src/HOL/Library/Float.thy
author immler
Mon, 16 Dec 2013 17:08:22 +0100
changeset 54782 cd8f55c358c5
parent 54489 03ff4d1e6784
child 54783 25860d89a044
permissions -rw-r--r--
additional definitions and lemmas for Float
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(*  Title:      HOL/Library/Float.thy
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    Author:     Johannes Hölzl, Fabian Immler
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    Copyright   2012  TU München
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*)
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header {* Floating-Point Numbers *}
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theory Float
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imports Complex_Main Lattice_Algebras
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begin
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definition "float = {m * 2 powr e | (m :: int) (e :: int). True}"
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typedef float = float
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  morphisms real_of_float float_of
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  unfolding float_def by auto
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defs (overloaded)
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  real_of_float_def[code_unfold]: "real \<equiv> real_of_float"
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lemma type_definition_float': "type_definition real float_of float"
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  using type_definition_float unfolding real_of_float_def .
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setup_lifting (no_code) type_definition_float'
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lemmas float_of_inject[simp]
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declare [[coercion "real :: float \<Rightarrow> real"]]
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lemma real_of_float_eq:
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  fixes f1 f2 :: float shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"
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  unfolding real_of_float_def real_of_float_inject ..
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lemma float_of_real[simp]: "float_of (real x) = x"
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  unfolding real_of_float_def by (rule real_of_float_inverse)
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lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
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  unfolding real_of_float_def by (rule float_of_inverse)
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subsection {* Real operations preserving the representation as floating point number *}
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lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"
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  by (auto simp: float_def)
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lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)
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lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp
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lemma numeral_float[simp]: "numeral i \<in> float" by (intro floatI[of "numeral i" 0]) simp
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lemma neg_numeral_float[simp]: "- numeral i \<in> float" by (intro floatI[of "- numeral i" 0]) simp
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lemma real_of_int_float[simp]: "real (x :: int) \<in> float" by (intro floatI[of x 0]) simp
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lemma real_of_nat_float[simp]: "real (x :: nat) \<in> float" by (intro floatI[of x 0]) simp
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lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float" by (intro floatI[of 1 i]) simp
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lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) \<in> float" by (intro floatI[of 1 i]) simp
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lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float" by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float" by (intro floatI[of 1 "-i"]) simp
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lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" by (intro floatI[of 1 "numeral i"]) simp
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lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i \<in> float" by (intro floatI[of 1 "- numeral i"]) simp
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lemma two_pow_float[simp]: "2 ^ n \<in> float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
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lemma real_of_float_float[simp]: "real (f::float) \<in> float" by (cases f) simp
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lemma plus_float[simp]: "r \<in> float \<Longrightarrow> p \<in> float \<Longrightarrow> r + p \<in> float"
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  unfolding float_def
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proof (safe, simp)
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  fix e1 m1 e2 m2 :: int
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  { fix e1 m1 e2 m2 :: int assume "e1 \<le> e2"
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    then have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
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      by (simp add: powr_realpow[symmetric] powr_divide2[symmetric] field_simps)
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    then have "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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      by blast }
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  note * = this
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  show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
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  proof (cases e1 e2 rule: linorder_le_cases)
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    assume "e2 \<le> e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)
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  qed (rule *)
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qed
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lemma uminus_float[simp]: "x \<in> float \<Longrightarrow> -x \<in> float"
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  apply (auto simp: float_def)
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  apply (rule_tac x="-x" in exI)
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  apply (rule_tac x="xa" in exI)
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  apply (simp add: field_simps)
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  done
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lemma times_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x * y \<in> float"
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  apply (auto simp: float_def)
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  apply (rule_tac x="x * xa" in exI)
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  apply (rule_tac x="xb + xc" in exI)
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  apply (simp add: powr_add)
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  done
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lemma minus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x - y \<in> float"
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  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 (rule_tac x="x" in exI)
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  apply (rule_tac x="xa - d" in exI)
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  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
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  done
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lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"
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  apply (auto simp add: float_def)
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  apply (rule_tac x="x" in exI)
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  apply (rule_tac x="xa - d" in exI)
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  apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
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  done
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lemma div_numeral_Bit0_float[simp]:
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  assumes x: "x / numeral n \<in> float" shows "x / (numeral (Num.Bit0 n)) \<in> float"
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proof -
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  have "(x / numeral n) / 2^1 \<in> float"
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    by (intro x div_power_2_float)
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  also have "(x / numeral n) / 2^1 = x / (numeral (Num.Bit0 n))"
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    by (induct n) auto
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  finally show ?thesis .
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qed
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lemma div_neg_numeral_Bit0_float[simp]:
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  assumes x: "x / numeral n \<in> float" shows "x / (- 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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  "fun_rel (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
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  unfolding fun_rel_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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  "fun_rel (op =) pcr_float (- numeral :: _ \<Rightarrow> real) (- numeral :: _ \<Rightarrow> float)"
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  unfolding fun_rel_def float.pcr_cr_eq cr_float_def by simp
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lemma
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  shows float_of_numeral[simp]: "numeral k = float_of (numeral k)"
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    and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)"
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  unfolding real_of_float_eq by simp_all
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subsection {* Represent floats as unique mantissa and exponent *}
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lemma int_induct_abs[case_names less]:
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  fixes j :: int
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  assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
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  shows "P j"
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proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)
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  case less show ?case by (rule H[OF less]) simp
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qed
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lemma int_cancel_factors:
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  fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
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proof (induct n rule: int_induct_abs)
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  case (less n)
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  { fix m assume n: "n \<noteq> 0" "n = m * r"
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    then have "\<bar>m \<bar> < \<bar>n\<bar>"
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      by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7)
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                dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le
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                mult_eq_0_iff zdvd_mult_cancel1)
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    from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
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  then show ?case
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    by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0)
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qed
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lemma mult_powr_eq_mult_powr_iff_asym:
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  fixes m1 m2 e1 e2 :: int
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  assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"
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  shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
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proof
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  have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
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  assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
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  with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)"
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    by (simp add: powr_divide2[symmetric] field_simps)
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  also have "\<dots> = m2 * 2^nat (e2 - e1)"
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    by (simp add: powr_realpow)
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  finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
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   277
    unfolding real_of_int_inject .
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   278
  with m1 have "m1 = m2"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   279
    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   280
  then show "m1 = m2 \<and> e1 = e2"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   281
    using eq `m1 \<noteq> 0` by (simp add: powr_inj)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   282
qed simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   283
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   284
lemma mult_powr_eq_mult_powr_iff:
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hoelzl
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diff changeset
   285
  fixes m1 m2 e1 e2 :: int
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   286
  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"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   287
  using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   288
  using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   289
  by (cases e1 e2 rule: linorder_le_cases) auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   290
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   291
lemma floatE_normed:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   292
  assumes x: "x \<in> float"
400b158f1589 replace the float datatype by a type with unique representation
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diff changeset
   293
  obtains (zero) "x = 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   294
   | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   295
proof atomize_elim
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   296
  { assume "x \<noteq> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   297
    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   298
    with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   299
      by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   300
    with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   301
      by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   302
         (simp add: powr_add powr_realpow) }
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   303
  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)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   304
    by blast
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   305
qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   306
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   307
lemma float_normed_cases:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   308
  fixes f :: float
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   309
  obtains (zero) "f = 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   310
   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   311
proof (atomize_elim, induct f)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   312
  case (float_of y) then show ?case
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   313
    by (cases rule: floatE_normed) (auto simp: zero_float_def)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   314
qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   315
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   316
definition mantissa :: "float \<Rightarrow> int" where
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   317
  "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   318
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   319
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
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diff changeset
   320
definition exponent :: "float \<Rightarrow> int" where
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   321
  "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   322
   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   323
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   324
lemma
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   325
  shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   326
    and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   327
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   328
  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   329
  then show ?E ?M
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   330
    by (auto simp add: mantissa_def exponent_def zero_float_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   331
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   332
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   333
lemma
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   334
  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   335
    and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   336
proof cases
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   337
  assume [simp]: "f \<noteq> (float_of 0)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   338
  have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   339
  proof (cases f rule: float_normed_cases)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   340
    case (powr m e)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   341
    then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   342
     \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   343
      by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   344
    then show ?thesis
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   345
      unfolding exponent_def mantissa_def
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   346
      by (rule someI2_ex) (simp add: zero_float_def)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   347
  qed (simp add: zero_float_def)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   348
  then show ?E ?D by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   349
qed simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   350
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   351
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   352
  using mantissa_not_dvd[of f] by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   353
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   354
lemma
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   355
  fixes m e :: int
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   356
  defines "f \<equiv> float_of (m * 2 powr e)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   357
  assumes dvd: "\<not> 2 dvd m"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   358
  shows mantissa_float: "mantissa f = m" (is "?M")
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   359
    and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   360
proof cases
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   361
  assume "m = 0" with dvd show "mantissa f = m" by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   362
next
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   363
  assume "m \<noteq> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   364
  then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   365
  from mantissa_exponent[of f]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   366
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   367
    by (auto simp add: f_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   368
  then show "?M" "?E"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   369
    using mantissa_not_dvd[OF f_not_0] dvd
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   370
    by (auto simp: mult_powr_eq_mult_powr_iff)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   371
qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   372
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   373
subsection {* Compute arithmetic operations *}
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   374
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   375
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   376
  unfolding real_of_float_eq mantissa_exponent[of f] by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   377
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   378
lemma Float_cases[case_names Float, cases type: float]:
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   379
  fixes f :: float
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   380
  obtains (Float) m e :: int where "f = Float m e"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   381
  using Float_mantissa_exponent[symmetric]
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   382
  by (atomize_elim) auto
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   383
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   384
lemma denormalize_shift:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   385
  assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   386
  obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   387
proof
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   388
  from mantissa_exponent[of f] f_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   389
  have "m * 2 powr e = mantissa f * 2 powr exponent f"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   390
    by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   391
  then have eq: "m = mantissa f * 2 powr (exponent f - e)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   392
    by (simp add: powr_divide2[symmetric] field_simps)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   393
  moreover
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   394
  have "e \<le> exponent f"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   395
  proof (rule ccontr)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   396
    assume "\<not> e \<le> exponent f"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   397
    then have pos: "exponent f < e" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   398
    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   399
      by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   400
    also have "\<dots> = 1 / 2^nat (e - exponent f)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   401
      using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   402
    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   403
      using eq by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   404
    then have "mantissa f = m * 2^nat (e - exponent f)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   405
      unfolding real_of_int_inject by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   406
    with `exponent f < e` have "2 dvd mantissa f"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   407
      apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   408
      apply (cases "nat (e - exponent f)")
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   409
      apply auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   410
      done
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   411
    then show False using mantissa_not_dvd[OF not_0] by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   412
  qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   413
  ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   414
    by (simp add: powr_realpow[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   415
  with `e \<le> exponent f`
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   416
  show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   417
    unfolding real_of_int_inject by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   418
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   419
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   420
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   421
  by transfer simp
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   422
hide_fact (open) compute_float_zero
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   423
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   424
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   425
  by transfer simp
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   426
hide_fact (open) compute_float_one
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   427
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   428
definition normfloat :: "float \<Rightarrow> float" where
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   429
  [simp]: "normfloat x = x"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   430
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   431
lemma compute_normfloat[code]: "normfloat (Float m e) =
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   432
  (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   433
                           else if m = 0 then 0 else Float m e)"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   434
  unfolding normfloat_def
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   435
  by transfer (auto simp add: powr_add zmod_eq_0_iff)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   436
hide_fact (open) compute_normfloat
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   437
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   438
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   439
  by transfer simp
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   440
hide_fact (open) compute_float_numeral
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   441
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54230
diff changeset
   442
lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   443
  by transfer simp
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   444
hide_fact (open) compute_float_neg_numeral
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   445
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   446
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   447
  by transfer simp
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   448
hide_fact (open) compute_float_uminus
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   449
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   450
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   451
  by transfer (simp add: field_simps powr_add)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   452
hide_fact (open) compute_float_times
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   453
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   454
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   455
  (if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   456
              else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   457
  by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   458
hide_fact (open) compute_float_plus
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   459
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   460
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   461
  by simp
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   462
hide_fact (open) compute_float_minus
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   463
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   464
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   465
  by transfer (simp add: sgn_times)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   466
hide_fact (open) compute_float_sgn
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   467
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   468
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" ..
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   469
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   470
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   471
  by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   472
hide_fact (open) compute_is_float_pos
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   473
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   474
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   475
  by transfer (simp add: field_simps)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   476
hide_fact (open) compute_float_less
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   477
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   478
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" ..
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   479
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   480
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   481
  by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   482
hide_fact (open) compute_is_float_nonneg
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   483
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   484
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   485
  by transfer (simp add: field_simps)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   486
hide_fact (open) compute_float_le
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   487
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   488
lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" by simp
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   489
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   490
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   491
  by transfer (auto simp add: is_float_zero_def)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   492
hide_fact (open) compute_is_float_zero
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   493
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   494
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   495
  by transfer (simp add: abs_mult)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   496
hide_fact (open) compute_float_abs
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   497
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   498
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   499
  by transfer simp
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   500
hide_fact (open) compute_float_eq
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   501
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   502
subsection {* Rounding Real numbers *}
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   503
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   504
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   505
  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   506
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   507
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   508
  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   509
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   510
lemma round_down_float[simp]: "round_down prec x \<in> float"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   511
  unfolding round_down_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   512
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   513
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   514
lemma round_up_float[simp]: "round_up prec x \<in> float"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   515
  unfolding round_up_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   516
  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   517
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   518
lemma round_up: "x \<le> round_up prec x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   519
  by (simp add: powr_minus_divide le_divide_eq round_up_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   520
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   521
lemma round_down: "round_down prec x \<le> x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   522
  by (simp add: powr_minus_divide divide_le_eq round_down_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   523
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   524
lemma round_up_0[simp]: "round_up p 0 = 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   525
  unfolding round_up_def by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   526
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   527
lemma round_down_0[simp]: "round_down p 0 = 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   528
  unfolding round_down_def by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   529
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   530
lemma round_up_diff_round_down:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   531
  "round_up prec x - round_down prec x \<le> 2 powr -prec"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   532
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   533
  have "round_up prec x - round_down prec x =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   534
    (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   535
    by (simp add: round_up_def round_down_def field_simps)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   536
  also have "\<dots> \<le> 1 * 2 powr -prec"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   537
    by (rule mult_mono)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   538
       (auto simp del: real_of_int_diff
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   539
             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   540
  finally show ?thesis by simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   541
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   542
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   543
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   544
  unfolding round_down_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   545
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   546
    (simp add: powr_add[symmetric])
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   547
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   548
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   549
  unfolding round_up_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   550
  by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   551
    (simp add: powr_add[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   552
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   553
subsection {* Rounding Floats *}
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   554
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   555
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
   556
declare float_up.rep_eq[simp]
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   557
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   558
lemma round_up_correct:
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   559
  shows "round_up e f - f \<in> {0..2 powr -e}"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   560
unfolding atLeastAtMost_iff
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   561
proof
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   562
  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   563
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   564
  finally show "round_up e f - f \<le> 2 powr real (- e)"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   565
    by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   566
qed (simp add: algebra_simps round_up)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   567
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   568
lemma float_up_correct:
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   569
  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   570
  by transfer (rule round_up_correct)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   571
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   572
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
   573
declare float_down.rep_eq[simp]
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   574
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   575
lemma round_down_correct:
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   576
  shows "f - (round_down e f) \<in> {0..2 powr -e}"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   577
unfolding atLeastAtMost_iff
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   578
proof
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   579
  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   580
  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   581
  finally show "f - round_down e f \<le> 2 powr real (- e)"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   582
    by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   583
qed (simp add: algebra_simps round_down)
24301
6c7f226b24c3 changed floatarith lemmas
obua
parents: 24124
diff changeset
   584
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   585
lemma float_down_correct:
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   586
  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   587
  by transfer (rule round_down_correct)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   588
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   589
lemma compute_float_down[code]:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   590
  "float_down p (Float m e) =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   591
    (if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   592
proof cases
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   593
  assume "p + e < 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   594
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   595
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   596
  also have "... = 1 / 2 powr p / 2 powr e"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   597
    unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   598
  finally show ?thesis
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   599
    using `p + e < 0`
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   600
    by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   601
next
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   602
  assume "\<not> p + e < 0"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   603
  then have r: "real e + real p = real (nat (e + p))" by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   604
  have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   605
    by (auto intro: exI[where x="m*2^nat (e+p)"]
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   606
             simp add: ac_simps powr_add[symmetric] r powr_realpow)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   607
  with `\<not> p + e < 0` show ?thesis
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   608
    by transfer
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   609
       (auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   610
qed
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   611
hide_fact (open) compute_float_down
24301
6c7f226b24c3 changed floatarith lemmas
obua
parents: 24124
diff changeset
   612
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   613
lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   614
  using round_down_correct[of f e] by simp
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   615
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   616
lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   617
  using round_up_correct[of e f] by simp
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   618
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   619
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   620
  by (auto simp: round_down_def intro!: mult_nonneg_nonneg)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
   621
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   622
lemma ceil_divide_floor_conv:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   623
assumes "b \<noteq> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   624
shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   625
proof cases
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   626
  assume "\<not> b dvd a"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   627
  hence "a mod b \<noteq> 0" by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   628
  hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   629
  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   630
  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   631
  proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   632
    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   633
    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   634
    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   635
    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   636
  qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   637
  thus ?thesis using `\<not> b dvd a` by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   638
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   639
  floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
19765
dfe940911617 misc cleanup;
wenzelm
parents: 16890
diff changeset
   640
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   641
lemma compute_float_up[code]:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   642
  "float_up p (Float m e) =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   643
    (let P = 2^nat (-(p + e)); r = m mod P in
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   644
      if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (-p) else Float m e)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   645
proof cases
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   646
  assume "p + e < 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   647
  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   648
    using powr_realpow[of 2 "nat (-(p + e))"] by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   649
  also have "... = 1 / 2 powr p / 2 powr e"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   650
  unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   651
  finally have twopow_rewrite:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   652
    "real ((2::int) ^ nat (- (p + e))) = 1 / 2 powr real p / 2 powr real e" .
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   653
  with `p + e < 0` have powr_rewrite:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   654
    "2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat (- (p + e)))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   655
    unfolding powr_divide2 by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   656
  show ?thesis
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   657
  proof cases
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   658
    assume "2^nat (-(p + e)) dvd m"
47615
341fd902ef1c transfer now handles Let
hoelzl
parents: 47608
diff changeset
   659
    with `p + e < 0` twopow_rewrite show ?thesis
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   660
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div dvd_eq_mod_eq_0)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   661
  next
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   662
    assume ndvd: "\<not> 2 ^ nat (- (p + e)) dvd m"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   663
    have one_div: "real m * (1 / real ((2::int) ^ nat (- (p + e)))) =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   664
      real m / real ((2::int) ^ nat (- (p + e)))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   665
      by (simp add: field_simps)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   666
    have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   667
      real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   668
      using ndvd unfolding powr_rewrite one_div
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   669
      by (subst ceil_divide_floor_conv) (auto simp: field_simps)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   670
    thus ?thesis using `p + e < 0` twopow_rewrite
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   671
      by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div[symmetric])
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   672
  qed
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   673
next
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   674
  assume "\<not> p + e < 0"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   675
  then have r1: "real e + real p = real (nat (e + p))" by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   676
  have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   677
    by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   678
      intro: exI[where x="m*2^nat (e+p)"])
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   679
  then show ?thesis using `\<not> p + e < 0`
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   680
    by transfer
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   681
       (simp add: round_up_def floor_divide_eq_div field_simps powr_add powr_minus inverse_eq_divide)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   682
qed
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   683
hide_fact (open) compute_float_up
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   684
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   685
lemmas real_of_ints =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   686
  real_of_int_zero
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   687
  real_of_one
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   688
  real_of_int_add
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   689
  real_of_int_minus
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   690
  real_of_int_diff
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   691
  real_of_int_mult
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   692
  real_of_int_power
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   693
  real_numeral
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   694
lemmas real_of_nats =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   695
  real_of_nat_zero
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   696
  real_of_nat_one
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   697
  real_of_nat_1
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   698
  real_of_nat_add
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   699
  real_of_nat_mult
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   700
  real_of_nat_power
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   701
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   702
lemmas int_of_reals = real_of_ints[symmetric]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   703
lemmas nat_of_reals = real_of_nats[symmetric]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   704
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   705
lemma two_real_int: "(2::real) = real (2::int)" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   706
lemma two_real_nat: "(2::real) = real (2::nat)" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   707
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   708
lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   709
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   710
subsection {* Compute bitlen of integers *}
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   711
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   712
definition bitlen :: "int \<Rightarrow> int" where
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   713
  "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   714
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   715
lemma bitlen_nonneg: "0 \<le> bitlen x"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   716
proof -
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   717
  {
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   718
    assume "0 > x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   719
    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   720
    also have "... < log 2 (-x)" using `0 > x` by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   721
    finally have "-1 < log 2 (-x)" .
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   722
  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   723
qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   724
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   725
lemma bitlen_bounds:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   726
  assumes "x > 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   727
  shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   728
proof
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   729
  have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   730
    using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   731
    using real_nat_eq_real[of "floor (log 2 (real x))"]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   732
    by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   733
  also have "... \<le> 2 powr log 2 (real x)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   734
    by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   735
  also have "... = real x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   736
    using `0 < x` by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   737
  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   738
  thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   739
    by (simp add: bitlen_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   740
next
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   741
  have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   742
  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   743
    apply (simp add: powr_realpow[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   744
    using `x > 0` by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   745
  finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   746
    by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   747
qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   748
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   749
lemma bitlen_pow2[simp]:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   750
  assumes "b > 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   751
  shows "bitlen (b * 2 ^ c) = bitlen b + c"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   752
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   753
  from assms have "b * 2 ^ c > 0" by (auto intro: mult_pos_pos)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   754
  thus ?thesis
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   755
    using floor_add[of "log 2 b" c] assms
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   756
    by (auto simp add: log_mult log_nat_power bitlen_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   757
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   758
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   759
lemma bitlen_Float:
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   760
  fixes m e
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   761
  defines "f \<equiv> Float m e"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   762
  shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   763
proof (cases "m = 0")
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   764
  case True
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   765
  then show ?thesis by (simp add: f_def bitlen_def Float_def)
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   766
next
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   767
  case False
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   768
  hence "f \<noteq> float_of 0"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   769
    unfolding real_of_float_eq by (simp add: f_def)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   770
  hence "mantissa f \<noteq> 0"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   771
    by (simp add: mantissa_noteq_0)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   772
  moreover
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   773
  obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   774
    by (rule f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`])
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   775
  ultimately show ?thesis by (simp add: abs_mult)
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   776
qed
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   777
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   778
lemma compute_bitlen[code]:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   779
  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   780
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   781
  { assume "2 \<le> x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   782
    then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   783
      by (simp add: log_mult zmod_zdiv_equality')
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   784
    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   785
    proof cases
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   786
      assume "x mod 2 = 0" then show ?thesis by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   787
    next
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   788
      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   789
      then have "0 \<le> n"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   790
        using `2 \<le> x` by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   791
      assume "x mod 2 \<noteq> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   792
      with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   793
      with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   794
      moreover
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   795
      { have "real (2^nat n :: int) = 2 powr (nat n)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   796
          by (simp add: powr_realpow)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   797
        also have "\<dots> \<le> 2 powr (log 2 x)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   798
          using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   799
        finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   800
      ultimately have "2^nat n \<le> x - 1" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   801
      then have "2^nat n \<le> real (x - 1)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   802
        unfolding real_of_int_le_iff[symmetric] by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   803
      { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   804
          using `0 \<le> n` by (simp add: log_nat_power)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   805
        also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   806
          using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   807
        finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   808
      moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   809
        using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   810
      ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   811
        unfolding n_def `x mod 2 = 1` by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   812
    qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   813
    finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   814
  moreover
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   815
  { assume "x < 2" "0 < x"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   816
    then have "x = 1" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   817
    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   818
  ultimately show ?thesis
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   819
    unfolding bitlen_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   820
    by (auto simp: pos_imp_zdiv_pos_iff not_le)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   821
qed
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   822
hide_fact (open) compute_bitlen
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   823
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   824
lemma float_gt1_scale: assumes "1 \<le> Float m e"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   825
  shows "0 \<le> e + (bitlen m - 1)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   826
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   827
  have "0 < Float m e" using assms by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   828
  hence "0 < m" using powr_gt_zero[of 2 e]
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   829
    by (auto simp: zero_less_mult_iff)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   830
  hence "m \<noteq> 0" by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   831
  show ?thesis
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   832
  proof (cases "0 \<le> e")
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   833
    case True thus ?thesis using `0 < m`  by (simp add: bitlen_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   834
  next
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   835
    have "(1::int) < 2" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   836
    case False let ?S = "2^(nat (-e))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   837
    have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   838
      by (auto simp: powr_minus field_simps inverse_eq_divide)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   839
    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   840
      by (auto simp: powr_minus)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   841
    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   842
    hence "?S \<le> real m" unfolding mult_assoc by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   843
    hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   844
    from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   845
    have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   846
    hence "-e < bitlen m" using False by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   847
    thus ?thesis by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   848
  qed
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   849
qed
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   850
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   851
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"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   852
proof -
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   853
  let ?B = "2^nat(bitlen m - 1)"
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   854
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   855
  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   856
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   857
  thus "1 \<le> real m / ?B" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   858
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   859
  have "m \<noteq> 0" using assms by auto
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   860
  have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   861
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   862
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   863
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   864
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   865
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   866
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   867
  thus "real m / ?B < 2" by auto
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   868
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   869
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   870
subsection {* Approximation of positive rationals *}
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   871
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   872
lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   873
by (simp add: zdiv_zmult2_eq)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   874
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   875
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   876
  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   877
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   878
lemma real_div_nat_eq_floor_of_divide:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   879
  fixes a b::nat
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   880
  shows "a div b = real (floor (a/b))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   881
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   882
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   883
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   884
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   885
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   886
  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   887
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   888
lemma compute_lapprox_posrat[code]:
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   889
  fixes prec x y
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   890
  shows "lapprox_posrat prec x y =
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   891
   (let
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   892
       l = rat_precision prec x y;
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   893
       d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   894
    in normfloat (Float d (- l)))"
47615
341fd902ef1c transfer now handles Let
hoelzl
parents: 47608
diff changeset
   895
    unfolding div_mult_twopow_eq normfloat_def
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   896
    by transfer
47615
341fd902ef1c transfer now handles Let
hoelzl
parents: 47608
diff changeset
   897
       (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   898
             del: two_powr_minus_int_float)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   899
hide_fact (open) compute_lapprox_posrat
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   900
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   901
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   902
  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   903
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   904
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   905
lemma compute_rapprox_posrat[code]:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   906
  fixes prec x y
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   907
  defines "l \<equiv> rat_precision prec x y"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   908
  shows "rapprox_posrat prec x y = (let
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   909
     l = l ;
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   910
     X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   911
     d = fst X div snd X ;
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   912
     m = fst X mod snd X
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   913
   in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   914
proof (cases "y = 0")
47615
341fd902ef1c transfer now handles Let
hoelzl
parents: 47608
diff changeset
   915
  assume "y = 0" thus ?thesis unfolding normfloat_def by transfer simp
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   916
next
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   917
  assume "y \<noteq> 0"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   918
  show ?thesis
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   919
  proof (cases "0 \<le> l")
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   920
    assume "0 \<le> l"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   921
    def x' == "x * 2 ^ nat l"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   922
    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   923
    moreover have "real x * 2 powr real l = real x'"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   924
      by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   925
    ultimately show ?thesis
47615
341fd902ef1c transfer now handles Let
hoelzl
parents: 47608
diff changeset
   926
      unfolding normfloat_def
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   927
      using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   928
        l_def[symmetric, THEN meta_eq_to_obj_eq]
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   929
      by transfer
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   930
         (simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   931
   next
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   932
    assume "\<not> 0 \<le> l"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   933
    def y' == "y * 2 ^ nat (- l)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   934
    from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   935
    have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   936
    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   937
      using `\<not> 0 \<le> l`
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   938
      by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps inverse_eq_divide)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   939
    ultimately show ?thesis
47615
341fd902ef1c transfer now handles Let
hoelzl
parents: 47608
diff changeset
   940
      unfolding normfloat_def
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   941
      using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   942
        l_def[symmetric, THEN meta_eq_to_obj_eq]
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   943
      by transfer
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   944
         (simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   945
  qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   946
qed
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
   947
hide_fact (open) compute_rapprox_posrat
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   948
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   949
lemma rat_precision_pos:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   950
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   951
  shows "rat_precision n (int x) (int y) > 0"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   952
proof -
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   953
  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   954
  hence "bitlen (int x) < bitlen (int y)" using assms
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   955
    by (simp add: bitlen_def del: floor_add_one)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   956
      (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   957
  thus ?thesis
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   958
    using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   959
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   960
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   961
lemma power_aux: assumes "x > 0" shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   962
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   963
  def y \<equiv> "nat (x - 1)" moreover
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   964
  have "(2::int) ^ y \<le> (2 ^ (y + 1)) - 1" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   965
  ultimately show ?thesis using assms by simp
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   966
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   967
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
   968
lemma rapprox_posrat_less1:
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
   969
  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
   970
  shows "real (rapprox_posrat n x y) < 1"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   971
proof -
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
   972
  have powr1: "2 powr real (rat_precision n (int x) (int y)) =
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   973
    2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   974
    by (simp add: powr_realpow[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   975
  have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) *
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   976
     2 powr real (rat_precision n (int x) (int y))" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   977
  also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   978
    apply (rule mult_strict_right_mono) by (insert assms) auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   979
  also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54230
diff changeset
   980
    using powr_add [of 2 _ "- 1", simplified add_uminus_conv_diff] by (simp add: powr_minus)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   981
  also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   982
    using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   983
  also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   984
    unfolding int_of_reals real_of_int_le_iff
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   985
    using rat_precision_pos[OF assms] by (rule power_aux)
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   986
  finally show ?thesis
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
   987
    apply (transfer fixing: n x y)
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
   988
    apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide powr1)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   989
    unfolding int_of_reals real_of_int_less_iff
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
   990
    apply (simp add: ceiling_less_eq)
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   991
    done
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   992
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   993
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   994
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
   995
  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
   996
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
   997
lemma compute_lapprox_rat[code]:
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   998
  "lapprox_rat prec x y =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
   999
    (if y = 0 then 0
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1000
    else if 0 \<le> x then
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1001
      (if 0 < y then lapprox_posrat prec (nat x) (nat y)
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
  1002
      else - (rapprox_posrat prec (nat x) (nat (-y))))
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1003
      else (if 0 < y
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1004
        then - (rapprox_posrat prec (nat (-x)) (nat y))
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1005
        else lapprox_posrat prec (nat (-x)) (nat (-y))))"
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1006
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
  1007
hide_fact (open) compute_lapprox_rat
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1008
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1009
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1010
  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1011
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1012
lemma compute_rapprox_rat[code]:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1013
  "rapprox_rat prec x y =
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1014
    (if y = 0 then 0
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1015
    else if 0 \<le> x then
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1016
      (if 0 < y then rapprox_posrat prec (nat x) (nat y)
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
  1017
      else - (lapprox_posrat prec (nat x) (nat (-y))))
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1018
      else (if 0 < y
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1019
        then - (lapprox_posrat prec (nat (-x)) (nat y))
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1020
        else rapprox_posrat prec (nat (-x)) (nat (-y))))"
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1021
  by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
  1022
hide_fact (open) compute_rapprox_rat
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1023
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1024
subsection {* Division *}
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1025
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1026
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)"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1027
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1028
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)"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1029
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1030
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1031
  by (simp add: real_divl_def)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1032
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1033
lemma compute_float_divl[code]:
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1034
  "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1035
proof cases
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1036
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1037
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1038
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1039
  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)"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1040
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1041
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1042
    by (simp add: field_simps powr_divide2[symmetric])
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1043
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1044
  show ?thesis
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
  1045
    using not_0
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1046
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1047
      simp add: field_simps)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1048
qed (transfer, auto simp: real_divl_def)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
  1049
hide_fact (open) compute_float_divl
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1050
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1051
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1052
  by (simp add: real_divr_def)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1053
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1054
lemma compute_float_divr[code]:
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1055
  "float_divr prec (Float m1 s1) (Float m2 s2) = rapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1056
proof cases
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1057
  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1058
  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1059
  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1060
  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)"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1061
    by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1062
  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1063
    by (simp add: field_simps powr_divide2[symmetric])
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1064
47601
050718fe6eee use real :: float => real as lifting-morphism so we can directlry use the rep_eq theorems
hoelzl
parents: 47600
diff changeset
  1065
  show ?thesis
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
  1066
    using not_0
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1067
    by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift real_divr_def,
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1068
      simp add: field_simps)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1069
qed (transfer, auto simp: real_divr_def)
47621
4cf6011fb884 hide code generation facts in the Float theory, they are only exported for Approximation
hoelzl
parents: 47615
diff changeset
  1070
hide_fact (open) compute_float_divr
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1071
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1072
subsection {* Lemmas needed by Approximate *}
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1073
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1074
lemma Float_num[simp]: shows
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1075
   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1076
   "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1077
   "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1078
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"]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1079
using powr_realpow[of 2 2] powr_realpow[of 2 3]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1080
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1081
by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1082
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1083
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1084
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1085
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1086
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1087
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1088
by arith
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1089
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1090
lemma lapprox_rat:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1091
  shows "real (lapprox_rat prec x y) \<le> real x / real y"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1092
  using round_down by (simp add: lapprox_rat_def)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1093
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1094
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1095
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1096
  from zmod_zdiv_equality'[of a b]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1097
  have "a = b * (a div b) + a mod b" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1098
  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1099
  using assms by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1100
  finally show ?thesis by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1101
qed
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1102
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1103
lemma lapprox_rat_nonneg:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1104
  fixes n x y
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1105
  defines "p == int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1106
  assumes "0 \<le> x" "0 < y"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1107
  shows "0 \<le> real (lapprox_rat n x y)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1108
using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1109
   powr_int[of 2, simplified]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1110
  by (auto simp add: inverse_eq_divide intro!: mult_nonneg_nonneg divide_nonneg_pos mult_pos_pos)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1111
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
  1112
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1113
  using round_up by (simp add: rapprox_rat_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1114
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1115
lemma rapprox_rat_le1:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1116
  fixes n x y
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1117
  assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1118
  shows "real (rapprox_rat n x y) \<le> 1"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1119
proof -
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1120
  have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1121
    using xy unfolding bitlen_def by (auto intro!: floor_mono)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1122
  then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1123
  have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1124
      \<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1125
    using xy by (auto intro!: ceiling_mono simp: field_simps)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1126
  also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1127
    using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>`
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1128
    by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1129
  finally show ?thesis
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1130
    by (simp add: rapprox_rat_def round_up_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1131
       (simp add: powr_minus inverse_eq_divide)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1132
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1133
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
  1134
lemma rapprox_rat_nonneg_neg:
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1135
  "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1136
  unfolding rapprox_rat_def round_up_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1137
  by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1138
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1139
lemma rapprox_rat_neg:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1140
  "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1141
  unfolding rapprox_rat_def round_up_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1142
  by (auto simp: field_simps mult_le_0_iff)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1143
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1144
lemma rapprox_rat_nonpos_pos:
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1145
  "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1146
  unfolding rapprox_rat_def round_up_def
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1147
  by (auto simp: field_simps mult_le_0_iff)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1148
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1149
lemma real_divl: "real_divl prec x y \<le> x / y"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1150
  by (simp add: real_divl_def round_down)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1151
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1152
lemma real_divr: "x / y \<le> real_divr prec x y"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1153
  using round_up by (simp add: real_divr_def)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1154
31098
73dd67adf90a replaced Ifloat => real_of_float and real, renamed ApproxEq => inequality, uneq => interpret_inequality, uneq' => approx_inequality, Ifloatarith => interpret_floatarith
hoelzl
parents: 31021
diff changeset
  1155
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1156
  by transfer (rule real_divl)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1157
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1158
lemma real_divl_lower_bound:
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1159
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1160
  by (simp add: real_divl_def round_down_def zero_le_mult_iff zero_le_divide_iff)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1161
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1162
lemma float_divl_lower_bound:
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1163
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1164
  by transfer (rule real_divl_lower_bound)
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1165
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1166
lemma exponent_1: "exponent 1 = 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1167
  using exponent_float[of 1 0] by (simp add: one_float_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1168
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1169
lemma mantissa_1: "mantissa 1 = 1"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1170
  using mantissa_float[of 1 0] by (simp add: one_float_def)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1171
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1172
lemma bitlen_1: "bitlen 1 = 1"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1173
  by (simp add: bitlen_def)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1174
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1175
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1176
proof
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1177
  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1178
  show "x = 0" by (simp add: zero_float_def z)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1179
qed (simp add: zero_float_def)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1180
47599
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1181
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1182
proof (cases "x = 0", simp)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1183
  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1184
  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1185
  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1186
  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1187
    using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1188
    by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1189
      real_of_int_le_iff less_imp_le)
400b158f1589 replace the float datatype by a type with unique representation
hoelzl
parents: 47230
diff changeset
  1190
  finally show ?thesis by (simp add: powr_add)
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1191
qed
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1192
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1193
lemma real_divl_pos_less1_bound:
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1194
  "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real_divl prec 1 x"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1195
proof (unfold real_divl_def)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1196
  fix prec :: nat and x :: real assume x: "0 < x" "x < 1" and prec: "1 \<le> prec"
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53215
diff changeset
  1197
  def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>"
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1198
  show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) "
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1199
  proof cases
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1200
    assume nonneg: "0 \<le> p"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1201
    hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1202
      by (simp add: powr_int del: real_of_int_power) simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1203
    also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1204
    also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le>
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1205
      floor (real ((2::int) ^ nat p) * (1 / x))"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1206
      by (rule le_mult_floor) (auto simp: x prec less_imp_le)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1207
    finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1208
    thus ?thesis unfolding p_def[symmetric]
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1209
      using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1210
  next
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1211
    assume neg: "\<not> 0 \<le> p"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1212
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1213
    have "x = 2 powr (log 2 x)"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1214
      using x by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1215
    also have "2 powr (log 2 x) \<le> 2 powr p"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1216
    proof (rule powr_mono)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1217
      have "log 2 x \<le> \<lceil>log 2 x\<rceil>"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1218
        by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1219
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1220
        using ceiling_diff_floor_le_1[of "log 2 x"] by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1221
      also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1222
        using prec by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1223
      finally show "log 2 x \<le> real p"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1224
        using x by (simp add: p_def)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1225
    qed simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1226
    finally have x_le: "x \<le> 2 powr p" .
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1227
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1228
    from neg have "2 powr real p \<le> 2 powr 0"
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1229
      by (intro powr_mono) auto
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1230
    also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1231
    also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1232
      using x x_le by (intro floor_mono) (simp add:  pos_le_divide_eq mult_pos_pos)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1233
    finally show ?thesis
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1234
      using prec x unfolding p_def[symmetric]
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1235
      by (simp add: round_down_def powr_minus_divide pos_le_divide_eq mult_pos_pos)
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1236
  qed
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1237
qed
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1238
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1239
lemma float_divl_pos_less1_bound:
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1240
  "0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1241
  by (transfer, rule real_divl_pos_less1_bound)
16782
b214f21ae396 - use TableFun instead of homebrew binary tree in am_interpreter.ML
obua
parents:
diff changeset
  1242
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1243
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1244
  by transfer (rule real_divr)
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1245
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1246
lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> real_divr prec 1 x"
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1247
proof -
54782
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1248
  have "1 \<le> 1 / x" using `0 < x` and `x < 1` by auto
cd8f55c358c5 additional definitions and lemmas for Float
immler
parents: 54489
diff changeset
  1249
  also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto
47600
e12289b5796b use lifting to introduce floating point numbers
hoelzl
parents: 47599
diff changeset
  1250
  finally show ?thesis by auto
29804
e15b74577368 Added new Float theory and moved old Library/Float.thy to ComputeFloat
hoelzl
parents: 29667
diff changeset
  1251