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(* Title: HOL/Archimedean_Field.thy 
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Author: Brian Huffman 

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*) 
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header {* 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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then have 

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"of_int y \<le> x" "x < of_int (y + 1)" 

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"of_int z \<le> x" "x < of_int (z + 1)" 

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by simp_all 

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from le_less_trans [OF `of_int y \<le> x` `x < of_int (z + 1)`] 

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le_less_trans [OF `of_int z \<le> x` `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_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 (neg_numeral v) = neg_numeral v" 
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using floor_of_int [of "neg_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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"neg_numeral v \<le> floor x \<longleftrightarrow> neg_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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"neg_numeral v < floor x \<longleftrightarrow> neg_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> neg_numeral v \<longleftrightarrow> x < neg_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 < neg_numeral v \<longleftrightarrow> x < neg_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_neg_numeral [simp]: 
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"floor (x + neg_numeral v) = floor x + neg_numeral v" 
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using floor_add_of_int [of x "neg_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_neg_numeral [simp]: 
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"floor (x  neg_numeral v) = floor x  neg_numeral v" 
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using floor_diff_of_int [of x "neg_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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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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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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unfolding ceiling_def using floor_correct [of " x"] by simp 

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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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lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)" 

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using ceiling_correct .. 

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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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lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z  1" 

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using ceiling_le_iff [of x "z  1"] by simp 

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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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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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using ceiling_of_int [of "of_nat n"] by simp 

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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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30096  349 
text {* Ceiling with numerals *} 
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lemma ceiling_zero [simp]: "ceiling 0 = 0" 

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using ceiling_of_int [of 0] by simp 

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lemma ceiling_one [simp]: "ceiling 1 = 1" 

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using ceiling_of_int [of 1] by simp 

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lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v" 
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using ceiling_of_int [of "numeral v"] by simp 
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359 

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lemma ceiling_neg_numeral [simp]: "ceiling (neg_numeral v) = neg_numeral v" 
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using ceiling_of_int [of "neg_numeral v"] by simp 
30096  362 

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lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0" 

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by (simp add: ceiling_le_iff) 

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lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1" 

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by (simp add: ceiling_le_iff) 

368 

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lemma ceiling_le_numeral [simp]: 
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"ceiling x \<le> numeral v \<longleftrightarrow> x \<le> numeral v" 
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by (simp add: ceiling_le_iff) 
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372 

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lemma ceiling_le_neg_numeral [simp]: 
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"ceiling x \<le> neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v" 
30096  375 
by (simp add: ceiling_le_iff) 
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lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> 1" 

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by (simp add: ceiling_less_iff) 

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lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0" 

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by (simp add: ceiling_less_iff) 

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lemma ceiling_less_numeral [simp]: 
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"ceiling x < numeral v \<longleftrightarrow> x \<le> numeral v  1" 
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by (simp add: ceiling_less_iff) 
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386 

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lemma ceiling_less_neg_numeral [simp]: 
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"ceiling x < neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v  1" 
30096  389 
by (simp add: ceiling_less_iff) 
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lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> 1 < x" 

392 
by (simp add: le_ceiling_iff) 

393 

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lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x" 

395 
by (simp add: le_ceiling_iff) 

396 

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lemma numeral_le_ceiling [simp]: 
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"numeral v \<le> ceiling x \<longleftrightarrow> numeral v  1 < x" 
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by (simp add: le_ceiling_iff) 
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400 

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lemma neg_numeral_le_ceiling [simp]: 
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"neg_numeral v \<le> ceiling x \<longleftrightarrow> neg_numeral v  1 < x" 
30096  403 
by (simp add: le_ceiling_iff) 
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lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x" 

406 
by (simp add: less_ceiling_iff) 

407 

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lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x" 

409 
by (simp add: less_ceiling_iff) 

410 

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lemma numeral_less_ceiling [simp]: 
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"numeral v < ceiling x \<longleftrightarrow> numeral v < x" 
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by (simp add: less_ceiling_iff) 
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lemma neg_numeral_less_ceiling [simp]: 
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"neg_numeral v < ceiling x \<longleftrightarrow> neg_numeral v < x" 
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by (simp add: less_ceiling_iff) 
418 

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text {* Addition and subtraction of integers *} 

420 

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lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z" 

422 
using ceiling_correct [of x] by (simp add: ceiling_unique) 

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lemma ceiling_add_numeral [simp]: 
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"ceiling (x + numeral v) = ceiling x + numeral v" 
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using ceiling_add_of_int [of x "numeral v"] by simp 
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427 

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lemma ceiling_add_neg_numeral [simp]: 
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"ceiling (x + neg_numeral v) = ceiling x + neg_numeral v" 
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430 
using ceiling_add_of_int [of x "neg_numeral v"] by simp 
30096  431 

432 
lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1" 

433 
using ceiling_add_of_int [of x 1] by simp 

434 

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lemma ceiling_diff_of_int [simp]: "ceiling (x  of_int z) = ceiling x  z" 

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using ceiling_add_of_int [of x " z"] by (simp add: algebra_simps) 

437 

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lemma ceiling_diff_numeral [simp]: 
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"ceiling (x  numeral v) = ceiling x  numeral v" 
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440 
using ceiling_diff_of_int [of x "numeral v"] by simp 
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441 

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lemma ceiling_diff_neg_numeral [simp]: 
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"ceiling (x  neg_numeral v) = ceiling x  neg_numeral v" 
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using ceiling_diff_of_int [of x "neg_numeral v"] by simp 
30096  445 

446 
lemma ceiling_diff_one [simp]: "ceiling (x  1) = ceiling x  1" 

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using ceiling_diff_of_int [of x 1] by simp 

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449 

450 
subsection {* Negation *} 

451 

30102  452 
lemma floor_minus: "floor ( x) =  ceiling x" 
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unfolding ceiling_def by simp 
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30102  455 
lemma ceiling_minus: "ceiling ( x) =  floor x" 
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unfolding ceiling_def by simp 
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458 
end 