src/HOL/Archimedean_Field.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58097 cfd3cff9387b
child 59613 7103019278f0
permissions -rw-r--r--
modernized header uniformly as section;
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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 {* Archimedean Fields, Floor and Ceiling Functions *}
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theory Archimedean_Field
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imports Main
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begin
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subsection {* Class of Archimedean fields *}
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text {* Archimedean fields have no infinite elements. *}
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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:
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  fixes x :: "'a::archimedean_field" shows "\<exists>z. x < of_int z"
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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:
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  fixes x :: "'a::archimedean_field" shows "\<exists>z. of_int z < x"
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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 ex_less_of_nat:
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  fixes x :: "'a::archimedean_field" shows "\<exists>n. x < of_nat n"
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proof -
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  obtain z where "x < of_int z" using ex_less_of_int ..
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  also have "\<dots> \<le> of_int (int (nat z))" by simp
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  also have "\<dots> = of_nat (nat z)" 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 ex_le_of_nat:
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  fixes x :: "'a::archimedean_field" shows "\<exists>n. x \<le> of_nat n"
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proof -
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  obtain n where "x < of_nat n" using ex_less_of_nat ..
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  then have "x \<le> of_nat n" by simp
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  then show ?thesis ..
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qed
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text {* Archimedean fields have no infinitesimal elements. *}
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lemma ex_inverse_of_nat_Suc_less:
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  fixes x :: "'a::archimedean_field"
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  assumes "0 < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
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proof -
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  from `0 < x` 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 ex_less_of_nat ..
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  then obtain m where "inverse x < of_nat (Suc m)"
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    using `0 < inverse x` 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 `0 < inverse x` by (rule less_imp_inverse_less)
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  then have "inverse (of_nat (Suc m)) < x"
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    using `0 < x` 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" shows "\<exists>n>0. inverse (of_nat n) < x"
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  using ex_inverse_of_nat_Suc_less [OF `0 < x`] 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" 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" using ex_less_of_nat ..
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  with `0 < x` have "y < of_nat n * x" by (simp add: pos_divide_less_eq)
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  then show ?thesis ..
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qed
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subsection {* Existence and uniqueness of floor function *}
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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 `\<exists>n. P n` have "P (Least P)" by (rule LeastI_ex)
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  with `\<not> P 0` 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" by simp
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  then have "\<not> P n" by (rule not_less_Least)
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  then have "\<not> P n \<and> P (Suc n)"
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    using `P (Least P)` `Least P = Suc n` 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)
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  assume "0 \<le> x"
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  then have "\<not> x < of_nat 0" 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 ex_less_of_nat 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)" by simp
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  then show ?thesis ..
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next
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  assume "\<not> 0 \<le> x"
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  then have "\<not> - x \<le> of_nat 0" 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 ex_le_of_nat 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)" by simp
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  then show ?thesis ..
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qed
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lemma floor_exists1:
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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 (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 assume
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    "of_int y \<le> x \<and> x < of_int (y + 1)"
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    "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)"]
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  show "y = z" by (simp del: of_int_add)
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qed
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subsection {* Floor function *}
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class floor_ceiling = archimedean_field +
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  fixes floor :: "'a \<Rightarrow> int"
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  assumes floor_correct: "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
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notation (xsymbols)
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  floor  ("\<lfloor>_\<rfloor>")
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notation (HTML output)
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  floor  ("\<lfloor>_\<rfloor>")
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lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
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  using floor_correct [of x] floor_exists1 [of x] by auto
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lemma of_int_floor_le: "of_int (floor x) \<le> x"
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  using floor_correct ..
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lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
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proof
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  assume "z \<le> floor x"
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  then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
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  also have "of_int (floor x) \<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 (floor x + 1)" using floor_correct ..
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  finally show "z \<le> floor x" by (simp del: of_int_add)
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qed
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lemma floor_less_iff: "floor x < 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 < floor x \<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: "floor x \<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 (floor t) \<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: assumes "x \<le> y" shows "floor x \<le> floor y"
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proof -
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  have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
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  also note `x \<le> y`
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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: "floor x < floor y \<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]: "floor (of_int z) = z"
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  by (rule floor_unique) simp_all
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lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
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  using floor_of_int [of "of_nat n"] by simp
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lemma le_floor_add: "floor x + floor y \<le> floor (x + y)"
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  by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
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text {* Floor with numerals *}
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lemma floor_zero [simp]: "floor 0 = 0"
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  using floor_of_int [of 0] by simp
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lemma floor_one [simp]: "floor 1 = 1"
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  using floor_of_int [of 1] by simp
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lemma floor_numeral [simp]: "floor (numeral v) = numeral v"
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  using floor_of_int [of "numeral v"] by simp
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lemma floor_neg_numeral [simp]: "floor (- numeral v) = - 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> floor x \<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> floor x \<longleftrightarrow> 1 \<le> x"
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  by (simp add: le_floor_iff)
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lemma numeral_le_floor [simp]:
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  "numeral v \<le> floor x \<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]:
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  "- numeral v \<le> floor x \<longleftrightarrow> - numeral v \<le> x"
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  by (simp add: le_floor_iff)
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lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
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  by (simp add: less_floor_iff)
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lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
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  by (simp add: less_floor_iff)
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lemma numeral_less_floor [simp]:
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  "numeral v < floor x \<longleftrightarrow> numeral v + 1 \<le> x"
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  by (simp add: less_floor_iff)
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lemma neg_numeral_less_floor [simp]:
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  "- numeral v < floor x \<longleftrightarrow> - numeral v + 1 \<le> x"
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  by (simp add: less_floor_iff)
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lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
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  by (simp add: floor_le_iff)
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lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
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  by (simp add: floor_le_iff)
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lemma floor_le_numeral [simp]:
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  "floor x \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
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  by (simp add: floor_le_iff)
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lemma floor_le_neg_numeral [simp]:
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  "floor x \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1"
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  by (simp add: floor_le_iff)
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lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
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  by (simp add: floor_less_iff)
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lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
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  by (simp add: floor_less_iff)
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lemma floor_less_numeral [simp]:
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  "floor x < numeral v \<longleftrightarrow> x < numeral v"
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  by (simp add: floor_less_iff)
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lemma floor_less_neg_numeral [simp]:
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  "floor x < - numeral v \<longleftrightarrow> x < - numeral v"
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  by (simp add: floor_less_iff)
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text {* Addition and subtraction of integers *}
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lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
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  using floor_correct [of x] by (simp add: floor_unique)
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lemma floor_add_numeral [simp]:
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    "floor (x + numeral v) = floor x + numeral v"
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  using floor_add_of_int [of x "numeral v"] by simp
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lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
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  using floor_add_of_int [of x 1] by simp
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lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
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  using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
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lemma floor_diff_numeral [simp]:
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  "floor (x - numeral v) = floor x - numeral v"
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  using floor_diff_of_int [of x "numeral v"] by simp
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lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
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  using floor_diff_of_int [of x 1] by simp
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lemma le_mult_floor:
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  assumes "0 \<le> a" and "0 \<le> b"
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  shows "floor a * floor b \<le> floor (a * b)"
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proof -
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  have "of_int (floor a) \<le> a"
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    and "of_int (floor b) \<le> b" by (auto intro: of_int_floor_le)
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  hence "of_int (floor a * floor b) \<le> a * b"
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    using assms by (auto intro!: mult_mono)
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  also have "a * b < of_int (floor (a * b) + 1)"  
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    using floor_correct[of "a * b"] by auto
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  finally show ?thesis unfolding of_int_less_iff by simp
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qed
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subsection {* Ceiling function *}
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definition
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  ceiling :: "'a::floor_ceiling \<Rightarrow> int" where
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  "ceiling x = - floor (- x)"
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   309
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notation (xsymbols)
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  ceiling  ("\<lceil>_\<rceil>")
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notation (HTML output)
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  ceiling  ("\<lceil>_\<rceil>")
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lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
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   317
  unfolding ceiling_def using floor_correct [of "- x"] by simp
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   319
lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
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  unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
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   321
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lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
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   323
  using ceiling_correct ..
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   324
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lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
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  unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
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lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
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  by (simp add: not_le [symmetric] ceiling_le_iff)
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   330
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lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
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   332
  using ceiling_le_iff [of x "z - 1"] by simp
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   333
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lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
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  by (simp add: not_less [symmetric] ceiling_less_iff)
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lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
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   338
  unfolding ceiling_def by (simp add: floor_mono)
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lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
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  by (auto simp add: not_le [symmetric] ceiling_mono)
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lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
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  by (rule ceiling_unique) simp_all
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lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
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   347
  using ceiling_of_int [of "of_nat n"] by simp
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   348
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lemma ceiling_add_le: "ceiling (x + y) \<le> ceiling x + ceiling y"
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  by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
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   351
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   352
text {* Ceiling with numerals *}
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   353
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   354
lemma ceiling_zero [simp]: "ceiling 0 = 0"
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   355
  using ceiling_of_int [of 0] by simp
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   357
lemma ceiling_one [simp]: "ceiling 1 = 1"
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   358
  using ceiling_of_int [of 1] by simp
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   359
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   360
lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v"
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   361
  using ceiling_of_int [of "numeral v"] by simp
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   362
haftmann@54489
   363
lemma ceiling_neg_numeral [simp]: "ceiling (- numeral v) = - numeral v"
haftmann@54489
   364
  using ceiling_of_int [of "- numeral v"] by simp
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   365
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   366
lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
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   367
  by (simp add: ceiling_le_iff)
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   368
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   369
lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
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   370
  by (simp add: ceiling_le_iff)
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   371
huffman@47108
   372
lemma ceiling_le_numeral [simp]:
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   373
  "ceiling x \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
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   374
  by (simp add: ceiling_le_iff)
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   375
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   376
lemma ceiling_le_neg_numeral [simp]:
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  "ceiling x \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v"
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   378
  by (simp add: ceiling_le_iff)
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   379
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   380
lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
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   381
  by (simp add: ceiling_less_iff)
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   382
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   383
lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
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   384
  by (simp add: ceiling_less_iff)
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   385
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   386
lemma ceiling_less_numeral [simp]:
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   387
  "ceiling x < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
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   388
  by (simp add: ceiling_less_iff)
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   389
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   390
lemma ceiling_less_neg_numeral [simp]:
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   391
  "ceiling x < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1"
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   392
  by (simp add: ceiling_less_iff)
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   393
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   394
lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
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   395
  by (simp add: le_ceiling_iff)
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   396
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   397
lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
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   398
  by (simp add: le_ceiling_iff)
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   399
huffman@47108
   400
lemma numeral_le_ceiling [simp]:
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   401
  "numeral v \<le> ceiling x \<longleftrightarrow> numeral v - 1 < x"
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   402
  by (simp add: le_ceiling_iff)
huffman@47108
   403
huffman@47108
   404
lemma neg_numeral_le_ceiling [simp]:
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   405
  "- numeral v \<le> ceiling x \<longleftrightarrow> - numeral v - 1 < x"
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   406
  by (simp add: le_ceiling_iff)
huffman@30096
   407
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   408
lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
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   409
  by (simp add: less_ceiling_iff)
huffman@30096
   410
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   411
lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
huffman@30096
   412
  by (simp add: less_ceiling_iff)
huffman@30096
   413
huffman@47108
   414
lemma numeral_less_ceiling [simp]:
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   415
  "numeral v < ceiling x \<longleftrightarrow> numeral v < x"
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   416
  by (simp add: less_ceiling_iff)
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   417
huffman@47108
   418
lemma neg_numeral_less_ceiling [simp]:
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   419
  "- numeral v < ceiling x \<longleftrightarrow> - numeral v < x"
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   420
  by (simp add: less_ceiling_iff)
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   421
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   422
text {* Addition and subtraction of integers *}
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   423
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   424
lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z"
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   425
  using ceiling_correct [of x] by (simp add: ceiling_unique)
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   426
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   427
lemma ceiling_add_numeral [simp]:
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   428
    "ceiling (x + numeral v) = ceiling x + numeral v"
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   429
  using ceiling_add_of_int [of x "numeral v"] by simp
huffman@47108
   430
huffman@30096
   431
lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
huffman@30096
   432
  using ceiling_add_of_int [of x 1] by simp
huffman@30096
   433
huffman@30096
   434
lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
huffman@30096
   435
  using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
huffman@30096
   436
huffman@47108
   437
lemma ceiling_diff_numeral [simp]:
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   438
  "ceiling (x - numeral v) = ceiling x - numeral v"
huffman@47108
   439
  using ceiling_diff_of_int [of x "numeral v"] by simp
huffman@47108
   440
huffman@30096
   441
lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
huffman@30096
   442
  using ceiling_diff_of_int [of x 1] by simp
huffman@30096
   443
hoelzl@58040
   444
lemma ceiling_split[arith_split]: "P (ceiling t) \<longleftrightarrow> (\<forall>i. of_int i - 1 < t \<and> t \<le> of_int i \<longrightarrow> P i)"
hoelzl@58040
   445
  by (auto simp add: ceiling_unique ceiling_correct)
hoelzl@58040
   446
hoelzl@47592
   447
lemma ceiling_diff_floor_le_1: "ceiling x - floor x \<le> 1"
hoelzl@47592
   448
proof -
hoelzl@47592
   449
  have "of_int \<lceil>x\<rceil> - 1 < x" 
hoelzl@47592
   450
    using ceiling_correct[of x] by simp
hoelzl@47592
   451
  also have "x < of_int \<lfloor>x\<rfloor> + 1"
hoelzl@47592
   452
    using floor_correct[of x] by simp_all
hoelzl@47592
   453
  finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
hoelzl@47592
   454
    by simp
hoelzl@47592
   455
  then show ?thesis
hoelzl@47592
   456
    unfolding of_int_less_iff by simp
hoelzl@47592
   457
qed
huffman@30096
   458
huffman@30096
   459
subsection {* Negation *}
huffman@30096
   460
huffman@30102
   461
lemma floor_minus: "floor (- x) = - ceiling x"
huffman@30096
   462
  unfolding ceiling_def by simp
huffman@30096
   463
huffman@30102
   464
lemma ceiling_minus: "ceiling (- x) = - floor x"
huffman@30096
   465
  unfolding ceiling_def by simp
huffman@30096
   466
huffman@30096
   467
end