src/HOL/Library/Float.thy
author hoelzl
Mon Sep 06 15:01:37 2010 +0200 (2010-09-06)
changeset 39161 75849a560c09
parent 36778 739a9379e29b
child 41024 ba961a606c67
permissions -rw-r--r--
When comparing Floats use integers instead of reals (represented as rationals), generates less code when Floats are used.
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(*  Title:      HOL/Library/Float.thy
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    Author:     Steven Obua 2008
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    Author:     Johannes Hoelzl, TU Muenchen <hoelzl@in.tum.de> 2008 / 2009
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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
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  pow2 :: "int \<Rightarrow> real" where
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  [simp]: "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
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datatype float = Float int int
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primrec of_float :: "float \<Rightarrow> real" where
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  "of_float (Float a b) = real a * pow2 b"
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defs (overloaded)
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  real_of_float_def [code_unfold]: "real == of_float"
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primrec mantissa :: "float \<Rightarrow> int" where
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  "mantissa (Float a b) = a"
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primrec scale :: "float \<Rightarrow> int" where
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  "scale (Float a b) = b"
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instantiation float :: zero begin
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definition zero_float where "0 = Float 0 0"
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instance ..
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end
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instantiation float :: one begin
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definition one_float where "1 = Float 1 0"
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instance ..
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end
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instantiation float :: number begin
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definition number_of_float where "number_of n = Float n 0"
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instance ..
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end
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lemma number_of_float_Float [code_unfold_post]:
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  "number_of k = Float (number_of k) 0"
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  by (simp add: number_of_float_def number_of_is_id)
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lemma real_of_float_simp[simp]: "real (Float a b) = real a * pow2 b"
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  unfolding real_of_float_def using of_float.simps .
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lemma real_of_float_neg_exp: "e < 0 \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto
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lemma real_of_float_nge0_exp: "\<not> 0 \<le> e \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto
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lemma real_of_float_ge0_exp: "0 \<le> e \<Longrightarrow> real (Float m e) = real m * (2^nat e)" by auto
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lemma Float_num[simp]: shows
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   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
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   "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
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   "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"
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  by auto
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lemma float_number_of[simp]: "real (number_of x :: float) = number_of x"
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  by (simp only:number_of_float_def Float_num[unfolded number_of_is_id])
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lemma float_number_of_int[simp]: "real (Float n 0) = real n"
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  by (simp add: Float_num[unfolded number_of_is_id] real_of_float_simp pow2_def)
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lemma pow2_0[simp]: "pow2 0 = 1" by simp
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lemma pow2_1[simp]: "pow2 1 = 2" by simp
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lemma pow2_neg: "pow2 x = inverse (pow2 (-x))" by simp
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declare pow2_def[simp del]
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lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
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proof -
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  have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
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  have g: "! a b. a - -1 = a + (1::int)" by arith
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  have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
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    apply (auto, induct_tac n)
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    apply (simp_all add: pow2_def)
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    apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
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    by (auto simp add: h)
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  show ?thesis
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  proof (induct a)
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    case (1 n)
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    from pos show ?case by (simp add: algebra_simps)
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  next
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    case (2 n)
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    show ?case
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      apply (auto)
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      apply (subst pow2_neg[of "- int n"])
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      apply (subst pow2_neg[of "-1 - int n"])
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      apply (auto simp add: g pos)
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      done
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  qed
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qed
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lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
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proof (induct b)
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  case (1 n)
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  show ?case
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  proof (induct n)
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    case 0
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    show ?case by simp
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  next
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    case (Suc m)
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    show ?case by (auto simp add: algebra_simps pow2_add1 prems)
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  qed
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next
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  case (2 n)
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  show ?case
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  proof (induct n)
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    case 0
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    show ?case
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      apply (auto)
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      apply (subst pow2_neg[of "a + -1"])
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      apply (subst pow2_neg[of "-1"])
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      apply (simp)
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      apply (insert pow2_add1[of "-a"])
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      apply (simp add: algebra_simps)
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      apply (subst pow2_neg[of "-a"])
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      apply (simp)
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      done
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    case (Suc m)
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    have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
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    have b: "int m - -2 = 1 + (int m + 1)" by arith
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    show ?case
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      apply (auto)
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      apply (subst pow2_neg[of "a + (-2 - int m)"])
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      apply (subst pow2_neg[of "-2 - int m"])
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      apply (auto simp add: algebra_simps)
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      apply (subst a)
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      apply (subst b)
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      apply (simp only: pow2_add1)
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      apply (subst pow2_neg[of "int m - a + 1"])
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      apply (subst pow2_neg[of "int m + 1"])
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      apply auto
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      apply (insert prems)
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      apply (auto simp add: algebra_simps)
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      done
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  qed
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qed
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lemma float_components[simp]: "Float (mantissa f) (scale f) = f" by (cases f, auto)
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lemma float_split: "\<exists> a b. x = Float a b" by (cases x, auto)
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lemma float_split2: "(\<forall> a b. x \<noteq> Float a b) = False" by (auto simp add: float_split)
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lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
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lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
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by arith
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function normfloat :: "float \<Rightarrow> float" where
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"normfloat (Float a b) = (if a \<noteq> 0 \<and> even a then normfloat (Float (a div 2) (b+1)) else if a=0 then Float 0 0 else Float a b)"
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by pat_completeness auto
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termination by (relation "measure (nat o abs o mantissa)") (auto intro: abs_div_2_less)
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declare normfloat.simps[simp del]
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theorem normfloat[symmetric, simp]: "real f = real (normfloat f)"
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proof (induct f rule: normfloat.induct)
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  case (1 a b)
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  have real2: "2 = real (2::int)"
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    by auto
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  show ?case
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    apply (subst normfloat.simps)
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    apply (auto simp add: float_zero)
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    apply (subst 1[symmetric])
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    apply (auto simp add: pow2_add even_def)
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    done
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qed
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lemma pow2_neq_zero[simp]: "pow2 x \<noteq> 0"
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  by (auto simp add: pow2_def)
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lemma pow2_int: "pow2 (int c) = 2^c"
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by (simp add: pow2_def)
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lemma zero_less_pow2[simp]:
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  "0 < pow2 x"
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proof -
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  {
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    fix y
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    have "0 <= y \<Longrightarrow> 0 < pow2 y"
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      by (induct y, induct_tac n, simp_all add: pow2_add)
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  }
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  note helper=this
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  show ?thesis
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    apply (case_tac "0 <= x")
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    apply (simp add: helper)
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    apply (subst pow2_neg)
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    apply (simp add: helper)
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    done
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qed
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lemma normfloat_imp_odd_or_zero: "normfloat f = Float a b \<Longrightarrow> odd a \<or> (a = 0 \<and> b = 0)"
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proof (induct f rule: normfloat.induct)
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  case (1 u v)
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  from 1 have ab: "normfloat (Float u v) = Float a b" by auto
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  {
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    assume eu: "even u"
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    assume z: "u \<noteq> 0"
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    have "normfloat (Float u v) = normfloat (Float (u div 2) (v + 1))"
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      apply (subst normfloat.simps)
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      by (simp add: eu z)
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    with ab have "normfloat (Float (u div 2) (v + 1)) = Float a b" by simp
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    with 1 eu z have ?case by auto
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  }
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  note case1 = this
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  {
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    assume "odd u \<or> u = 0"
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    then have ou: "\<not> (u \<noteq> 0 \<and> even u)" by auto
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    have "normfloat (Float u v) = (if u = 0 then Float 0 0 else Float u v)"
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      apply (subst normfloat.simps)
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      apply (simp add: ou)
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      done
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    with ab have "Float a b = (if u = 0 then Float 0 0 else Float u v)" by auto
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    then have ?case
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      apply (case_tac "u=0")
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      apply (auto)
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      by (insert ou, auto)
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  }
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  note case2 = this
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  show ?case
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    apply (case_tac "odd u \<or> u = 0")
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    apply (rule case2)
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    apply simp
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    apply (rule case1)
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    apply auto
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    done
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qed
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lemma float_eq_odd_helper: 
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  assumes odd: "odd a'"
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  and floateq: "real (Float a b) = real (Float a' b')"
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  shows "b \<le> b'"
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proof - 
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  {
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    assume bcmp: "b > b'"
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    from floateq have eq: "real a * pow2 b = real a' * pow2 b'" by simp
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    {
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      fix x y z :: real
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      assume "y \<noteq> 0"
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      then have "(x * inverse y = z) = (x = z * y)"
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        by auto
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    }
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    note inverse = this
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    have eq': "real a * (pow2 (b - b')) = real a'"
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      apply (subst diff_int_def)
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      apply (subst pow2_add)
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      apply (subst pow2_neg[where x = "-b'"])
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      apply simp
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      apply (subst mult_assoc[symmetric])
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      apply (subst inverse)
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      apply (simp_all add: eq)
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      done
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    have "\<exists> z > 0. pow2 (b-b') = 2^z"
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      apply (rule exI[where x="nat (b - b')"])
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      apply (auto)
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      apply (insert bcmp)
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      apply simp
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      apply (subst pow2_int[symmetric])
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      apply auto
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      done
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    then obtain z where z: "z > 0 \<and> pow2 (b-b') = 2^z" by auto
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    with eq' have "real a * 2^z = real a'"
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      by auto
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    then have "real a * real ((2::int)^z) = real a'"
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      by auto
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    then have "real (a * 2^z) = real a'"
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      apply (subst real_of_int_mult)
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      apply simp
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      done
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    then have a'_rep: "a * 2^z = a'" by arith
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    then have "a' = a*2^z" by simp
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    with z have "even a'" by simp
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    with odd have False by auto
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  }
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  then show ?thesis by arith
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qed
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lemma float_eq_odd: 
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  assumes odd1: "odd a"
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  and odd2: "odd a'"
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  and floateq: "real (Float a b) = real (Float a' b')"
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  shows "a = a' \<and> b = b'"
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proof -
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  from 
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     float_eq_odd_helper[OF odd2 floateq] 
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     float_eq_odd_helper[OF odd1 floateq[symmetric]]
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  have beq: "b = b'"  by arith
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  with floateq show ?thesis by auto
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qed
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theorem normfloat_unique:
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  assumes real_of_float_eq: "real f = real g"
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  shows "normfloat f = normfloat g"
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proof - 
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  from float_split[of "normfloat f"] obtain a b where normf:"normfloat f = Float a b" by auto
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  from float_split[of "normfloat g"] obtain a' b' where normg:"normfloat g = Float a' b'" by auto
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  have "real (normfloat f) = real (normfloat g)"
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    by (simp add: real_of_float_eq)
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  then have float_eq: "real (Float a b) = real (Float a' b')"
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    by (simp add: normf normg)
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  have ab: "odd a \<or> (a = 0 \<and> b = 0)" by (rule normfloat_imp_odd_or_zero[OF normf])
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  have ab': "odd a' \<or> (a' = 0 \<and> b' = 0)" by (rule normfloat_imp_odd_or_zero[OF normg])
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  {
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    assume odd: "odd a"
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    then have "a \<noteq> 0" by (simp add: even_def, arith)
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    with float_eq have "a' \<noteq> 0" by auto
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    with ab' have "odd a'" by simp
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    from odd this float_eq have "a = a' \<and> b = b'" by (rule float_eq_odd)
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  }
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  note odd_case = this
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  {
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    assume even: "even a"
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    with ab have a0: "a = 0" by simp
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    with float_eq have a0': "a' = 0" by auto 
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    from a0 a0' ab ab' have "a = a' \<and> b = b'" by auto
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  }
hoelzl@29804
   322
  note even_case = this
hoelzl@29804
   323
  from odd_case even_case show ?thesis
hoelzl@29804
   324
    apply (simp add: normf normg)
hoelzl@29804
   325
    apply (case_tac "even a")
hoelzl@29804
   326
    apply auto
hoelzl@29804
   327
    done
hoelzl@29804
   328
qed
hoelzl@29804
   329
hoelzl@29804
   330
instantiation float :: plus begin
hoelzl@29804
   331
fun plus_float where
hoelzl@29804
   332
[simp del]: "(Float a_m a_e) + (Float b_m b_e) = 
hoelzl@29804
   333
     (if a_e \<le> b_e then Float (a_m + b_m * 2^(nat(b_e - a_e))) a_e 
hoelzl@29804
   334
                   else Float (a_m * 2^(nat (a_e - b_e)) + b_m) b_e)"
hoelzl@29804
   335
instance ..
hoelzl@29804
   336
end
hoelzl@29804
   337
hoelzl@29804
   338
instantiation float :: uminus begin
haftmann@30960
   339
primrec uminus_float where [simp del]: "uminus_float (Float m e) = Float (-m) e"
hoelzl@29804
   340
instance ..
hoelzl@29804
   341
end
hoelzl@29804
   342
hoelzl@29804
   343
instantiation float :: minus begin
haftmann@30960
   344
definition minus_float where [simp del]: "(z::float) - w = z + (- w)"
hoelzl@29804
   345
instance ..
hoelzl@29804
   346
end
hoelzl@29804
   347
hoelzl@29804
   348
instantiation float :: times begin
hoelzl@29804
   349
fun times_float where [simp del]: "(Float a_m a_e) * (Float b_m b_e) = Float (a_m * b_m) (a_e + b_e)"
hoelzl@29804
   350
instance ..
hoelzl@29804
   351
end
hoelzl@29804
   352
haftmann@30960
   353
primrec float_pprt :: "float \<Rightarrow> float" where
haftmann@30960
   354
  "float_pprt (Float a e) = (if 0 <= a then (Float a e) else 0)"
hoelzl@29804
   355
haftmann@30960
   356
primrec float_nprt :: "float \<Rightarrow> float" where
haftmann@30960
   357
  "float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))" 
hoelzl@29804
   358
hoelzl@29804
   359
instantiation float :: ord begin
hoelzl@31098
   360
definition le_float_def: "z \<le> (w :: float) \<equiv> real z \<le> real w"
hoelzl@31098
   361
definition less_float_def: "z < (w :: float) \<equiv> real z < real w"
hoelzl@29804
   362
instance ..
hoelzl@29804
   363
end
hoelzl@29804
   364
hoelzl@31098
   365
lemma real_of_float_add[simp]: "real (a + b) = real a + real (b :: float)"
hoelzl@29804
   366
  by (cases a, cases b, simp add: algebra_simps plus_float.simps, 
hoelzl@29804
   367
      auto simp add: pow2_int[symmetric] pow2_add[symmetric])
hoelzl@29804
   368
hoelzl@31098
   369
lemma real_of_float_minus[simp]: "real (- a) = - real (a :: float)"
hoelzl@29804
   370
  by (cases a, simp add: uminus_float.simps)
hoelzl@29804
   371
hoelzl@31098
   372
lemma real_of_float_sub[simp]: "real (a - b) = real a - real (b :: float)"
haftmann@30960
   373
  by (cases a, cases b, simp add: minus_float_def)
hoelzl@29804
   374
hoelzl@31098
   375
lemma real_of_float_mult[simp]: "real (a*b) = real a * real (b :: float)"
hoelzl@29804
   376
  by (cases a, cases b, simp add: times_float.simps pow2_add)
hoelzl@29804
   377
hoelzl@31098
   378
lemma real_of_float_0[simp]: "real (0 :: float) = 0"
hoelzl@29804
   379
  by (auto simp add: zero_float_def float_zero)
hoelzl@29804
   380
hoelzl@31098
   381
lemma real_of_float_1[simp]: "real (1 :: float) = 1"
hoelzl@29804
   382
  by (auto simp add: one_float_def)
hoelzl@29804
   383
obua@16782
   384
lemma zero_le_float:
hoelzl@31098
   385
  "(0 <= real (Float a b)) = (0 <= a)"
hoelzl@29804
   386
  apply auto
hoelzl@29804
   387
  apply (auto simp add: zero_le_mult_iff)
obua@16782
   388
  apply (insert zero_less_pow2[of b])
obua@16782
   389
  apply (simp_all)
obua@16782
   390
  done
obua@16782
   391
obua@16782
   392
lemma float_le_zero:
hoelzl@31098
   393
  "(real (Float a b) <= 0) = (a <= 0)"
hoelzl@29804
   394
  apply auto
obua@16782
   395
  apply (auto simp add: mult_le_0_iff)
obua@16782
   396
  apply (insert zero_less_pow2[of b])
obua@16782
   397
  apply auto
obua@16782
   398
  done
obua@16782
   399
hoelzl@39161
   400
lemma zero_less_float:
hoelzl@39161
   401
  "(0 < real (Float a b)) = (0 < a)"
hoelzl@39161
   402
  apply auto
hoelzl@39161
   403
  apply (auto simp add: zero_less_mult_iff)
hoelzl@39161
   404
  apply (insert zero_less_pow2[of b])
hoelzl@39161
   405
  apply (simp_all)
hoelzl@39161
   406
  done
hoelzl@39161
   407
hoelzl@39161
   408
lemma float_less_zero:
hoelzl@39161
   409
  "(real (Float a b) < 0) = (a < 0)"
hoelzl@39161
   410
  apply auto
hoelzl@39161
   411
  apply (auto simp add: mult_less_0_iff)
hoelzl@39161
   412
  apply (insert zero_less_pow2[of b])
hoelzl@39161
   413
  apply (simp_all)
hoelzl@39161
   414
  done
hoelzl@39161
   415
hoelzl@31098
   416
declare real_of_float_simp[simp del]
hoelzl@29804
   417
hoelzl@31098
   418
lemma real_of_float_pprt[simp]: "real (float_pprt a) = pprt (real a)"
hoelzl@29804
   419
  by (cases a, auto simp add: float_pprt.simps zero_le_float float_le_zero float_zero)
hoelzl@29804
   420
hoelzl@31098
   421
lemma real_of_float_nprt[simp]: "real (float_nprt a) = nprt (real a)"
hoelzl@29804
   422
  by (cases a,  auto simp add: float_nprt.simps zero_le_float float_le_zero float_zero)
hoelzl@29804
   423
hoelzl@29804
   424
instance float :: ab_semigroup_add
hoelzl@29804
   425
proof (intro_classes)
hoelzl@29804
   426
  fix a b c :: float
hoelzl@29804
   427
  show "a + b + c = a + (b + c)"
hoelzl@29804
   428
    by (cases a, cases b, cases c, auto simp add: algebra_simps power_add[symmetric] plus_float.simps)
hoelzl@29804
   429
next
hoelzl@29804
   430
  fix a b :: float
hoelzl@29804
   431
  show "a + b = b + a"
hoelzl@29804
   432
    by (cases a, cases b, simp add: plus_float.simps)
hoelzl@29804
   433
qed
hoelzl@29804
   434
hoelzl@29804
   435
instance float :: comm_monoid_mult
hoelzl@29804
   436
proof (intro_classes)
hoelzl@29804
   437
  fix a b c :: float
hoelzl@29804
   438
  show "a * b * c = a * (b * c)"
hoelzl@29804
   439
    by (cases a, cases b, cases c, simp add: times_float.simps)
hoelzl@29804
   440
next
hoelzl@29804
   441
  fix a b :: float
hoelzl@29804
   442
  show "a * b = b * a"
hoelzl@29804
   443
    by (cases a, cases b, simp add: times_float.simps)
hoelzl@29804
   444
next
hoelzl@29804
   445
  fix a :: float
hoelzl@29804
   446
  show "1 * a = a"
hoelzl@29804
   447
    by (cases a, simp add: times_float.simps one_float_def)
hoelzl@29804
   448
qed
hoelzl@29804
   449
hoelzl@29804
   450
(* Floats do NOT form a cancel_semigroup_add: *)
hoelzl@29804
   451
lemma "0 + Float 0 1 = 0 + Float 0 2"
hoelzl@29804
   452
  by (simp add: plus_float.simps zero_float_def)
hoelzl@29804
   453
hoelzl@29804
   454
instance float :: comm_semiring
hoelzl@29804
   455
proof (intro_classes)
hoelzl@29804
   456
  fix a b c :: float
hoelzl@29804
   457
  show "(a + b) * c = a * c + b * c"
hoelzl@29804
   458
    by (cases a, cases b, cases c, simp, simp add: plus_float.simps times_float.simps algebra_simps)
hoelzl@29804
   459
qed
hoelzl@29804
   460
hoelzl@29804
   461
(* Floats do NOT form an order, because "(x < y) = (x <= y & x <> y)" does NOT hold *)
hoelzl@29804
   462
hoelzl@29804
   463
instance float :: zero_neq_one
hoelzl@29804
   464
proof (intro_classes)
hoelzl@29804
   465
  show "(0::float) \<noteq> 1"
hoelzl@29804
   466
    by (simp add: zero_float_def one_float_def)
hoelzl@29804
   467
qed
hoelzl@29804
   468
hoelzl@29804
   469
lemma float_le_simp: "((x::float) \<le> y) = (0 \<le> y - x)"
hoelzl@29804
   470
  by (auto simp add: le_float_def)
hoelzl@29804
   471
hoelzl@29804
   472
lemma float_less_simp: "((x::float) < y) = (0 < y - x)"
hoelzl@29804
   473
  by (auto simp add: less_float_def)
hoelzl@29804
   474
hoelzl@31098
   475
lemma real_of_float_min: "real (min x y :: float) = min (real x) (real y)" unfolding min_def le_float_def by auto
hoelzl@31098
   476
lemma real_of_float_max: "real (max a b :: float) = max (real a) (real b)" unfolding max_def le_float_def by auto
hoelzl@29804
   477
hoelzl@31098
   478
lemma float_power: "real (x ^ n :: float) = real x ^ n"
haftmann@30960
   479
  by (induct n) simp_all
hoelzl@29804
   480
hoelzl@29804
   481
lemma zero_le_pow2[simp]: "0 \<le> pow2 s"
hoelzl@29804
   482
  apply (subgoal_tac "0 < pow2 s")
hoelzl@29804
   483
  apply (auto simp only:)
hoelzl@29804
   484
  apply auto
obua@16782
   485
  done
obua@16782
   486
hoelzl@29804
   487
lemma pow2_less_0_eq_False[simp]: "(pow2 s < 0) = False"
hoelzl@29804
   488
  apply auto
hoelzl@29804
   489
  apply (subgoal_tac "0 \<le> pow2 s")
hoelzl@29804
   490
  apply simp
hoelzl@29804
   491
  apply simp
obua@24301
   492
  done
obua@24301
   493
hoelzl@29804
   494
lemma pow2_le_0_eq_False[simp]: "(pow2 s \<le> 0) = False"
hoelzl@29804
   495
  apply auto
hoelzl@29804
   496
  apply (subgoal_tac "0 < pow2 s")
hoelzl@29804
   497
  apply simp
hoelzl@29804
   498
  apply simp
obua@24301
   499
  done
obua@24301
   500
hoelzl@29804
   501
lemma float_pos_m_pos: "0 < Float m e \<Longrightarrow> 0 < m"
hoelzl@31098
   502
  unfolding less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff
obua@16782
   503
  by auto
wenzelm@19765
   504
hoelzl@29804
   505
lemma float_pos_less1_e_neg: assumes "0 < Float m e" and "Float m e < 1" shows "e < 0"
hoelzl@29804
   506
proof -
hoelzl@29804
   507
  have "0 < m" using float_pos_m_pos `0 < Float m e` by auto
hoelzl@29804
   508
  hence "0 \<le> real m" and "1 \<le> real m" by auto
hoelzl@29804
   509
  
hoelzl@29804
   510
  show "e < 0"
hoelzl@29804
   511
  proof (rule ccontr)
hoelzl@29804
   512
    assume "\<not> e < 0" hence "0 \<le> e" by auto
hoelzl@29804
   513
    hence "1 \<le> pow2 e" unfolding pow2_def by auto
hoelzl@29804
   514
    from mult_mono[OF `1 \<le> real m` this `0 \<le> real m`]
hoelzl@31098
   515
    have "1 \<le> Float m e" by (simp add: le_float_def real_of_float_simp)
hoelzl@29804
   516
    thus False using `Float m e < 1` unfolding less_float_def le_float_def by auto
hoelzl@29804
   517
  qed
hoelzl@29804
   518
qed
hoelzl@29804
   519
hoelzl@29804
   520
lemma float_less1_mantissa_bound: assumes "0 < Float m e" "Float m e < 1" shows "m < 2^(nat (-e))"
hoelzl@29804
   521
proof -
hoelzl@29804
   522
  have "e < 0" using float_pos_less1_e_neg assms by auto
hoelzl@29804
   523
  have "\<And>x. (0::real) < 2^x" by auto
hoelzl@29804
   524
  have "real m < 2^(nat (-e))" using `Float m e < 1`
hoelzl@31098
   525
    unfolding less_float_def real_of_float_neg_exp[OF `e < 0`] real_of_float_1
hoelzl@29804
   526
          real_mult_less_iff1[of _ _ 1, OF `0 < 2^(nat (-e))`, symmetric] 
huffman@36778
   527
          mult_assoc by auto
hoelzl@29804
   528
  thus ?thesis unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   529
qed
hoelzl@29804
   530
hoelzl@29804
   531
function bitlen :: "int \<Rightarrow> int" where
hoelzl@29804
   532
"bitlen 0 = 0" | 
hoelzl@29804
   533
"bitlen -1 = 1" | 
hoelzl@29804
   534
"0 < x \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))" | 
hoelzl@29804
   535
"x < -1 \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))"
hoelzl@29804
   536
  apply (case_tac "x = 0 \<or> x = -1 \<or> x < -1 \<or> x > 0")
hoelzl@29804
   537
  apply auto
hoelzl@29804
   538
  done
hoelzl@29804
   539
termination by (relation "measure (nat o abs)", auto)
hoelzl@29804
   540
hoelzl@29804
   541
lemma bitlen_ge0: "0 \<le> bitlen x" by (induct x rule: bitlen.induct, auto)
hoelzl@29804
   542
lemma bitlen_ge1: "x \<noteq> 0 \<Longrightarrow> 1 \<le> bitlen x" by (induct x rule: bitlen.induct, auto simp add: bitlen_ge0)
hoelzl@29804
   543
hoelzl@29804
   544
lemma bitlen_bounds': assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x + 1 \<le> 2^nat (bitlen x)" (is "?P x")
hoelzl@29804
   545
  using `0 < x`
hoelzl@29804
   546
proof (induct x rule: bitlen.induct)
hoelzl@29804
   547
  fix x
hoelzl@29804
   548
  assume "0 < x" and hyp: "0 < x div 2 \<Longrightarrow> ?P (x div 2)" hence "0 \<le> x" and "x \<noteq> 0" by auto
hoelzl@29804
   549
  { fix x have "0 \<le> 1 + bitlen x" using bitlen_ge0[of x] by auto } note gt0_pls1 = this
hoelzl@29804
   550
hoelzl@29804
   551
  have "0 < (2::int)" by auto
obua@16782
   552
hoelzl@29804
   553
  show "?P x"
hoelzl@29804
   554
  proof (cases "x = 1")
hoelzl@29804
   555
    case True show "?P x" unfolding True by auto
hoelzl@29804
   556
  next
hoelzl@29804
   557
    case False hence "2 \<le> x" using `0 < x` `x \<noteq> 1` by auto
hoelzl@29804
   558
    hence "2 div 2 \<le> x div 2" by (rule zdiv_mono1, auto)
hoelzl@29804
   559
    hence "0 < x div 2" and "x div 2 \<noteq> 0" by auto
hoelzl@29804
   560
    hence bitlen_s1_ge0: "0 \<le> bitlen (x div 2) - 1" using bitlen_ge1[OF `x div 2 \<noteq> 0`] by auto
obua@16782
   561
hoelzl@29804
   562
    { from hyp[OF `0 < x div 2`]
hoelzl@29804
   563
      have "2 ^ nat (bitlen (x div 2) - 1) \<le> x div 2" by auto
hoelzl@29804
   564
      hence "2 ^ nat (bitlen (x div 2) - 1) * 2 \<le> x div 2 * 2" by (rule mult_right_mono, auto)
hoelzl@29804
   565
      also have "\<dots> \<le> x" using `0 < x` by auto
hoelzl@29804
   566
      finally have "2^nat (1 + bitlen (x div 2) - 1) \<le> x" unfolding power_Suc2[symmetric] Suc_nat_eq_nat_zadd1[OF bitlen_s1_ge0] by auto
hoelzl@29804
   567
    } moreover
hoelzl@29804
   568
    { have "x + 1 \<le> x - x mod 2 + 2"
hoelzl@29804
   569
      proof -
wenzelm@32960
   570
        have "x mod 2 < 2" using `0 < x` by auto
wenzelm@32960
   571
        hence "x < x - x mod 2 +  2" unfolding algebra_simps by auto
wenzelm@32960
   572
        thus ?thesis by auto
hoelzl@29804
   573
      qed
hoelzl@29804
   574
      also have "x - x mod 2 + 2 = (x div 2 + 1) * 2" unfolding algebra_simps using `0 < x` zdiv_zmod_equality2[of x 2 0] by auto
hoelzl@29804
   575
      also have "\<dots> \<le> 2^nat (bitlen (x div 2)) * 2" using hyp[OF `0 < x div 2`, THEN conjunct2] by (rule mult_right_mono, auto)
hoelzl@29804
   576
      also have "\<dots> = 2^(1 + nat (bitlen (x div 2)))" unfolding power_Suc2[symmetric] by auto
hoelzl@29804
   577
      finally have "x + 1 \<le> 2^(1 + nat (bitlen (x div 2)))" .
hoelzl@29804
   578
    }
hoelzl@29804
   579
    ultimately show ?thesis
hoelzl@29804
   580
      unfolding bitlen.simps(3)[OF `0 < x`] nat_add_distrib[OF zero_le_one bitlen_ge0]
hoelzl@29804
   581
      unfolding add_commute nat_add_distrib[OF zero_le_one gt0_pls1]
hoelzl@29804
   582
      by auto
hoelzl@29804
   583
  qed
hoelzl@29804
   584
next
hoelzl@29804
   585
  fix x :: int assume "x < -1" and "0 < x" hence False by auto
hoelzl@29804
   586
  thus "?P x" by auto
hoelzl@29804
   587
qed auto
hoelzl@29804
   588
hoelzl@29804
   589
lemma bitlen_bounds: assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x < 2^nat (bitlen x)"
hoelzl@29804
   590
  using bitlen_bounds'[OF `0<x`] by auto
hoelzl@29804
   591
hoelzl@29804
   592
lemma bitlen_div: assumes "0 < m" shows "1 \<le> real m / 2^nat (bitlen m - 1)" and "real m / 2^nat (bitlen m - 1) < 2"
hoelzl@29804
   593
proof -
hoelzl@29804
   594
  let ?B = "2^nat(bitlen m - 1)"
hoelzl@29804
   595
hoelzl@29804
   596
  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   597
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   598
  thus "1 \<le> real m / ?B" by auto
hoelzl@29804
   599
hoelzl@29804
   600
  have "m \<noteq> 0" using assms by auto
hoelzl@29804
   601
  have "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
obua@16782
   602
hoelzl@29804
   603
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   604
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
hoelzl@29804
   605
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
hoelzl@29804
   606
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   607
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
hoelzl@29804
   608
  thus "real m / ?B < 2" by auto
hoelzl@29804
   609
qed
hoelzl@29804
   610
hoelzl@29804
   611
lemma float_gt1_scale: assumes "1 \<le> Float m e"
hoelzl@29804
   612
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@29804
   613
proof (cases "0 \<le> e")
hoelzl@29804
   614
  have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
hoelzl@29804
   615
  hence "0 < m" using float_pos_m_pos by auto
hoelzl@29804
   616
  hence "m \<noteq> 0" by auto
hoelzl@29804
   617
  case True with bitlen_ge1[OF `m \<noteq> 0`] show ?thesis by auto
hoelzl@29804
   618
next
hoelzl@29804
   619
  have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
hoelzl@29804
   620
  hence "0 < m" using float_pos_m_pos by auto
hoelzl@29804
   621
  hence "m \<noteq> 0" and "1 < (2::int)" by auto
hoelzl@29804
   622
  case False let ?S = "2^(nat (-e))"
hoelzl@31098
   623
  have "1 \<le> real m * inverse ?S" using assms unfolding le_float_def real_of_float_nge0_exp[OF False] by auto
hoelzl@29804
   624
  hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hoelzl@29804
   625
  hence "?S \<le> real m" unfolding mult_assoc by auto
hoelzl@29804
   626
  hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   627
  from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
hoelzl@29804
   628
  have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
hoelzl@29804
   629
  hence "-e < bitlen m" using False bitlen_ge0 by auto
hoelzl@29804
   630
  thus ?thesis by auto
hoelzl@29804
   631
qed
hoelzl@29804
   632
hoelzl@31098
   633
lemma normalized_float: assumes "m \<noteq> 0" shows "real (Float m (- (bitlen m - 1))) = real m / 2^nat (bitlen m - 1)"
hoelzl@29804
   634
proof (cases "- (bitlen m - 1) = 0")
hoelzl@31098
   635
  case True show ?thesis unfolding real_of_float_simp pow2_def using True by auto
hoelzl@29804
   636
next
hoelzl@29804
   637
  case False hence P: "\<not> 0 \<le> - (bitlen m - 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
huffman@36778
   638
  show ?thesis unfolding real_of_float_nge0_exp[OF P] divide_inverse by auto
hoelzl@29804
   639
qed
hoelzl@29804
   640
hoelzl@29804
   641
lemma bitlen_Pls: "bitlen (Int.Pls) = Int.Pls" by (subst Pls_def, subst Pls_def, simp)
hoelzl@29804
   642
hoelzl@29804
   643
lemma bitlen_Min: "bitlen (Int.Min) = Int.Bit1 Int.Pls" by (subst Min_def, simp add: Bit1_def) 
hoelzl@29804
   644
hoelzl@29804
   645
lemma bitlen_B0: "bitlen (Int.Bit0 b) = (if iszero b then Int.Pls else Int.succ (bitlen b))"
hoelzl@29804
   646
  apply (auto simp add: iszero_def succ_def)
hoelzl@29804
   647
  apply (simp add: Bit0_def Pls_def)
hoelzl@29804
   648
  apply (subst Bit0_def)
hoelzl@29804
   649
  apply simp
hoelzl@29804
   650
  apply (subgoal_tac "0 < 2 * b \<or> 2 * b < -1")
hoelzl@29804
   651
  apply auto
hoelzl@29804
   652
  done
obua@16782
   653
hoelzl@29804
   654
lemma bitlen_B1: "bitlen (Int.Bit1 b) = (if iszero (Int.succ b) then Int.Bit1 Int.Pls else Int.succ (bitlen b))"
hoelzl@29804
   655
proof -
hoelzl@29804
   656
  have h: "! x. (2*x + 1) div 2 = (x::int)"
hoelzl@29804
   657
    by arith    
hoelzl@29804
   658
  show ?thesis
hoelzl@29804
   659
    apply (auto simp add: iszero_def succ_def)
hoelzl@29804
   660
    apply (subst Bit1_def)+
hoelzl@29804
   661
    apply simp
hoelzl@29804
   662
    apply (subgoal_tac "2 * b + 1 = -1")
hoelzl@29804
   663
    apply (simp only:)
hoelzl@29804
   664
    apply simp_all
hoelzl@29804
   665
    apply (subst Bit1_def)
hoelzl@29804
   666
    apply simp
hoelzl@29804
   667
    apply (subgoal_tac "0 < 2 * b + 1 \<or> 2 * b + 1 < -1")
hoelzl@29804
   668
    apply (auto simp add: h)
hoelzl@29804
   669
    done
hoelzl@29804
   670
qed
hoelzl@29804
   671
hoelzl@29804
   672
lemma bitlen_number_of: "bitlen (number_of w) = number_of (bitlen w)"
hoelzl@29804
   673
  by (simp add: number_of_is_id)
obua@16782
   674
hoelzl@29804
   675
lemma [code]: "bitlen x = 
hoelzl@29804
   676
     (if x = 0  then 0 
hoelzl@29804
   677
 else if x = -1 then 1 
hoelzl@29804
   678
                else (1 + (bitlen (x div 2))))"
hoelzl@29804
   679
  by (cases "x = 0 \<or> x = -1 \<or> 0 < x") auto
hoelzl@29804
   680
hoelzl@29804
   681
definition lapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   682
where
hoelzl@29804
   683
  "lapprox_posrat prec x y = 
hoelzl@29804
   684
   (let 
hoelzl@29804
   685
       l = nat (int prec + bitlen y - bitlen x) ;
hoelzl@29804
   686
       d = (x * 2^l) div y
hoelzl@29804
   687
    in normfloat (Float d (- (int l))))"
hoelzl@29804
   688
hoelzl@29804
   689
lemma pow2_minus: "pow2 (-x) = inverse (pow2 x)"
hoelzl@29804
   690
  unfolding pow2_neg[of "-x"] by auto
hoelzl@29804
   691
hoelzl@29804
   692
lemma lapprox_posrat: 
hoelzl@29804
   693
  assumes x: "0 \<le> x"
hoelzl@29804
   694
  and y: "0 < y"
hoelzl@31098
   695
  shows "real (lapprox_posrat prec x y) \<le> real x / real y"
hoelzl@29804
   696
proof -
hoelzl@29804
   697
  let ?l = "nat (int prec + bitlen y - bitlen x)"
hoelzl@29804
   698
  
hoelzl@29804
   699
  have "real (x * 2^?l div y) * inverse (2^?l) \<le> (real (x * 2^?l) / real y) * inverse (2^?l)" 
hoelzl@29804
   700
    by (rule mult_right_mono, fact real_of_int_div4, simp)
hoelzl@29804
   701
  also have "\<dots> \<le> (real x / real y) * 2^?l * inverse (2^?l)" by auto
huffman@36778
   702
  finally have "real (x * 2^?l div y) * inverse (2^?l) \<le> real x / real y" unfolding mult_assoc by auto
hoelzl@31098
   703
  thus ?thesis unfolding lapprox_posrat_def Let_def normfloat real_of_float_simp
hoelzl@29804
   704
    unfolding pow2_minus pow2_int minus_minus .
hoelzl@29804
   705
qed
obua@16782
   706
hoelzl@29804
   707
lemma real_of_int_div_mult: 
hoelzl@29804
   708
  fixes x y c :: int assumes "0 < y" and "0 < c"
hoelzl@29804
   709
  shows "real (x div y) \<le> real (x * c div y) * inverse (real c)"
hoelzl@29804
   710
proof -
hoelzl@29804
   711
  have "c * (x div y) + 0 \<le> c * x div y" unfolding zdiv_zmult1_eq[of c x y]
hoelzl@29804
   712
    by (rule zadd_left_mono, 
hoelzl@29804
   713
        auto intro!: mult_nonneg_nonneg 
hoelzl@29804
   714
             simp add: pos_imp_zdiv_nonneg_iff[OF `0 < y`] `0 < c`[THEN less_imp_le] pos_mod_sign[OF `0 < y`])
hoelzl@29804
   715
  hence "real (x div y) * real c \<le> real (x * c div y)" 
hoelzl@29804
   716
    unfolding real_of_int_mult[symmetric] real_of_int_le_iff zmult_commute by auto
hoelzl@29804
   717
  hence "real (x div y) * real c * inverse (real c) \<le> real (x * c div y) * inverse (real c)"
hoelzl@29804
   718
    using `0 < c` by auto
huffman@36778
   719
  thus ?thesis unfolding mult_assoc using `0 < c` by auto
hoelzl@29804
   720
qed
hoelzl@29804
   721
hoelzl@29804
   722
lemma lapprox_posrat_bottom: assumes "0 < y"
hoelzl@31098
   723
  shows "real (x div y) \<le> real (lapprox_posrat n x y)" 
hoelzl@29804
   724
proof -
hoelzl@29804
   725
  have pow: "\<And>x. (0::int) < 2^x" by auto
hoelzl@29804
   726
  show ?thesis
hoelzl@31098
   727
    unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
hoelzl@29804
   728
    using real_of_int_div_mult[OF `0 < y` pow] by auto
hoelzl@29804
   729
qed
hoelzl@29804
   730
hoelzl@29804
   731
lemma lapprox_posrat_nonneg: assumes "0 \<le> x" and "0 < y"
hoelzl@31098
   732
  shows "0 \<le> real (lapprox_posrat n x y)" 
hoelzl@29804
   733
proof -
hoelzl@29804
   734
  show ?thesis
hoelzl@31098
   735
    unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
hoelzl@29804
   736
    using pos_imp_zdiv_nonneg_iff[OF `0 < y`] assms by (auto intro!: mult_nonneg_nonneg)
hoelzl@29804
   737
qed
hoelzl@29804
   738
hoelzl@29804
   739
definition rapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   740
where
hoelzl@29804
   741
  "rapprox_posrat prec x y = (let
hoelzl@29804
   742
     l = nat (int prec + bitlen y - bitlen x) ;
hoelzl@29804
   743
     X = x * 2^l ;
hoelzl@29804
   744
     d = X div y ;
hoelzl@29804
   745
     m = X mod y
hoelzl@29804
   746
   in normfloat (Float (d + (if m = 0 then 0 else 1)) (- (int l))))"
obua@16782
   747
hoelzl@29804
   748
lemma rapprox_posrat:
hoelzl@29804
   749
  assumes x: "0 \<le> x"
hoelzl@29804
   750
  and y: "0 < y"
hoelzl@31098
   751
  shows "real x / real y \<le> real (rapprox_posrat prec x y)"
hoelzl@29804
   752
proof -
hoelzl@29804
   753
  let ?l = "nat (int prec + bitlen y - bitlen x)" let ?X = "x * 2^?l"
hoelzl@29804
   754
  show ?thesis 
hoelzl@29804
   755
  proof (cases "?X mod y = 0")
hoelzl@29804
   756
    case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
hoelzl@29804
   757
    from real_of_int_div[OF this]
hoelzl@29804
   758
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
hoelzl@29804
   759
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
hoelzl@29804
   760
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
hoelzl@29804
   761
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True] 
hoelzl@31098
   762
      unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   763
  next
hoelzl@29804
   764
    case False
hoelzl@29804
   765
    have "0 \<le> real y" and "real y \<noteq> 0" using `0 < y` by auto
hoelzl@29804
   766
    have "0 \<le> real y * 2^?l" by (rule mult_nonneg_nonneg, rule `0 \<le> real y`, auto)
obua@16782
   767
hoelzl@29804
   768
    have "?X = y * (?X div y) + ?X mod y" by auto
hoelzl@29804
   769
    also have "\<dots> \<le> y * (?X div y) + y" by (rule add_mono, auto simp add: pos_mod_bound[OF `0 < y`, THEN less_imp_le])
hoelzl@29804
   770
    also have "\<dots> = y * (?X div y + 1)" unfolding zadd_zmult_distrib2 by auto
hoelzl@29804
   771
    finally have "real ?X \<le> real y * real (?X div y + 1)" unfolding real_of_int_le_iff real_of_int_mult[symmetric] .
hoelzl@29804
   772
    hence "real ?X / (real y * 2^?l) \<le> real y * real (?X div y + 1) / (real y * 2^?l)" 
hoelzl@29804
   773
      by (rule divide_right_mono, simp only: `0 \<le> real y * 2^?l`)
hoelzl@29804
   774
    also have "\<dots> = real y * real (?X div y + 1) / real y / 2^?l" by auto
hoelzl@29804
   775
    also have "\<dots> = real (?X div y + 1) * inverse (2^?l)" unfolding nonzero_mult_divide_cancel_left[OF `real y \<noteq> 0`] 
huffman@36778
   776
      unfolding divide_inverse ..
hoelzl@31098
   777
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
hoelzl@29804
   778
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   779
  qed
hoelzl@29804
   780
qed
hoelzl@29804
   781
hoelzl@29804
   782
lemma rapprox_posrat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
hoelzl@31098
   783
  shows "real (rapprox_posrat n x y) \<le> 1"
hoelzl@29804
   784
proof -
hoelzl@29804
   785
  let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
hoelzl@29804
   786
  show ?thesis
hoelzl@29804
   787
  proof (cases "?X mod y = 0")
hoelzl@29804
   788
    case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
hoelzl@29804
   789
    from real_of_int_div[OF this]
hoelzl@29804
   790
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
hoelzl@29804
   791
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
hoelzl@29804
   792
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
hoelzl@29804
   793
    also have "real x / real y \<le> 1" using `0 \<le> x` and `0 < y` and `x \<le> y` by auto
hoelzl@29804
   794
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True]
hoelzl@31098
   795
      unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   796
  next
hoelzl@29804
   797
    case False
hoelzl@29804
   798
    have "x \<noteq> y"
hoelzl@29804
   799
    proof (rule ccontr)
hoelzl@29804
   800
      assume "\<not> x \<noteq> y" hence "x = y" by auto
nipkow@30034
   801
      have "?X mod y = 0" unfolding `x = y` using mod_mult_self1_is_0 by auto
hoelzl@29804
   802
      thus False using False by auto
hoelzl@29804
   803
    qed
hoelzl@29804
   804
    hence "x < y" using `x \<le> y` by auto
hoelzl@29804
   805
    hence "real x / real y < 1" using `0 < y` and `0 \<le> x` by auto
obua@16782
   806
hoelzl@29804
   807
    from real_of_int_div4[of "?X" y]
huffman@35344
   808
    have "real (?X div y) \<le> (real x / real y) * 2^?l" unfolding real_of_int_mult times_divide_eq_left real_of_int_power real_number_of .
hoelzl@29804
   809
    also have "\<dots> < 1 * 2^?l" using `real x / real y < 1` by (rule mult_strict_right_mono, auto)
hoelzl@29804
   810
    finally have "?X div y < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
hoelzl@29804
   811
    hence "?X div y + 1 \<le> 2^?l" by auto
hoelzl@29804
   812
    hence "real (?X div y + 1) * inverse (2^?l) \<le> 2^?l * inverse (2^?l)"
huffman@35344
   813
      unfolding real_of_int_le_iff[of _ "2^?l", symmetric] real_of_int_power real_number_of
hoelzl@29804
   814
      by (rule mult_right_mono, auto)
hoelzl@29804
   815
    hence "real (?X div y + 1) * inverse (2^?l) \<le> 1" by auto
hoelzl@31098
   816
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
hoelzl@29804
   817
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   818
  qed
hoelzl@29804
   819
qed
obua@16782
   820
hoelzl@29804
   821
lemma zdiv_greater_zero: fixes a b :: int assumes "0 < a" and "a \<le> b"
hoelzl@29804
   822
  shows "0 < b div a"
hoelzl@29804
   823
proof (rule ccontr)
hoelzl@29804
   824
  have "0 \<le> b" using assms by auto
hoelzl@29804
   825
  assume "\<not> 0 < b div a" hence "b div a = 0" using `0 \<le> b`[unfolded pos_imp_zdiv_nonneg_iff[OF `0<a`, of b, symmetric]] by auto
hoelzl@29804
   826
  have "b = a * (b div a) + b mod a" by auto
hoelzl@29804
   827
  hence "b = b mod a" unfolding `b div a = 0` by auto
hoelzl@29804
   828
  hence "b < a" using `0 < a`[THEN pos_mod_bound, of b] by auto
hoelzl@29804
   829
  thus False using `a \<le> b` by auto
hoelzl@29804
   830
qed
hoelzl@29804
   831
hoelzl@29804
   832
lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@31098
   833
  shows "real (rapprox_posrat n x y) < 1"
hoelzl@29804
   834
proof (cases "x = 0")
hoelzl@31098
   835
  case True thus ?thesis unfolding rapprox_posrat_def True Let_def normfloat real_of_float_simp by auto
hoelzl@29804
   836
next
hoelzl@29804
   837
  case False hence "0 < x" using `0 \<le> x` by auto
hoelzl@29804
   838
  hence "x < y" using assms by auto
hoelzl@29804
   839
  
hoelzl@29804
   840
  let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
hoelzl@29804
   841
  show ?thesis
hoelzl@29804
   842
  proof (cases "?X mod y = 0")
hoelzl@29804
   843
    case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
hoelzl@29804
   844
    from real_of_int_div[OF this]
hoelzl@29804
   845
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
hoelzl@29804
   846
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
hoelzl@29804
   847
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
hoelzl@29804
   848
    also have "real x / real y < 1" using `0 \<le> x` and `0 < y` and `x < y` by auto
hoelzl@31098
   849
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_P[OF True]
hoelzl@29804
   850
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   851
  next
hoelzl@29804
   852
    case False
hoelzl@29804
   853
    hence "(real x / real y) < 1 / 2" using `0 < y` and `0 \<le> x` `2 * x < y` by auto
obua@16782
   854
hoelzl@29804
   855
    have "0 < ?X div y"
hoelzl@29804
   856
    proof -
hoelzl@29804
   857
      have "2^nat (bitlen x - 1) \<le> y" and "y < 2^nat (bitlen y)"
wenzelm@32960
   858
        using bitlen_bounds[OF `0 < x`, THEN conjunct1] bitlen_bounds[OF `0 < y`, THEN conjunct2] `x < y` by auto
hoelzl@29804
   859
      hence "(2::int)^nat (bitlen x - 1) < 2^nat (bitlen y)" by (rule order_le_less_trans)
hoelzl@29804
   860
      hence "bitlen x \<le> bitlen y" by auto
hoelzl@29804
   861
      hence len_less: "nat (bitlen x - 1) \<le> nat (int (n - 1) + bitlen y)" by auto
hoelzl@29804
   862
hoelzl@29804
   863
      have "x \<noteq> 0" and "y \<noteq> 0" using `0 < x` `0 < y` by auto
hoelzl@29804
   864
hoelzl@29804
   865
      have exp_eq: "nat (int (n - 1) + bitlen y) - nat (bitlen x - 1) = ?l"
wenzelm@32960
   866
        using `bitlen x \<le> bitlen y` bitlen_ge1[OF `x \<noteq> 0`] bitlen_ge1[OF `y \<noteq> 0`] `0 < n` by auto
hoelzl@29804
   867
hoelzl@29804
   868
      have "y * 2^nat (bitlen x - 1) \<le> y * x" 
wenzelm@32960
   869
        using bitlen_bounds[OF `0 < x`, THEN conjunct1] `0 < y`[THEN less_imp_le] by (rule mult_left_mono)
hoelzl@29804
   870
      also have "\<dots> \<le> 2^nat (bitlen y) * x" using bitlen_bounds[OF `0 < y`, THEN conjunct2, THEN less_imp_le] `0 \<le> x` by (rule mult_right_mono)
hoelzl@29804
   871
      also have "\<dots> \<le> x * 2^nat (int (n - 1) + bitlen y)" unfolding mult_commute[of x] by (rule mult_right_mono, auto simp add: `0 \<le> x`)
hoelzl@29804
   872
      finally have "real y * 2^nat (bitlen x - 1) * inverse (2^nat (bitlen x - 1)) \<le> real x * 2^nat (int (n - 1) + bitlen y) * inverse (2^nat (bitlen x - 1))"
wenzelm@32960
   873
        unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   874
      hence "real y \<le> real x * (2^nat (int (n - 1) + bitlen y) / (2^nat (bitlen x - 1)))" 
huffman@36778
   875
        unfolding mult_assoc divide_inverse by auto
hoelzl@29804
   876
      also have "\<dots> = real x * (2^(nat (int (n - 1) + bitlen y) - nat (bitlen x - 1)))" using power_diff[of "2::real", OF _ len_less] by auto
hoelzl@29804
   877
      finally have "y \<le> x * 2^?l" unfolding exp_eq unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   878
      thus ?thesis using zdiv_greater_zero[OF `0 < y`] by auto
hoelzl@29804
   879
    qed
hoelzl@29804
   880
hoelzl@29804
   881
    from real_of_int_div4[of "?X" y]
huffman@35344
   882
    have "real (?X div y) \<le> (real x / real y) * 2^?l" unfolding real_of_int_mult times_divide_eq_left real_of_int_power real_number_of .
hoelzl@29804
   883
    also have "\<dots> < 1/2 * 2^?l" using `real x / real y < 1/2` by (rule mult_strict_right_mono, auto)
hoelzl@29804
   884
    finally have "?X div y * 2 < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
hoelzl@29804
   885
    hence "?X div y + 1 < 2^?l" using `0 < ?X div y` by auto
hoelzl@29804
   886
    hence "real (?X div y + 1) * inverse (2^?l) < 2^?l * inverse (2^?l)"
huffman@35344
   887
      unfolding real_of_int_less_iff[of _ "2^?l", symmetric] real_of_int_power real_number_of
hoelzl@29804
   888
      by (rule mult_strict_right_mono, auto)
hoelzl@29804
   889
    hence "real (?X div y + 1) * inverse (2^?l) < 1" by auto
hoelzl@31098
   890
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
hoelzl@29804
   891
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   892
  qed
hoelzl@29804
   893
qed
hoelzl@29804
   894
hoelzl@29804
   895
lemma approx_rat_pattern: fixes P and ps :: "nat * int * int"
hoelzl@29804
   896
  assumes Y: "\<And>y prec x. \<lbrakk>y = 0; ps = (prec, x, 0)\<rbrakk> \<Longrightarrow> P" 
hoelzl@29804
   897
  and A: "\<And>x y prec. \<lbrakk>0 \<le> x; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   898
  and B: "\<And>x y prec. \<lbrakk>x < 0; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   899
  and C: "\<And>x y prec. \<lbrakk>x < 0; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   900
  and D: "\<And>x y prec. \<lbrakk>0 \<le> x; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   901
  shows P
obua@16782
   902
proof -
hoelzl@29804
   903
  obtain prec x y where [simp]: "ps = (prec, x, y)" by (cases ps, auto)
hoelzl@29804
   904
  from Y have "y = 0 \<Longrightarrow> P" by auto
hoelzl@29804
   905
  moreover { assume "0 < y" have P proof (cases "0 \<le> x") case True with A and `0 < y` show P by auto next case False with B and `0 < y` show P by auto qed } 
hoelzl@29804
   906
  moreover { assume "y < 0" have P proof (cases "0 \<le> x") case True with D and `y < 0` show P by auto next case False with C and `y < 0` show P by auto qed }
hoelzl@29804
   907
  ultimately show P by (cases "y = 0 \<or> 0 < y \<or> y < 0", auto)
obua@16782
   908
qed
obua@16782
   909
hoelzl@29804
   910
function lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   911
where
hoelzl@29804
   912
  "y = 0 \<Longrightarrow> lapprox_rat prec x y = 0"
hoelzl@29804
   913
| "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec x y"
hoelzl@29804
   914
| "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec (-x) y)"
hoelzl@29804
   915
| "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec (-x) (-y)"
hoelzl@29804
   916
| "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec x (-y))"
hoelzl@29804
   917
apply simp_all by (rule approx_rat_pattern)
hoelzl@29804
   918
termination by lexicographic_order
obua@16782
   919
hoelzl@29804
   920
lemma compute_lapprox_rat[code]:
hoelzl@29804
   921
      "lapprox_rat prec x y = (if y = 0 then 0 else if 0 \<le> x then (if 0 < y then lapprox_posrat prec x y else - (rapprox_posrat prec x (-y))) 
hoelzl@29804
   922
                                                             else (if 0 < y then - (rapprox_posrat prec (-x) y) else lapprox_posrat prec (-x) (-y)))"
hoelzl@29804
   923
  by auto
hoelzl@29804
   924
            
hoelzl@31098
   925
lemma lapprox_rat: "real (lapprox_rat prec x y) \<le> real x / real y"
hoelzl@29804
   926
proof -      
hoelzl@29804
   927
  have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
hoelzl@29804
   928
  show ?thesis
hoelzl@29804
   929
    apply (case_tac "y = 0")
hoelzl@29804
   930
    apply simp
hoelzl@29804
   931
    apply (case_tac "0 \<le> x \<and> 0 < y")
hoelzl@29804
   932
    apply (simp add: lapprox_posrat)
hoelzl@29804
   933
    apply (case_tac "x < 0 \<and> 0 < y")
hoelzl@29804
   934
    apply simp
hoelzl@29804
   935
    apply (subst minus_le_iff)   
hoelzl@29804
   936
    apply (rule h[OF rapprox_posrat])
hoelzl@29804
   937
    apply (simp_all)
hoelzl@29804
   938
    apply (case_tac "x < 0 \<and> y < 0")
hoelzl@29804
   939
    apply simp
hoelzl@29804
   940
    apply (rule h[OF _ lapprox_posrat])
hoelzl@29804
   941
    apply (simp_all)
hoelzl@29804
   942
    apply (case_tac "0 \<le> x \<and> y < 0")
hoelzl@29804
   943
    apply (simp)
hoelzl@29804
   944
    apply (subst minus_le_iff)   
hoelzl@29804
   945
    apply (rule h[OF rapprox_posrat])
hoelzl@29804
   946
    apply simp_all
hoelzl@29804
   947
    apply arith
hoelzl@29804
   948
    done
hoelzl@29804
   949
qed
obua@16782
   950
hoelzl@29804
   951
lemma lapprox_rat_bottom: assumes "0 \<le> x" and "0 < y"
hoelzl@31098
   952
  shows "real (x div y) \<le> real (lapprox_rat n x y)" 
hoelzl@29804
   953
  unfolding lapprox_rat.simps(2)[OF assms]  using lapprox_posrat_bottom[OF `0<y`] .
hoelzl@29804
   954
hoelzl@29804
   955
function rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   956
where
hoelzl@29804
   957
  "y = 0 \<Longrightarrow> rapprox_rat prec x y = 0"
hoelzl@29804
   958
| "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec x y"
hoelzl@29804
   959
| "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec (-x) y)"
hoelzl@29804
   960
| "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec (-x) (-y)"
hoelzl@29804
   961
| "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec x (-y))"
hoelzl@29804
   962
apply simp_all by (rule approx_rat_pattern)
hoelzl@29804
   963
termination by lexicographic_order
obua@16782
   964
hoelzl@29804
   965
lemma compute_rapprox_rat[code]:
hoelzl@29804
   966
      "rapprox_rat prec x y = (if y = 0 then 0 else if 0 \<le> x then (if 0 < y then rapprox_posrat prec x y else - (lapprox_posrat prec x (-y))) else 
hoelzl@29804
   967
                                                                  (if 0 < y then - (lapprox_posrat prec (-x) y) else rapprox_posrat prec (-x) (-y)))"
hoelzl@29804
   968
  by auto
obua@16782
   969
hoelzl@31098
   970
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
hoelzl@29804
   971
proof -      
hoelzl@29804
   972
  have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
hoelzl@29804
   973
  show ?thesis
hoelzl@29804
   974
    apply (case_tac "y = 0")
hoelzl@29804
   975
    apply simp
hoelzl@29804
   976
    apply (case_tac "0 \<le> x \<and> 0 < y")
hoelzl@29804
   977
    apply (simp add: rapprox_posrat)
hoelzl@29804
   978
    apply (case_tac "x < 0 \<and> 0 < y")
hoelzl@29804
   979
    apply simp
hoelzl@29804
   980
    apply (subst le_minus_iff)   
hoelzl@29804
   981
    apply (rule h[OF _ lapprox_posrat])
hoelzl@29804
   982
    apply (simp_all)
hoelzl@29804
   983
    apply (case_tac "x < 0 \<and> y < 0")
hoelzl@29804
   984
    apply simp
hoelzl@29804
   985
    apply (rule h[OF rapprox_posrat])
hoelzl@29804
   986
    apply (simp_all)
hoelzl@29804
   987
    apply (case_tac "0 \<le> x \<and> y < 0")
hoelzl@29804
   988
    apply (simp)
hoelzl@29804
   989
    apply (subst le_minus_iff)   
hoelzl@29804
   990
    apply (rule h[OF _ lapprox_posrat])
hoelzl@29804
   991
    apply simp_all
hoelzl@29804
   992
    apply arith
hoelzl@29804
   993
    done
hoelzl@29804
   994
qed
obua@16782
   995
hoelzl@29804
   996
lemma rapprox_rat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
hoelzl@31098
   997
  shows "real (rapprox_rat n x y) \<le> 1"
hoelzl@29804
   998
  unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`] using rapprox_posrat_le1[OF assms] .
hoelzl@29804
   999
hoelzl@29804
  1000
lemma rapprox_rat_neg: assumes "x < 0" and "0 < y"
hoelzl@31098
  1001
  shows "real (rapprox_rat n x y) \<le> 0"
hoelzl@29804
  1002
  unfolding rapprox_rat.simps(3)[OF assms] using lapprox_posrat_nonneg[of "-x" y n] assms by auto
hoelzl@29804
  1003
hoelzl@29804
  1004
lemma rapprox_rat_nonneg_neg: assumes "0 \<le> x" and "y < 0"
hoelzl@31098
  1005
  shows "real (rapprox_rat n x y) \<le> 0"
hoelzl@29804
  1006
  unfolding rapprox_rat.simps(5)[OF assms] using lapprox_posrat_nonneg[of x "-y" n] assms by auto
obua@16782
  1007
hoelzl@29804
  1008
lemma rapprox_rat_nonpos_pos: assumes "x \<le> 0" and "0 < y"
hoelzl@31098
  1009
  shows "real (rapprox_rat n x y) \<le> 0"
hoelzl@29804
  1010
proof (cases "x = 0") 
hoelzl@29804
  1011
  case True hence "0 \<le> x" by auto show ?thesis unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`]
hoelzl@29804
  1012
    unfolding True rapprox_posrat_def Let_def by auto
hoelzl@29804
  1013
next
hoelzl@29804
  1014
  case False hence "x < 0" using assms by auto
hoelzl@29804
  1015
  show ?thesis using rapprox_rat_neg[OF `x < 0` `0 < y`] .
hoelzl@29804
  1016
qed
hoelzl@29804
  1017
hoelzl@29804
  1018
fun float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
hoelzl@29804
  1019
where
hoelzl@29804
  1020
  "float_divl prec (Float m1 s1) (Float m2 s2) = 
hoelzl@29804
  1021
    (let
hoelzl@29804
  1022
       l = lapprox_rat prec m1 m2;
hoelzl@29804
  1023
       f = Float 1 (s1 - s2)
hoelzl@29804
  1024
     in
hoelzl@29804
  1025
       f * l)"     
obua@16782
  1026
hoelzl@31098
  1027
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
hoelzl@29804
  1028
proof - 
hoelzl@29804
  1029
  from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto
hoelzl@29804
  1030
  from float_split[of y] obtain my sy where y: "y = Float my sy" by auto
hoelzl@29804
  1031
  have "real mx / real my \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
hoelzl@29804
  1032
    apply (case_tac "my = 0")
hoelzl@29804
  1033
    apply simp
hoelzl@29804
  1034
    apply (case_tac "my > 0")       
hoelzl@29804
  1035
    apply (subst pos_le_divide_eq)
hoelzl@29804
  1036
    apply simp
hoelzl@29804
  1037
    apply (subst pos_le_divide_eq)
hoelzl@29804
  1038
    apply (simp add: mult_pos_pos)
hoelzl@29804
  1039
    apply simp
hoelzl@29804
  1040
    apply (subst pow2_add[symmetric])
hoelzl@29804
  1041
    apply simp
hoelzl@29804
  1042
    apply (subgoal_tac "my < 0")
hoelzl@29804
  1043
    apply auto
hoelzl@29804
  1044
    apply (simp add: field_simps)
hoelzl@29804
  1045
    apply (subst pow2_add[symmetric])
hoelzl@29804
  1046
    apply (simp add: field_simps)
hoelzl@29804
  1047
    done
hoelzl@31098
  1048
  then have "real (lapprox_rat prec mx my) \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
hoelzl@29804
  1049
    by (rule order_trans[OF lapprox_rat])
hoelzl@31098
  1050
  then have "real (lapprox_rat prec mx my) * pow2 (sx - sy) \<le> real mx * pow2 sx / (real my * pow2 sy)"
hoelzl@29804
  1051
    apply (subst pos_le_divide_eq[symmetric])
hoelzl@29804
  1052
    apply simp_all
hoelzl@29804
  1053
    done
hoelzl@31098
  1054
  then have "pow2 (sx - sy) * real (lapprox_rat prec mx my) \<le> real mx * pow2 sx / (real my * pow2 sy)"
hoelzl@29804
  1055
    by (simp add: algebra_simps)
hoelzl@29804
  1056
  then show ?thesis
hoelzl@31098
  1057
    by (simp add: x y Let_def real_of_float_simp)
hoelzl@29804
  1058
qed
obua@16782
  1059
hoelzl@29804
  1060
lemma float_divl_lower_bound: assumes "0 \<le> x" and "0 < y" shows "0 \<le> float_divl prec x y"
hoelzl@29804
  1061
proof (cases x, cases y)
hoelzl@29804
  1062
  fix xm xe ym ye :: int
hoelzl@29804
  1063
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
hoelzl@31098
  1064
  have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff] by auto
hoelzl@31098
  1065
  have "0 < ym" using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff] by auto
obua@16782
  1066
hoelzl@31098
  1067
  have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
hoelzl@31098
  1068
  moreover have "0 \<le> real (lapprox_rat prec xm ym)" by (rule order_trans[OF _ lapprox_rat_bottom[OF `0 \<le> xm` `0 < ym`]], auto simp add: `0 \<le> xm` pos_imp_zdiv_nonneg_iff[OF `0 < ym`])
hoelzl@29804
  1069
  ultimately show "0 \<le> float_divl prec x y"
hoelzl@31098
  1070
    unfolding x_eq y_eq float_divl.simps Let_def le_float_def real_of_float_0 by (auto intro!: mult_nonneg_nonneg)
hoelzl@29804
  1071
qed
hoelzl@29804
  1072
hoelzl@29804
  1073
lemma float_divl_pos_less1_bound: assumes "0 < x" and "x < 1" and "0 < prec" shows "1 \<le> float_divl prec 1 x"
hoelzl@29804
  1074
proof (cases x)
hoelzl@29804
  1075
  case (Float m e)
hoelzl@29804
  1076
  from `0 < x` `x < 1` have "0 < m" "e < 0" using float_pos_m_pos float_pos_less1_e_neg unfolding Float by auto
hoelzl@29804
  1077
  let ?b = "nat (bitlen m)" and ?e = "nat (-e)"
hoelzl@29804
  1078
  have "1 \<le> m" and "m \<noteq> 0" using `0 < m` by auto
hoelzl@29804
  1079
  with bitlen_bounds[OF `0 < m`] have "m < 2^?b" and "(2::int) \<le> 2^?b" by auto
hoelzl@29804
  1080
  hence "1 \<le> bitlen m" using power_le_imp_le_exp[of "2::int" 1 ?b] by auto
hoelzl@29804
  1081
  hence pow_split: "nat (int prec + bitlen m - 1) = (prec - 1) + ?b" using `0 < prec` by auto
hoelzl@29804
  1082
  
hoelzl@29804
  1083
  have pow_not0: "\<And>x. (2::real)^x \<noteq> 0" by auto
obua@16782
  1084
hoelzl@29804
  1085
  from float_less1_mantissa_bound `0 < x` `x < 1` Float 
hoelzl@29804
  1086
  have "m < 2^?e" by auto
hoelzl@29804
  1087
  with bitlen_bounds[OF `0 < m`, THEN conjunct1]
hoelzl@29804
  1088
  have "(2::int)^nat (bitlen m - 1) < 2^?e" by (rule order_le_less_trans)
hoelzl@29804
  1089
  from power_less_imp_less_exp[OF _ this]
hoelzl@29804
  1090
  have "bitlen m \<le> - e" by auto
hoelzl@29804
  1091
  hence "(2::real)^?b \<le> 2^?e" by auto
hoelzl@29804
  1092
  hence "(2::real)^?b * inverse (2^?b) \<le> 2^?e * inverse (2^?b)" by (rule mult_right_mono, auto)
hoelzl@29804
  1093
  hence "(1::real) \<le> 2^?e * inverse (2^?b)" by auto
hoelzl@29804
  1094
  also
hoelzl@29804
  1095
  let ?d = "real (2 ^ nat (int prec + bitlen m - 1) div m) * inverse (2 ^ nat (int prec + bitlen m - 1))"
hoelzl@29804
  1096
  { have "2^(prec - 1) * m \<le> 2^(prec - 1) * 2^?b" using `m < 2^?b`[THEN less_imp_le] by (rule mult_left_mono, auto)
hoelzl@29804
  1097
    also have "\<dots> = 2 ^ nat (int prec + bitlen m - 1)" unfolding pow_split zpower_zadd_distrib by auto
hoelzl@29804
  1098
    finally have "2^(prec - 1) * m div m \<le> 2 ^ nat (int prec + bitlen m - 1) div m" using `0 < m` by (rule zdiv_mono1)
nipkow@30181
  1099
    hence "2^(prec - 1) \<le> 2 ^ nat (int prec + bitlen m - 1) div m" unfolding div_mult_self2_is_id[OF `m \<noteq> 0`] .
hoelzl@29804
  1100
    hence "2^(prec - 1) * inverse (2 ^ nat (int prec + bitlen m - 1)) \<le> ?d"
hoelzl@29804
  1101
      unfolding real_of_int_le_iff[of "2^(prec - 1)", symmetric] by auto }
huffman@36778
  1102
  from mult_left_mono[OF this[unfolded pow_split power_add inverse_mult_distrib mult_assoc[symmetric] right_inverse[OF pow_not0] mult_1_left], of "2^?e"]
hoelzl@29804
  1103
  have "2^?e * inverse (2^?b) \<le> 2^?e * ?d" unfolding pow_split power_add by auto
hoelzl@29804
  1104
  finally have "1 \<le> 2^?e * ?d" .
hoelzl@29804
  1105
  
hoelzl@29804
  1106
  have e_nat: "0 - e = int (nat (-e))" using `e < 0` by auto
hoelzl@29804
  1107
  have "bitlen 1 = 1" using bitlen.simps by auto
hoelzl@29804
  1108
  
hoelzl@29804
  1109
  show ?thesis 
hoelzl@29804
  1110
    unfolding one_float_def Float float_divl.simps Let_def lapprox_rat.simps(2)[OF zero_le_one `0 < m`] lapprox_posrat_def `bitlen 1 = 1`
hoelzl@31098
  1111
    unfolding le_float_def real_of_float_mult normfloat real_of_float_simp pow2_minus pow2_int e_nat
hoelzl@29804
  1112
    using `1 \<le> 2^?e * ?d` by (auto simp add: pow2_def)
hoelzl@29804
  1113
qed
obua@16782
  1114
hoelzl@29804
  1115
fun float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
hoelzl@29804
  1116
where
hoelzl@29804
  1117
  "float_divr prec (Float m1 s1) (Float m2 s2) = 
hoelzl@29804
  1118
    (let
hoelzl@29804
  1119
       r = rapprox_rat prec m1 m2;
hoelzl@29804
  1120
       f = Float 1 (s1 - s2)
hoelzl@29804
  1121
     in
hoelzl@29804
  1122
       f * r)"  
obua@16782
  1123
hoelzl@31098
  1124
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
hoelzl@29804
  1125
proof - 
hoelzl@29804
  1126
  from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto
hoelzl@29804
  1127
  from float_split[of y] obtain my sy where y: "y = Float my sy" by auto
hoelzl@29804
  1128
  have "real mx / real my \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
hoelzl@29804
  1129
    apply (case_tac "my = 0")
hoelzl@29804
  1130
    apply simp
hoelzl@29804
  1131
    apply (case_tac "my > 0")
hoelzl@29804
  1132
    apply auto
hoelzl@29804
  1133
    apply (subst pos_divide_le_eq)
hoelzl@29804
  1134
    apply (rule mult_pos_pos)+
hoelzl@29804
  1135
    apply simp_all
hoelzl@29804
  1136
    apply (subst pow2_add[symmetric])
hoelzl@29804
  1137
    apply simp
hoelzl@29804
  1138
    apply (subgoal_tac "my < 0")
hoelzl@29804
  1139
    apply auto
hoelzl@29804
  1140
    apply (simp add: field_simps)
hoelzl@29804
  1141
    apply (subst pow2_add[symmetric])
hoelzl@29804
  1142
    apply (simp add: field_simps)
hoelzl@29804
  1143
    done
hoelzl@31098
  1144
  then have "real (rapprox_rat prec mx my) \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
hoelzl@29804
  1145
    by (rule order_trans[OF _ rapprox_rat])
hoelzl@31098
  1146
  then have "real (rapprox_rat prec mx my) * pow2 (sx - sy) \<ge> real mx * pow2 sx / (real my * pow2 sy)"
hoelzl@29804
  1147
    apply (subst pos_divide_le_eq[symmetric])
hoelzl@29804
  1148
    apply simp_all
hoelzl@29804
  1149
    done
hoelzl@29804
  1150
  then show ?thesis
hoelzl@31098
  1151
    by (simp add: x y Let_def algebra_simps real_of_float_simp)
hoelzl@29804
  1152
qed
obua@16782
  1153
hoelzl@29804
  1154
lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
hoelzl@29804
  1155
proof -
hoelzl@31098
  1156
  have "1 \<le> 1 / real x" using `0 < x` and `x < 1` unfolding less_float_def by auto
hoelzl@31098
  1157
  also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
hoelzl@29804
  1158
  finally show ?thesis unfolding le_float_def by auto
hoelzl@29804
  1159
qed
hoelzl@29804
  1160
hoelzl@29804
  1161
lemma float_divr_nonpos_pos_upper_bound: assumes "x \<le> 0" and "0 < y" shows "float_divr prec x y \<le> 0"
hoelzl@29804
  1162
proof (cases x, cases y)
hoelzl@29804
  1163
  fix xm xe ym ye :: int
hoelzl@29804
  1164
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
hoelzl@31098
  1165
  have "xm \<le> 0" using `x \<le> 0`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 mult_le_0_iff] by auto
hoelzl@31098
  1166
  have "0 < ym" using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff] by auto
hoelzl@29804
  1167
hoelzl@31098
  1168
  have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
hoelzl@31098
  1169
  moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonpos_pos[OF `xm \<le> 0` `0 < ym`] .
hoelzl@29804
  1170
  ultimately show "float_divr prec x y \<le> 0"
hoelzl@31098
  1171
    unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)
hoelzl@29804
  1172
qed
obua@16782
  1173
hoelzl@29804
  1174
lemma float_divr_nonneg_neg_upper_bound: assumes "0 \<le> x" and "y < 0" shows "float_divr prec x y \<le> 0"
hoelzl@29804
  1175
proof (cases x, cases y)
hoelzl@29804
  1176
  fix xm xe ym ye :: int
hoelzl@29804
  1177
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
hoelzl@31098
  1178
  have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff] by auto
hoelzl@31098
  1179
  have "ym < 0" using `y < 0`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 mult_less_0_iff] by auto
hoelzl@29804
  1180
  hence "0 < - ym" by auto
hoelzl@29804
  1181
hoelzl@31098
  1182
  have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
hoelzl@31098
  1183
  moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonneg_neg[OF `0 \<le> xm` `ym < 0`] .
hoelzl@29804
  1184
  ultimately show "float_divr prec x y \<le> 0"
hoelzl@31098
  1185
    unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)
hoelzl@29804
  1186
qed
hoelzl@29804
  1187
haftmann@30960
  1188
primrec round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1189
"round_down prec (Float m e) = (let d = bitlen m - int prec in
hoelzl@29804
  1190
     if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
hoelzl@29804
  1191
              else Float m e)"
hoelzl@29804
  1192
haftmann@30960
  1193
primrec round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1194
"round_up prec (Float m e) = (let d = bitlen m - int prec in
hoelzl@29804
  1195
  if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P in Float (n + (if r = 0 then 0 else 1)) (e + d) 
hoelzl@29804
  1196
           else Float m e)"
obua@16782
  1197
hoelzl@31098
  1198
lemma round_up: "real x \<le> real (round_up prec x)"
hoelzl@29804
  1199
proof (cases x)
hoelzl@29804
  1200
  case (Float m e)
hoelzl@29804
  1201
  let ?d = "bitlen m - int prec"
hoelzl@29804
  1202
  let ?p = "(2::int)^nat ?d"
hoelzl@29804
  1203
  have "0 < ?p" by auto
hoelzl@29804
  1204
  show "?thesis"
hoelzl@29804
  1205
  proof (cases "0 < ?d")
hoelzl@29804
  1206
    case True
huffman@35344
  1207
    hence pow_d: "pow2 ?d = real ?p" using pow2_int[symmetric] by simp
hoelzl@29804
  1208
    show ?thesis
hoelzl@29804
  1209
    proof (cases "m mod ?p = 0")
hoelzl@29804
  1210
      case True
hoelzl@29804
  1211
      have m: "m = m div ?p * ?p + 0" unfolding True[symmetric] using zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right, symmetric] .
hoelzl@31098
  1212
      have "real (Float m e) = real (Float (m div ?p) (e + ?d))" unfolding real_of_float_simp arg_cong[OF m, of real]
wenzelm@32960
  1213
        by (auto simp add: pow2_add `0 < ?d` pow_d)
hoelzl@29804
  1214
      thus ?thesis
wenzelm@32960
  1215
        unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]
wenzelm@32960
  1216
        by auto
hoelzl@29804
  1217
    next
hoelzl@29804
  1218
      case False
hoelzl@29804
  1219
      have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
hoelzl@29804
  1220
      also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib zmult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
hoelzl@31098
  1221
      finally have "real (Float m e) \<le> real (Float (m div ?p + 1) (e + ?d))" unfolding real_of_float_simp add_commute[of e]
wenzelm@32960
  1222
        unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
wenzelm@32960
  1223
        by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
hoelzl@29804
  1224
      thus ?thesis
wenzelm@32960
  1225
        unfolding Float round_up.simps Let_def if_not_P[OF `\<not> m mod ?p = 0`] if_P[OF `0 < ?d`] .
hoelzl@29804
  1226
    qed
hoelzl@29804
  1227
  next
hoelzl@29804
  1228
    case False
hoelzl@29804
  1229
    show ?thesis
hoelzl@29804
  1230
      unfolding Float round_up.simps Let_def if_not_P[OF False] .. 
hoelzl@29804
  1231
  qed
hoelzl@29804
  1232
qed
obua@16782
  1233
hoelzl@31098
  1234
lemma round_down: "real (round_down prec x) \<le> real x"
hoelzl@29804
  1235
proof (cases x)
hoelzl@29804
  1236
  case (Float m e)
hoelzl@29804
  1237
  let ?d = "bitlen m - int prec"
hoelzl@29804
  1238
  let ?p = "(2::int)^nat ?d"
hoelzl@29804
  1239
  have "0 < ?p" by auto
hoelzl@29804
  1240
  show "?thesis"
hoelzl@29804
  1241
  proof (cases "0 < ?d")
hoelzl@29804
  1242
    case True
huffman@35344
  1243
    hence pow_d: "pow2 ?d = real ?p" using pow2_int[symmetric] by simp
hoelzl@29804
  1244
    have "m div ?p * ?p \<le> m div ?p * ?p + m mod ?p" by (auto simp add: pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
hoelzl@29804
  1245
    also have "\<dots> \<le> m" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
hoelzl@31098
  1246
    finally have "real (Float (m div ?p) (e + ?d)) \<le> real (Float m e)" unfolding real_of_float_simp add_commute[of e]
hoelzl@29804
  1247
      unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of _ m, symmetric]
hoelzl@29804
  1248
      by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
hoelzl@29804
  1249
    thus ?thesis
hoelzl@29804
  1250
      unfolding Float round_down.simps Let_def if_P[OF `0 < ?d`] .
hoelzl@29804
  1251
  next
hoelzl@29804
  1252
    case False
hoelzl@29804
  1253
    show ?thesis
hoelzl@29804
  1254
      unfolding Float round_down.simps Let_def if_not_P[OF False] .. 
hoelzl@29804
  1255
  qed
hoelzl@29804
  1256
qed
hoelzl@29804
  1257
hoelzl@29804
  1258
definition lb_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1259
"lb_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let
hoelzl@29804
  1260
    l = bitlen m - int prec
hoelzl@29804
  1261
  in if l > 0 then Float (m div (2^nat l)) (e + l)
hoelzl@29804
  1262
              else Float m e)"
obua@16782
  1263
hoelzl@29804
  1264
definition ub_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1265
"ub_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let
hoelzl@29804
  1266
    l = bitlen m - int prec
hoelzl@29804
  1267
  in if l > 0 then Float (m div (2^nat l) + 1) (e + l)
hoelzl@29804
  1268
              else Float m e)"
obua@16782
  1269
hoelzl@31098
  1270
lemma lb_mult: "real (lb_mult prec x y) \<le> real (x * y)"
hoelzl@29804
  1271
proof (cases "normfloat (x * y)")
hoelzl@29804
  1272
  case (Float m e)
hoelzl@29804
  1273
  hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
hoelzl@29804
  1274
  let ?l = "bitlen m - int prec"
hoelzl@31098
  1275
  have "real (lb_mult prec x y) \<le> real (normfloat (x * y))"
hoelzl@29804
  1276
  proof (cases "?l > 0")
hoelzl@29804
  1277
    case False thus ?thesis unfolding lb_mult_def Float Let_def float.cases by auto
hoelzl@29804
  1278
  next
hoelzl@29804
  1279
    case True
hoelzl@29804
  1280
    have "real (m div 2^(nat ?l)) * pow2 ?l \<le> real m"
hoelzl@29804
  1281
    proof -
huffman@35344
  1282
      have "real (m div 2^(nat ?l)) * pow2 ?l = real (2^(nat ?l) * (m div 2^(nat ?l)))" unfolding real_of_int_mult real_of_int_power real_number_of unfolding pow2_int[symmetric] 
wenzelm@32960
  1283
        using `?l > 0` by auto
hoelzl@29804
  1284
      also have "\<dots> \<le> real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding real_of_int_add by auto
hoelzl@29804
  1285
      also have "\<dots> = real m" unfolding zmod_zdiv_equality[symmetric] ..
hoelzl@29804
  1286
      finally show ?thesis by auto
hoelzl@29804
  1287
    qed
huffman@36778
  1288
    thus ?thesis unfolding lb_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add mult_commute mult_assoc by auto
hoelzl@29804
  1289
  qed
hoelzl@31098
  1290
  also have "\<dots> = real (x * y)" unfolding normfloat ..
hoelzl@29804
  1291
  finally show ?thesis .
hoelzl@29804
  1292
qed
obua@16782
  1293
hoelzl@31098
  1294
lemma ub_mult: "real (x * y) \<le> real (ub_mult prec x y)"
hoelzl@29804
  1295
proof (cases "normfloat (x * y)")
hoelzl@29804
  1296
  case (Float m e)
hoelzl@29804
  1297
  hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
hoelzl@29804
  1298
  let ?l = "bitlen m - int prec"
hoelzl@31098
  1299
  have "real (x * y) = real (normfloat (x * y))" unfolding normfloat ..
hoelzl@31098
  1300
  also have "\<dots> \<le> real (ub_mult prec x y)"
hoelzl@29804
  1301
  proof (cases "?l > 0")
hoelzl@29804
  1302
    case False thus ?thesis unfolding ub_mult_def Float Let_def float.cases by auto
hoelzl@29804
  1303
  next
hoelzl@29804
  1304
    case True
hoelzl@29804
  1305
    have "real m \<le> real (m div 2^(nat ?l) + 1) * pow2 ?l"
hoelzl@29804
  1306
    proof -
hoelzl@29804
  1307
      have "m mod 2^(nat ?l) < 2^(nat ?l)" by (rule pos_mod_bound) auto
hoelzl@29804
  1308
      hence mod_uneq: "real (m mod 2^(nat ?l)) \<le> 1 * 2^(nat ?l)" unfolding zmult_1 real_of_int_less_iff[symmetric] by auto
hoelzl@29804
  1309
      
hoelzl@29804
  1310
      have "real m = real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding zmod_zdiv_equality[symmetric] ..
hoelzl@29804
  1311
      also have "\<dots> = real (m div 2^(nat ?l)) * 2^(nat ?l) + real (m mod 2^(nat ?l))" unfolding real_of_int_add by auto
huffman@36778
  1312
      also have "\<dots> \<le> (real (m div 2^(nat ?l)) + 1) * 2^(nat ?l)" unfolding left_distrib using mod_uneq by auto
hoelzl@29804
  1313
      finally show ?thesis unfolding pow2_int[symmetric] using True by auto
hoelzl@29804
  1314
    qed
huffman@36778
  1315
    thus ?thesis unfolding ub_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add mult_commute mult_assoc by auto
hoelzl@29804
  1316
  qed
hoelzl@29804
  1317
  finally show ?thesis .
hoelzl@29804
  1318
qed
hoelzl@29804
  1319
haftmann@30960
  1320
primrec float_abs :: "float \<Rightarrow> float" where
haftmann@30960
  1321
  "float_abs (Float m e) = Float \<bar>m\<bar> e"
hoelzl@29804
  1322
hoelzl@29804
  1323
instantiation float :: abs begin
hoelzl@29804
  1324
definition abs_float_def: "\<bar>x\<bar> = float_abs x"
hoelzl@29804
  1325
instance ..
hoelzl@29804
  1326
end
obua@16782
  1327
hoelzl@31098
  1328
lemma real_of_float_abs: "real \<bar>x :: float\<bar> = \<bar>real x\<bar>" 
hoelzl@29804
  1329
proof (cases x)
hoelzl@29804
  1330
  case (Float m e)
hoelzl@29804
  1331
  have "\<bar>real m\<bar> * pow2 e = \<bar>real m * pow2 e\<bar>" unfolding abs_mult by auto
hoelzl@31098
  1332
  thus ?thesis unfolding Float abs_float_def float_abs.simps real_of_float_simp by auto
hoelzl@29804
  1333
qed
hoelzl@29804
  1334
haftmann@30960
  1335
primrec floor_fl :: "float \<Rightarrow> float" where
haftmann@30960
  1336
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e
hoelzl@29804
  1337
                                  else Float (m div (2 ^ (nat (-e)))) 0)"
obua@16782
  1338
hoelzl@31098
  1339
lemma floor_fl: "real (floor_fl x) \<le> real x"
hoelzl@29804
  1340
proof (cases x)
hoelzl@29804
  1341
  case (Float m e)
hoelzl@29804
  1342
  show ?thesis
hoelzl@29804
  1343
  proof (cases "0 \<le> e")
hoelzl@29804
  1344
    case False
hoelzl@29804
  1345
    hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
hoelzl@31098
  1346
    have "real (Float (m div (2 ^ (nat (-e)))) 0) = real (m div 2 ^ (nat (-e)))" unfolding real_of_float_simp by auto
hoelzl@29804
  1347
    also have "\<dots> \<le> real m / real ((2::int) ^ (nat (-e)))" using real_of_int_div4 .
huffman@36778
  1348
    also have "\<dots> = real m * inverse (2 ^ (nat (-e)))" unfolding real_of_int_power real_number_of divide_inverse ..
hoelzl@31098
  1349
    also have "\<dots> = real (Float m e)" unfolding real_of_float_simp me_eq pow2_int pow2_neg[of e] ..
hoelzl@29804
  1350
    finally show ?thesis unfolding Float floor_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
hoelzl@29804
  1351
  next
hoelzl@29804
  1352
    case True thus ?thesis unfolding Float by auto
hoelzl@29804
  1353
  qed
hoelzl@29804
  1354
qed
obua@16782
  1355
hoelzl@29804
  1356
lemma floor_pos_exp: assumes floor: "Float m e = floor_fl x" shows "0 \<le> e"
hoelzl@29804
  1357
proof (cases x)
hoelzl@29804
  1358
  case (Float mx me)
hoelzl@29804
  1359
  from floor[unfolded Float floor_fl.simps] show ?thesis by (cases "0 \<le> me", auto)
hoelzl@29804
  1360
qed
hoelzl@29804
  1361
hoelzl@29804
  1362
declare floor_fl.simps[simp del]
obua@16782
  1363
haftmann@30960
  1364
primrec ceiling_fl :: "float \<Rightarrow> float" where
haftmann@30960
  1365
  "ceiling_fl (Float m e) = (if 0 \<le> e then Float m e
hoelzl@29804
  1366
                                    else Float (m div (2 ^ (nat (-e))) + 1) 0)"
obua@16782
  1367
hoelzl@31098
  1368
lemma ceiling_fl: "real x \<le> real (ceiling_fl x)"
hoelzl@29804
  1369
proof (cases x)
hoelzl@29804
  1370
  case (Float m e)
hoelzl@29804
  1371
  show ?thesis
hoelzl@29804
  1372
  proof (cases "0 \<le> e")
hoelzl@29804
  1373
    case False
hoelzl@29804
  1374
    hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
hoelzl@31098
  1375
    have "real (Float m e) = real m * inverse (2 ^ (nat (-e)))" unfolding real_of_float_simp me_eq pow2_int pow2_neg[of e] ..
huffman@36778
  1376
    also have "\<dots> = real m / real ((2::int) ^ (nat (-e)))" unfolding real_of_int_power real_number_of divide_inverse ..
hoelzl@29804
  1377
    also have "\<dots> \<le> 1 + real (m div 2 ^ (nat (-e)))" using real_of_int_div3[unfolded diff_le_eq] .
hoelzl@31098
  1378
    also have "\<dots> = real (Float (m div (2 ^ (nat (-e))) + 1) 0)" unfolding real_of_float_simp by auto
hoelzl@29804
  1379
    finally show ?thesis unfolding Float ceiling_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
hoelzl@29804
  1380
  next
hoelzl@29804
  1381
    case True thus ?thesis unfolding Float by auto
hoelzl@29804
  1382
  qed
hoelzl@29804
  1383
qed
hoelzl@29804
  1384
hoelzl@29804
  1385
declare ceiling_fl.simps[simp del]
hoelzl@29804
  1386
hoelzl@29804
  1387
definition lb_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1388
"lb_mod prec x ub lb = x - ceiling_fl (float_divr prec x lb) * ub"
hoelzl@29804
  1389
hoelzl@29804
  1390
definition ub_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1391
"ub_mod prec x ub lb = x - floor_fl (float_divl prec x ub) * lb"
obua@16782
  1392
hoelzl@31098
  1393
lemma lb_mod: fixes k :: int assumes "0 \<le> real x" and "real k * y \<le> real x" (is "?k * y \<le> ?x")
hoelzl@31098
  1394
  assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
hoelzl@31098
  1395
  shows "real (lb_mod prec x ub lb) \<le> ?x - ?k * y"
hoelzl@29804
  1396
proof -
wenzelm@33555
  1397
  have "?lb \<le> ?ub" using assms by auto
wenzelm@33555
  1398
  have "0 \<le> ?lb" and "?lb \<noteq> 0" using assms by auto
hoelzl@29804
  1399
  have "?k * y \<le> ?x" using assms by auto
hoelzl@29804
  1400
  also have "\<dots> \<le> ?x / ?lb * ?ub" by (metis mult_left_mono[OF `?lb \<le> ?ub` `0 \<le> ?x`] divide_right_mono[OF _ `0 \<le> ?lb` ] times_divide_eq_left nonzero_mult_divide_cancel_right[OF `?lb \<noteq> 0`])
hoelzl@31098
  1401
  also have "\<dots> \<le> real (ceiling_fl (float_divr prec x lb)) * ?ub" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divr ceiling_fl)
hoelzl@31098
  1402
  finally show ?thesis unfolding lb_mod_def real_of_float_sub real_of_float_mult by auto
hoelzl@29804
  1403
qed
obua@16782
  1404
hoelzl@31098
  1405
lemma ub_mod: fixes k :: int and x :: float assumes "0 \<le> real x" and "real x \<le> real k * y" (is "?x \<le> ?k * y")
hoelzl@31098
  1406
  assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
hoelzl@31098
  1407
  shows "?x - ?k * y \<le> real (ub_mod prec x ub lb)"
hoelzl@29804
  1408
proof -
wenzelm@33555
  1409
  have "?lb \<le> ?ub" using assms by auto
wenzelm@33555
  1410
  hence "0 \<le> ?lb" and "0 \<le> ?ub" and "?ub \<noteq> 0" using assms by auto
hoelzl@31098
  1411
  have "real (floor_fl (float_divl prec x ub)) * ?lb \<le> ?x / ?ub * ?lb" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divl floor_fl)
hoelzl@29804
  1412
  also have "\<dots> \<le> ?x" by (metis mult_left_mono[OF `?lb \<le> ?ub` `0 \<le> ?x`] divide_right_mono[OF _ `0 \<le> ?ub` ] times_divide_eq_left nonzero_mult_divide_cancel_right[OF `?ub \<noteq> 0`])
hoelzl@29804
  1413
  also have "\<dots> \<le> ?k * y" using assms by auto
hoelzl@31098
  1414
  finally show ?thesis unfolding ub_mod_def real_of_float_sub real_of_float_mult by auto
hoelzl@29804
  1415
qed
obua@16782
  1416
hoelzl@39161
  1417
lemma le_float_def'[code]: "f \<le> g = (case f - g of Float a b \<Rightarrow> a \<le> 0)"
hoelzl@29804
  1418
proof -
hoelzl@31098
  1419
  have le_transfer: "(f \<le> g) = (real (f - g) \<le> 0)" by (auto simp add: le_float_def)
hoelzl@29804
  1420
  from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
hoelzl@31098
  1421
  with le_transfer have le_transfer': "f \<le> g = (real (Float a b) \<le> 0)" by simp
hoelzl@29804
  1422
  show ?thesis by (simp add: le_transfer' f_diff_g float_le_zero)
hoelzl@29804
  1423
qed
hoelzl@29804
  1424
hoelzl@39161
  1425
lemma less_float_def'[code]: "f < g = (case f - g of Float a b \<Rightarrow> a < 0)"
hoelzl@29804
  1426
proof -
hoelzl@31098
  1427
  have less_transfer: "(f < g) = (real (f - g) < 0)" by (auto simp add: less_float_def)
hoelzl@29804
  1428
  from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
hoelzl@31098
  1429
  with less_transfer have less_transfer': "f < g = (real (Float a b) < 0)" by simp
hoelzl@29804
  1430
  show ?thesis by (simp add: less_transfer' f_diff_g float_less_zero)
hoelzl@29804
  1431
qed
wenzelm@20771
  1432
obua@16782
  1433
end