Approximate *}
hoelzl@47599
  1080
hoelzl@47599
  1081
lemma Float_num[simp]: shows
hoelzl@47599
  1082
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
haftmann@58410
  1083
   "real (Float 1 (- 1)) = 1/2" and "real (Float 1 (- 2)) = 1/4" and "real (Float 1 (- 3)) = 1/8" and
haftmann@58410
  1084
   "real (Float (- 1) 0) = -1" and "real (Float (number_of n) 0) = number_of n"
hoelzl@47599
  1085
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
  1086
using powr_realpow[of 2 2] powr_realpow[of 2 3]
hoelzl@47599
  1087
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
hoelzl@47599
  1088
by auto
hoelzl@47599
  1089
hoelzl@47599
  1090
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp
hoelzl@47599
  1091
hoelzl@47599
  1092
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
hoelzl@47599
  1093
hoelzl@47599
  1094
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
hoelzl@47599
  1095
by arith
hoelzl@29804
  1096
hoelzl@47599
  1097
lemma lapprox_rat:
hoelzl@47599
  1098
  shows "real (lapprox_rat prec x y) \<le> real x / real y"
hoelzl@47599
  1099
  using round_down by (simp add: lapprox_rat_def)
obua@16782
  1100
hoelzl@47599
  1101
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
hoelzl@47599
  1102
proof -
hoelzl@47599
  1103
  from zmod_zdiv_equality'[of a b]
hoelzl@47599
  1104
  have "a = b * (a div b) + a mod b" by simp
hoelzl@47599
  1105
  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
hoelzl@47599
  1106
  using assms by simp
hoelzl@47599
  1107
  finally show ?thesis by simp
hoelzl@47599
  1108
qed
hoelzl@47599
  1109
hoelzl@47599
  1110
lemma lapprox_rat_nonneg:
hoelzl@47599
  1111
  fixes n x y
immler@58982
  1112
  assumes "0 \<le> x" and "0 \<le> y"
hoelzl@47599
  1113
  shows "0 \<le> real (lapprox_rat n x y)"
immler@58982
  1114
  using assms by (auto simp: lapprox_rat_def simp: round_down_nonneg)
obua@16782
  1115
hoelzl@31098
  1116
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
hoelzl@47599
  1117
  using round_up by (simp add: rapprox_rat_def)
hoelzl@47599
  1118
hoelzl@47599
  1119
lemma rapprox_rat_le1:
hoelzl@47599
  1120
  fixes n x y
hoelzl@47599
  1121
  assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
hoelzl@47599
  1122
  shows "real (rapprox_rat n x y) \<le> 1"
hoelzl@47599
  1123
proof -
hoelzl@47599
  1124
  have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
hoelzl@47599
  1125
    using xy unfolding bitlen_def by (auto intro!: floor_mono)
immler@58982
  1126
  from this assms show ?thesis
immler@58982
  1127
    by transfer (auto intro!: round_up_le1 simp: rat_precision_def)
hoelzl@29804
  1128
qed
obua@16782
  1129
immler@58982
  1130
lemma rapprox_rat_nonneg_nonpos:
immler@58982
  1131
  "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
immler@58982
  1132
  by transfer (simp add: round_up_le0 divide_nonneg_nonpos)
obua@16782
  1133
immler@58982
  1134
lemma rapprox_rat_nonpos_nonneg:
immler@58982
  1135
  "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
immler@58982
  1136
  by transfer (simp add: round_up_le0 divide_nonpos_nonneg)
obua@16782
  1137
immler@54782
  1138
lemma real_divl: "real_divl prec x y \<le> x / y"
immler@54782
  1139
  by (simp add: real_divl_def round_down)
immler@54782
  1140
immler@54782
  1141
lemma real_divr: "x / y \<le> real_divr prec x y"
immler@54782
  1142
  using round_up by (simp add: real_divr_def)
immler@54782
  1143
hoelzl@31098
  1144
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
immler@54782
  1145
  by transfer (rule real_divl)
immler@54782
  1146
immler@54782
  1147
lemma real_divl_lower_bound:
immler@54782
  1148
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y"
immler@58982
  1149
  by (simp add: real_divl_def round_down_nonneg)
hoelzl@47599
  1150
hoelzl@47599
  1151
lemma float_divl_lower_bound:
immler@54782
  1152
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
immler@54782
  1153
  by transfer (rule real_divl_lower_bound)
hoelzl@47599
  1154
hoelzl@47599
  1155
lemma exponent_1: "exponent 1 = 0"
hoelzl@47599
  1156
  using exponent_float[of 1 0] by (simp add: one_float_def)
hoelzl@47599
  1157
hoelzl@47599
  1158
lemma mantissa_1: "mantissa 1 = 1"
hoelzl@47599
  1159
  using mantissa_float[of 1 0] by (simp add: one_float_def)
obua@16782
  1160
hoelzl@47599
  1161
lemma bitlen_1: "bitlen 1 = 1"
hoelzl@47599
  1162
  by (simp add: bitlen_def)
hoelzl@47599
  1163
hoelzl@47599
  1164
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
hoelzl@47599
  1165
proof
hoelzl@47599
  1166
  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
hoelzl@47599
  1167
  show "x = 0" by (simp add: zero_float_def z)
hoelzl@47599
  1168
qed (simp add: zero_float_def)
obua@16782
  1169
hoelzl@47599
  1170
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
hoelzl@47599
  1171
proof (cases "x = 0", simp)
hoelzl@47599
  1172
  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
hoelzl@47599
  1173
  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
hoelzl@47599
  1174
  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
hoelzl@47599
  1175
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
hoelzl@47599
  1176
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
hoelzl@47599
  1177
    by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]
hoelzl@47599
  1178
      real_of_int_le_iff less_imp_le)
hoelzl@47599
  1179
  finally show ?thesis by (simp add: powr_add)
hoelzl@29804
  1180
qed
hoelzl@29804
  1181
immler@54782
  1182
lemma real_divl_pos_less1_bound:
immler@58982
  1183
  assumes "0 < x" "x \<le> 1" "prec \<ge> 1"
immler@58982
  1184
  shows "1 \<le> real_divl prec 1 x"
immler@58982
  1185
proof -
immler@58982
  1186
  have "log 2 x \<le> real prec + real \<lfloor>log 2 x\<rfloor>" using `prec \<ge> 1` by arith
immler@58982
  1187
  from this assms show ?thesis
immler@58982
  1188
    by (simp add: real_divl_def log_divide round_down_ge1)
hoelzl@29804
  1189
qed
obua@16782
  1190
immler@54782
  1191
lemma float_divl_pos_less1_bound:
immler@58982
  1192
  "0 < real x \<Longrightarrow> real x \<le> 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
immler@54782
  1193
  by (transfer, rule real_divl_pos_less1_bound)
obua@16782
  1194
immler@54782
  1195
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
immler@54782
  1196
  by transfer (rule real_divr)
immler@54782
  1197
immler@58982
  1198
lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x \<le> 1" shows "1 \<le> real_divr prec 1 x"
hoelzl@29804
  1199
proof -
immler@58982
  1200
  have "1 \<le> 1 / x" using `0 < x` and `x <= 1` by auto
immler@54782
  1201
  also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto
hoelzl@47600
  1202
  finally show ?thesis by auto
hoelzl@29804
  1203
qed
hoelzl@29804
  1204
immler@58982
  1205
lemma float_divr_pos_less1_lower_bound: "0 < x \<Longrightarrow> x \<le> 1 \<Longrightarrow> 1 \<le> float_divr prec 1 x"
immler@54782
  1206
  by transfer (rule real_divr_pos_less1_lower_bound)
immler@54782
  1207
immler@54782
  1208
lemma real_divr_nonpos_pos_upper_bound:
immler@58982
  1209
  "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real_divr prec x y \<le> 0"
immler@58982
  1210
  by (simp add: real_divr_def round_up_le0 divide_le_0_iff)
immler@54782
  1211
hoelzl@47599
  1212
lemma float_divr_nonpos_pos_upper_bound:
immler@58982
  1213
  "real x \<le> 0 \<Longrightarrow> 0 \<le> real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"
immler@54782
  1214
  by transfer (rule real_divr_nonpos_pos_upper_bound)
immler@54782
  1215
immler@54782
  1216
lemma real_divr_nonneg_neg_upper_bound:
immler@58982
  1217
  "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real_divr prec x y \<le> 0"
immler@58982
  1218
  by (simp add: real_divr_def round_up_le0 divide_le_0_iff)
obua@16782
  1219
hoelzl@47599
  1220
lemma float_divr_nonneg_neg_upper_bound:
immler@58982
  1221
  "0 \<le> real x \<Longrightarrow> real y \<le> 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"
immler@54782
  1222
  by transfer (rule real_divr_nonneg_neg_upper_bound)
immler@54782
  1223
immler@54782
  1224
definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where
immler@54782
  1225
  "truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
immler@54782
  1226
immler@54782
  1227
lemma truncate_down: "truncate_down prec x \<le> x"
immler@54782
  1228
  using round_down by (simp add: truncate_down_def)
immler@54782
  1229
immler@54782
  1230
lemma truncate_down_le: "x \<le> y \<Longrightarrow> truncate_down prec x \<le> y"
immler@54782
  1231
  by (rule order_trans[OF truncate_down])
hoelzl@47600
  1232
immler@54782
  1233
definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real" where
immler@54782
  1234
  "truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
immler@54782
  1235
immler@54782
  1236
lemma truncate_up: "x \<le> truncate_up prec x"
immler@54782
  1237
  using round_up by (simp add: truncate_up_def)
immler@54782
  1238
immler@54782
  1239
lemma truncate_up_le: "x \<le> y \<Longrightarrow> x \<le> truncate_up prec y"
immler@54782
  1240
  by (rule order_trans[OF _ truncate_up])
immler@54782
  1241
immler@54782
  1242
lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0"
immler@54782
  1243
  by (simp add: truncate_up_def)
immler@54782
  1244
immler@58982
  1245
lemma truncate_up_uminus_eq: "truncate_up prec (-x) = - truncate_down prec x"
immler@58982
  1246
  and truncate_down_uminus_eq: "truncate_down prec (-x) = - truncate_up prec x"
immler@58982
  1247
  by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
immler@58982
  1248
immler@54782
  1249
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
immler@54782
  1250
  by (simp add: truncate_up_def)
hoelzl@47600
  1251
hoelzl@47600
  1252
lemma float_round_up: "real x \<le> real (float_round_up prec x)"
immler@54782
  1253
  using truncate_up by transfer simp
hoelzl@47599
  1254
immler@54782
  1255
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down
immler@54782
  1256
  by (simp add: truncate_down_def)
hoelzl@47599
  1257
hoelzl@47600
  1258
lemma float_round_down: "real (float_round_down prec x) \<le> real x"
immler@54782
  1259
  using truncate_down by transfer simp
hoelzl@47599
  1260
hoelzl@47600
  1261
lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>"
hoelzl@47600
  1262
  using floor_add[of x i] by (simp del: floor_add add: ac_simps)
obua@16782
  1263
hoelzl@47599
  1264
lemma compute_float_round_down[code]:
hoelzl@47600
  1265
  "float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
hoelzl@47600
  1266
    if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
hoelzl@47600
  1267
             else Float m e)"
hoelzl@47621
  1268
  using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
immler@54782
  1269
  by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_down_def
immler@54782
  1270
    cong del: if_weak_cong)
hoelzl@47621
  1271
hide_fact (open) compute_float_round_down
hoelzl@47599
  1272
hoelzl@47600
  1273
lemma compute_float_round_up[code]:
immler@58982
  1274
  "float_round_up prec x = - float_round_down prec (-x)"
immler@58982
  1275
  by transfer (simp add: truncate_down_uminus_eq)
hoelzl@47621
  1276
hide_fact (open) compute_float_round_up
obua@16782
  1277
immler@54784
  1278
lemma truncate_up_nonneg_mono:
immler@54784
  1279
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  1280
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  1281
proof -
immler@54784
  1282
  {
immler@54784
  1283
    assume "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>"
immler@54784
  1284
    hence ?thesis
immler@54784
  1285
      using assms
immler@54784
  1286
      by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
immler@54784
  1287
  } moreover {
immler@54784
  1288
    assume "0 < x"
immler@54784
  1289
    hence "log 2 x \<le> log 2 y" using assms by auto
immler@54784
  1290
    moreover
immler@54784
  1291
    assume "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>"
immler@54784
  1292
    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
immler@54784
  1293
      unfolding atomize_conj
immler@54784
  1294
      by (metis floor_less_cancel linorder_cases not_le)
immler@54784
  1295
    have "truncate_up prec x =
immler@54784
  1296
      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
  1297
      using assms by (simp add: truncate_up_def round_up_def)
immler@54784
  1298
    also have "\<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)"
immler@54784
  1299
    proof (unfold ceiling_le_eq)
immler@54784
  1300
      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
  1301
        using real_of_int_floor_add_one_ge[of "log 2 x"] assms
immler@54784
  1302
        by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono)
immler@54784
  1303
      thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
immler@54784
  1304
        using `0 < x` by (simp add: powr_realpow)
immler@54784
  1305
    qed
immler@54784
  1306
    hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
immler@54784
  1307
      by (auto simp: powr_realpow)
immler@54784
  1308
    also
immler@54784
  1309
    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
  1310
      using logless flogless by (auto intro!: floor_mono)
immler@54784
  1311
    also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))"
immler@54784
  1312
      using assms `0 < x`
immler@54784
  1313
      by (auto simp: algebra_simps)
immler@54784
  1314
    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
  1315
      by simp
immler@54784
  1316
    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
  1317
      by (subst powr_add[symmetric]) simp
immler@54784
  1318
    also have "\<dots> = y"
immler@54784
  1319
      using `0 < x` assms
immler@54784
  1320
      by (simp add: powr_add)
immler@54784
  1321
    also have "\<dots> \<le> truncate_up prec y"
immler@54784
  1322
      by (rule truncate_up)
immler@54784
  1323
    finally have ?thesis .
immler@54784
  1324
  } moreover {
immler@54784
  1325
    assume "~ 0 < x"
immler@54784
  1326
    hence ?thesis
immler@54784
  1327
      using assms
immler@54784
  1328
      by (auto intro!: truncate_up_le)
immler@54784
  1329
  } ultimately show ?thesis
immler@54784
  1330
    by blast
immler@54784
  1331
qed
immler@54784
  1332
immler@54784
  1333
lemma truncate_up_nonpos: "x \<le> 0 \<Longrightarrow> truncate_up prec x \<le> 0"
immler@54784
  1334
  by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)
immler@54784
  1335
immler@54784
  1336
lemma truncate_down_nonpos: "x \<le> 0 \<Longrightarrow> truncate_down prec x \<le> 0"
immler@54784
  1337
  by (auto simp: truncate_down_def round_down_def intro!: mult_nonpos_nonneg
immler@54784
  1338
    order_le_less_trans[of _ 0, OF mult_nonpos_nonneg])
immler@54784
  1339
immler@54784
  1340
lemma truncate_up_switch_sign_mono:
immler@54784
  1341
  assumes "x \<le> 0" "0 \<le> y"
immler@54784
  1342
  shows "truncate_up prec x \<le> truncate_up prec y"
immler@54784
  1343
proof -
immler@54784
  1344
  note truncate_up_nonpos[OF `x \<le> 0`]
immler@54784
  1345
  also note truncate_up_le[OF `0 \<le> y`]
immler@54784
  1346
  finally show ?thesis .
immler@54784
  1347
qed
immler@54784
  1348
immler@54784
  1349
lemma truncate_down_zeroprec_mono:
immler@54784
  1350
  assumes "0 < x" "x \<le> y"
immler@54784
  1351
  shows "truncate_down 0 x \<le> truncate_down 0 y"
immler@54784
  1352
proof -
immler@54784
  1353
  have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))"
immler@54784
  1354
    by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide)
immler@54784
  1355
  also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)"
immler@54784
  1356
    using `0 < x`
wenzelm@57862
  1357
    by (auto simp: field_simps powr_add powr_divide2[symmetric])
immler@54784
  1358
  also have "\<dots> < 2 powr 0"
immler@54784
  1359
    using real_of_int_floor_add_one_gt
immler@54784
  1360
    unfolding neg_less_iff_less
immler@54784
  1361
    by (intro powr_less_mono) (auto simp: algebra_simps)
immler@54784
  1362
  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1"
immler@54784
  1363
    unfolding less_ceiling_eq real_of_int_minus real_of_one
immler@54784
  1364
    by simp
immler@54784
  1365
  moreover
immler@54784
  1366
  have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"
nipkow@56536
  1367
    using `x > 0` by auto
immler@54784
  1368
  ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"
immler@54784
  1369
    by simp
immler@54784
  1370
  also have "\<dots> \<subseteq> {0}" by auto
immler@54784
  1371
  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0" by simp
immler@54784
  1372
  with assms show ?thesis
nipkow@56536
  1373
    by (auto simp: truncate_down_def round_down_def)
immler@54784
  1374
qed
immler@54784
  1375
immler@54784
  1376
lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y"
nipkow@56536
  1377
  by (auto simp: truncate_down_def round_down_def)
immler@54784
  1378
immler@54784
  1379
lemma truncate_down_zero: "truncate_down prec 0 = 0"
nipkow@56536
  1380
  by (auto simp: truncate_down_def round_down_def)
immler@54784
  1381
immler@54784
  1382
lemma truncate_down_switch_sign_mono:
immler@54784
  1383
  assumes "x \<le> 0" "0 \<le> y"
immler@54784
  1384
  assumes "x \<le> y"
immler@54784
  1385
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  1386
proof -
immler@54784
  1387
  note truncate_down_nonpos[OF `x \<le> 0`]
immler@54784
  1388
  also note truncate_down_nonneg[OF `0 \<le> y`]
immler@54784
  1389
  finally show ?thesis .
immler@54784
  1390
qed
immler@54784
  1391
immler@54784
  1392
lemma truncate_down_nonneg_mono:
immler@54784
  1393
  assumes "0 \<le> x" "x \<le> y"
immler@54784
  1394
  shows "truncate_down prec x \<le> truncate_down prec y"
immler@54784
  1395
proof -
immler@54784
  1396
  {
immler@54784
  1397
    assume "0 < x" "prec = 0"
immler@54784
  1398
    with assms have ?thesis
immler@54784
  1399
      by (simp add: truncate_down_zeroprec_mono)
immler@54784
  1400
  } moreover {
immler@54784
  1401
    assume "~ 0 < x"
immler@54784
  1402
    with assms have "x = 0" "0 \<le> y" by simp_all
immler@54784
  1403
    hence ?thesis
immler@54784
  1404
      by (auto simp add: truncate_down_zero intro!: truncate_down_nonneg)
immler@54784
  1405
  } moreover {
immler@54784
  1406
    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@54784
  1407
    hence ?thesis
immler@54784
  1408
      using assms
immler@54784
  1409
      by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
immler@54784
  1410
  } moreover {
immler@54784
  1411
    assume "0 < x"
immler@54784
  1412
    hence "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y" using assms by auto
immler@54784
  1413
    moreover
immler@54784
  1414
    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
immler@54784
  1415
    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
immler@54784
  1416
      unfolding atomize_conj abs_of_pos[OF `0 < x`] abs_of_pos[OF `0 < y`]
immler@54784
  1417
      by (metis floor_less_cancel linorder_cases not_le)
immler@54784
  1418
    assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto
immler@54784
  1419
    have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)"
immler@54784
  1420
      using `0 < y`
immler@54784
  1421
      by simp
immler@54784
  1422
    also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))"
immler@54784
  1423
      using `0 \<le> y` `0 \<le> x` assms(2)
nipkow@56544
  1424
      by (auto intro!: powr_mono divide_left_mono
immler@54784
  1425
        simp: real_of_nat_diff powr_add
immler@54784
  1426
        powr_divide2[symmetric])
immler@54784
  1427
    also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)"
immler@54784
  1428
      by (auto simp: powr_add)
immler@54784
  1429
    finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
immler@54784
  1430
      using `0 \<le> y`
immler@54784
  1431
      by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow)
immler@54784
  1432
    hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
immler@54784
  1433
      by (auto simp: truncate_down_def round_down_def)
immler@54784
  1434
    moreover
immler@54784
  1435
    {
immler@54784
  1436
      have "x = 2 powr (log 2 \<bar>x\<bar>)" using `0 < x` by simp
immler@54784
  1437
      also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"
immler@54784
  1438
        using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"]
immler@54784
  1439
        by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps)
immler@54784
  1440
      also
immler@54784
  1441
      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
  1442
        using logless flogless `x > 0` `y > 0`
immler@54784
  1443
        by (auto intro!: floor_mono)
immler@54784
  1444
      finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"
immler@54784
  1445
        by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff)
immler@54784
  1446
    } ultimately have ?thesis
immler@54784
  1447
      by (metis dual_order.trans truncate_down)
immler@54784
  1448
  } ultimately show ?thesis by blast
immler@54784
  1449
qed
immler@54784
  1450
immler@58982
  1451
lemma truncate_down_eq_truncate_up: "truncate_down p x = - truncate_up p (-x)"
immler@58982
  1452
  and truncate_up_eq_truncate_down: "truncate_up p x = - truncate_down p (-x)"
immler@58982
  1453
  by (auto simp: truncate_up_uminus_eq truncate_down_uminus_eq)
immler@58982
  1454
immler@54784
  1455
lemma truncate_down_mono: "x \<le> y \<Longrightarrow> truncate_down p x \<le> truncate_down p y"
immler@54784
  1456
  apply (cases "0 \<le> x")
immler@54784
  1457
  apply (rule truncate_down_nonneg_mono, assumption+)
immler@58982
  1458
  apply (simp add: truncate_down_eq_truncate_up)
immler@54784
  1459
  apply (cases "0 \<le> y")
immler@54784
  1460
  apply (auto intro: truncate_up_nonneg_mono truncate_up_switch_sign_mono)
immler@54784
  1461
  done
immler@54784
  1462
immler@54784
  1463
lemma truncate_up_mono: "x \<le> y \<Longrightarrow> truncate_up p x \<le> truncate_up p y"
immler@58982
  1464
  by (simp add: truncate_up_eq_truncate_down truncate_down_mono)
immler@54784
  1465
hoelzl@47599
  1466
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
hoelzl@47599
  1467
 apply (auto simp: zero_float_def mult_le_0_iff)
hoelzl@47599
  1468
 using powr_gt_zero[of 2 b] by simp
hoelzl@47599
  1469
hoelzl@47621
  1470
lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
hoelzl@47600
  1471
  unfolding pprt_def sup_float_def max_def sup_real_def by auto
hoelzl@47599
  1472
hoelzl@47621
  1473
lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
hoelzl@47600
  1474
  unfolding nprt_def inf_float_def min_def inf_real_def by auto
hoelzl@47599
  1475
kuncar@55565
  1476
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor .
obua@16782
  1477
hoelzl@47599
  1478
lemma compute_int_floor_fl[code]:
hoelzl@47601
  1479
  "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
hoelzl@47600
  1480
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  1481
hide_fact (open) compute_int_floor_fl
hoelzl@47599
  1482
hoelzl@47600
  1483
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp
hoelzl@47599
  1484
hoelzl@47599
  1485
lemma compute_floor_fl[code]:
hoelzl@47601
  1486
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
hoelzl@47600
  1487
  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hoelzl@47621
  1488
hide_fact (open) compute_floor_fl
obua@16782
  1489
hoelzl@47600
  1490
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp
hoelzl@47600
  1491
hoelzl@47600
  1492
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp
hoelzl@29804
  1493
hoelzl@47599
  1494
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
wenzelm@53381
  1495
proof (cases "floor_fl x = float_of 0")
wenzelm@53381
  1496
  case True
wenzelm@53381
  1497
  then show ?thesis by (simp add: floor_fl_def)
wenzelm@53381
  1498
next
wenzelm@53381
  1499
  case False
wenzelm@53381
  1500
  have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp
wenzelm@53381
  1501
  obtain i where "\<lfloor>real x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
wenzelm@53381
  1502
    by (rule denormalize_shift[OF eq[THEN eq_reflection] False])
wenzelm@53381
  1503
  then show ?thesis by simp
wenzelm@53381
  1504
qed
obua@16782
  1505
obua@16782
  1506
end
hoelzl@47599
  1507