wenzelm@41959: (*  Title:      HOL/Archimedean_Field.thy
wenzelm@41959:     Author:     Brian Huffman
huffman@30096: *)
huffman@30096: 
huffman@30096: header {* Archimedean Fields, Floor and Ceiling Functions *}
huffman@30096: 
huffman@30096: theory Archimedean_Field
huffman@30096: imports Main
huffman@30096: begin
huffman@30096: 
huffman@30096: subsection {* Class of Archimedean fields *}
huffman@30096: 
huffman@30096: text {* Archimedean fields have no infinite elements. *}
huffman@30096: 
huffman@47108: class archimedean_field = linordered_field +
huffman@30096:   assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
huffman@30096: 
huffman@30096: lemma ex_less_of_int:
huffman@30096:   fixes x :: "'a::archimedean_field" shows "\<exists>z. x < of_int z"
huffman@30096: proof -
huffman@30096:   from ex_le_of_int obtain z where "x \<le> of_int z" ..
huffman@30096:   then have "x < of_int (z + 1)" by simp
huffman@30096:   then show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma ex_of_int_less:
huffman@30096:   fixes x :: "'a::archimedean_field" shows "\<exists>z. of_int z < x"
huffman@30096: proof -
huffman@30096:   from ex_less_of_int obtain z where "- x < of_int z" ..
huffman@30096:   then have "of_int (- z) < x" by simp
huffman@30096:   then show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma ex_less_of_nat:
huffman@30096:   fixes x :: "'a::archimedean_field" shows "\<exists>n. x < of_nat n"
huffman@30096: proof -
huffman@30096:   obtain z where "x < of_int z" using ex_less_of_int ..
huffman@30096:   also have "\<dots> \<le> of_int (int (nat z))" by simp
huffman@30096:   also have "\<dots> = of_nat (nat z)" by (simp only: of_int_of_nat_eq)
huffman@30096:   finally show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma ex_le_of_nat:
huffman@30096:   fixes x :: "'a::archimedean_field" shows "\<exists>n. x \<le> of_nat n"
huffman@30096: proof -
huffman@30096:   obtain n where "x < of_nat n" using ex_less_of_nat ..
huffman@30096:   then have "x \<le> of_nat n" by simp
huffman@30096:   then show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: text {* Archimedean fields have no infinitesimal elements. *}
huffman@30096: 
huffman@30096: lemma ex_inverse_of_nat_Suc_less:
huffman@30096:   fixes x :: "'a::archimedean_field"
huffman@30096:   assumes "0 < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
huffman@30096: proof -
huffman@30096:   from `0 < x` have "0 < inverse x"
huffman@30096:     by (rule positive_imp_inverse_positive)
huffman@30096:   obtain n where "inverse x < of_nat n"
huffman@30096:     using ex_less_of_nat ..
huffman@30096:   then obtain m where "inverse x < of_nat (Suc m)"
huffman@30096:     using `0 < inverse x` by (cases n) (simp_all del: of_nat_Suc)
huffman@30096:   then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
huffman@30096:     using `0 < inverse x` by (rule less_imp_inverse_less)
huffman@30096:   then have "inverse (of_nat (Suc m)) < x"
huffman@30096:     using `0 < x` by (simp add: nonzero_inverse_inverse_eq)
huffman@30096:   then show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma ex_inverse_of_nat_less:
huffman@30096:   fixes x :: "'a::archimedean_field"
huffman@30096:   assumes "0 < x" shows "\<exists>n>0. inverse (of_nat n) < x"
huffman@30096:   using ex_inverse_of_nat_Suc_less [OF `0 < x`] by auto
huffman@30096: 
huffman@30096: lemma ex_less_of_nat_mult:
huffman@30096:   fixes x :: "'a::archimedean_field"
huffman@30096:   assumes "0 < x" shows "\<exists>n. y < of_nat n * x"
huffman@30096: proof -
huffman@30096:   obtain n where "y / x < of_nat n" using ex_less_of_nat ..
huffman@30096:   with `0 < x` have "y < of_nat n * x" by (simp add: pos_divide_less_eq)
huffman@30096:   then show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: 
huffman@30096: subsection {* Existence and uniqueness of floor function *}
huffman@30096: 
huffman@30096: lemma exists_least_lemma:
huffman@30096:   assumes "\<not> P 0" and "\<exists>n. P n"
huffman@30096:   shows "\<exists>n. \<not> P n \<and> P (Suc n)"
huffman@30096: proof -
huffman@30096:   from `\<exists>n. P n` have "P (Least P)" by (rule LeastI_ex)
huffman@30096:   with `\<not> P 0` obtain n where "Least P = Suc n"
huffman@30096:     by (cases "Least P") auto
huffman@30096:   then have "n < Least P" by simp
huffman@30096:   then have "\<not> P n" by (rule not_less_Least)
huffman@30096:   then have "\<not> P n \<and> P (Suc n)"
huffman@30096:     using `P (Least P)` `Least P = Suc n` by simp
huffman@30096:   then show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma floor_exists:
huffman@30096:   fixes x :: "'a::archimedean_field"
huffman@30096:   shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
huffman@30096: proof (cases)
huffman@30096:   assume "0 \<le> x"
huffman@30096:   then have "\<not> x < of_nat 0" by simp
huffman@30096:   then have "\<exists>n. \<not> x < of_nat n \<and> x < of_nat (Suc n)"
huffman@30096:     using ex_less_of_nat by (rule exists_least_lemma)
huffman@30096:   then obtain n where "\<not> x < of_nat n \<and> x < of_nat (Suc n)" ..
huffman@30096:   then have "of_int (int n) \<le> x \<and> x < of_int (int n + 1)" by simp
huffman@30096:   then show ?thesis ..
huffman@30096: next
huffman@30096:   assume "\<not> 0 \<le> x"
huffman@30096:   then have "\<not> - x \<le> of_nat 0" by simp
huffman@30096:   then have "\<exists>n. \<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)"
huffman@30096:     using ex_le_of_nat by (rule exists_least_lemma)
huffman@30096:   then obtain n where "\<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" ..
huffman@30096:   then have "of_int (- int n - 1) \<le> x \<and> x < of_int (- int n - 1 + 1)" by simp
huffman@30096:   then show ?thesis ..
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma floor_exists1:
huffman@30096:   fixes x :: "'a::archimedean_field"
huffman@30096:   shows "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
huffman@30096: proof (rule ex_ex1I)
huffman@30096:   show "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
huffman@30096:     by (rule floor_exists)
huffman@30096: next
huffman@30096:   fix y z assume
huffman@30096:     "of_int y \<le> x \<and> x < of_int (y + 1)"
huffman@30096:     "of_int z \<le> x \<and> x < of_int (z + 1)"
huffman@30096:   then have
huffman@30096:     "of_int y \<le> x" "x < of_int (y + 1)"
huffman@30096:     "of_int z \<le> x" "x < of_int (z + 1)"
huffman@30096:     by simp_all
huffman@30096:   from le_less_trans [OF `of_int y \<le> x` `x < of_int (z + 1)`]
huffman@30096:        le_less_trans [OF `of_int z \<le> x` `x < of_int (y + 1)`]
huffman@30096:   show "y = z" by (simp del: of_int_add)
huffman@30096: qed
huffman@30096: 
huffman@30096: 
huffman@30096: subsection {* Floor function *}
huffman@30096: 
bulwahn@43732: class floor_ceiling = archimedean_field +
bulwahn@43732:   fixes floor :: "'a \<Rightarrow> int"
bulwahn@43732:   assumes floor_correct: "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
huffman@30096: 
huffman@30096: notation (xsymbols)
huffman@30096:   floor  ("\<lfloor>_\<rfloor>")
huffman@30096: 
huffman@30096: notation (HTML output)
huffman@30096:   floor  ("\<lfloor>_\<rfloor>")
huffman@30096: 
huffman@30096: lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
huffman@30096:   using floor_correct [of x] floor_exists1 [of x] by auto
huffman@30096: 
huffman@30096: lemma of_int_floor_le: "of_int (floor x) \<le> x"
huffman@30096:   using floor_correct ..
huffman@30096: 
huffman@30096: lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
huffman@30096: proof
huffman@30096:   assume "z \<le> floor x"
huffman@30096:   then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
huffman@30096:   also have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
huffman@30096:   finally show "of_int z \<le> x" .
huffman@30096: next
huffman@30096:   assume "of_int z \<le> x"
huffman@30096:   also have "x < of_int (floor x + 1)" using floor_correct ..
huffman@30096:   finally show "z \<le> floor x" by (simp del: of_int_add)
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
huffman@30096:   by (simp add: not_le [symmetric] le_floor_iff)
huffman@30096: 
huffman@30096: lemma less_floor_iff: "z < floor x \<longleftrightarrow> of_int z + 1 \<le> x"
huffman@30096:   using le_floor_iff [of "z + 1" x] by auto
huffman@30096: 
huffman@30096: lemma floor_le_iff: "floor x \<le> z \<longleftrightarrow> x < of_int z + 1"
huffman@30096:   by (simp add: not_less [symmetric] less_floor_iff)
huffman@30096: 
huffman@30096: lemma floor_mono: assumes "x \<le> y" shows "floor x \<le> floor y"
huffman@30096: proof -
huffman@30096:   have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
huffman@30096:   also note `x \<le> y`
huffman@30096:   finally show ?thesis by (simp add: le_floor_iff)
huffman@30096: qed
huffman@30096: 
huffman@30096: lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
huffman@30096:   by (auto simp add: not_le [symmetric] floor_mono)
huffman@30096: 
huffman@30096: lemma floor_of_int [simp]: "floor (of_int z) = z"
huffman@30096:   by (rule floor_unique) simp_all
huffman@30096: 
huffman@30096: lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
huffman@30096:   using floor_of_int [of "of_nat n"] by simp
huffman@30096: 
huffman@47307: lemma le_floor_add: "floor x + floor y \<le> floor (x + y)"
huffman@47307:   by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
huffman@47307: 
huffman@30096: text {* Floor with numerals *}
huffman@30096: 
huffman@30096: lemma floor_zero [simp]: "floor 0 = 0"
huffman@30096:   using floor_of_int [of 0] by simp
huffman@30096: 
huffman@30096: lemma floor_one [simp]: "floor 1 = 1"
huffman@30096:   using floor_of_int [of 1] by simp
huffman@30096: 
huffman@47108: lemma floor_numeral [simp]: "floor (numeral v) = numeral v"
huffman@47108:   using floor_of_int [of "numeral v"] by simp
huffman@47108: 
huffman@47108: lemma floor_neg_numeral [simp]: "floor (neg_numeral v) = neg_numeral v"
huffman@47108:   using floor_of_int [of "neg_numeral v"] by simp
huffman@30096: 
huffman@30096: lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
huffman@30096:   by (simp add: le_floor_iff)
huffman@30096: 
huffman@30096: lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
huffman@30096:   by (simp add: le_floor_iff)
huffman@30096: 
huffman@47108: lemma numeral_le_floor [simp]:
huffman@47108:   "numeral v \<le> floor x \<longleftrightarrow> numeral v \<le> x"
huffman@47108:   by (simp add: le_floor_iff)
huffman@47108: 
huffman@47108: lemma neg_numeral_le_floor [simp]:
huffman@47108:   "neg_numeral v \<le> floor x \<longleftrightarrow> neg_numeral v \<le> x"
huffman@30096:   by (simp add: le_floor_iff)
huffman@30096: 
huffman@30096: lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
huffman@30096:   by (simp add: less_floor_iff)
huffman@30096: 
huffman@30096: lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
huffman@30096:   by (simp add: less_floor_iff)
huffman@30096: 
huffman@47108: lemma numeral_less_floor [simp]:
huffman@47108:   "numeral v < floor x \<longleftrightarrow> numeral v + 1 \<le> x"
huffman@47108:   by (simp add: less_floor_iff)
huffman@47108: 
huffman@47108: lemma neg_numeral_less_floor [simp]:
huffman@47108:   "neg_numeral v < floor x \<longleftrightarrow> neg_numeral v + 1 \<le> x"
huffman@30096:   by (simp add: less_floor_iff)
huffman@30096: 
huffman@30096: lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
huffman@30096:   by (simp add: floor_le_iff)
huffman@30096: 
huffman@30096: lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
huffman@30096:   by (simp add: floor_le_iff)
huffman@30096: 
huffman@47108: lemma floor_le_numeral [simp]:
huffman@47108:   "floor x \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
huffman@47108:   by (simp add: floor_le_iff)
huffman@47108: 
huffman@47108: lemma floor_le_neg_numeral [simp]:
huffman@47108:   "floor x \<le> neg_numeral v \<longleftrightarrow> x < neg_numeral v + 1"
huffman@30096:   by (simp add: floor_le_iff)
huffman@30096: 
huffman@30096: lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
huffman@30096:   by (simp add: floor_less_iff)
huffman@30096: 
huffman@30096: lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
huffman@30096:   by (simp add: floor_less_iff)
huffman@30096: 
huffman@47108: lemma floor_less_numeral [simp]:
huffman@47108:   "floor x < numeral v \<longleftrightarrow> x < numeral v"
huffman@47108:   by (simp add: floor_less_iff)
huffman@47108: 
huffman@47108: lemma floor_less_neg_numeral [simp]:
huffman@47108:   "floor x < neg_numeral v \<longleftrightarrow> x < neg_numeral v"
huffman@30096:   by (simp add: floor_less_iff)
huffman@30096: 
huffman@30096: text {* Addition and subtraction of integers *}
huffman@30096: 
huffman@30096: lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
huffman@30096:   using floor_correct [of x] by (simp add: floor_unique)
huffman@30096: 
huffman@47108: lemma floor_add_numeral [simp]:
huffman@47108:     "floor (x + numeral v) = floor x + numeral v"
huffman@47108:   using floor_add_of_int [of x "numeral v"] by simp
huffman@47108: 
huffman@47108: lemma floor_add_neg_numeral [simp]:
huffman@47108:     "floor (x + neg_numeral v) = floor x + neg_numeral v"
huffman@47108:   using floor_add_of_int [of x "neg_numeral v"] by simp
huffman@30096: 
huffman@30096: lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
huffman@30096:   using floor_add_of_int [of x 1] by simp
huffman@30096: 
huffman@30096: lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
huffman@30096:   using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
huffman@30096: 
huffman@47108: lemma floor_diff_numeral [simp]:
huffman@47108:   "floor (x - numeral v) = floor x - numeral v"
huffman@47108:   using floor_diff_of_int [of x "numeral v"] by simp
huffman@47108: 
huffman@47108: lemma floor_diff_neg_numeral [simp]:
huffman@47108:   "floor (x - neg_numeral v) = floor x - neg_numeral v"
huffman@47108:   using floor_diff_of_int [of x "neg_numeral v"] by simp
huffman@30096: 
huffman@30096: lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
huffman@30096:   using floor_diff_of_int [of x 1] by simp
huffman@30096: 
huffman@30096: 
huffman@30096: subsection {* Ceiling function *}
huffman@30096: 
huffman@30096: definition
bulwahn@43732:   ceiling :: "'a::floor_ceiling \<Rightarrow> int" where
bulwahn@43733:   "ceiling x = - floor (- x)"
huffman@30096: 
huffman@30096: notation (xsymbols)
huffman@30096:   ceiling  ("\<lceil>_\<rceil>")
huffman@30096: 
huffman@30096: notation (HTML output)
huffman@30096:   ceiling  ("\<lceil>_\<rceil>")
huffman@30096: 
huffman@30096: lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
huffman@30096:   unfolding ceiling_def using floor_correct [of "- x"] by simp
huffman@30096: 
huffman@30096: lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
huffman@30096:   unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
huffman@30096: 
huffman@30096: lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
huffman@30096:   using ceiling_correct ..
huffman@30096: 
huffman@30096: lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
huffman@30096:   unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
huffman@30096: 
huffman@30096: lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
huffman@30096:   by (simp add: not_le [symmetric] ceiling_le_iff)
huffman@30096: 
huffman@30096: lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
huffman@30096:   using ceiling_le_iff [of x "z - 1"] by simp
huffman@30096: 
huffman@30096: lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
huffman@30096:   by (simp add: not_less [symmetric] ceiling_less_iff)
huffman@30096: 
huffman@30096: lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
huffman@30096:   unfolding ceiling_def by (simp add: floor_mono)
huffman@30096: 
huffman@30096: lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
huffman@30096:   by (auto simp add: not_le [symmetric] ceiling_mono)
huffman@30096: 
huffman@30096: lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
huffman@30096:   by (rule ceiling_unique) simp_all
huffman@30096: 
huffman@30096: lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
huffman@30096:   using ceiling_of_int [of "of_nat n"] by simp
huffman@30096: 
huffman@47307: lemma ceiling_add_le: "ceiling (x + y) \<le> ceiling x + ceiling y"
huffman@47307:   by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
huffman@47307: 
huffman@30096: text {* Ceiling with numerals *}
huffman@30096: 
huffman@30096: lemma ceiling_zero [simp]: "ceiling 0 = 0"
huffman@30096:   using ceiling_of_int [of 0] by simp
huffman@30096: 
huffman@30096: lemma ceiling_one [simp]: "ceiling 1 = 1"
huffman@30096:   using ceiling_of_int [of 1] by simp
huffman@30096: 
huffman@47108: lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v"
huffman@47108:   using ceiling_of_int [of "numeral v"] by simp
huffman@47108: 
huffman@47108: lemma ceiling_neg_numeral [simp]: "ceiling (neg_numeral v) = neg_numeral v"
huffman@47108:   using ceiling_of_int [of "neg_numeral v"] by simp
huffman@30096: 
huffman@30096: lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
huffman@30096:   by (simp add: ceiling_le_iff)
huffman@30096: 
huffman@30096: lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
huffman@30096:   by (simp add: ceiling_le_iff)
huffman@30096: 
huffman@47108: lemma ceiling_le_numeral [simp]:
huffman@47108:   "ceiling x \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
huffman@47108:   by (simp add: ceiling_le_iff)
huffman@47108: 
huffman@47108: lemma ceiling_le_neg_numeral [simp]:
huffman@47108:   "ceiling x \<le> neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v"
huffman@30096:   by (simp add: ceiling_le_iff)
huffman@30096: 
huffman@30096: lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
huffman@30096:   by (simp add: ceiling_less_iff)
huffman@30096: 
huffman@30096: lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
huffman@30096:   by (simp add: ceiling_less_iff)
huffman@30096: 
huffman@47108: lemma ceiling_less_numeral [simp]:
huffman@47108:   "ceiling x < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
huffman@47108:   by (simp add: ceiling_less_iff)
huffman@47108: 
huffman@47108: lemma ceiling_less_neg_numeral [simp]:
huffman@47108:   "ceiling x < neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v - 1"
huffman@30096:   by (simp add: ceiling_less_iff)
huffman@30096: 
huffman@30096: lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
huffman@30096:   by (simp add: le_ceiling_iff)
huffman@30096: 
huffman@30096: lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
huffman@30096:   by (simp add: le_ceiling_iff)
huffman@30096: 
huffman@47108: lemma numeral_le_ceiling [simp]:
huffman@47108:   "numeral v \<le> ceiling x \<longleftrightarrow> numeral v - 1 < x"
huffman@47108:   by (simp add: le_ceiling_iff)
huffman@47108: 
huffman@47108: lemma neg_numeral_le_ceiling [simp]:
huffman@47108:   "neg_numeral v \<le> ceiling x \<longleftrightarrow> neg_numeral v - 1 < x"
huffman@30096:   by (simp add: le_ceiling_iff)
huffman@30096: 
huffman@30096: lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
huffman@30096:   by (simp add: less_ceiling_iff)
huffman@30096: 
huffman@30096: lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
huffman@30096:   by (simp add: less_ceiling_iff)
huffman@30096: 
huffman@47108: lemma numeral_less_ceiling [simp]:
huffman@47108:   "numeral v < ceiling x \<longleftrightarrow> numeral v < x"
huffman@47108:   by (simp add: less_ceiling_iff)
huffman@47108: 
huffman@47108: lemma neg_numeral_less_ceiling [simp]:
huffman@47108:   "neg_numeral v < ceiling x \<longleftrightarrow> neg_numeral v < x"
huffman@30096:   by (simp add: less_ceiling_iff)
huffman@30096: 
huffman@30096: text {* Addition and subtraction of integers *}
huffman@30096: 
huffman@30096: lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z"
huffman@30096:   using ceiling_correct [of x] by (simp add: ceiling_unique)
huffman@30096: 
huffman@47108: lemma ceiling_add_numeral [simp]:
huffman@47108:     "ceiling (x + numeral v) = ceiling x + numeral v"
huffman@47108:   using ceiling_add_of_int [of x "numeral v"] by simp
huffman@47108: 
huffman@47108: lemma ceiling_add_neg_numeral [simp]:
huffman@47108:     "ceiling (x + neg_numeral v) = ceiling x + neg_numeral v"
huffman@47108:   using ceiling_add_of_int [of x "neg_numeral v"] by simp
huffman@30096: 
huffman@30096: lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
huffman@30096:   using ceiling_add_of_int [of x 1] by simp
huffman@30096: 
huffman@30096: lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
huffman@30096:   using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
huffman@30096: 
huffman@47108: lemma ceiling_diff_numeral [simp]:
huffman@47108:   "ceiling (x - numeral v) = ceiling x - numeral v"
huffman@47108:   using ceiling_diff_of_int [of x "numeral v"] by simp
huffman@47108: 
huffman@47108: lemma ceiling_diff_neg_numeral [simp]:
huffman@47108:   "ceiling (x - neg_numeral v) = ceiling x - neg_numeral v"
huffman@47108:   using ceiling_diff_of_int [of x "neg_numeral v"] by simp
huffman@30096: 
huffman@30096: lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
huffman@30096:   using ceiling_diff_of_int [of x 1] by simp
huffman@30096: 
hoelzl@47592: lemma ceiling_diff_floor_le_1: "ceiling x - floor x \<le> 1"
hoelzl@47592: proof -
hoelzl@47592:   have "of_int \<lceil>x\<rceil> - 1 < x" 
hoelzl@47592:     using ceiling_correct[of x] by simp
hoelzl@47592:   also have "x < of_int \<lfloor>x\<rfloor> + 1"
hoelzl@47592:     using floor_correct[of x] by simp_all
hoelzl@47592:   finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
hoelzl@47592:     by simp
hoelzl@47592:   then show ?thesis
hoelzl@47592:     unfolding of_int_less_iff by simp
hoelzl@47592: qed
huffman@30096: 
huffman@30096: subsection {* Negation *}
huffman@30096: 
huffman@30102: lemma floor_minus: "floor (- x) = - ceiling x"
huffman@30096:   unfolding ceiling_def by simp
huffman@30096: 
huffman@30102: lemma ceiling_minus: "ceiling (- x) = - floor x"
huffman@30096:   unfolding ceiling_def by simp
huffman@30096: 
huffman@30096: end