src/HOL/Library/Float.thy
author haftmann
Thu Dec 29 10:47:55 2011 +0100 (2011-12-29)
changeset 46028 9f113cdf3d66
parent 45772 8a8f78ce0dcf
child 46573 8c4c5c8dcf7a
permissions -rw-r--r--
attribute code_abbrev superseedes code_unfold_post
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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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declare [[coercion "% x . Float x 0"]]
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declare [[coercion "real::float\<Rightarrow>real"]]
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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:
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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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declare number_of_float_Float [symmetric, code_abbrev]
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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
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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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lemma pow2_powr: "pow2 a = 2 powr a"
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  by (simp add: powr_realpow[symmetric] powr_minus)
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declare pow2_def[simp del]
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lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
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  by (simp add: pow2_powr powr_add)
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lemma pow2_diff: "pow2 (a - b) = pow2 a / pow2 b"
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  by (simp add: pow2_powr powr_divide2)
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lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
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  by (simp add: pow2_add)
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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) =
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    (if a \<noteq> 0 \<and> even a then normfloat (Float (a div 2) (b+1))
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     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
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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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  by (simp add: pow2_powr)
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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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  from odd have "a' \<noteq> 0" by auto
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  with floateq have a': "real a' = real a * pow2 (b - b')"
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    by (simp add: pow2_diff field_simps)
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  {
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    assume bcmp: "b > b'"
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    then have "\<exists>c::nat. b - b' = int c + 1"
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      by arith
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    then guess c ..
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    with a' have "real a' = real (a * 2^c * 2)"
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      by (simp add: pow2_def nat_add_distrib)
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    with odd have False
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      unfolding real_of_int_inject by simp
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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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  }
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  note even_case = this
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  from odd_case even_case show ?thesis
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    apply (simp add: normf normg)
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    apply (case_tac "even a")
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    apply auto
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    done
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qed
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instantiation float :: plus begin
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fun plus_float where
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[simp del]: "(Float a_m a_e) + (Float b_m b_e) = 
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     (if a_e \<le> b_e then Float (a_m + b_m * 2^(nat(b_e - a_e))) a_e 
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                   else Float (a_m * 2^(nat (a_e - b_e)) + b_m) b_e)"
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instance ..
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end
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instantiation float :: uminus begin
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primrec uminus_float where [simp del]: "uminus_float (Float m e) = Float (-m) e"
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instance ..
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end
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instantiation float :: minus begin
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definition minus_float where [simp del]: "(z::float) - w = z + (- w)"
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instance ..
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end
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instantiation float :: times begin
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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)"
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instance ..
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end
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primrec float_pprt :: "float \<Rightarrow> float" where
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  "float_pprt (Float a e) = (if 0 <= a then (Float a e) else 0)"
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primrec float_nprt :: "float \<Rightarrow> float" where
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  "float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))" 
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instantiation float :: ord begin
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definition le_float_def: "z \<le> (w :: float) \<equiv> real z \<le> real w"
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definition less_float_def: "z < (w :: float) \<equiv> real z < real w"
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instance ..
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end
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lemma real_of_float_add[simp]: "real (a + b) = real a + real (b :: float)"
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  by (cases a, cases b) (simp add: algebra_simps plus_float.simps, 
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      auto simp add: pow2_int[symmetric] pow2_add[symmetric])
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lemma real_of_float_minus[simp]: "real (- a) = - real (a :: float)"
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  by (cases a) (simp add: uminus_float.simps)
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lemma real_of_float_sub[simp]: "real (a - b) = real a - real (b :: float)"
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  by (cases a, cases b) (simp add: minus_float_def)
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lemma real_of_float_mult[simp]: "real (a*b) = real a * real (b :: float)"
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  by (cases a, cases b) (simp add: times_float.simps pow2_add)
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lemma real_of_float_0[simp]: "real (0 :: float) = 0"
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  by (auto simp add: zero_float_def float_zero)
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lemma real_of_float_1[simp]: "real (1 :: float) = 1"
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  by (auto simp add: one_float_def)
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lemma zero_le_float:
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  "(0 <= real (Float a b)) = (0 <= a)"
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  apply auto
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  apply (auto simp add: zero_le_mult_iff)
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  apply (insert zero_less_pow2[of b])
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  apply (simp_all)
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  done
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lemma float_le_zero:
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  "(real (Float a b) <= 0) = (a <= 0)"
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  apply auto
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  apply (auto simp add: mult_le_0_iff)
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  apply (insert zero_less_pow2[of b])
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  apply auto
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  done
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lemma zero_less_float:
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  "(0 < real (Float a b)) = (0 < a)"
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  apply auto
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  apply (auto simp add: zero_less_mult_iff)
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  apply (insert zero_less_pow2[of b])
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  apply (simp_all)
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  done
hoelzl@39161
   316
hoelzl@39161
   317
lemma float_less_zero:
hoelzl@39161
   318
  "(real (Float a b) < 0) = (a < 0)"
hoelzl@39161
   319
  apply auto
hoelzl@39161
   320
  apply (auto simp add: mult_less_0_iff)
hoelzl@39161
   321
  apply (insert zero_less_pow2[of b])
hoelzl@39161
   322
  apply (simp_all)
hoelzl@39161
   323
  done
hoelzl@39161
   324
hoelzl@31098
   325
declare real_of_float_simp[simp del]
hoelzl@29804
   326
hoelzl@31098
   327
lemma real_of_float_pprt[simp]: "real (float_pprt a) = pprt (real a)"
wenzelm@41528
   328
  by (cases a) (auto simp add: zero_le_float float_le_zero)
hoelzl@29804
   329
hoelzl@31098
   330
lemma real_of_float_nprt[simp]: "real (float_nprt a) = nprt (real a)"
wenzelm@41528
   331
  by (cases a) (auto simp add: zero_le_float float_le_zero)
hoelzl@29804
   332
hoelzl@29804
   333
instance float :: ab_semigroup_add
hoelzl@29804
   334
proof (intro_classes)
hoelzl@29804
   335
  fix a b c :: float
hoelzl@29804
   336
  show "a + b + c = a + (b + c)"
wenzelm@41528
   337
    by (cases a, cases b, cases c)
wenzelm@41528
   338
      (auto simp add: algebra_simps power_add[symmetric] plus_float.simps)
hoelzl@29804
   339
next
hoelzl@29804
   340
  fix a b :: float
hoelzl@29804
   341
  show "a + b = b + a"
wenzelm@41528
   342
    by (cases a, cases b) (simp add: plus_float.simps)
hoelzl@29804
   343
qed
hoelzl@29804
   344
hoelzl@29804
   345
instance float :: comm_monoid_mult
hoelzl@29804
   346
proof (intro_classes)
hoelzl@29804
   347
  fix a b c :: float
hoelzl@29804
   348
  show "a * b * c = a * (b * c)"
wenzelm@41528
   349
    by (cases a, cases b, cases c) (simp add: times_float.simps)
hoelzl@29804
   350
next
hoelzl@29804
   351
  fix a b :: float
hoelzl@29804
   352
  show "a * b = b * a"
wenzelm@41528
   353
    by (cases a, cases b) (simp add: times_float.simps)
hoelzl@29804
   354
next
hoelzl@29804
   355
  fix a :: float
hoelzl@29804
   356
  show "1 * a = a"
wenzelm@41528
   357
    by (cases a) (simp add: times_float.simps one_float_def)
hoelzl@29804
   358
qed
hoelzl@29804
   359
hoelzl@29804
   360
(* Floats do NOT form a cancel_semigroup_add: *)
hoelzl@29804
   361
lemma "0 + Float 0 1 = 0 + Float 0 2"
hoelzl@29804
   362
  by (simp add: plus_float.simps zero_float_def)
hoelzl@29804
   363
hoelzl@29804
   364
instance float :: comm_semiring
hoelzl@29804
   365
proof (intro_classes)
hoelzl@29804
   366
  fix a b c :: float
hoelzl@29804
   367
  show "(a + b) * c = a * c + b * c"
wenzelm@41528
   368
    by (cases a, cases b, cases c) (simp add: plus_float.simps times_float.simps algebra_simps)
hoelzl@29804
   369
qed
hoelzl@29804
   370
hoelzl@29804
   371
(* Floats do NOT form an order, because "(x < y) = (x <= y & x <> y)" does NOT hold *)
hoelzl@29804
   372
hoelzl@29804
   373
instance float :: zero_neq_one
hoelzl@29804
   374
proof (intro_classes)
hoelzl@29804
   375
  show "(0::float) \<noteq> 1"
hoelzl@29804
   376
    by (simp add: zero_float_def one_float_def)
hoelzl@29804
   377
qed
hoelzl@29804
   378
hoelzl@29804
   379
lemma float_le_simp: "((x::float) \<le> y) = (0 \<le> y - x)"
hoelzl@29804
   380
  by (auto simp add: le_float_def)
hoelzl@29804
   381
hoelzl@29804
   382
lemma float_less_simp: "((x::float) < y) = (0 < y - x)"
hoelzl@29804
   383
  by (auto simp add: less_float_def)
hoelzl@29804
   384
hoelzl@31098
   385
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
   386
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
   387
hoelzl@31098
   388
lemma float_power: "real (x ^ n :: float) = real x ^ n"
haftmann@30960
   389
  by (induct n) simp_all
hoelzl@29804
   390
hoelzl@29804
   391
lemma zero_le_pow2[simp]: "0 \<le> pow2 s"
hoelzl@29804
   392
  apply (subgoal_tac "0 < pow2 s")
hoelzl@29804
   393
  apply (auto simp only:)
hoelzl@29804
   394
  apply auto
obua@16782
   395
  done
obua@16782
   396
hoelzl@29804
   397
lemma pow2_less_0_eq_False[simp]: "(pow2 s < 0) = False"
hoelzl@29804
   398
  apply auto
hoelzl@29804
   399
  apply (subgoal_tac "0 \<le> pow2 s")
hoelzl@29804
   400
  apply simp
hoelzl@29804
   401
  apply simp
obua@24301
   402
  done
obua@24301
   403
hoelzl@29804
   404
lemma pow2_le_0_eq_False[simp]: "(pow2 s \<le> 0) = False"
hoelzl@29804
   405
  apply auto
hoelzl@29804
   406
  apply (subgoal_tac "0 < pow2 s")
hoelzl@29804
   407
  apply simp
hoelzl@29804
   408
  apply simp
obua@24301
   409
  done
obua@24301
   410
hoelzl@29804
   411
lemma float_pos_m_pos: "0 < Float m e \<Longrightarrow> 0 < m"
hoelzl@31098
   412
  unfolding less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff
obua@16782
   413
  by auto
wenzelm@19765
   414
hoelzl@29804
   415
lemma float_pos_less1_e_neg: assumes "0 < Float m e" and "Float m e < 1" shows "e < 0"
hoelzl@29804
   416
proof -
hoelzl@29804
   417
  have "0 < m" using float_pos_m_pos `0 < Float m e` by auto
hoelzl@29804
   418
  hence "0 \<le> real m" and "1 \<le> real m" by auto
hoelzl@29804
   419
  
hoelzl@29804
   420
  show "e < 0"
hoelzl@29804
   421
  proof (rule ccontr)
hoelzl@29804
   422
    assume "\<not> e < 0" hence "0 \<le> e" by auto
hoelzl@29804
   423
    hence "1 \<le> pow2 e" unfolding pow2_def by auto
hoelzl@29804
   424
    from mult_mono[OF `1 \<le> real m` this `0 \<le> real m`]
hoelzl@31098
   425
    have "1 \<le> Float m e" by (simp add: le_float_def real_of_float_simp)
hoelzl@29804
   426
    thus False using `Float m e < 1` unfolding less_float_def le_float_def by auto
hoelzl@29804
   427
  qed
hoelzl@29804
   428
qed
hoelzl@29804
   429
hoelzl@29804
   430
lemma float_less1_mantissa_bound: assumes "0 < Float m e" "Float m e < 1" shows "m < 2^(nat (-e))"
hoelzl@29804
   431
proof -
hoelzl@29804
   432
  have "e < 0" using float_pos_less1_e_neg assms by auto
hoelzl@29804
   433
  have "\<And>x. (0::real) < 2^x" by auto
hoelzl@29804
   434
  have "real m < 2^(nat (-e))" using `Float m e < 1`
hoelzl@31098
   435
    unfolding less_float_def real_of_float_neg_exp[OF `e < 0`] real_of_float_1
hoelzl@29804
   436
          real_mult_less_iff1[of _ _ 1, OF `0 < 2^(nat (-e))`, symmetric] 
huffman@36778
   437
          mult_assoc by auto
hoelzl@29804
   438
  thus ?thesis unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   439
qed
hoelzl@29804
   440
hoelzl@29804
   441
function bitlen :: "int \<Rightarrow> int" where
hoelzl@29804
   442
"bitlen 0 = 0" | 
hoelzl@29804
   443
"bitlen -1 = 1" | 
hoelzl@29804
   444
"0 < x \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))" | 
hoelzl@29804
   445
"x < -1 \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))"
hoelzl@29804
   446
  apply (case_tac "x = 0 \<or> x = -1 \<or> x < -1 \<or> x > 0")
hoelzl@29804
   447
  apply auto
hoelzl@29804
   448
  done
hoelzl@29804
   449
termination by (relation "measure (nat o abs)", auto)
hoelzl@29804
   450
hoelzl@29804
   451
lemma bitlen_ge0: "0 \<le> bitlen x" by (induct x rule: bitlen.induct, auto)
hoelzl@29804
   452
lemma bitlen_ge1: "x \<noteq> 0 \<Longrightarrow> 1 \<le> bitlen x" by (induct x rule: bitlen.induct, auto simp add: bitlen_ge0)
hoelzl@29804
   453
hoelzl@29804
   454
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
   455
  using `0 < x`
hoelzl@29804
   456
proof (induct x rule: bitlen.induct)
hoelzl@29804
   457
  fix x
hoelzl@29804
   458
  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
   459
  { fix x have "0 \<le> 1 + bitlen x" using bitlen_ge0[of x] by auto } note gt0_pls1 = this
hoelzl@29804
   460
hoelzl@29804
   461
  have "0 < (2::int)" by auto
obua@16782
   462
hoelzl@29804
   463
  show "?P x"
hoelzl@29804
   464
  proof (cases "x = 1")
hoelzl@29804
   465
    case True show "?P x" unfolding True by auto
hoelzl@29804
   466
  next
hoelzl@29804
   467
    case False hence "2 \<le> x" using `0 < x` `x \<noteq> 1` by auto
hoelzl@29804
   468
    hence "2 div 2 \<le> x div 2" by (rule zdiv_mono1, auto)
hoelzl@29804
   469
    hence "0 < x div 2" and "x div 2 \<noteq> 0" by auto
hoelzl@29804
   470
    hence bitlen_s1_ge0: "0 \<le> bitlen (x div 2) - 1" using bitlen_ge1[OF `x div 2 \<noteq> 0`] by auto
obua@16782
   471
hoelzl@29804
   472
    { from hyp[OF `0 < x div 2`]
hoelzl@29804
   473
      have "2 ^ nat (bitlen (x div 2) - 1) \<le> x div 2" by auto
hoelzl@29804
   474
      hence "2 ^ nat (bitlen (x div 2) - 1) * 2 \<le> x div 2 * 2" by (rule mult_right_mono, auto)
hoelzl@29804
   475
      also have "\<dots> \<le> x" using `0 < x` by auto
hoelzl@29804
   476
      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
   477
    } moreover
hoelzl@29804
   478
    { have "x + 1 \<le> x - x mod 2 + 2"
hoelzl@29804
   479
      proof -
wenzelm@32960
   480
        have "x mod 2 < 2" using `0 < x` by auto
wenzelm@32960
   481
        hence "x < x - x mod 2 +  2" unfolding algebra_simps by auto
wenzelm@32960
   482
        thus ?thesis by auto
hoelzl@29804
   483
      qed
hoelzl@29804
   484
      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
   485
      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
   486
      also have "\<dots> = 2^(1 + nat (bitlen (x div 2)))" unfolding power_Suc2[symmetric] by auto
hoelzl@29804
   487
      finally have "x + 1 \<le> 2^(1 + nat (bitlen (x div 2)))" .
hoelzl@29804
   488
    }
hoelzl@29804
   489
    ultimately show ?thesis
hoelzl@29804
   490
      unfolding bitlen.simps(3)[OF `0 < x`] nat_add_distrib[OF zero_le_one bitlen_ge0]
hoelzl@29804
   491
      unfolding add_commute nat_add_distrib[OF zero_le_one gt0_pls1]
hoelzl@29804
   492
      by auto
hoelzl@29804
   493
  qed
hoelzl@29804
   494
next
hoelzl@29804
   495
  fix x :: int assume "x < -1" and "0 < x" hence False by auto
hoelzl@29804
   496
  thus "?P x" by auto
hoelzl@29804
   497
qed auto
hoelzl@29804
   498
hoelzl@29804
   499
lemma bitlen_bounds: assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x < 2^nat (bitlen x)"
hoelzl@29804
   500
  using bitlen_bounds'[OF `0<x`] by auto
hoelzl@29804
   501
hoelzl@29804
   502
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
   503
proof -
hoelzl@29804
   504
  let ?B = "2^nat(bitlen m - 1)"
hoelzl@29804
   505
hoelzl@29804
   506
  have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   507
  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   508
  thus "1 \<le> real m / ?B" by auto
hoelzl@29804
   509
hoelzl@29804
   510
  have "m \<noteq> 0" using assms by auto
hoelzl@29804
   511
  have "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
obua@16782
   512
hoelzl@29804
   513
  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
hoelzl@29804
   514
  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
hoelzl@29804
   515
  also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
hoelzl@29804
   516
  finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hoelzl@29804
   517
  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
hoelzl@29804
   518
  thus "real m / ?B < 2" by auto
hoelzl@29804
   519
qed
hoelzl@29804
   520
hoelzl@29804
   521
lemma float_gt1_scale: assumes "1 \<le> Float m e"
hoelzl@29804
   522
  shows "0 \<le> e + (bitlen m - 1)"
hoelzl@29804
   523
proof (cases "0 \<le> e")
hoelzl@29804
   524
  have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
hoelzl@29804
   525
  hence "0 < m" using float_pos_m_pos by auto
hoelzl@29804
   526
  hence "m \<noteq> 0" by auto
hoelzl@29804
   527
  case True with bitlen_ge1[OF `m \<noteq> 0`] show ?thesis by auto
hoelzl@29804
   528
next
hoelzl@29804
   529
  have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
hoelzl@29804
   530
  hence "0 < m" using float_pos_m_pos by auto
hoelzl@29804
   531
  hence "m \<noteq> 0" and "1 < (2::int)" by auto
hoelzl@29804
   532
  case False let ?S = "2^(nat (-e))"
hoelzl@31098
   533
  have "1 \<le> real m * inverse ?S" using assms unfolding le_float_def real_of_float_nge0_exp[OF False] by auto
hoelzl@29804
   534
  hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hoelzl@29804
   535
  hence "?S \<le> real m" unfolding mult_assoc by auto
hoelzl@29804
   536
  hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   537
  from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
hoelzl@29804
   538
  have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
hoelzl@29804
   539
  hence "-e < bitlen m" using False bitlen_ge0 by auto
hoelzl@29804
   540
  thus ?thesis by auto
hoelzl@29804
   541
qed
hoelzl@29804
   542
hoelzl@31098
   543
lemma normalized_float: assumes "m \<noteq> 0" shows "real (Float m (- (bitlen m - 1))) = real m / 2^nat (bitlen m - 1)"
hoelzl@29804
   544
proof (cases "- (bitlen m - 1) = 0")
hoelzl@31098
   545
  case True show ?thesis unfolding real_of_float_simp pow2_def using True by auto
hoelzl@29804
   546
next
hoelzl@29804
   547
  case False hence P: "\<not> 0 \<le> - (bitlen m - 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
huffman@36778
   548
  show ?thesis unfolding real_of_float_nge0_exp[OF P] divide_inverse by auto
hoelzl@29804
   549
qed
hoelzl@29804
   550
hoelzl@29804
   551
lemma bitlen_Pls: "bitlen (Int.Pls) = Int.Pls" by (subst Pls_def, subst Pls_def, simp)
hoelzl@29804
   552
hoelzl@29804
   553
lemma bitlen_Min: "bitlen (Int.Min) = Int.Bit1 Int.Pls" by (subst Min_def, simp add: Bit1_def) 
hoelzl@29804
   554
hoelzl@29804
   555
lemma bitlen_B0: "bitlen (Int.Bit0 b) = (if iszero b then Int.Pls else Int.succ (bitlen b))"
hoelzl@29804
   556
  apply (auto simp add: iszero_def succ_def)
hoelzl@29804
   557
  apply (simp add: Bit0_def Pls_def)
hoelzl@29804
   558
  apply (subst Bit0_def)
hoelzl@29804
   559
  apply simp
hoelzl@29804
   560
  apply (subgoal_tac "0 < 2 * b \<or> 2 * b < -1")
hoelzl@29804
   561
  apply auto
hoelzl@29804
   562
  done
obua@16782
   563
hoelzl@29804
   564
lemma bitlen_B1: "bitlen (Int.Bit1 b) = (if iszero (Int.succ b) then Int.Bit1 Int.Pls else Int.succ (bitlen b))"
hoelzl@29804
   565
proof -
hoelzl@29804
   566
  have h: "! x. (2*x + 1) div 2 = (x::int)"
hoelzl@29804
   567
    by arith    
hoelzl@29804
   568
  show ?thesis
hoelzl@29804
   569
    apply (auto simp add: iszero_def succ_def)
hoelzl@29804
   570
    apply (subst Bit1_def)+
hoelzl@29804
   571
    apply simp
hoelzl@29804
   572
    apply (subgoal_tac "2 * b + 1 = -1")
hoelzl@29804
   573
    apply (simp only:)
hoelzl@29804
   574
    apply simp_all
hoelzl@29804
   575
    apply (subst Bit1_def)
hoelzl@29804
   576
    apply simp
hoelzl@29804
   577
    apply (subgoal_tac "0 < 2 * b + 1 \<or> 2 * b + 1 < -1")
hoelzl@29804
   578
    apply (auto simp add: h)
hoelzl@29804
   579
    done
hoelzl@29804
   580
qed
hoelzl@29804
   581
hoelzl@29804
   582
lemma bitlen_number_of: "bitlen (number_of w) = number_of (bitlen w)"
hoelzl@29804
   583
  by (simp add: number_of_is_id)
obua@16782
   584
hoelzl@29804
   585
lemma [code]: "bitlen x = 
hoelzl@29804
   586
     (if x = 0  then 0 
hoelzl@29804
   587
 else if x = -1 then 1 
hoelzl@29804
   588
                else (1 + (bitlen (x div 2))))"
hoelzl@29804
   589
  by (cases "x = 0 \<or> x = -1 \<or> 0 < x") auto
hoelzl@29804
   590
hoelzl@29804
   591
definition lapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   592
where
hoelzl@29804
   593
  "lapprox_posrat prec x y = 
hoelzl@29804
   594
   (let 
hoelzl@29804
   595
       l = nat (int prec + bitlen y - bitlen x) ;
hoelzl@29804
   596
       d = (x * 2^l) div y
hoelzl@29804
   597
    in normfloat (Float d (- (int l))))"
hoelzl@29804
   598
hoelzl@29804
   599
lemma pow2_minus: "pow2 (-x) = inverse (pow2 x)"
hoelzl@29804
   600
  unfolding pow2_neg[of "-x"] by auto
hoelzl@29804
   601
hoelzl@29804
   602
lemma lapprox_posrat: 
hoelzl@29804
   603
  assumes x: "0 \<le> x"
hoelzl@29804
   604
  and y: "0 < y"
hoelzl@31098
   605
  shows "real (lapprox_posrat prec x y) \<le> real x / real y"
hoelzl@29804
   606
proof -
hoelzl@29804
   607
  let ?l = "nat (int prec + bitlen y - bitlen x)"
hoelzl@29804
   608
  
hoelzl@29804
   609
  have "real (x * 2^?l div y) * inverse (2^?l) \<le> (real (x * 2^?l) / real y) * inverse (2^?l)" 
hoelzl@29804
   610
    by (rule mult_right_mono, fact real_of_int_div4, simp)
hoelzl@29804
   611
  also have "\<dots> \<le> (real x / real y) * 2^?l * inverse (2^?l)" by auto
huffman@36778
   612
  finally have "real (x * 2^?l div y) * inverse (2^?l) \<le> real x / real y" unfolding mult_assoc by auto
hoelzl@31098
   613
  thus ?thesis unfolding lapprox_posrat_def Let_def normfloat real_of_float_simp
hoelzl@29804
   614
    unfolding pow2_minus pow2_int minus_minus .
hoelzl@29804
   615
qed
obua@16782
   616
hoelzl@29804
   617
lemma real_of_int_div_mult: 
hoelzl@29804
   618
  fixes x y c :: int assumes "0 < y" and "0 < c"
hoelzl@29804
   619
  shows "real (x div y) \<le> real (x * c div y) * inverse (real c)"
hoelzl@29804
   620
proof -
hoelzl@29804
   621
  have "c * (x div y) + 0 \<le> c * x div y" unfolding zdiv_zmult1_eq[of c x y]
huffman@44766
   622
    by (rule add_left_mono, 
hoelzl@29804
   623
        auto intro!: mult_nonneg_nonneg 
hoelzl@29804
   624
             simp add: pos_imp_zdiv_nonneg_iff[OF `0 < y`] `0 < c`[THEN less_imp_le] pos_mod_sign[OF `0 < y`])
hoelzl@29804
   625
  hence "real (x div y) * real c \<le> real (x * c div y)" 
huffman@44766
   626
    unfolding real_of_int_mult[symmetric] real_of_int_le_iff mult_commute by auto
hoelzl@29804
   627
  hence "real (x div y) * real c * inverse (real c) \<le> real (x * c div y) * inverse (real c)"
hoelzl@29804
   628
    using `0 < c` by auto
huffman@36778
   629
  thus ?thesis unfolding mult_assoc using `0 < c` by auto
hoelzl@29804
   630
qed
hoelzl@29804
   631
hoelzl@29804
   632
lemma lapprox_posrat_bottom: assumes "0 < y"
hoelzl@31098
   633
  shows "real (x div y) \<le> real (lapprox_posrat n x y)" 
hoelzl@29804
   634
proof -
hoelzl@29804
   635
  have pow: "\<And>x. (0::int) < 2^x" by auto
hoelzl@29804
   636
  show ?thesis
hoelzl@31098
   637
    unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
hoelzl@29804
   638
    using real_of_int_div_mult[OF `0 < y` pow] by auto
hoelzl@29804
   639
qed
hoelzl@29804
   640
hoelzl@29804
   641
lemma lapprox_posrat_nonneg: assumes "0 \<le> x" and "0 < y"
hoelzl@31098
   642
  shows "0 \<le> real (lapprox_posrat n x y)" 
hoelzl@29804
   643
proof -
hoelzl@29804
   644
  show ?thesis
hoelzl@31098
   645
    unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
hoelzl@29804
   646
    using pos_imp_zdiv_nonneg_iff[OF `0 < y`] assms by (auto intro!: mult_nonneg_nonneg)
hoelzl@29804
   647
qed
hoelzl@29804
   648
hoelzl@29804
   649
definition rapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   650
where
hoelzl@29804
   651
  "rapprox_posrat prec x y = (let
hoelzl@29804
   652
     l = nat (int prec + bitlen y - bitlen x) ;
hoelzl@29804
   653
     X = x * 2^l ;
hoelzl@29804
   654
     d = X div y ;
hoelzl@29804
   655
     m = X mod y
hoelzl@29804
   656
   in normfloat (Float (d + (if m = 0 then 0 else 1)) (- (int l))))"
obua@16782
   657
hoelzl@29804
   658
lemma rapprox_posrat:
hoelzl@29804
   659
  assumes x: "0 \<le> x"
hoelzl@29804
   660
  and y: "0 < y"
hoelzl@31098
   661
  shows "real x / real y \<le> real (rapprox_posrat prec x y)"
hoelzl@29804
   662
proof -
hoelzl@29804
   663
  let ?l = "nat (int prec + bitlen y - bitlen x)" let ?X = "x * 2^?l"
hoelzl@29804
   664
  show ?thesis 
hoelzl@29804
   665
  proof (cases "?X mod y = 0")
hoelzl@29804
   666
    case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
hoelzl@29804
   667
    from real_of_int_div[OF this]
hoelzl@29804
   668
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
hoelzl@29804
   669
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
hoelzl@29804
   670
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
hoelzl@29804
   671
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True] 
hoelzl@31098
   672
      unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   673
  next
hoelzl@29804
   674
    case False
hoelzl@29804
   675
    have "0 \<le> real y" and "real y \<noteq> 0" using `0 < y` by auto
hoelzl@29804
   676
    have "0 \<le> real y * 2^?l" by (rule mult_nonneg_nonneg, rule `0 \<le> real y`, auto)
obua@16782
   677
hoelzl@29804
   678
    have "?X = y * (?X div y) + ?X mod y" by auto
hoelzl@29804
   679
    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])
huffman@44766
   680
    also have "\<dots> = y * (?X div y + 1)" unfolding right_distrib by auto
hoelzl@29804
   681
    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
   682
    hence "real ?X / (real y * 2^?l) \<le> real y * real (?X div y + 1) / (real y * 2^?l)" 
hoelzl@29804
   683
      by (rule divide_right_mono, simp only: `0 \<le> real y * 2^?l`)
hoelzl@29804
   684
    also have "\<dots> = real y * real (?X div y + 1) / real y / 2^?l" by auto
hoelzl@29804
   685
    also have "\<dots> = real (?X div y + 1) * inverse (2^?l)" unfolding nonzero_mult_divide_cancel_left[OF `real y \<noteq> 0`] 
huffman@36778
   686
      unfolding divide_inverse ..
hoelzl@31098
   687
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
hoelzl@29804
   688
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   689
  qed
hoelzl@29804
   690
qed
hoelzl@29804
   691
hoelzl@29804
   692
lemma rapprox_posrat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
hoelzl@31098
   693
  shows "real (rapprox_posrat n x y) \<le> 1"
hoelzl@29804
   694
proof -
hoelzl@29804
   695
  let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
hoelzl@29804
   696
  show ?thesis
hoelzl@29804
   697
  proof (cases "?X mod y = 0")
hoelzl@29804
   698
    case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
hoelzl@29804
   699
    from real_of_int_div[OF this]
hoelzl@29804
   700
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
hoelzl@29804
   701
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
hoelzl@29804
   702
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
hoelzl@29804
   703
    also have "real x / real y \<le> 1" using `0 \<le> x` and `0 < y` and `x \<le> y` by auto
hoelzl@29804
   704
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True]
hoelzl@31098
   705
      unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   706
  next
hoelzl@29804
   707
    case False
hoelzl@29804
   708
    have "x \<noteq> y"
hoelzl@29804
   709
    proof (rule ccontr)
hoelzl@29804
   710
      assume "\<not> x \<noteq> y" hence "x = y" by auto
nipkow@30034
   711
      have "?X mod y = 0" unfolding `x = y` using mod_mult_self1_is_0 by auto
hoelzl@29804
   712
      thus False using False by auto
hoelzl@29804
   713
    qed
hoelzl@29804
   714
    hence "x < y" using `x \<le> y` by auto
hoelzl@29804
   715
    hence "real x / real y < 1" using `0 < y` and `0 \<le> x` by auto
obua@16782
   716
hoelzl@29804
   717
    from real_of_int_div4[of "?X" y]
huffman@35344
   718
    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
   719
    also have "\<dots> < 1 * 2^?l" using `real x / real y < 1` by (rule mult_strict_right_mono, auto)
hoelzl@29804
   720
    finally have "?X div y < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
hoelzl@29804
   721
    hence "?X div y + 1 \<le> 2^?l" by auto
hoelzl@29804
   722
    hence "real (?X div y + 1) * inverse (2^?l) \<le> 2^?l * inverse (2^?l)"
huffman@35344
   723
      unfolding real_of_int_le_iff[of _ "2^?l", symmetric] real_of_int_power real_number_of
hoelzl@29804
   724
      by (rule mult_right_mono, auto)
hoelzl@29804
   725
    hence "real (?X div y + 1) * inverse (2^?l) \<le> 1" by auto
hoelzl@31098
   726
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
hoelzl@29804
   727
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   728
  qed
hoelzl@29804
   729
qed
obua@16782
   730
hoelzl@29804
   731
lemma zdiv_greater_zero: fixes a b :: int assumes "0 < a" and "a \<le> b"
hoelzl@29804
   732
  shows "0 < b div a"
hoelzl@29804
   733
proof (rule ccontr)
hoelzl@29804
   734
  have "0 \<le> b" using assms by auto
hoelzl@29804
   735
  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
   736
  have "b = a * (b div a) + b mod a" by auto
hoelzl@29804
   737
  hence "b = b mod a" unfolding `b div a = 0` by auto
hoelzl@29804
   738
  hence "b < a" using `0 < a`[THEN pos_mod_bound, of b] by auto
hoelzl@29804
   739
  thus False using `a \<le> b` by auto
hoelzl@29804
   740
qed
hoelzl@29804
   741
hoelzl@29804
   742
lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
hoelzl@31098
   743
  shows "real (rapprox_posrat n x y) < 1"
hoelzl@29804
   744
proof (cases "x = 0")
hoelzl@31098
   745
  case True thus ?thesis unfolding rapprox_posrat_def True Let_def normfloat real_of_float_simp by auto
hoelzl@29804
   746
next
hoelzl@29804
   747
  case False hence "0 < x" using `0 \<le> x` by auto
hoelzl@29804
   748
  hence "x < y" using assms by auto
hoelzl@29804
   749
  
hoelzl@29804
   750
  let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
hoelzl@29804
   751
  show ?thesis
hoelzl@29804
   752
  proof (cases "?X mod y = 0")
hoelzl@29804
   753
    case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
hoelzl@29804
   754
    from real_of_int_div[OF this]
hoelzl@29804
   755
    have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
hoelzl@29804
   756
    also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
hoelzl@29804
   757
    finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
hoelzl@29804
   758
    also have "real x / real y < 1" using `0 \<le> x` and `0 < y` and `x < y` by auto
hoelzl@31098
   759
    finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_P[OF True]
hoelzl@29804
   760
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   761
  next
hoelzl@29804
   762
    case False
hoelzl@29804
   763
    hence "(real x / real y) < 1 / 2" using `0 < y` and `0 \<le> x` `2 * x < y` by auto
obua@16782
   764
hoelzl@29804
   765
    have "0 < ?X div y"
hoelzl@29804
   766
    proof -
hoelzl@29804
   767
      have "2^nat (bitlen x - 1) \<le> y" and "y < 2^nat (bitlen y)"
wenzelm@32960
   768
        using bitlen_bounds[OF `0 < x`, THEN conjunct1] bitlen_bounds[OF `0 < y`, THEN conjunct2] `x < y` by auto
hoelzl@29804
   769
      hence "(2::int)^nat (bitlen x - 1) < 2^nat (bitlen y)" by (rule order_le_less_trans)
hoelzl@29804
   770
      hence "bitlen x \<le> bitlen y" by auto
hoelzl@29804
   771
      hence len_less: "nat (bitlen x - 1) \<le> nat (int (n - 1) + bitlen y)" by auto
hoelzl@29804
   772
hoelzl@29804
   773
      have "x \<noteq> 0" and "y \<noteq> 0" using `0 < x` `0 < y` by auto
hoelzl@29804
   774
hoelzl@29804
   775
      have exp_eq: "nat (int (n - 1) + bitlen y) - nat (bitlen x - 1) = ?l"
wenzelm@32960
   776
        using `bitlen x \<le> bitlen y` bitlen_ge1[OF `x \<noteq> 0`] bitlen_ge1[OF `y \<noteq> 0`] `0 < n` by auto
hoelzl@29804
   777
hoelzl@29804
   778
      have "y * 2^nat (bitlen x - 1) \<le> y * x" 
wenzelm@32960
   779
        using bitlen_bounds[OF `0 < x`, THEN conjunct1] `0 < y`[THEN less_imp_le] by (rule mult_left_mono)
hoelzl@29804
   780
      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
   781
      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
   782
      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
   783
        unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   784
      hence "real y \<le> real x * (2^nat (int (n - 1) + bitlen y) / (2^nat (bitlen x - 1)))" 
huffman@36778
   785
        unfolding mult_assoc divide_inverse by auto
hoelzl@29804
   786
      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
   787
      finally have "y \<le> x * 2^?l" unfolding exp_eq unfolding real_of_int_le_iff[symmetric] by auto
hoelzl@29804
   788
      thus ?thesis using zdiv_greater_zero[OF `0 < y`] by auto
hoelzl@29804
   789
    qed
hoelzl@29804
   790
hoelzl@29804
   791
    from real_of_int_div4[of "?X" y]
huffman@35344
   792
    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
   793
    also have "\<dots> < 1/2 * 2^?l" using `real x / real y < 1/2` by (rule mult_strict_right_mono, auto)
hoelzl@29804
   794
    finally have "?X div y * 2 < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
hoelzl@29804
   795
    hence "?X div y + 1 < 2^?l" using `0 < ?X div y` by auto
hoelzl@29804
   796
    hence "real (?X div y + 1) * inverse (2^?l) < 2^?l * inverse (2^?l)"
huffman@35344
   797
      unfolding real_of_int_less_iff[of _ "2^?l", symmetric] real_of_int_power real_number_of
hoelzl@29804
   798
      by (rule mult_strict_right_mono, auto)
hoelzl@29804
   799
    hence "real (?X div y + 1) * inverse (2^?l) < 1" by auto
hoelzl@31098
   800
    thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
hoelzl@29804
   801
      unfolding pow2_minus pow2_int minus_minus by auto
hoelzl@29804
   802
  qed
hoelzl@29804
   803
qed
hoelzl@29804
   804
hoelzl@29804
   805
lemma approx_rat_pattern: fixes P and ps :: "nat * int * int"
hoelzl@29804
   806
  assumes Y: "\<And>y prec x. \<lbrakk>y = 0; ps = (prec, x, 0)\<rbrakk> \<Longrightarrow> P" 
hoelzl@29804
   807
  and A: "\<And>x y prec. \<lbrakk>0 \<le> x; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   808
  and B: "\<And>x y prec. \<lbrakk>x < 0; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   809
  and C: "\<And>x y prec. \<lbrakk>x < 0; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   810
  and D: "\<And>x y prec. \<lbrakk>0 \<le> x; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
hoelzl@29804
   811
  shows P
obua@16782
   812
proof -
wenzelm@41528
   813
  obtain prec x y where [simp]: "ps = (prec, x, y)" by (cases ps) auto
hoelzl@29804
   814
  from Y have "y = 0 \<Longrightarrow> P" by auto
wenzelm@41528
   815
  moreover {
wenzelm@41528
   816
    assume "0 < y"
wenzelm@41528
   817
    have P
wenzelm@41528
   818
    proof (cases "0 \<le> x")
wenzelm@41528
   819
      case True
wenzelm@41528
   820
      with A and `0 < y` show P by auto
wenzelm@41528
   821
    next
wenzelm@41528
   822
      case False
wenzelm@41528
   823
      with B and `0 < y` show P by auto
wenzelm@41528
   824
    qed
wenzelm@41528
   825
  } 
wenzelm@41528
   826
  moreover {
wenzelm@41528
   827
    assume "y < 0"
wenzelm@41528
   828
    have P
wenzelm@41528
   829
    proof (cases "0 \<le> x")
wenzelm@41528
   830
      case True
wenzelm@41528
   831
      with D and `y < 0` show P by auto
wenzelm@41528
   832
    next
wenzelm@41528
   833
      case False
wenzelm@41528
   834
      with C and `y < 0` show P by auto
wenzelm@41528
   835
    qed
wenzelm@41528
   836
  }
wenzelm@41528
   837
  ultimately show P by (cases "y = 0 \<or> 0 < y \<or> y < 0") auto
obua@16782
   838
qed
obua@16782
   839
hoelzl@29804
   840
function lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   841
where
hoelzl@29804
   842
  "y = 0 \<Longrightarrow> lapprox_rat prec x y = 0"
hoelzl@29804
   843
| "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec x y"
hoelzl@29804
   844
| "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec (-x) y)"
hoelzl@29804
   845
| "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec (-x) (-y)"
hoelzl@29804
   846
| "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec x (-y))"
hoelzl@29804
   847
apply simp_all by (rule approx_rat_pattern)
hoelzl@29804
   848
termination by lexicographic_order
obua@16782
   849
hoelzl@29804
   850
lemma compute_lapprox_rat[code]:
hoelzl@29804
   851
      "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
   852
                                                             else (if 0 < y then - (rapprox_posrat prec (-x) y) else lapprox_posrat prec (-x) (-y)))"
hoelzl@29804
   853
  by auto
hoelzl@29804
   854
            
hoelzl@31098
   855
lemma lapprox_rat: "real (lapprox_rat prec x y) \<le> real x / real y"
hoelzl@29804
   856
proof -      
hoelzl@29804
   857
  have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
hoelzl@29804
   858
  show ?thesis
hoelzl@29804
   859
    apply (case_tac "y = 0")
hoelzl@29804
   860
    apply simp
hoelzl@29804
   861
    apply (case_tac "0 \<le> x \<and> 0 < y")
hoelzl@29804
   862
    apply (simp add: lapprox_posrat)
hoelzl@29804
   863
    apply (case_tac "x < 0 \<and> 0 < y")
hoelzl@29804
   864
    apply simp
hoelzl@29804
   865
    apply (subst minus_le_iff)   
hoelzl@29804
   866
    apply (rule h[OF rapprox_posrat])
hoelzl@29804
   867
    apply (simp_all)
hoelzl@29804
   868
    apply (case_tac "x < 0 \<and> y < 0")
hoelzl@29804
   869
    apply simp
hoelzl@29804
   870
    apply (rule h[OF _ lapprox_posrat])
hoelzl@29804
   871
    apply (simp_all)
hoelzl@29804
   872
    apply (case_tac "0 \<le> x \<and> y < 0")
hoelzl@29804
   873
    apply (simp)
hoelzl@29804
   874
    apply (subst minus_le_iff)   
hoelzl@29804
   875
    apply (rule h[OF rapprox_posrat])
hoelzl@29804
   876
    apply simp_all
hoelzl@29804
   877
    apply arith
hoelzl@29804
   878
    done
hoelzl@29804
   879
qed
obua@16782
   880
hoelzl@29804
   881
lemma lapprox_rat_bottom: assumes "0 \<le> x" and "0 < y"
hoelzl@31098
   882
  shows "real (x div y) \<le> real (lapprox_rat n x y)" 
hoelzl@29804
   883
  unfolding lapprox_rat.simps(2)[OF assms]  using lapprox_posrat_bottom[OF `0<y`] .
hoelzl@29804
   884
hoelzl@29804
   885
function rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
hoelzl@29804
   886
where
hoelzl@29804
   887
  "y = 0 \<Longrightarrow> rapprox_rat prec x y = 0"
hoelzl@29804
   888
| "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec x y"
hoelzl@29804
   889
| "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec (-x) y)"
hoelzl@29804
   890
| "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec (-x) (-y)"
hoelzl@29804
   891
| "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec x (-y))"
hoelzl@29804
   892
apply simp_all by (rule approx_rat_pattern)
hoelzl@29804
   893
termination by lexicographic_order
obua@16782
   894
hoelzl@29804
   895
lemma compute_rapprox_rat[code]:
hoelzl@29804
   896
      "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
   897
                                                                  (if 0 < y then - (lapprox_posrat prec (-x) y) else rapprox_posrat prec (-x) (-y)))"
hoelzl@29804
   898
  by auto
obua@16782
   899
hoelzl@31098
   900
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
hoelzl@29804
   901
proof -      
hoelzl@29804
   902
  have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
hoelzl@29804
   903
  show ?thesis
hoelzl@29804
   904
    apply (case_tac "y = 0")
hoelzl@29804
   905
    apply simp
hoelzl@29804
   906
    apply (case_tac "0 \<le> x \<and> 0 < y")
hoelzl@29804
   907
    apply (simp add: rapprox_posrat)
hoelzl@29804
   908
    apply (case_tac "x < 0 \<and> 0 < y")
hoelzl@29804
   909
    apply simp
hoelzl@29804
   910
    apply (subst le_minus_iff)   
hoelzl@29804
   911
    apply (rule h[OF _ lapprox_posrat])
hoelzl@29804
   912
    apply (simp_all)
hoelzl@29804
   913
    apply (case_tac "x < 0 \<and> y < 0")
hoelzl@29804
   914
    apply simp
hoelzl@29804
   915
    apply (rule h[OF rapprox_posrat])
hoelzl@29804
   916
    apply (simp_all)
hoelzl@29804
   917
    apply (case_tac "0 \<le> x \<and> y < 0")
hoelzl@29804
   918
    apply (simp)
hoelzl@29804
   919
    apply (subst le_minus_iff)   
hoelzl@29804
   920
    apply (rule h[OF _ lapprox_posrat])
hoelzl@29804
   921
    apply simp_all
hoelzl@29804
   922
    apply arith
hoelzl@29804
   923
    done
hoelzl@29804
   924
qed
obua@16782
   925
hoelzl@29804
   926
lemma rapprox_rat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
hoelzl@31098
   927
  shows "real (rapprox_rat n x y) \<le> 1"
hoelzl@29804
   928
  unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`] using rapprox_posrat_le1[OF assms] .
hoelzl@29804
   929
hoelzl@29804
   930
lemma rapprox_rat_neg: assumes "x < 0" and "0 < y"
hoelzl@31098
   931
  shows "real (rapprox_rat n x y) \<le> 0"
hoelzl@29804
   932
  unfolding rapprox_rat.simps(3)[OF assms] using lapprox_posrat_nonneg[of "-x" y n] assms by auto
hoelzl@29804
   933
hoelzl@29804
   934
lemma rapprox_rat_nonneg_neg: assumes "0 \<le> x" and "y < 0"
hoelzl@31098
   935
  shows "real (rapprox_rat n x y) \<le> 0"
hoelzl@29804
   936
  unfolding rapprox_rat.simps(5)[OF assms] using lapprox_posrat_nonneg[of x "-y" n] assms by auto
obua@16782
   937
hoelzl@29804
   938
lemma rapprox_rat_nonpos_pos: assumes "x \<le> 0" and "0 < y"
hoelzl@31098
   939
  shows "real (rapprox_rat n x y) \<le> 0"
hoelzl@29804
   940
proof (cases "x = 0") 
wenzelm@41528
   941
  case True
wenzelm@41528
   942
  hence "0 \<le> x" by auto show ?thesis
wenzelm@41528
   943
    unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`]
wenzelm@41528
   944
    unfolding True rapprox_posrat_def Let_def
wenzelm@41528
   945
    by auto
hoelzl@29804
   946
next
wenzelm@41528
   947
  case False
wenzelm@41528
   948
  hence "x < 0" using assms by auto
hoelzl@29804
   949
  show ?thesis using rapprox_rat_neg[OF `x < 0` `0 < y`] .
hoelzl@29804
   950
qed
hoelzl@29804
   951
hoelzl@29804
   952
fun float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
hoelzl@29804
   953
where
hoelzl@29804
   954
  "float_divl prec (Float m1 s1) (Float m2 s2) = 
hoelzl@29804
   955
    (let
hoelzl@29804
   956
       l = lapprox_rat prec m1 m2;
hoelzl@29804
   957
       f = Float 1 (s1 - s2)
hoelzl@29804
   958
     in
hoelzl@29804
   959
       f * l)"     
obua@16782
   960
hoelzl@31098
   961
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
hoelzl@45772
   962
  using lapprox_rat[of prec "mantissa x" "mantissa y"]
hoelzl@45772
   963
  by (cases x y rule: float.exhaust[case_product float.exhaust])
hoelzl@45772
   964
     (simp split: split_if_asm
hoelzl@45772
   965
           add: real_of_float_simp pow2_diff field_simps le_divide_eq mult_less_0_iff zero_less_mult_iff)
obua@16782
   966
hoelzl@29804
   967
lemma float_divl_lower_bound: assumes "0 \<le> x" and "0 < y" shows "0 \<le> float_divl prec x y"
hoelzl@29804
   968
proof (cases x, cases y)
hoelzl@29804
   969
  fix xm xe ym ye :: int
hoelzl@29804
   970
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
wenzelm@41528
   971
  have "0 \<le> xm"
wenzelm@41528
   972
    using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff]
wenzelm@41528
   973
    by auto
wenzelm@41528
   974
  have "0 < ym"
wenzelm@41528
   975
    using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff]
wenzelm@41528
   976
    by auto
obua@16782
   977
wenzelm@41528
   978
  have "\<And>n. 0 \<le> real (Float 1 n)"
wenzelm@41528
   979
    unfolding real_of_float_simp using zero_le_pow2 by auto
wenzelm@41528
   980
  moreover have "0 \<le> real (lapprox_rat prec xm ym)"
wenzelm@41528
   981
    apply (rule order_trans[OF _ lapprox_rat_bottom[OF `0 \<le> xm` `0 < ym`]])
wenzelm@41528
   982
    apply (auto simp add: `0 \<le> xm` pos_imp_zdiv_nonneg_iff[OF `0 < ym`])
wenzelm@41528
   983
    done
hoelzl@29804
   984
  ultimately show "0 \<le> float_divl prec x y"
wenzelm@41528
   985
    unfolding x_eq y_eq float_divl.simps Let_def le_float_def real_of_float_0
wenzelm@41528
   986
    by (auto intro!: mult_nonneg_nonneg)
hoelzl@29804
   987
qed
hoelzl@29804
   988
wenzelm@41528
   989
lemma float_divl_pos_less1_bound:
wenzelm@41528
   990
  assumes "0 < x" and "x < 1" and "0 < prec"
wenzelm@41528
   991
  shows "1 \<le> float_divl prec 1 x"
hoelzl@29804
   992
proof (cases x)
hoelzl@29804
   993
  case (Float m e)
wenzelm@41528
   994
  from `0 < x` `x < 1` have "0 < m" "e < 0"
wenzelm@41528
   995
    using float_pos_m_pos float_pos_less1_e_neg unfolding Float by auto
hoelzl@29804
   996
  let ?b = "nat (bitlen m)" and ?e = "nat (-e)"
hoelzl@29804
   997
  have "1 \<le> m" and "m \<noteq> 0" using `0 < m` by auto
hoelzl@29804
   998
  with bitlen_bounds[OF `0 < m`] have "m < 2^?b" and "(2::int) \<le> 2^?b" by auto
hoelzl@29804
   999
  hence "1 \<le> bitlen m" using power_le_imp_le_exp[of "2::int" 1 ?b] by auto
hoelzl@29804
  1000
  hence pow_split: "nat (int prec + bitlen m - 1) = (prec - 1) + ?b" using `0 < prec` by auto
hoelzl@29804
  1001
  
hoelzl@29804
  1002
  have pow_not0: "\<And>x. (2::real)^x \<noteq> 0" by auto
obua@16782
  1003
hoelzl@29804
  1004
  from float_less1_mantissa_bound `0 < x` `x < 1` Float 
hoelzl@29804
  1005
  have "m < 2^?e" by auto
wenzelm@41528
  1006
  with bitlen_bounds[OF `0 < m`, THEN conjunct1] have "(2::int)^nat (bitlen m - 1) < 2^?e"
wenzelm@41528
  1007
    by (rule order_le_less_trans)
hoelzl@29804
  1008
  from power_less_imp_less_exp[OF _ this]
hoelzl@29804
  1009
  have "bitlen m \<le> - e" by auto
hoelzl@29804
  1010
  hence "(2::real)^?b \<le> 2^?e" by auto
wenzelm@41528
  1011
  hence "(2::real)^?b * inverse (2^?b) \<le> 2^?e * inverse (2^?b)"
wenzelm@41528
  1012
    by (rule mult_right_mono) auto
hoelzl@29804
  1013
  hence "(1::real) \<le> 2^?e * inverse (2^?b)" by auto
hoelzl@29804
  1014
  also
hoelzl@29804
  1015
  let ?d = "real (2 ^ nat (int prec + bitlen m - 1) div m) * inverse (2 ^ nat (int prec + bitlen m - 1))"
wenzelm@41528
  1016
  {
wenzelm@41528
  1017
    have "2^(prec - 1) * m \<le> 2^(prec - 1) * 2^?b"
wenzelm@41528
  1018
      using `m < 2^?b`[THEN less_imp_le] by (rule mult_left_mono) auto
wenzelm@41528
  1019
    also have "\<dots> = 2 ^ nat (int prec + bitlen m - 1)"
huffman@44766
  1020
      unfolding pow_split power_add by auto
wenzelm@41528
  1021
    finally have "2^(prec - 1) * m div m \<le> 2 ^ nat (int prec + bitlen m - 1) div m"
wenzelm@41528
  1022
      using `0 < m` by (rule zdiv_mono1)
wenzelm@41528
  1023
    hence "2^(prec - 1) \<le> 2 ^ nat (int prec + bitlen m - 1) div m"
wenzelm@41528
  1024
      unfolding div_mult_self2_is_id[OF `m \<noteq> 0`] .
hoelzl@29804
  1025
    hence "2^(prec - 1) * inverse (2 ^ nat (int prec + bitlen m - 1)) \<le> ?d"
wenzelm@41528
  1026
      unfolding real_of_int_le_iff[of "2^(prec - 1)", symmetric] by auto
wenzelm@41528
  1027
  }
wenzelm@41528
  1028
  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
  1029
  have "2^?e * inverse (2^?b) \<le> 2^?e * ?d" unfolding pow_split power_add by auto
hoelzl@29804
  1030
  finally have "1 \<le> 2^?e * ?d" .
hoelzl@29804
  1031
  
hoelzl@29804
  1032
  have e_nat: "0 - e = int (nat (-e))" using `e < 0` by auto
hoelzl@29804
  1033
  have "bitlen 1 = 1" using bitlen.simps by auto
hoelzl@29804
  1034
  
hoelzl@29804
  1035
  show ?thesis 
wenzelm@41528
  1036
    unfolding one_float_def Float float_divl.simps Let_def
wenzelm@41528
  1037
      lapprox_rat.simps(2)[OF zero_le_one `0 < m`]
wenzelm@41528
  1038
      lapprox_posrat_def `bitlen 1 = 1`
wenzelm@41528
  1039
    unfolding le_float_def real_of_float_mult normfloat real_of_float_simp
wenzelm@41528
  1040
      pow2_minus pow2_int e_nat
hoelzl@29804
  1041
    using `1 \<le> 2^?e * ?d` by (auto simp add: pow2_def)
hoelzl@29804
  1042
qed
obua@16782
  1043
hoelzl@29804
  1044
fun float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
hoelzl@29804
  1045
where
hoelzl@29804
  1046
  "float_divr prec (Float m1 s1) (Float m2 s2) = 
hoelzl@29804
  1047
    (let
hoelzl@29804
  1048
       r = rapprox_rat prec m1 m2;
hoelzl@29804
  1049
       f = Float 1 (s1 - s2)
hoelzl@29804
  1050
     in
hoelzl@29804
  1051
       f * r)"  
obua@16782
  1052
hoelzl@31098
  1053
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
hoelzl@45772
  1054
  using rapprox_rat[of "mantissa x" "mantissa y" prec]
hoelzl@45772
  1055
  by (cases x y rule: float.exhaust[case_product float.exhaust])
hoelzl@45772
  1056
     (simp split: split_if_asm
hoelzl@45772
  1057
           add: real_of_float_simp pow2_diff field_simps divide_le_eq mult_less_0_iff zero_less_mult_iff)
obua@16782
  1058
hoelzl@29804
  1059
lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
hoelzl@29804
  1060
proof -
hoelzl@31098
  1061
  have "1 \<le> 1 / real x" using `0 < x` and `x < 1` unfolding less_float_def by auto
hoelzl@31098
  1062
  also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
hoelzl@29804
  1063
  finally show ?thesis unfolding le_float_def by auto
hoelzl@29804
  1064
qed
hoelzl@29804
  1065
hoelzl@29804
  1066
lemma float_divr_nonpos_pos_upper_bound: assumes "x \<le> 0" and "0 < y" shows "float_divr prec x y \<le> 0"
hoelzl@29804
  1067
proof (cases x, cases y)
hoelzl@29804
  1068
  fix xm xe ym ye :: int
hoelzl@29804
  1069
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
hoelzl@31098
  1070
  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
  1071
  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
  1072
hoelzl@31098
  1073
  have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
hoelzl@31098
  1074
  moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonpos_pos[OF `xm \<le> 0` `0 < ym`] .
hoelzl@29804
  1075
  ultimately show "float_divr prec x y \<le> 0"
hoelzl@31098
  1076
    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
  1077
qed
obua@16782
  1078
hoelzl@29804
  1079
lemma float_divr_nonneg_neg_upper_bound: assumes "0 \<le> x" and "y < 0" shows "float_divr prec x y \<le> 0"
hoelzl@29804
  1080
proof (cases x, cases y)
hoelzl@29804
  1081
  fix xm xe ym ye :: int
hoelzl@29804
  1082
  assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
hoelzl@31098
  1083
  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
  1084
  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
  1085
  hence "0 < - ym" by auto
hoelzl@29804
  1086
hoelzl@31098
  1087
  have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
hoelzl@31098
  1088
  moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonneg_neg[OF `0 \<le> xm` `ym < 0`] .
hoelzl@29804
  1089
  ultimately show "float_divr prec x y \<le> 0"
hoelzl@31098
  1090
    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
  1091
qed
hoelzl@29804
  1092
haftmann@30960
  1093
primrec round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1094
"round_down prec (Float m e) = (let d = bitlen m - int prec in
hoelzl@29804
  1095
     if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
hoelzl@29804
  1096
              else Float m e)"
hoelzl@29804
  1097
haftmann@30960
  1098
primrec round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1099
"round_up prec (Float m e) = (let d = bitlen m - int prec in
hoelzl@29804
  1100
  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
  1101
           else Float m e)"
obua@16782
  1102
hoelzl@31098
  1103
lemma round_up: "real x \<le> real (round_up prec x)"
hoelzl@29804
  1104
proof (cases x)
hoelzl@29804
  1105
  case (Float m e)
hoelzl@29804
  1106
  let ?d = "bitlen m - int prec"
hoelzl@29804
  1107
  let ?p = "(2::int)^nat ?d"
hoelzl@29804
  1108
  have "0 < ?p" by auto
hoelzl@29804
  1109
  show "?thesis"
hoelzl@29804
  1110
  proof (cases "0 < ?d")
hoelzl@29804
  1111
    case True
huffman@35344
  1112
    hence pow_d: "pow2 ?d = real ?p" using pow2_int[symmetric] by simp
hoelzl@29804
  1113
    show ?thesis
hoelzl@29804
  1114
    proof (cases "m mod ?p = 0")
hoelzl@29804
  1115
      case True
hoelzl@29804
  1116
      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
  1117
      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
  1118
        by (auto simp add: pow2_add `0 < ?d` pow_d)
hoelzl@29804
  1119
      thus ?thesis
wenzelm@32960
  1120
        unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]
wenzelm@32960
  1121
        by auto
hoelzl@29804
  1122
    next
hoelzl@29804
  1123
      case False
hoelzl@29804
  1124
      have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
huffman@44766
  1125
      also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib mult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
hoelzl@31098
  1126
      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
  1127
        unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
wenzelm@32960
  1128
        by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
hoelzl@29804
  1129
      thus ?thesis
wenzelm@32960
  1130
        unfolding Float round_up.simps Let_def if_not_P[OF `\<not> m mod ?p = 0`] if_P[OF `0 < ?d`] .
hoelzl@29804
  1131
    qed
hoelzl@29804
  1132
  next
hoelzl@29804
  1133
    case False
hoelzl@29804
  1134
    show ?thesis
hoelzl@29804
  1135
      unfolding Float round_up.simps Let_def if_not_P[OF False] .. 
hoelzl@29804
  1136
  qed
hoelzl@29804
  1137
qed
obua@16782
  1138
hoelzl@31098
  1139
lemma round_down: "real (round_down prec x) \<le> real x"
hoelzl@29804
  1140
proof (cases x)
hoelzl@29804
  1141
  case (Float m e)
hoelzl@29804
  1142
  let ?d = "bitlen m - int prec"
hoelzl@29804
  1143
  let ?p = "(2::int)^nat ?d"
hoelzl@29804
  1144
  have "0 < ?p" by auto
hoelzl@29804
  1145
  show "?thesis"
hoelzl@29804
  1146
  proof (cases "0 < ?d")
hoelzl@29804
  1147
    case True
huffman@35344
  1148
    hence pow_d: "pow2 ?d = real ?p" using pow2_int[symmetric] by simp
hoelzl@29804
  1149
    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
  1150
    also have "\<dots> \<le> m" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
hoelzl@31098
  1151
    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
  1152
      unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of _ m, symmetric]
hoelzl@29804
  1153
      by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
hoelzl@29804
  1154
    thus ?thesis
hoelzl@29804
  1155
      unfolding Float round_down.simps Let_def if_P[OF `0 < ?d`] .
hoelzl@29804
  1156
  next
hoelzl@29804
  1157
    case False
hoelzl@29804
  1158
    show ?thesis
hoelzl@29804
  1159
      unfolding Float round_down.simps Let_def if_not_P[OF False] .. 
hoelzl@29804
  1160
  qed
hoelzl@29804
  1161
qed
hoelzl@29804
  1162
hoelzl@29804
  1163
definition lb_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1164
"lb_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let
hoelzl@29804
  1165
    l = bitlen m - int prec
hoelzl@29804
  1166
  in if l > 0 then Float (m div (2^nat l)) (e + l)
hoelzl@29804
  1167
              else Float m e)"
obua@16782
  1168
hoelzl@29804
  1169
definition ub_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1170
"ub_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let
hoelzl@29804
  1171
    l = bitlen m - int prec
hoelzl@29804
  1172
  in if l > 0 then Float (m div (2^nat l) + 1) (e + l)
hoelzl@29804
  1173
              else Float m e)"
obua@16782
  1174
hoelzl@31098
  1175
lemma lb_mult: "real (lb_mult prec x y) \<le> real (x * y)"
hoelzl@29804
  1176
proof (cases "normfloat (x * y)")
hoelzl@29804
  1177
  case (Float m e)
hoelzl@29804
  1178
  hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
hoelzl@29804
  1179
  let ?l = "bitlen m - int prec"
hoelzl@31098
  1180
  have "real (lb_mult prec x y) \<le> real (normfloat (x * y))"
hoelzl@29804
  1181
  proof (cases "?l > 0")
hoelzl@29804
  1182
    case False thus ?thesis unfolding lb_mult_def Float Let_def float.cases by auto
hoelzl@29804
  1183
  next
hoelzl@29804
  1184
    case True
hoelzl@29804
  1185
    have "real (m div 2^(nat ?l)) * pow2 ?l \<le> real m"
hoelzl@29804
  1186
    proof -
huffman@35344
  1187
      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
  1188
        using `?l > 0` by auto
hoelzl@29804
  1189
      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
  1190
      also have "\<dots> = real m" unfolding zmod_zdiv_equality[symmetric] ..
hoelzl@29804
  1191
      finally show ?thesis by auto
hoelzl@29804
  1192
    qed
huffman@36778
  1193
    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
  1194
  qed
hoelzl@31098
  1195
  also have "\<dots> = real (x * y)" unfolding normfloat ..
hoelzl@29804
  1196
  finally show ?thesis .
hoelzl@29804
  1197
qed
obua@16782
  1198
hoelzl@31098
  1199
lemma ub_mult: "real (x * y) \<le> real (ub_mult prec x y)"
hoelzl@29804
  1200
proof (cases "normfloat (x * y)")
hoelzl@29804
  1201
  case (Float m e)
hoelzl@29804
  1202
  hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
hoelzl@29804
  1203
  let ?l = "bitlen m - int prec"
hoelzl@31098
  1204
  have "real (x * y) = real (normfloat (x * y))" unfolding normfloat ..
hoelzl@31098
  1205
  also have "\<dots> \<le> real (ub_mult prec x y)"
hoelzl@29804
  1206
  proof (cases "?l > 0")
hoelzl@29804
  1207
    case False thus ?thesis unfolding ub_mult_def Float Let_def float.cases by auto
hoelzl@29804
  1208
  next
hoelzl@29804
  1209
    case True
hoelzl@29804
  1210
    have "real m \<le> real (m div 2^(nat ?l) + 1) * pow2 ?l"
hoelzl@29804
  1211
    proof -
hoelzl@29804
  1212
      have "m mod 2^(nat ?l) < 2^(nat ?l)" by (rule pos_mod_bound) auto
huffman@44766
  1213
      hence mod_uneq: "real (m mod 2^(nat ?l)) \<le> 1 * 2^(nat ?l)" unfolding mult_1 real_of_int_less_iff[symmetric] by auto
hoelzl@29804
  1214
      
hoelzl@29804
  1215
      have "real m = real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding zmod_zdiv_equality[symmetric] ..
hoelzl@29804
  1216
      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
  1217
      also have "\<dots> \<le> (real (m div 2^(nat ?l)) + 1) * 2^(nat ?l)" unfolding left_distrib using mod_uneq by auto
hoelzl@29804
  1218
      finally show ?thesis unfolding pow2_int[symmetric] using True by auto
hoelzl@29804
  1219
    qed
huffman@36778
  1220
    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
  1221
  qed
hoelzl@29804
  1222
  finally show ?thesis .
hoelzl@29804
  1223
qed
hoelzl@29804
  1224
haftmann@30960
  1225
primrec float_abs :: "float \<Rightarrow> float" where
haftmann@30960
  1226
  "float_abs (Float m e) = Float \<bar>m\<bar> e"
hoelzl@29804
  1227
hoelzl@29804
  1228
instantiation float :: abs begin
hoelzl@29804
  1229
definition abs_float_def: "\<bar>x\<bar> = float_abs x"
hoelzl@29804
  1230
instance ..
hoelzl@29804
  1231
end
obua@16782
  1232
hoelzl@31098
  1233
lemma real_of_float_abs: "real \<bar>x :: float\<bar> = \<bar>real x\<bar>" 
hoelzl@29804
  1234
proof (cases x)
hoelzl@29804
  1235
  case (Float m e)
hoelzl@29804
  1236
  have "\<bar>real m\<bar> * pow2 e = \<bar>real m * pow2 e\<bar>" unfolding abs_mult by auto
hoelzl@31098
  1237
  thus ?thesis unfolding Float abs_float_def float_abs.simps real_of_float_simp by auto
hoelzl@29804
  1238
qed
hoelzl@29804
  1239
haftmann@30960
  1240
primrec floor_fl :: "float \<Rightarrow> float" where
haftmann@30960
  1241
  "floor_fl (Float m e) = (if 0 \<le> e then Float m e
hoelzl@29804
  1242
                                  else Float (m div (2 ^ (nat (-e)))) 0)"
obua@16782
  1243
hoelzl@31098
  1244
lemma floor_fl: "real (floor_fl x) \<le> real x"
hoelzl@29804
  1245
proof (cases x)
hoelzl@29804
  1246
  case (Float m e)
hoelzl@29804
  1247
  show ?thesis
hoelzl@29804
  1248
  proof (cases "0 \<le> e")
hoelzl@29804
  1249
    case False
hoelzl@29804
  1250
    hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
hoelzl@31098
  1251
    have "real (Float (m div (2 ^ (nat (-e)))) 0) = real (m div 2 ^ (nat (-e)))" unfolding real_of_float_simp by auto
hoelzl@29804
  1252
    also have "\<dots> \<le> real m / real ((2::int) ^ (nat (-e)))" using real_of_int_div4 .
huffman@36778
  1253
    also have "\<dots> = real m * inverse (2 ^ (nat (-e)))" unfolding real_of_int_power real_number_of divide_inverse ..
hoelzl@31098
  1254
    also have "\<dots> = real (Float m e)" unfolding real_of_float_simp me_eq pow2_int pow2_neg[of e] ..
hoelzl@29804
  1255
    finally show ?thesis unfolding Float floor_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
hoelzl@29804
  1256
  next
hoelzl@29804
  1257
    case True thus ?thesis unfolding Float by auto
hoelzl@29804
  1258
  qed
hoelzl@29804
  1259
qed
obua@16782
  1260
hoelzl@29804
  1261
lemma floor_pos_exp: assumes floor: "Float m e = floor_fl x" shows "0 \<le> e"
hoelzl@29804
  1262
proof (cases x)
hoelzl@29804
  1263
  case (Float mx me)
hoelzl@29804
  1264
  from floor[unfolded Float floor_fl.simps] show ?thesis by (cases "0 \<le> me", auto)
hoelzl@29804
  1265
qed
hoelzl@29804
  1266
hoelzl@29804
  1267
declare floor_fl.simps[simp del]
obua@16782
  1268
haftmann@30960
  1269
primrec ceiling_fl :: "float \<Rightarrow> float" where
haftmann@30960
  1270
  "ceiling_fl (Float m e) = (if 0 \<le> e then Float m e
hoelzl@29804
  1271
                                    else Float (m div (2 ^ (nat (-e))) + 1) 0)"
obua@16782
  1272
hoelzl@31098
  1273
lemma ceiling_fl: "real x \<le> real (ceiling_fl x)"
hoelzl@29804
  1274
proof (cases x)
hoelzl@29804
  1275
  case (Float m e)
hoelzl@29804
  1276
  show ?thesis
hoelzl@29804
  1277
  proof (cases "0 \<le> e")
hoelzl@29804
  1278
    case False
hoelzl@29804
  1279
    hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
hoelzl@31098
  1280
    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
  1281
    also have "\<dots> = real m / real ((2::int) ^ (nat (-e)))" unfolding real_of_int_power real_number_of divide_inverse ..
hoelzl@29804
  1282
    also have "\<dots> \<le> 1 + real (m div 2 ^ (nat (-e)))" using real_of_int_div3[unfolded diff_le_eq] .
hoelzl@31098
  1283
    also have "\<dots> = real (Float (m div (2 ^ (nat (-e))) + 1) 0)" unfolding real_of_float_simp by auto
hoelzl@29804
  1284
    finally show ?thesis unfolding Float ceiling_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
hoelzl@29804
  1285
  next
hoelzl@29804
  1286
    case True thus ?thesis unfolding Float by auto
hoelzl@29804
  1287
  qed
hoelzl@29804
  1288
qed
hoelzl@29804
  1289
hoelzl@29804
  1290
declare ceiling_fl.simps[simp del]
hoelzl@29804
  1291
hoelzl@29804
  1292
definition lb_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1293
"lb_mod prec x ub lb = x - ceiling_fl (float_divr prec x lb) * ub"
hoelzl@29804
  1294
hoelzl@29804
  1295
definition ub_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29804
  1296
"ub_mod prec x ub lb = x - floor_fl (float_divl prec x ub) * lb"
obua@16782
  1297
hoelzl@31098
  1298
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
  1299
  assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
hoelzl@31098
  1300
  shows "real (lb_mod prec x ub lb) \<le> ?x - ?k * y"
hoelzl@29804
  1301
proof -
wenzelm@33555
  1302
  have "?lb \<le> ?ub" using assms by auto
wenzelm@33555
  1303
  have "0 \<le> ?lb" and "?lb \<noteq> 0" using assms by auto
hoelzl@29804
  1304
  have "?k * y \<le> ?x" using assms by auto
hoelzl@29804
  1305
  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
  1306
  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
  1307
  finally show ?thesis unfolding lb_mod_def real_of_float_sub real_of_float_mult by auto
hoelzl@29804
  1308
qed
obua@16782
  1309
hoelzl@31098
  1310
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
  1311
  assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
hoelzl@31098
  1312
  shows "?x - ?k * y \<le> real (ub_mod prec x ub lb)"
hoelzl@29804
  1313
proof -
wenzelm@33555
  1314
  have "?lb \<le> ?ub" using assms by auto
wenzelm@33555
  1315
  hence "0 \<le> ?lb" and "0 \<le> ?ub" and "?ub \<noteq> 0" using assms by auto
hoelzl@31098
  1316
  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
  1317
  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
  1318
  also have "\<dots> \<le> ?k * y" using assms by auto
hoelzl@31098
  1319
  finally show ?thesis unfolding ub_mod_def real_of_float_sub real_of_float_mult by auto
hoelzl@29804
  1320
qed
obua@16782
  1321
hoelzl@39161
  1322
lemma le_float_def'[code]: "f \<le> g = (case f - g of Float a b \<Rightarrow> a \<le> 0)"
hoelzl@29804
  1323
proof -
hoelzl@31098
  1324
  have le_transfer: "(f \<le> g) = (real (f - g) \<le> 0)" by (auto simp add: le_float_def)
hoelzl@29804
  1325
  from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
hoelzl@31098
  1326
  with le_transfer have le_transfer': "f \<le> g = (real (Float a b) \<le> 0)" by simp
hoelzl@29804
  1327
  show ?thesis by (simp add: le_transfer' f_diff_g float_le_zero)
hoelzl@29804
  1328
qed
hoelzl@29804
  1329
hoelzl@39161
  1330
lemma less_float_def'[code]: "f < g = (case f - g of Float a b \<Rightarrow> a < 0)"
hoelzl@29804
  1331
proof -
hoelzl@31098
  1332
  have less_transfer: "(f < g) = (real (f - g) < 0)" by (auto simp add: less_float_def)
hoelzl@29804
  1333
  from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
hoelzl@31098
  1334
  with less_transfer have less_transfer': "f < g = (real (Float a b) < 0)" by simp
hoelzl@29804
  1335
  show ?thesis by (simp add: less_transfer' f_diff_g float_less_zero)
hoelzl@29804
  1336
qed
wenzelm@20771
  1337
obua@16782
  1338
end