src/HOL/Archimedean_Field.thy
author wenzelm
Fri Jul 22 11:00:43 2016 +0200 (2016-07-22)
changeset 63540 f8652d0534fa
parent 63489 cd540c8031a4
child 63597 bef0277ec73b
permissions -rw-r--r--
tuned proofs -- avoid unstructured calculation;
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(*  Title:      HOL/Archimedean_Field.thy
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    Author:     Brian Huffman
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*)
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section \<open>Archimedean Fields, Floor and Ceiling Functions\<close>
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theory Archimedean_Field
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imports Main
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begin
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lemma cInf_abs_ge:
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  fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
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  assumes "S \<noteq> {}"
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    and bdd: "\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a"
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  shows "\<bar>Inf S\<bar> \<le> a"
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proof -
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  have "Sup (uminus ` S) = - (Inf S)"
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  proof (rule antisym)
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    show "- (Inf S) \<le> Sup (uminus ` S)"
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      apply (subst minus_le_iff)
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      apply (rule cInf_greatest [OF \<open>S \<noteq> {}\<close>])
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      apply (subst minus_le_iff)
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      apply (rule cSup_upper)
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       apply force
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      using bdd
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      apply (force simp: abs_le_iff bdd_above_def)
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      done
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  next
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    show "Sup (uminus ` S) \<le> - Inf S"
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      apply (rule cSup_least)
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      using \<open>S \<noteq> {}\<close>
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       apply force
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      apply clarsimp
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      apply (rule cInf_lower)
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       apply assumption
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      using bdd
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      apply (simp add: bdd_below_def)
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      apply (rule_tac x = "- a" in exI)
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      apply force
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      done
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  qed
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  with cSup_abs_le [of "uminus ` S"] assms show ?thesis
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    by fastforce
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qed
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lemma cSup_asclose:
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  fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
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  assumes S: "S \<noteq> {}"
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    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
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  shows "\<bar>Sup S - l\<bar> \<le> e"
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proof -
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  have *: "\<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" for x l e :: 'a
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    by arith
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  have "bdd_above S"
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    using b by (auto intro!: bdd_aboveI[of _ "l + e"])
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  with S b show ?thesis
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    unfolding * by (auto intro!: cSup_upper2 cSup_least)
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qed
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lemma cInf_asclose:
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  fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
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  assumes S: "S \<noteq> {}"
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    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
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  shows "\<bar>Inf S - l\<bar> \<le> e"
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proof -
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  have *: "\<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" for x l e :: 'a
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    by arith
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  have "bdd_below S"
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    using b by (auto intro!: bdd_belowI[of _ "l - e"])
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  with S b show ?thesis
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    unfolding * by (auto intro!: cInf_lower2 cInf_greatest)
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qed
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subsection \<open>Class of Archimedean fields\<close>
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text \<open>Archimedean fields have no infinite elements.\<close>
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class archimedean_field = linordered_field +
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  assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
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lemma ex_less_of_int: "\<exists>z. x < of_int z"
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  for x :: "'a::archimedean_field"
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proof -
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  from ex_le_of_int obtain z where "x \<le> of_int z" ..
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  then have "x < of_int (z + 1)" by simp
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  then show ?thesis ..
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qed
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lemma ex_of_int_less: "\<exists>z. of_int z < x"
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  for x :: "'a::archimedean_field"
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proof -
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  from ex_less_of_int obtain z where "- x < of_int z" ..
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  then have "of_int (- z) < x" by simp
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  then show ?thesis ..
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qed
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lemma reals_Archimedean2: "\<exists>n. x < of_nat n"
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  for x :: "'a::archimedean_field"
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proof -
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  obtain z where "x < of_int z"
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    using ex_less_of_int ..
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  also have "\<dots> \<le> of_int (int (nat z))"
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    by simp
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  also have "\<dots> = of_nat (nat z)"
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    by (simp only: of_int_of_nat_eq)
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  finally show ?thesis ..
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qed
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lemma real_arch_simple: "\<exists>n. x \<le> of_nat n"
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  for x :: "'a::archimedean_field"
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proof -
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  obtain n where "x < of_nat n"
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    using reals_Archimedean2 ..
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  then have "x \<le> of_nat n"
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    by simp
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  then show ?thesis ..
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qed
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text \<open>Archimedean fields have no infinitesimal elements.\<close>
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lemma reals_Archimedean:
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  fixes x :: "'a::archimedean_field"
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  assumes "0 < x"
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  shows "\<exists>n. inverse (of_nat (Suc n)) < x"
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proof -
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  from \<open>0 < x\<close> have "0 < inverse x"
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    by (rule positive_imp_inverse_positive)
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  obtain n where "inverse x < of_nat n"
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    using reals_Archimedean2 ..
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  then obtain m where "inverse x < of_nat (Suc m)"
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    using \<open>0 < inverse x\<close> by (cases n) (simp_all del: of_nat_Suc)
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  then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
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    using \<open>0 < inverse x\<close> by (rule less_imp_inverse_less)
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  then have "inverse (of_nat (Suc m)) < x"
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    using \<open>0 < x\<close> by (simp add: nonzero_inverse_inverse_eq)
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  then show ?thesis ..
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qed
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lemma ex_inverse_of_nat_less:
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  fixes x :: "'a::archimedean_field"
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  assumes "0 < x"
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  shows "\<exists>n>0. inverse (of_nat n) < x"
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  using reals_Archimedean [OF \<open>0 < x\<close>] by auto
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lemma ex_less_of_nat_mult:
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  fixes x :: "'a::archimedean_field"
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  assumes "0 < x"
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  shows "\<exists>n. y < of_nat n * x"
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proof -
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  obtain n where "y / x < of_nat n"
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    using reals_Archimedean2 ..
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  with \<open>0 < x\<close> have "y < of_nat n * x"
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    by (simp add: pos_divide_less_eq)
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  then show ?thesis ..
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qed
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subsection \<open>Existence and uniqueness of floor function\<close>
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lemma exists_least_lemma:
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  assumes "\<not> P 0" and "\<exists>n. P n"
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  shows "\<exists>n. \<not> P n \<and> P (Suc n)"
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proof -
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  from \<open>\<exists>n. P n\<close> have "P (Least P)"
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    by (rule LeastI_ex)
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  with \<open>\<not> P 0\<close> obtain n where "Least P = Suc n"
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    by (cases "Least P") auto
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  then have "n < Least P"
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    by simp
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  then have "\<not> P n"
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    by (rule not_less_Least)
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  then have "\<not> P n \<and> P (Suc n)"
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    using \<open>P (Least P)\<close> \<open>Least P = Suc n\<close> by simp
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  then show ?thesis ..
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qed
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lemma floor_exists:
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  fixes x :: "'a::archimedean_field"
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  shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
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proof (cases "0 \<le> x")
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  case True
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  then have "\<not> x < of_nat 0"
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    by simp
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  then have "\<exists>n. \<not> x < of_nat n \<and> x < of_nat (Suc n)"
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    using reals_Archimedean2 by (rule exists_least_lemma)
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  then obtain n where "\<not> x < of_nat n \<and> x < of_nat (Suc n)" ..
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  then have "of_int (int n) \<le> x \<and> x < of_int (int n + 1)"
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    by simp
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  then show ?thesis ..
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next
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  case False
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  then have "\<not> - x \<le> of_nat 0"
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    by simp
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  then have "\<exists>n. \<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)"
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    using real_arch_simple by (rule exists_least_lemma)
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  then obtain n where "\<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" ..
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  then have "of_int (- int n - 1) \<le> x \<and> x < of_int (- int n - 1 + 1)"
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    by simp
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  then show ?thesis ..
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qed
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lemma floor_exists1: "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
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  for x :: "'a::archimedean_field"
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proof (rule ex_ex1I)
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  show "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
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    by (rule floor_exists)
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next
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  fix y z
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  assume "of_int y \<le> x \<and> x < of_int (y + 1)"
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    and "of_int z \<le> x \<and> x < of_int (z + 1)"
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  with le_less_trans [of "of_int y" "x" "of_int (z + 1)"]
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       le_less_trans [of "of_int z" "x" "of_int (y + 1)"] show "y = z"
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    by (simp del: of_int_add)
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qed
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subsection \<open>Floor function\<close>
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class floor_ceiling = archimedean_field +
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  fixes floor :: "'a \<Rightarrow> int"  ("\<lfloor>_\<rfloor>")
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  assumes floor_correct: "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)"
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lemma floor_unique: "of_int z \<le> x \<Longrightarrow> x < of_int z + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = z"
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  using floor_correct [of x] floor_exists1 [of x] by auto
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lemma floor_unique_iff: "\<lfloor>x\<rfloor> = a \<longleftrightarrow> of_int a \<le> x \<and> x < of_int a + 1"
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  for x :: "'a::floor_ceiling"
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  using floor_correct floor_unique by auto
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lemma of_int_floor_le [simp]: "of_int \<lfloor>x\<rfloor> \<le> x"
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  using floor_correct ..
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lemma le_floor_iff: "z \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> of_int z \<le> x"
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proof
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  assume "z \<le> \<lfloor>x\<rfloor>"
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  then have "(of_int z :: 'a) \<le> of_int \<lfloor>x\<rfloor>" by simp
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  also have "of_int \<lfloor>x\<rfloor> \<le> x" by (rule of_int_floor_le)
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  finally show "of_int z \<le> x" .
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next
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  assume "of_int z \<le> x"
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  also have "x < of_int (\<lfloor>x\<rfloor> + 1)" using floor_correct ..
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  finally show "z \<le> \<lfloor>x\<rfloor>" by (simp del: of_int_add)
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qed
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lemma floor_less_iff: "\<lfloor>x\<rfloor> < z \<longleftrightarrow> x < of_int z"
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  by (simp add: not_le [symmetric] le_floor_iff)
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lemma less_floor_iff: "z < \<lfloor>x\<rfloor> \<longleftrightarrow> of_int z + 1 \<le> x"
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  using le_floor_iff [of "z + 1" x] by auto
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lemma floor_le_iff: "\<lfloor>x\<rfloor> \<le> z \<longleftrightarrow> x < of_int z + 1"
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  by (simp add: not_less [symmetric] less_floor_iff)
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lemma floor_split[arith_split]: "P \<lfloor>t\<rfloor> \<longleftrightarrow> (\<forall>i. of_int i \<le> t \<and> t < of_int i + 1 \<longrightarrow> P i)"
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  by (metis floor_correct floor_unique less_floor_iff not_le order_refl)
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lemma floor_mono:
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  assumes "x \<le> y"
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  shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"
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proof -
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  have "of_int \<lfloor>x\<rfloor> \<le> x" by (rule of_int_floor_le)
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  also note \<open>x \<le> y\<close>
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  finally show ?thesis by (simp add: le_floor_iff)
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qed
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lemma floor_less_cancel: "\<lfloor>x\<rfloor> < \<lfloor>y\<rfloor> \<Longrightarrow> x < y"
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  by (auto simp add: not_le [symmetric] floor_mono)
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lemma floor_of_int [simp]: "\<lfloor>of_int z\<rfloor> = z"
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  by (rule floor_unique) simp_all
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lemma floor_of_nat [simp]: "\<lfloor>of_nat n\<rfloor> = int n"
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  using floor_of_int [of "of_nat n"] by simp
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lemma le_floor_add: "\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> \<le> \<lfloor>x + y\<rfloor>"
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  by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
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text \<open>Floor with numerals.\<close>
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lemma floor_zero [simp]: "\<lfloor>0\<rfloor> = 0"
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  using floor_of_int [of 0] by simp
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lemma floor_one [simp]: "\<lfloor>1\<rfloor> = 1"
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  using floor_of_int [of 1] by simp
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lemma floor_numeral [simp]: "\<lfloor>numeral v\<rfloor> = numeral v"
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  using floor_of_int [of "numeral v"] by simp
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lemma floor_neg_numeral [simp]: "\<lfloor>- numeral v\<rfloor> = - numeral v"
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  using floor_of_int [of "- numeral v"] by simp
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lemma zero_le_floor [simp]: "0 \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> 0 \<le> x"
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  by (simp add: le_floor_iff)
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lemma one_le_floor [simp]: "1 \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> 1 \<le> x"
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  by (simp add: le_floor_iff)
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lemma numeral_le_floor [simp]: "numeral v \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> numeral v \<le> x"
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  by (simp add: le_floor_iff)
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lemma neg_numeral_le_floor [simp]: "- numeral v \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> - numeral v \<le> x"
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  by (simp add: le_floor_iff)
huffman@30096
   305
wenzelm@61942
   306
lemma zero_less_floor [simp]: "0 < \<lfloor>x\<rfloor> \<longleftrightarrow> 1 \<le> x"
huffman@30096
   307
  by (simp add: less_floor_iff)
huffman@30096
   308
wenzelm@61942
   309
lemma one_less_floor [simp]: "1 < \<lfloor>x\<rfloor> \<longleftrightarrow> 2 \<le> x"
huffman@30096
   310
  by (simp add: less_floor_iff)
huffman@30096
   311
wenzelm@63489
   312
lemma numeral_less_floor [simp]: "numeral v < \<lfloor>x\<rfloor> \<longleftrightarrow> numeral v + 1 \<le> x"
huffman@47108
   313
  by (simp add: less_floor_iff)
huffman@47108
   314
wenzelm@63489
   315
lemma neg_numeral_less_floor [simp]: "- numeral v < \<lfloor>x\<rfloor> \<longleftrightarrow> - numeral v + 1 \<le> x"
huffman@30096
   316
  by (simp add: less_floor_iff)
huffman@30096
   317
wenzelm@61942
   318
lemma floor_le_zero [simp]: "\<lfloor>x\<rfloor> \<le> 0 \<longleftrightarrow> x < 1"
huffman@30096
   319
  by (simp add: floor_le_iff)
huffman@30096
   320
wenzelm@61942
   321
lemma floor_le_one [simp]: "\<lfloor>x\<rfloor> \<le> 1 \<longleftrightarrow> x < 2"
huffman@30096
   322
  by (simp add: floor_le_iff)
huffman@30096
   323
wenzelm@63489
   324
lemma floor_le_numeral [simp]: "\<lfloor>x\<rfloor> \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
huffman@47108
   325
  by (simp add: floor_le_iff)
huffman@47108
   326
wenzelm@63489
   327
lemma floor_le_neg_numeral [simp]: "\<lfloor>x\<rfloor> \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1"
huffman@30096
   328
  by (simp add: floor_le_iff)
huffman@30096
   329
wenzelm@61942
   330
lemma floor_less_zero [simp]: "\<lfloor>x\<rfloor> < 0 \<longleftrightarrow> x < 0"
huffman@30096
   331
  by (simp add: floor_less_iff)
huffman@30096
   332
wenzelm@61942
   333
lemma floor_less_one [simp]: "\<lfloor>x\<rfloor> < 1 \<longleftrightarrow> x < 1"
huffman@30096
   334
  by (simp add: floor_less_iff)
huffman@30096
   335
wenzelm@63489
   336
lemma floor_less_numeral [simp]: "\<lfloor>x\<rfloor> < numeral v \<longleftrightarrow> x < numeral v"
huffman@47108
   337
  by (simp add: floor_less_iff)
huffman@47108
   338
wenzelm@63489
   339
lemma floor_less_neg_numeral [simp]: "\<lfloor>x\<rfloor> < - numeral v \<longleftrightarrow> x < - numeral v"
huffman@30096
   340
  by (simp add: floor_less_iff)
huffman@30096
   341
wenzelm@63489
   342
wenzelm@63489
   343
text \<open>Addition and subtraction of integers.\<close>
huffman@30096
   344
wenzelm@61942
   345
lemma floor_add_of_int [simp]: "\<lfloor>x + of_int z\<rfloor> = \<lfloor>x\<rfloor> + z"
huffman@30096
   346
  using floor_correct [of x] by (simp add: floor_unique)
huffman@30096
   347
wenzelm@63489
   348
lemma floor_add_numeral [simp]: "\<lfloor>x + numeral v\<rfloor> = \<lfloor>x\<rfloor> + numeral v"
huffman@47108
   349
  using floor_add_of_int [of x "numeral v"] by simp
huffman@47108
   350
wenzelm@61942
   351
lemma floor_add_one [simp]: "\<lfloor>x + 1\<rfloor> = \<lfloor>x\<rfloor> + 1"
huffman@30096
   352
  using floor_add_of_int [of x 1] by simp
huffman@30096
   353
wenzelm@61942
   354
lemma floor_diff_of_int [simp]: "\<lfloor>x - of_int z\<rfloor> = \<lfloor>x\<rfloor> - z"
huffman@30096
   355
  using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
huffman@30096
   356
wenzelm@61942
   357
lemma floor_uminus_of_int [simp]: "\<lfloor>- (of_int z)\<rfloor> = - z"
lp15@59613
   358
  by (metis floor_diff_of_int [of 0] diff_0 floor_zero)
lp15@59613
   359
wenzelm@63489
   360
lemma floor_diff_numeral [simp]: "\<lfloor>x - numeral v\<rfloor> = \<lfloor>x\<rfloor> - numeral v"
huffman@47108
   361
  using floor_diff_of_int [of x "numeral v"] by simp
huffman@47108
   362
wenzelm@61942
   363
lemma floor_diff_one [simp]: "\<lfloor>x - 1\<rfloor> = \<lfloor>x\<rfloor> - 1"
huffman@30096
   364
  using floor_diff_of_int [of x 1] by simp
huffman@30096
   365
hoelzl@58097
   366
lemma le_mult_floor:
hoelzl@58097
   367
  assumes "0 \<le> a" and "0 \<le> b"
wenzelm@61942
   368
  shows "\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor> \<le> \<lfloor>a * b\<rfloor>"
hoelzl@58097
   369
proof -
wenzelm@63489
   370
  have "of_int \<lfloor>a\<rfloor> \<le> a" and "of_int \<lfloor>b\<rfloor> \<le> b"
wenzelm@63489
   371
    by (auto intro: of_int_floor_le)
wenzelm@63489
   372
  then have "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> a * b"
hoelzl@58097
   373
    using assms by (auto intro!: mult_mono)
wenzelm@61942
   374
  also have "a * b < of_int (\<lfloor>a * b\<rfloor> + 1)"
hoelzl@58097
   375
    using floor_correct[of "a * b"] by auto
wenzelm@63489
   376
  finally show ?thesis
wenzelm@63489
   377
    unfolding of_int_less_iff by simp
hoelzl@58097
   378
qed
hoelzl@58097
   379
wenzelm@63489
   380
lemma floor_divide_of_int_eq: "\<lfloor>of_int k / of_int l\<rfloor> = k div l"
wenzelm@63489
   381
  for k l :: int
haftmann@59984
   382
proof (cases "l = 0")
wenzelm@63489
   383
  case True
wenzelm@63489
   384
  then show ?thesis by simp
haftmann@59984
   385
next
haftmann@59984
   386
  case False
haftmann@59984
   387
  have *: "\<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> = 0"
haftmann@59984
   388
  proof (cases "l > 0")
wenzelm@63489
   389
    case True
wenzelm@63489
   390
    then show ?thesis
haftmann@59984
   391
      by (auto intro: floor_unique)
haftmann@59984
   392
  next
haftmann@59984
   393
    case False
wenzelm@63489
   394
    obtain r where "r = - l"
wenzelm@63489
   395
      by blast
wenzelm@63489
   396
    then have l: "l = - r"
wenzelm@63489
   397
      by simp
wenzelm@63540
   398
    with \<open>l \<noteq> 0\<close> False have "r > 0"
wenzelm@63489
   399
      by simp
wenzelm@63540
   400
    with l show ?thesis
wenzelm@63489
   401
      using pos_mod_bound [of r]
haftmann@59984
   402
      by (auto simp add: zmod_zminus2_eq_if less_le field_simps intro: floor_unique)
haftmann@59984
   403
  qed
haftmann@59984
   404
  have "(of_int k :: 'a) = of_int (k div l * l + k mod l)"
haftmann@59984
   405
    by simp
haftmann@59984
   406
  also have "\<dots> = (of_int (k div l) + of_int (k mod l) / of_int l) * of_int l"
haftmann@59984
   407
    using False by (simp only: of_int_add) (simp add: field_simps)
haftmann@59984
   408
  finally have "(of_int k / of_int l :: 'a) = \<dots> / of_int l"
hoelzl@63331
   409
    by simp
haftmann@59984
   410
  then have "(of_int k / of_int l :: 'a) = of_int (k div l) + of_int (k mod l) / of_int l"
haftmann@59984
   411
    using False by (simp only:) (simp add: field_simps)
hoelzl@63331
   412
  then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k div l) + of_int (k mod l) / of_int l :: 'a\<rfloor>"
haftmann@59984
   413
    by simp
haftmann@59984
   414
  then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l + of_int (k div l) :: 'a\<rfloor>"
haftmann@59984
   415
    by (simp add: ac_simps)
haftmann@60128
   416
  then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> + k div l"
haftmann@59984
   417
    by simp
wenzelm@63489
   418
  with * show ?thesis
wenzelm@63489
   419
    by simp
haftmann@59984
   420
qed
haftmann@59984
   421
wenzelm@63489
   422
lemma floor_divide_of_nat_eq: "\<lfloor>of_nat m / of_nat n\<rfloor> = of_nat (m div n)"
wenzelm@63489
   423
  for m n :: nat
haftmann@59984
   424
proof (cases "n = 0")
wenzelm@63489
   425
  case True
wenzelm@63489
   426
  then show ?thesis by simp
haftmann@59984
   427
next
haftmann@59984
   428
  case False
haftmann@59984
   429
  then have *: "\<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> = 0"
haftmann@59984
   430
    by (auto intro: floor_unique)
haftmann@59984
   431
  have "(of_nat m :: 'a) = of_nat (m div n * n + m mod n)"
haftmann@59984
   432
    by simp
haftmann@59984
   433
  also have "\<dots> = (of_nat (m div n) + of_nat (m mod n) / of_nat n) * of_nat n"
wenzelm@63489
   434
    using False by (simp only: of_nat_add) (simp add: field_simps)
haftmann@59984
   435
  finally have "(of_nat m / of_nat n :: 'a) = \<dots> / of_nat n"
hoelzl@63331
   436
    by simp
haftmann@59984
   437
  then have "(of_nat m / of_nat n :: 'a) = of_nat (m div n) + of_nat (m mod n) / of_nat n"
haftmann@59984
   438
    using False by (simp only:) simp
hoelzl@63331
   439
  then have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m div n) + of_nat (m mod n) / of_nat n :: 'a\<rfloor>"
haftmann@59984
   440
    by simp
haftmann@59984
   441
  then have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m mod n) / of_nat n + of_nat (m div n) :: 'a\<rfloor>"
haftmann@59984
   442
    by (simp add: ac_simps)
haftmann@59984
   443
  moreover have "(of_nat (m div n) :: 'a) = of_int (of_nat (m div n))"
haftmann@59984
   444
    by simp
wenzelm@63489
   445
  ultimately have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> =
wenzelm@63489
   446
      \<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> + of_nat (m div n)"
haftmann@59984
   447
    by (simp only: floor_add_of_int)
wenzelm@63489
   448
  with * show ?thesis
wenzelm@63489
   449
    by simp
haftmann@59984
   450
qed
haftmann@59984
   451
haftmann@59984
   452
wenzelm@60758
   453
subsection \<open>Ceiling function\<close>
huffman@30096
   454
wenzelm@61942
   455
definition ceiling :: "'a::floor_ceiling \<Rightarrow> int"  ("\<lceil>_\<rceil>")
wenzelm@61942
   456
  where "\<lceil>x\<rceil> = - \<lfloor>- x\<rfloor>"
huffman@30096
   457
wenzelm@61942
   458
lemma ceiling_correct: "of_int \<lceil>x\<rceil> - 1 < x \<and> x \<le> of_int \<lceil>x\<rceil>"
wenzelm@63489
   459
  unfolding ceiling_def using floor_correct [of "- x"]
wenzelm@63489
   460
  by (simp add: le_minus_iff)
huffman@30096
   461
wenzelm@63489
   462
lemma ceiling_unique: "of_int z - 1 < x \<Longrightarrow> x \<le> of_int z \<Longrightarrow> \<lceil>x\<rceil> = z"
huffman@30096
   463
  unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
huffman@30096
   464
wenzelm@61942
   465
lemma le_of_int_ceiling [simp]: "x \<le> of_int \<lceil>x\<rceil>"
huffman@30096
   466
  using ceiling_correct ..
huffman@30096
   467
wenzelm@61942
   468
lemma ceiling_le_iff: "\<lceil>x\<rceil> \<le> z \<longleftrightarrow> x \<le> of_int z"
huffman@30096
   469
  unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
huffman@30096
   470
wenzelm@61942
   471
lemma less_ceiling_iff: "z < \<lceil>x\<rceil> \<longleftrightarrow> of_int z < x"
huffman@30096
   472
  by (simp add: not_le [symmetric] ceiling_le_iff)
huffman@30096
   473
wenzelm@61942
   474
lemma ceiling_less_iff: "\<lceil>x\<rceil> < z \<longleftrightarrow> x \<le> of_int z - 1"
huffman@30096
   475
  using ceiling_le_iff [of x "z - 1"] by simp
huffman@30096
   476
wenzelm@61942
   477
lemma le_ceiling_iff: "z \<le> \<lceil>x\<rceil> \<longleftrightarrow> of_int z - 1 < x"
huffman@30096
   478
  by (simp add: not_less [symmetric] ceiling_less_iff)
huffman@30096
   479
wenzelm@61942
   480
lemma ceiling_mono: "x \<ge> y \<Longrightarrow> \<lceil>x\<rceil> \<ge> \<lceil>y\<rceil>"
huffman@30096
   481
  unfolding ceiling_def by (simp add: floor_mono)
huffman@30096
   482
wenzelm@61942
   483
lemma ceiling_less_cancel: "\<lceil>x\<rceil> < \<lceil>y\<rceil> \<Longrightarrow> x < y"
huffman@30096
   484
  by (auto simp add: not_le [symmetric] ceiling_mono)
huffman@30096
   485
wenzelm@61942
   486
lemma ceiling_of_int [simp]: "\<lceil>of_int z\<rceil> = z"
huffman@30096
   487
  by (rule ceiling_unique) simp_all
huffman@30096
   488
wenzelm@61942
   489
lemma ceiling_of_nat [simp]: "\<lceil>of_nat n\<rceil> = int n"
huffman@30096
   490
  using ceiling_of_int [of "of_nat n"] by simp
huffman@30096
   491
wenzelm@61942
   492
lemma ceiling_add_le: "\<lceil>x + y\<rceil> \<le> \<lceil>x\<rceil> + \<lceil>y\<rceil>"
huffman@47307
   493
  by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
huffman@47307
   494
wenzelm@63489
   495
wenzelm@63489
   496
text \<open>Ceiling with numerals.\<close>
huffman@30096
   497
wenzelm@61942
   498
lemma ceiling_zero [simp]: "\<lceil>0\<rceil> = 0"
huffman@30096
   499
  using ceiling_of_int [of 0] by simp
huffman@30096
   500
wenzelm@61942
   501
lemma ceiling_one [simp]: "\<lceil>1\<rceil> = 1"
huffman@30096
   502
  using ceiling_of_int [of 1] by simp
huffman@30096
   503
wenzelm@61942
   504
lemma ceiling_numeral [simp]: "\<lceil>numeral v\<rceil> = numeral v"
huffman@47108
   505
  using ceiling_of_int [of "numeral v"] by simp
huffman@47108
   506
wenzelm@61942
   507
lemma ceiling_neg_numeral [simp]: "\<lceil>- numeral v\<rceil> = - numeral v"
haftmann@54489
   508
  using ceiling_of_int [of "- numeral v"] by simp
huffman@30096
   509
wenzelm@61942
   510
lemma ceiling_le_zero [simp]: "\<lceil>x\<rceil> \<le> 0 \<longleftrightarrow> x \<le> 0"
huffman@30096
   511
  by (simp add: ceiling_le_iff)
huffman@30096
   512
wenzelm@61942
   513
lemma ceiling_le_one [simp]: "\<lceil>x\<rceil> \<le> 1 \<longleftrightarrow> x \<le> 1"
huffman@30096
   514
  by (simp add: ceiling_le_iff)
huffman@30096
   515
wenzelm@63489
   516
lemma ceiling_le_numeral [simp]: "\<lceil>x\<rceil> \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
huffman@47108
   517
  by (simp add: ceiling_le_iff)
huffman@47108
   518
wenzelm@63489
   519
lemma ceiling_le_neg_numeral [simp]: "\<lceil>x\<rceil> \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v"
huffman@30096
   520
  by (simp add: ceiling_le_iff)
huffman@30096
   521
wenzelm@61942
   522
lemma ceiling_less_zero [simp]: "\<lceil>x\<rceil> < 0 \<longleftrightarrow> x \<le> -1"
huffman@30096
   523
  by (simp add: ceiling_less_iff)
huffman@30096
   524
wenzelm@61942
   525
lemma ceiling_less_one [simp]: "\<lceil>x\<rceil> < 1 \<longleftrightarrow> x \<le> 0"
huffman@30096
   526
  by (simp add: ceiling_less_iff)
huffman@30096
   527
wenzelm@63489
   528
lemma ceiling_less_numeral [simp]: "\<lceil>x\<rceil> < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
huffman@47108
   529
  by (simp add: ceiling_less_iff)
huffman@47108
   530
wenzelm@63489
   531
lemma ceiling_less_neg_numeral [simp]: "\<lceil>x\<rceil> < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1"
huffman@30096
   532
  by (simp add: ceiling_less_iff)
huffman@30096
   533
wenzelm@61942
   534
lemma zero_le_ceiling [simp]: "0 \<le> \<lceil>x\<rceil> \<longleftrightarrow> -1 < x"
huffman@30096
   535
  by (simp add: le_ceiling_iff)
huffman@30096
   536
wenzelm@61942
   537
lemma one_le_ceiling [simp]: "1 \<le> \<lceil>x\<rceil> \<longleftrightarrow> 0 < x"
huffman@30096
   538
  by (simp add: le_ceiling_iff)
huffman@30096
   539
wenzelm@63489
   540
lemma numeral_le_ceiling [simp]: "numeral v \<le> \<lceil>x\<rceil> \<longleftrightarrow> numeral v - 1 < x"
huffman@47108
   541
  by (simp add: le_ceiling_iff)
huffman@47108
   542
wenzelm@63489
   543
lemma neg_numeral_le_ceiling [simp]: "- numeral v \<le> \<lceil>x\<rceil> \<longleftrightarrow> - numeral v - 1 < x"
huffman@30096
   544
  by (simp add: le_ceiling_iff)
huffman@30096
   545
wenzelm@61942
   546
lemma zero_less_ceiling [simp]: "0 < \<lceil>x\<rceil> \<longleftrightarrow> 0 < x"
huffman@30096
   547
  by (simp add: less_ceiling_iff)
huffman@30096
   548
wenzelm@61942
   549
lemma one_less_ceiling [simp]: "1 < \<lceil>x\<rceil> \<longleftrightarrow> 1 < x"
huffman@30096
   550
  by (simp add: less_ceiling_iff)
huffman@30096
   551
wenzelm@63489
   552
lemma numeral_less_ceiling [simp]: "numeral v < \<lceil>x\<rceil> \<longleftrightarrow> numeral v < x"
huffman@47108
   553
  by (simp add: less_ceiling_iff)
huffman@47108
   554
wenzelm@63489
   555
lemma neg_numeral_less_ceiling [simp]: "- numeral v < \<lceil>x\<rceil> \<longleftrightarrow> - numeral v < x"
huffman@30096
   556
  by (simp add: less_ceiling_iff)
huffman@30096
   557
wenzelm@61942
   558
lemma ceiling_altdef: "\<lceil>x\<rceil> = (if x = of_int \<lfloor>x\<rfloor> then \<lfloor>x\<rfloor> else \<lfloor>x\<rfloor> + 1)"
wenzelm@63489
   559
  by (intro ceiling_unique; simp, linarith?)
eberlm@61531
   560
wenzelm@61942
   561
lemma floor_le_ceiling [simp]: "\<lfloor>x\<rfloor> \<le> \<lceil>x\<rceil>"
wenzelm@61942
   562
  by (simp add: ceiling_altdef)
eberlm@61531
   563
wenzelm@63489
   564
wenzelm@63489
   565
text \<open>Addition and subtraction of integers.\<close>
huffman@30096
   566
wenzelm@61942
   567
lemma ceiling_add_of_int [simp]: "\<lceil>x + of_int z\<rceil> = \<lceil>x\<rceil> + z"
lp15@61649
   568
  using ceiling_correct [of x] by (simp add: ceiling_def)
huffman@30096
   569
wenzelm@61942
   570
lemma ceiling_add_numeral [simp]: "\<lceil>x + numeral v\<rceil> = \<lceil>x\<rceil> + numeral v"
huffman@47108
   571
  using ceiling_add_of_int [of x "numeral v"] by simp
huffman@47108
   572
wenzelm@61942
   573
lemma ceiling_add_one [simp]: "\<lceil>x + 1\<rceil> = \<lceil>x\<rceil> + 1"
huffman@30096
   574
  using ceiling_add_of_int [of x 1] by simp
huffman@30096
   575
wenzelm@61942
   576
lemma ceiling_diff_of_int [simp]: "\<lceil>x - of_int z\<rceil> = \<lceil>x\<rceil> - z"
huffman@30096
   577
  using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
huffman@30096
   578
wenzelm@61942
   579
lemma ceiling_diff_numeral [simp]: "\<lceil>x - numeral v\<rceil> = \<lceil>x\<rceil> - numeral v"
huffman@47108
   580
  using ceiling_diff_of_int [of x "numeral v"] by simp
huffman@47108
   581
wenzelm@61942
   582
lemma ceiling_diff_one [simp]: "\<lceil>x - 1\<rceil> = \<lceil>x\<rceil> - 1"
huffman@30096
   583
  using ceiling_diff_of_int [of x 1] by simp
huffman@30096
   584
wenzelm@61942
   585
lemma ceiling_split[arith_split]: "P \<lceil>t\<rceil> \<longleftrightarrow> (\<forall>i. of_int i - 1 < t \<and> t \<le> of_int i \<longrightarrow> P i)"
hoelzl@58040
   586
  by (auto simp add: ceiling_unique ceiling_correct)
hoelzl@58040
   587
wenzelm@61942
   588
lemma ceiling_diff_floor_le_1: "\<lceil>x\<rceil> - \<lfloor>x\<rfloor> \<le> 1"
hoelzl@47592
   589
proof -
hoelzl@63331
   590
  have "of_int \<lceil>x\<rceil> - 1 < x"
hoelzl@47592
   591
    using ceiling_correct[of x] by simp
hoelzl@47592
   592
  also have "x < of_int \<lfloor>x\<rfloor> + 1"
hoelzl@47592
   593
    using floor_correct[of x] by simp_all
hoelzl@47592
   594
  finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
hoelzl@47592
   595
    by simp
hoelzl@47592
   596
  then show ?thesis
hoelzl@47592
   597
    unfolding of_int_less_iff by simp
hoelzl@47592
   598
qed
huffman@30096
   599
wenzelm@63489
   600
wenzelm@60758
   601
subsection \<open>Negation\<close>
huffman@30096
   602
wenzelm@61942
   603
lemma floor_minus: "\<lfloor>- x\<rfloor> = - \<lceil>x\<rceil>"
huffman@30096
   604
  unfolding ceiling_def by simp
huffman@30096
   605
wenzelm@61942
   606
lemma ceiling_minus: "\<lceil>- x\<rceil> = - \<lfloor>x\<rfloor>"
huffman@30096
   607
  unfolding ceiling_def by simp
huffman@30096
   608
wenzelm@61942
   609
wenzelm@60758
   610
subsection \<open>Frac Function\<close>
lp15@59613
   611
wenzelm@63489
   612
definition frac :: "'a \<Rightarrow> 'a::floor_ceiling"
wenzelm@63489
   613
  where "frac x \<equiv> x - of_int \<lfloor>x\<rfloor>"
lp15@59613
   614
lp15@59613
   615
lemma frac_lt_1: "frac x < 1"
wenzelm@63489
   616
  by (simp add: frac_def) linarith
lp15@59613
   617
wenzelm@61070
   618
lemma frac_eq_0_iff [simp]: "frac x = 0 \<longleftrightarrow> x \<in> \<int>"
lp15@59613
   619
  by (simp add: frac_def) (metis Ints_cases Ints_of_int floor_of_int )
lp15@59613
   620
lp15@59613
   621
lemma frac_ge_0 [simp]: "frac x \<ge> 0"
wenzelm@63489
   622
  unfolding frac_def by linarith
lp15@59613
   623
wenzelm@61070
   624
lemma frac_gt_0_iff [simp]: "frac x > 0 \<longleftrightarrow> x \<notin> \<int>"
lp15@59613
   625
  by (metis frac_eq_0_iff frac_ge_0 le_less less_irrefl)
lp15@59613
   626
lp15@59613
   627
lemma frac_of_int [simp]: "frac (of_int z) = 0"
lp15@59613
   628
  by (simp add: frac_def)
lp15@59613
   629
hoelzl@63331
   630
lemma floor_add: "\<lfloor>x + y\<rfloor> = (if frac x + frac y < 1 then \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> else (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>) + 1)"
lp15@59613
   631
proof -
wenzelm@63489
   632
  have "\<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>" if "x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>)"
wenzelm@63489
   633
    using that by (metis add.commute floor_unique le_floor_add le_floor_iff of_int_add)
lp15@59613
   634
  moreover
wenzelm@63489
   635
  have "\<lfloor>x + y\<rfloor> = 1 + (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>)" if "\<not> x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>)"
wenzelm@63489
   636
    using that
wenzelm@63489
   637
    apply (simp add: floor_unique_iff)
wenzelm@63489
   638
    apply (auto simp add: algebra_simps)
wenzelm@63489
   639
    apply linarith
wenzelm@63489
   640
    done
lp15@59613
   641
  ultimately show ?thesis
lp15@59613
   642
    by (auto simp add: frac_def algebra_simps)
lp15@59613
   643
qed
lp15@59613
   644
wenzelm@63489
   645
lemma frac_add:
wenzelm@63489
   646
  "frac (x + y) = (if frac x + frac y < 1 then frac x + frac y else (frac x + frac y) - 1)"
lp15@59613
   647
  by (simp add: frac_def floor_add)
lp15@59613
   648
wenzelm@63489
   649
lemma frac_unique_iff: "frac x = a \<longleftrightarrow> x - a \<in> \<int> \<and> 0 \<le> a \<and> a < 1"
wenzelm@63489
   650
  for x :: "'a::floor_ceiling"
haftmann@62348
   651
  apply (auto simp: Ints_def frac_def algebra_simps floor_unique)
wenzelm@63489
   652
   apply linarith+
haftmann@62348
   653
  done
lp15@59613
   654
wenzelm@63489
   655
lemma frac_eq: "frac x = x \<longleftrightarrow> 0 \<le> x \<and> x < 1"
lp15@59613
   656
  by (simp add: frac_unique_iff)
hoelzl@63331
   657
wenzelm@63489
   658
lemma frac_neg: "frac (- x) = (if x \<in> \<int> then 0 else 1 - frac x)"
wenzelm@63489
   659
  for x :: "'a::floor_ceiling"
lp15@59613
   660
  apply (auto simp add: frac_unique_iff)
wenzelm@63489
   661
   apply (simp add: frac_def)
wenzelm@63489
   662
  apply (meson frac_lt_1 less_iff_diff_less_0 not_le not_less_iff_gr_or_eq)
wenzelm@63489
   663
  done
lp15@59613
   664
eberlm@61531
   665
eberlm@61531
   666
subsection \<open>Rounding to the nearest integer\<close>
eberlm@61531
   667
wenzelm@63489
   668
definition round :: "'a::floor_ceiling \<Rightarrow> int"
wenzelm@63489
   669
  where "round x = \<lfloor>x + 1/2\<rfloor>"
eberlm@61531
   670
wenzelm@63489
   671
lemma of_int_round_ge: "of_int (round x) \<ge> x - 1/2"
wenzelm@63489
   672
  and of_int_round_le: "of_int (round x) \<le> x + 1/2"
eberlm@61531
   673
  and of_int_round_abs_le: "\<bar>of_int (round x) - x\<bar> \<le> 1/2"
wenzelm@63489
   674
  and of_int_round_gt: "of_int (round x) > x - 1/2"
eberlm@61531
   675
proof -
wenzelm@63489
   676
  from floor_correct[of "x + 1/2"] have "x + 1/2 < of_int (round x) + 1"
wenzelm@63489
   677
    by (simp add: round_def)
wenzelm@63489
   678
  from add_strict_right_mono[OF this, of "-1"] show A: "of_int (round x) > x - 1/2"
wenzelm@63489
   679
    by simp
wenzelm@63489
   680
  then show "of_int (round x) \<ge> x - 1/2"
wenzelm@63489
   681
    by simp
wenzelm@63489
   682
  from floor_correct[of "x + 1/2"] show "of_int (round x) \<le> x + 1/2"
wenzelm@63489
   683
    by (simp add: round_def)
wenzelm@63489
   684
  with A show "\<bar>of_int (round x) - x\<bar> \<le> 1/2"
wenzelm@63489
   685
    by linarith
eberlm@61531
   686
qed
eberlm@61531
   687
eberlm@61531
   688
lemma round_of_int [simp]: "round (of_int n) = n"
eberlm@61531
   689
  unfolding round_def by (subst floor_unique_iff) force
eberlm@61531
   690
eberlm@61531
   691
lemma round_0 [simp]: "round 0 = 0"
eberlm@61531
   692
  using round_of_int[of 0] by simp
eberlm@61531
   693
eberlm@61531
   694
lemma round_1 [simp]: "round 1 = 1"
eberlm@61531
   695
  using round_of_int[of 1] by simp
eberlm@61531
   696
eberlm@61531
   697
lemma round_numeral [simp]: "round (numeral n) = numeral n"
eberlm@61531
   698
  using round_of_int[of "numeral n"] by simp
eberlm@61531
   699
eberlm@61531
   700
lemma round_neg_numeral [simp]: "round (-numeral n) = -numeral n"
eberlm@61531
   701
  using round_of_int[of "-numeral n"] by simp
eberlm@61531
   702
eberlm@61531
   703
lemma round_of_nat [simp]: "round (of_nat n) = of_nat n"
eberlm@61531
   704
  using round_of_int[of "int n"] by simp
eberlm@61531
   705
eberlm@61531
   706
lemma round_mono: "x \<le> y \<Longrightarrow> round x \<le> round y"
eberlm@61531
   707
  unfolding round_def by (intro floor_mono) simp
eberlm@61531
   708
eberlm@61531
   709
lemma round_unique: "of_int y > x - 1/2 \<Longrightarrow> of_int y \<le> x + 1/2 \<Longrightarrow> round x = y"
wenzelm@63489
   710
  unfolding round_def
eberlm@61531
   711
proof (rule floor_unique)
eberlm@61531
   712
  assume "x - 1 / 2 < of_int y"
wenzelm@63489
   713
  from add_strict_left_mono[OF this, of 1] show "x + 1 / 2 < of_int y + 1"
wenzelm@63489
   714
    by simp
eberlm@61531
   715
qed
eberlm@61531
   716
wenzelm@61942
   717
lemma round_altdef: "round x = (if frac x \<ge> 1/2 then \<lceil>x\<rceil> else \<lfloor>x\<rfloor>)"
eberlm@61531
   718
  by (cases "frac x \<ge> 1/2")
wenzelm@63489
   719
    (rule round_unique, ((simp add: frac_def field_simps ceiling_altdef; linarith)+)[2])+
eberlm@61531
   720
eberlm@61531
   721
lemma floor_le_round: "\<lfloor>x\<rfloor> \<le> round x"
eberlm@61531
   722
  unfolding round_def by (intro floor_mono) simp
eberlm@61531
   723
wenzelm@63489
   724
lemma ceiling_ge_round: "\<lceil>x\<rceil> \<ge> round x"
wenzelm@63489
   725
  unfolding round_altdef by simp
hoelzl@63331
   726
wenzelm@63489
   727
lemma round_diff_minimal: "\<bar>z - of_int (round z)\<bar> \<le> \<bar>z - of_int m\<bar>"
wenzelm@63489
   728
  for z :: "'a::floor_ceiling"
eberlm@61531
   729
proof (cases "of_int m \<ge> z")
eberlm@61531
   730
  case True
wenzelm@63489
   731
  then have "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int \<lceil>z\<rceil> - z\<bar>"
wenzelm@63489
   732
    unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith
wenzelm@63489
   733
  also have "of_int \<lceil>z\<rceil> - z \<ge> 0"
wenzelm@63489
   734
    by linarith
wenzelm@61942
   735
  with True have "\<bar>of_int \<lceil>z\<rceil> - z\<bar> \<le> \<bar>z - of_int m\<bar>"
eberlm@61531
   736
    by (simp add: ceiling_le_iff)
eberlm@61531
   737
  finally show ?thesis .
eberlm@61531
   738
next
eberlm@61531
   739
  case False
wenzelm@63489
   740
  then have "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int \<lfloor>z\<rfloor> - z\<bar>"
wenzelm@63489
   741
    unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith
wenzelm@63489
   742
  also have "z - of_int \<lfloor>z\<rfloor> \<ge> 0"
wenzelm@63489
   743
    by linarith
wenzelm@61942
   744
  with False have "\<bar>of_int \<lfloor>z\<rfloor> - z\<bar> \<le> \<bar>z - of_int m\<bar>"
eberlm@61531
   745
    by (simp add: le_floor_iff)
eberlm@61531
   746
  finally show ?thesis .
eberlm@61531
   747
qed
eberlm@61531
   748
huffman@30096
   749
end