src/HOL/Archimedean_Field.thy
 author paulson Mon Oct 09 15:34:23 2017 +0100 (20 months ago) changeset 66793 deabce3ccf1f parent 66515 85c505c98332 child 68499 d4312962161a permissions -rw-r--r--
```     1 (*  Title:      HOL/Archimedean_Field.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 section \<open>Archimedean Fields, Floor and Ceiling Functions\<close>
```
```     6
```
```     7 theory Archimedean_Field
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 lemma cInf_abs_ge:
```
```    12   fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
```
```    13   assumes "S \<noteq> {}"
```
```    14     and bdd: "\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a"
```
```    15   shows "\<bar>Inf S\<bar> \<le> a"
```
```    16 proof -
```
```    17   have "Sup (uminus ` S) = - (Inf S)"
```
```    18   proof (rule antisym)
```
```    19     show "- (Inf S) \<le> Sup (uminus ` S)"
```
```    20       apply (subst minus_le_iff)
```
```    21       apply (rule cInf_greatest [OF \<open>S \<noteq> {}\<close>])
```
```    22       apply (subst minus_le_iff)
```
```    23       apply (rule cSup_upper)
```
```    24        apply force
```
```    25       using bdd
```
```    26       apply (force simp: abs_le_iff bdd_above_def)
```
```    27       done
```
```    28   next
```
```    29     show "Sup (uminus ` S) \<le> - Inf S"
```
```    30       apply (rule cSup_least)
```
```    31       using \<open>S \<noteq> {}\<close>
```
```    32        apply force
```
```    33       apply clarsimp
```
```    34       apply (rule cInf_lower)
```
```    35        apply assumption
```
```    36       using bdd
```
```    37       apply (simp add: bdd_below_def)
```
```    38       apply (rule_tac x = "- a" in exI)
```
```    39       apply force
```
```    40       done
```
```    41   qed
```
```    42   with cSup_abs_le [of "uminus ` S"] assms show ?thesis
```
```    43     by fastforce
```
```    44 qed
```
```    45
```
```    46 lemma cSup_asclose:
```
```    47   fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
```
```    48   assumes S: "S \<noteq> {}"
```
```    49     and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
```
```    50   shows "\<bar>Sup S - l\<bar> \<le> e"
```
```    51 proof -
```
```    52   have *: "\<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" for x l e :: 'a
```
```    53     by arith
```
```    54   have "bdd_above S"
```
```    55     using b by (auto intro!: bdd_aboveI[of _ "l + e"])
```
```    56   with S b show ?thesis
```
```    57     unfolding * by (auto intro!: cSup_upper2 cSup_least)
```
```    58 qed
```
```    59
```
```    60 lemma cInf_asclose:
```
```    61   fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set"
```
```    62   assumes S: "S \<noteq> {}"
```
```    63     and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
```
```    64   shows "\<bar>Inf S - l\<bar> \<le> e"
```
```    65 proof -
```
```    66   have *: "\<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" for x l e :: 'a
```
```    67     by arith
```
```    68   have "bdd_below S"
```
```    69     using b by (auto intro!: bdd_belowI[of _ "l - e"])
```
```    70   with S b show ?thesis
```
```    71     unfolding * by (auto intro!: cInf_lower2 cInf_greatest)
```
```    72 qed
```
```    73
```
```    74
```
```    75 subsection \<open>Class of Archimedean fields\<close>
```
```    76
```
```    77 text \<open>Archimedean fields have no infinite elements.\<close>
```
```    78
```
```    79 class archimedean_field = linordered_field +
```
```    80   assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
```
```    81
```
```    82 lemma ex_less_of_int: "\<exists>z. x < of_int z"
```
```    83   for x :: "'a::archimedean_field"
```
```    84 proof -
```
```    85   from ex_le_of_int obtain z where "x \<le> of_int z" ..
```
```    86   then have "x < of_int (z + 1)" by simp
```
```    87   then show ?thesis ..
```
```    88 qed
```
```    89
```
```    90 lemma ex_of_int_less: "\<exists>z. of_int z < x"
```
```    91   for x :: "'a::archimedean_field"
```
```    92 proof -
```
```    93   from ex_less_of_int obtain z where "- x < of_int z" ..
```
```    94   then have "of_int (- z) < x" by simp
```
```    95   then show ?thesis ..
```
```    96 qed
```
```    97
```
```    98 lemma reals_Archimedean2: "\<exists>n. x < of_nat n"
```
```    99   for x :: "'a::archimedean_field"
```
```   100 proof -
```
```   101   obtain z where "x < of_int z"
```
```   102     using ex_less_of_int ..
```
```   103   also have "\<dots> \<le> of_int (int (nat z))"
```
```   104     by simp
```
```   105   also have "\<dots> = of_nat (nat z)"
```
```   106     by (simp only: of_int_of_nat_eq)
```
```   107   finally show ?thesis ..
```
```   108 qed
```
```   109
```
```   110 lemma real_arch_simple: "\<exists>n. x \<le> of_nat n"
```
```   111   for x :: "'a::archimedean_field"
```
```   112 proof -
```
```   113   obtain n where "x < of_nat n"
```
```   114     using reals_Archimedean2 ..
```
```   115   then have "x \<le> of_nat n"
```
```   116     by simp
```
```   117   then show ?thesis ..
```
```   118 qed
```
```   119
```
```   120 text \<open>Archimedean fields have no infinitesimal elements.\<close>
```
```   121
```
```   122 lemma reals_Archimedean:
```
```   123   fixes x :: "'a::archimedean_field"
```
```   124   assumes "0 < x"
```
```   125   shows "\<exists>n. inverse (of_nat (Suc n)) < x"
```
```   126 proof -
```
```   127   from \<open>0 < x\<close> have "0 < inverse x"
```
```   128     by (rule positive_imp_inverse_positive)
```
```   129   obtain n where "inverse x < of_nat n"
```
```   130     using reals_Archimedean2 ..
```
```   131   then obtain m where "inverse x < of_nat (Suc m)"
```
```   132     using \<open>0 < inverse x\<close> by (cases n) (simp_all del: of_nat_Suc)
```
```   133   then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
```
```   134     using \<open>0 < inverse x\<close> by (rule less_imp_inverse_less)
```
```   135   then have "inverse (of_nat (Suc m)) < x"
```
```   136     using \<open>0 < x\<close> by (simp add: nonzero_inverse_inverse_eq)
```
```   137   then show ?thesis ..
```
```   138 qed
```
```   139
```
```   140 lemma ex_inverse_of_nat_less:
```
```   141   fixes x :: "'a::archimedean_field"
```
```   142   assumes "0 < x"
```
```   143   shows "\<exists>n>0. inverse (of_nat n) < x"
```
```   144   using reals_Archimedean [OF \<open>0 < x\<close>] by auto
```
```   145
```
```   146 lemma ex_less_of_nat_mult:
```
```   147   fixes x :: "'a::archimedean_field"
```
```   148   assumes "0 < x"
```
```   149   shows "\<exists>n. y < of_nat n * x"
```
```   150 proof -
```
```   151   obtain n where "y / x < of_nat n"
```
```   152     using reals_Archimedean2 ..
```
```   153   with \<open>0 < x\<close> have "y < of_nat n * x"
```
```   154     by (simp add: pos_divide_less_eq)
```
```   155   then show ?thesis ..
```
```   156 qed
```
```   157
```
```   158
```
```   159 subsection \<open>Existence and uniqueness of floor function\<close>
```
```   160
```
```   161 lemma exists_least_lemma:
```
```   162   assumes "\<not> P 0" and "\<exists>n. P n"
```
```   163   shows "\<exists>n. \<not> P n \<and> P (Suc n)"
```
```   164 proof -
```
```   165   from \<open>\<exists>n. P n\<close> have "P (Least P)"
```
```   166     by (rule LeastI_ex)
```
```   167   with \<open>\<not> P 0\<close> obtain n where "Least P = Suc n"
```
```   168     by (cases "Least P") auto
```
```   169   then have "n < Least P"
```
```   170     by simp
```
```   171   then have "\<not> P n"
```
```   172     by (rule not_less_Least)
```
```   173   then have "\<not> P n \<and> P (Suc n)"
```
```   174     using \<open>P (Least P)\<close> \<open>Least P = Suc n\<close> by simp
```
```   175   then show ?thesis ..
```
```   176 qed
```
```   177
```
```   178 lemma floor_exists:
```
```   179   fixes x :: "'a::archimedean_field"
```
```   180   shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   181 proof (cases "0 \<le> x")
```
```   182   case True
```
```   183   then have "\<not> x < of_nat 0"
```
```   184     by simp
```
```   185   then have "\<exists>n. \<not> x < of_nat n \<and> x < of_nat (Suc n)"
```
```   186     using reals_Archimedean2 by (rule exists_least_lemma)
```
```   187   then obtain n where "\<not> x < of_nat n \<and> x < of_nat (Suc n)" ..
```
```   188   then have "of_int (int n) \<le> x \<and> x < of_int (int n + 1)"
```
```   189     by simp
```
```   190   then show ?thesis ..
```
```   191 next
```
```   192   case False
```
```   193   then have "\<not> - x \<le> of_nat 0"
```
```   194     by simp
```
```   195   then have "\<exists>n. \<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)"
```
```   196     using real_arch_simple by (rule exists_least_lemma)
```
```   197   then obtain n where "\<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" ..
```
```   198   then have "of_int (- int n - 1) \<le> x \<and> x < of_int (- int n - 1 + 1)"
```
```   199     by simp
```
```   200   then show ?thesis ..
```
```   201 qed
```
```   202
```
```   203 lemma floor_exists1: "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   204   for x :: "'a::archimedean_field"
```
```   205 proof (rule ex_ex1I)
```
```   206   show "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   207     by (rule floor_exists)
```
```   208 next
```
```   209   fix y z
```
```   210   assume "of_int y \<le> x \<and> x < of_int (y + 1)"
```
```   211     and "of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   212   with le_less_trans [of "of_int y" "x" "of_int (z + 1)"]
```
```   213        le_less_trans [of "of_int z" "x" "of_int (y + 1)"] show "y = z"
```
```   214     by (simp del: of_int_add)
```
```   215 qed
```
```   216
```
```   217
```
```   218 subsection \<open>Floor function\<close>
```
```   219
```
```   220 class floor_ceiling = archimedean_field +
```
```   221   fixes floor :: "'a \<Rightarrow> int"  ("\<lfloor>_\<rfloor>")
```
```   222   assumes floor_correct: "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)"
```
```   223
```
```   224 lemma floor_unique: "of_int z \<le> x \<Longrightarrow> x < of_int z + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = z"
```
```   225   using floor_correct [of x] floor_exists1 [of x] by auto
```
```   226
```
```   227 lemma floor_eq_iff: "\<lfloor>x\<rfloor> = a \<longleftrightarrow> of_int a \<le> x \<and> x < of_int a + 1"
```
```   228 using floor_correct floor_unique by auto
```
```   229
```
```   230 lemma of_int_floor_le [simp]: "of_int \<lfloor>x\<rfloor> \<le> x"
```
```   231   using floor_correct ..
```
```   232
```
```   233 lemma le_floor_iff: "z \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> of_int z \<le> x"
```
```   234 proof
```
```   235   assume "z \<le> \<lfloor>x\<rfloor>"
```
```   236   then have "(of_int z :: 'a) \<le> of_int \<lfloor>x\<rfloor>" by simp
```
```   237   also have "of_int \<lfloor>x\<rfloor> \<le> x" by (rule of_int_floor_le)
```
```   238   finally show "of_int z \<le> x" .
```
```   239 next
```
```   240   assume "of_int z \<le> x"
```
```   241   also have "x < of_int (\<lfloor>x\<rfloor> + 1)" using floor_correct ..
```
```   242   finally show "z \<le> \<lfloor>x\<rfloor>" by (simp del: of_int_add)
```
```   243 qed
```
```   244
```
```   245 lemma floor_less_iff: "\<lfloor>x\<rfloor> < z \<longleftrightarrow> x < of_int z"
```
```   246   by (simp add: not_le [symmetric] le_floor_iff)
```
```   247
```
```   248 lemma less_floor_iff: "z < \<lfloor>x\<rfloor> \<longleftrightarrow> of_int z + 1 \<le> x"
```
```   249   using le_floor_iff [of "z + 1" x] by auto
```
```   250
```
```   251 lemma floor_le_iff: "\<lfloor>x\<rfloor> \<le> z \<longleftrightarrow> x < of_int z + 1"
```
```   252   by (simp add: not_less [symmetric] less_floor_iff)
```
```   253
```
```   254 lemma floor_split[arith_split]: "P \<lfloor>t\<rfloor> \<longleftrightarrow> (\<forall>i. of_int i \<le> t \<and> t < of_int i + 1 \<longrightarrow> P i)"
```
```   255   by (metis floor_correct floor_unique less_floor_iff not_le order_refl)
```
```   256
```
```   257 lemma floor_mono:
```
```   258   assumes "x \<le> y"
```
```   259   shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"
```
```   260 proof -
```
```   261   have "of_int \<lfloor>x\<rfloor> \<le> x" by (rule of_int_floor_le)
```
```   262   also note \<open>x \<le> y\<close>
```
```   263   finally show ?thesis by (simp add: le_floor_iff)
```
```   264 qed
```
```   265
```
```   266 lemma floor_less_cancel: "\<lfloor>x\<rfloor> < \<lfloor>y\<rfloor> \<Longrightarrow> x < y"
```
```   267   by (auto simp add: not_le [symmetric] floor_mono)
```
```   268
```
```   269 lemma floor_of_int [simp]: "\<lfloor>of_int z\<rfloor> = z"
```
```   270   by (rule floor_unique) simp_all
```
```   271
```
```   272 lemma floor_of_nat [simp]: "\<lfloor>of_nat n\<rfloor> = int n"
```
```   273   using floor_of_int [of "of_nat n"] by simp
```
```   274
```
```   275 lemma le_floor_add: "\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> \<le> \<lfloor>x + y\<rfloor>"
```
```   276   by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
```
```   277
```
```   278
```
```   279 text \<open>Floor with numerals.\<close>
```
```   280
```
```   281 lemma floor_zero [simp]: "\<lfloor>0\<rfloor> = 0"
```
```   282   using floor_of_int [of 0] by simp
```
```   283
```
```   284 lemma floor_one [simp]: "\<lfloor>1\<rfloor> = 1"
```
```   285   using floor_of_int [of 1] by simp
```
```   286
```
```   287 lemma floor_numeral [simp]: "\<lfloor>numeral v\<rfloor> = numeral v"
```
```   288   using floor_of_int [of "numeral v"] by simp
```
```   289
```
```   290 lemma floor_neg_numeral [simp]: "\<lfloor>- numeral v\<rfloor> = - numeral v"
```
```   291   using floor_of_int [of "- numeral v"] by simp
```
```   292
```
```   293 lemma zero_le_floor [simp]: "0 \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> 0 \<le> x"
```
```   294   by (simp add: le_floor_iff)
```
```   295
```
```   296 lemma one_le_floor [simp]: "1 \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> 1 \<le> x"
```
```   297   by (simp add: le_floor_iff)
```
```   298
```
```   299 lemma numeral_le_floor [simp]: "numeral v \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> numeral v \<le> x"
```
```   300   by (simp add: le_floor_iff)
```
```   301
```
```   302 lemma neg_numeral_le_floor [simp]: "- numeral v \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> - numeral v \<le> x"
```
```   303   by (simp add: le_floor_iff)
```
```   304
```
```   305 lemma zero_less_floor [simp]: "0 < \<lfloor>x\<rfloor> \<longleftrightarrow> 1 \<le> x"
```
```   306   by (simp add: less_floor_iff)
```
```   307
```
```   308 lemma one_less_floor [simp]: "1 < \<lfloor>x\<rfloor> \<longleftrightarrow> 2 \<le> x"
```
```   309   by (simp add: less_floor_iff)
```
```   310
```
```   311 lemma numeral_less_floor [simp]: "numeral v < \<lfloor>x\<rfloor> \<longleftrightarrow> numeral v + 1 \<le> x"
```
```   312   by (simp add: less_floor_iff)
```
```   313
```
```   314 lemma neg_numeral_less_floor [simp]: "- numeral v < \<lfloor>x\<rfloor> \<longleftrightarrow> - numeral v + 1 \<le> x"
```
```   315   by (simp add: less_floor_iff)
```
```   316
```
```   317 lemma floor_le_zero [simp]: "\<lfloor>x\<rfloor> \<le> 0 \<longleftrightarrow> x < 1"
```
```   318   by (simp add: floor_le_iff)
```
```   319
```
```   320 lemma floor_le_one [simp]: "\<lfloor>x\<rfloor> \<le> 1 \<longleftrightarrow> x < 2"
```
```   321   by (simp add: floor_le_iff)
```
```   322
```
```   323 lemma floor_le_numeral [simp]: "\<lfloor>x\<rfloor> \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
```
```   324   by (simp add: floor_le_iff)
```
```   325
```
```   326 lemma floor_le_neg_numeral [simp]: "\<lfloor>x\<rfloor> \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1"
```
```   327   by (simp add: floor_le_iff)
```
```   328
```
```   329 lemma floor_less_zero [simp]: "\<lfloor>x\<rfloor> < 0 \<longleftrightarrow> x < 0"
```
```   330   by (simp add: floor_less_iff)
```
```   331
```
```   332 lemma floor_less_one [simp]: "\<lfloor>x\<rfloor> < 1 \<longleftrightarrow> x < 1"
```
```   333   by (simp add: floor_less_iff)
```
```   334
```
```   335 lemma floor_less_numeral [simp]: "\<lfloor>x\<rfloor> < numeral v \<longleftrightarrow> x < numeral v"
```
```   336   by (simp add: floor_less_iff)
```
```   337
```
```   338 lemma floor_less_neg_numeral [simp]: "\<lfloor>x\<rfloor> < - numeral v \<longleftrightarrow> x < - numeral v"
```
```   339   by (simp add: floor_less_iff)
```
```   340
```
```   341 lemma le_mult_floor_Ints:
```
```   342   assumes "0 \<le> a" "a \<in> Ints"
```
```   343   shows "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> (of_int\<lfloor>a * b\<rfloor> :: 'a :: linordered_idom)"
```
```   344   by (metis Ints_cases assms floor_less_iff floor_of_int linorder_not_less mult_left_mono of_int_floor_le of_int_less_iff of_int_mult)
```
```   345
```
```   346
```
```   347 text \<open>Addition and subtraction of integers.\<close>
```
```   348
```
```   349 lemma floor_add_int: "\<lfloor>x\<rfloor> + z = \<lfloor>x + of_int z\<rfloor>"
```
```   350   using floor_correct [of x] by (simp add: floor_unique[symmetric])
```
```   351
```
```   352 lemma int_add_floor: "z + \<lfloor>x\<rfloor> = \<lfloor>of_int z + x\<rfloor>"
```
```   353   using floor_correct [of x] by (simp add: floor_unique[symmetric])
```
```   354
```
```   355 lemma one_add_floor: "\<lfloor>x\<rfloor> + 1 = \<lfloor>x + 1\<rfloor>"
```
```   356   using floor_add_int [of x 1] by simp
```
```   357
```
```   358 lemma floor_diff_of_int [simp]: "\<lfloor>x - of_int z\<rfloor> = \<lfloor>x\<rfloor> - z"
```
```   359   using floor_add_int [of x "- z"] by (simp add: algebra_simps)
```
```   360
```
```   361 lemma floor_uminus_of_int [simp]: "\<lfloor>- (of_int z)\<rfloor> = - z"
```
```   362   by (metis floor_diff_of_int [of 0] diff_0 floor_zero)
```
```   363
```
```   364 lemma floor_diff_numeral [simp]: "\<lfloor>x - numeral v\<rfloor> = \<lfloor>x\<rfloor> - numeral v"
```
```   365   using floor_diff_of_int [of x "numeral v"] by simp
```
```   366
```
```   367 lemma floor_diff_one [simp]: "\<lfloor>x - 1\<rfloor> = \<lfloor>x\<rfloor> - 1"
```
```   368   using floor_diff_of_int [of x 1] by simp
```
```   369
```
```   370 lemma le_mult_floor:
```
```   371   assumes "0 \<le> a" and "0 \<le> b"
```
```   372   shows "\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor> \<le> \<lfloor>a * b\<rfloor>"
```
```   373 proof -
```
```   374   have "of_int \<lfloor>a\<rfloor> \<le> a" and "of_int \<lfloor>b\<rfloor> \<le> b"
```
```   375     by (auto intro: of_int_floor_le)
```
```   376   then have "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> a * b"
```
```   377     using assms by (auto intro!: mult_mono)
```
```   378   also have "a * b < of_int (\<lfloor>a * b\<rfloor> + 1)"
```
```   379     using floor_correct[of "a * b"] by auto
```
```   380   finally show ?thesis
```
```   381     unfolding of_int_less_iff by simp
```
```   382 qed
```
```   383
```
```   384 lemma floor_divide_of_int_eq: "\<lfloor>of_int k / of_int l\<rfloor> = k div l"
```
```   385   for k l :: int
```
```   386 proof (cases "l = 0")
```
```   387   case True
```
```   388   then show ?thesis by simp
```
```   389 next
```
```   390   case False
```
```   391   have *: "\<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> = 0"
```
```   392   proof (cases "l > 0")
```
```   393     case True
```
```   394     then show ?thesis
```
```   395       by (auto intro: floor_unique)
```
```   396   next
```
```   397     case False
```
```   398     obtain r where "r = - l"
```
```   399       by blast
```
```   400     then have l: "l = - r"
```
```   401       by simp
```
```   402     with \<open>l \<noteq> 0\<close> False have "r > 0"
```
```   403       by simp
```
```   404     with l show ?thesis
```
```   405       using pos_mod_bound [of r]
```
```   406       by (auto simp add: zmod_zminus2_eq_if less_le field_simps intro: floor_unique)
```
```   407   qed
```
```   408   have "(of_int k :: 'a) = of_int (k div l * l + k mod l)"
```
```   409     by simp
```
```   410   also have "\<dots> = (of_int (k div l) + of_int (k mod l) / of_int l) * of_int l"
```
```   411     using False by (simp only: of_int_add) (simp add: field_simps)
```
```   412   finally have "(of_int k / of_int l :: 'a) = \<dots> / of_int l"
```
```   413     by simp
```
```   414   then have "(of_int k / of_int l :: 'a) = of_int (k div l) + of_int (k mod l) / of_int l"
```
```   415     using False by (simp only:) (simp add: field_simps)
```
```   416   then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k div l) + of_int (k mod l) / of_int l :: 'a\<rfloor>"
```
```   417     by simp
```
```   418   then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l + of_int (k div l) :: 'a\<rfloor>"
```
```   419     by (simp add: ac_simps)
```
```   420   then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> + k div l"
```
```   421     by (simp add: floor_add_int)
```
```   422   with * show ?thesis
```
```   423     by simp
```
```   424 qed
```
```   425
```
```   426 lemma floor_divide_of_nat_eq: "\<lfloor>of_nat m / of_nat n\<rfloor> = of_nat (m div n)"
```
```   427   for m n :: nat
```
```   428 proof (cases "n = 0")
```
```   429   case True
```
```   430   then show ?thesis by simp
```
```   431 next
```
```   432   case False
```
```   433   then have *: "\<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> = 0"
```
```   434     by (auto intro: floor_unique)
```
```   435   have "(of_nat m :: 'a) = of_nat (m div n * n + m mod n)"
```
```   436     by simp
```
```   437   also have "\<dots> = (of_nat (m div n) + of_nat (m mod n) / of_nat n) * of_nat n"
```
```   438     using False by (simp only: of_nat_add) (simp add: field_simps)
```
```   439   finally have "(of_nat m / of_nat n :: 'a) = \<dots> / of_nat n"
```
```   440     by simp
```
```   441   then have "(of_nat m / of_nat n :: 'a) = of_nat (m div n) + of_nat (m mod n) / of_nat n"
```
```   442     using False by (simp only:) simp
```
```   443   then have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m div n) + of_nat (m mod n) / of_nat n :: 'a\<rfloor>"
```
```   444     by simp
```
```   445   then have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m mod n) / of_nat n + of_nat (m div n) :: 'a\<rfloor>"
```
```   446     by (simp add: ac_simps)
```
```   447   moreover have "(of_nat (m div n) :: 'a) = of_int (of_nat (m div n))"
```
```   448     by simp
```
```   449   ultimately have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> =
```
```   450       \<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> + of_nat (m div n)"
```
```   451     by (simp only: floor_add_int)
```
```   452   with * show ?thesis
```
```   453     by simp
```
```   454 qed
```
```   455
```
```   456
```
```   457 subsection \<open>Ceiling function\<close>
```
```   458
```
```   459 definition ceiling :: "'a::floor_ceiling \<Rightarrow> int"  ("\<lceil>_\<rceil>")
```
```   460   where "\<lceil>x\<rceil> = - \<lfloor>- x\<rfloor>"
```
```   461
```
```   462 lemma ceiling_correct: "of_int \<lceil>x\<rceil> - 1 < x \<and> x \<le> of_int \<lceil>x\<rceil>"
```
```   463   unfolding ceiling_def using floor_correct [of "- x"]
```
```   464   by (simp add: le_minus_iff)
```
```   465
```
```   466 lemma ceiling_unique: "of_int z - 1 < x \<Longrightarrow> x \<le> of_int z \<Longrightarrow> \<lceil>x\<rceil> = z"
```
```   467   unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
```
```   468
```
```   469 lemma ceiling_eq_iff: "\<lceil>x\<rceil> = a \<longleftrightarrow> of_int a - 1 < x \<and> x \<le> of_int a"
```
```   470 using ceiling_correct ceiling_unique by auto
```
```   471
```
```   472 lemma le_of_int_ceiling [simp]: "x \<le> of_int \<lceil>x\<rceil>"
```
```   473   using ceiling_correct ..
```
```   474
```
```   475 lemma ceiling_le_iff: "\<lceil>x\<rceil> \<le> z \<longleftrightarrow> x \<le> of_int z"
```
```   476   unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
```
```   477
```
```   478 lemma less_ceiling_iff: "z < \<lceil>x\<rceil> \<longleftrightarrow> of_int z < x"
```
```   479   by (simp add: not_le [symmetric] ceiling_le_iff)
```
```   480
```
```   481 lemma ceiling_less_iff: "\<lceil>x\<rceil> < z \<longleftrightarrow> x \<le> of_int z - 1"
```
```   482   using ceiling_le_iff [of x "z - 1"] by simp
```
```   483
```
```   484 lemma le_ceiling_iff: "z \<le> \<lceil>x\<rceil> \<longleftrightarrow> of_int z - 1 < x"
```
```   485   by (simp add: not_less [symmetric] ceiling_less_iff)
```
```   486
```
```   487 lemma ceiling_mono: "x \<ge> y \<Longrightarrow> \<lceil>x\<rceil> \<ge> \<lceil>y\<rceil>"
```
```   488   unfolding ceiling_def by (simp add: floor_mono)
```
```   489
```
```   490 lemma ceiling_less_cancel: "\<lceil>x\<rceil> < \<lceil>y\<rceil> \<Longrightarrow> x < y"
```
```   491   by (auto simp add: not_le [symmetric] ceiling_mono)
```
```   492
```
```   493 lemma ceiling_of_int [simp]: "\<lceil>of_int z\<rceil> = z"
```
```   494   by (rule ceiling_unique) simp_all
```
```   495
```
```   496 lemma ceiling_of_nat [simp]: "\<lceil>of_nat n\<rceil> = int n"
```
```   497   using ceiling_of_int [of "of_nat n"] by simp
```
```   498
```
```   499 lemma ceiling_add_le: "\<lceil>x + y\<rceil> \<le> \<lceil>x\<rceil> + \<lceil>y\<rceil>"
```
```   500   by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
```
```   501
```
```   502 lemma mult_ceiling_le:
```
```   503   assumes "0 \<le> a" and "0 \<le> b"
```
```   504   shows "\<lceil>a * b\<rceil> \<le> \<lceil>a\<rceil> * \<lceil>b\<rceil>"
```
```   505   by (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult)
```
```   506
```
```   507 lemma mult_ceiling_le_Ints:
```
```   508   assumes "0 \<le> a" "a \<in> Ints"
```
```   509   shows "(of_int \<lceil>a * b\<rceil> :: 'a :: linordered_idom) \<le> of_int(\<lceil>a\<rceil> * \<lceil>b\<rceil>)"
```
```   510   by (metis Ints_cases assms ceiling_le_iff ceiling_of_int le_of_int_ceiling mult_left_mono of_int_le_iff of_int_mult)
```
```   511
```
```   512 lemma finite_int_segment:
```
```   513   fixes a :: "'a::floor_ceiling"
```
```   514   shows "finite {x \<in> \<int>. a \<le> x \<and> x \<le> b}"
```
```   515 proof -
```
```   516   have "finite {ceiling a..floor b}"
```
```   517     by simp
```
```   518   moreover have "{x \<in> \<int>. a \<le> x \<and> x \<le> b} = of_int ` {ceiling a..floor b}"
```
```   519     by (auto simp: le_floor_iff ceiling_le_iff elim!: Ints_cases)
```
```   520   ultimately show ?thesis
```
```   521     by simp
```
```   522 qed
```
```   523
```
```   524 corollary finite_abs_int_segment:
```
```   525   fixes a :: "'a::floor_ceiling"
```
```   526   shows "finite {k \<in> \<int>. \<bar>k\<bar> \<le> a}"
```
```   527   using finite_int_segment [of "-a" a] by (auto simp add: abs_le_iff conj_commute minus_le_iff)
```
```   528
```
```   529
```
```   530 subsubsection \<open>Ceiling with numerals.\<close>
```
```   531
```
```   532 lemma ceiling_zero [simp]: "\<lceil>0\<rceil> = 0"
```
```   533   using ceiling_of_int [of 0] by simp
```
```   534
```
```   535 lemma ceiling_one [simp]: "\<lceil>1\<rceil> = 1"
```
```   536   using ceiling_of_int [of 1] by simp
```
```   537
```
```   538 lemma ceiling_numeral [simp]: "\<lceil>numeral v\<rceil> = numeral v"
```
```   539   using ceiling_of_int [of "numeral v"] by simp
```
```   540
```
```   541 lemma ceiling_neg_numeral [simp]: "\<lceil>- numeral v\<rceil> = - numeral v"
```
```   542   using ceiling_of_int [of "- numeral v"] by simp
```
```   543
```
```   544 lemma ceiling_le_zero [simp]: "\<lceil>x\<rceil> \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```   545   by (simp add: ceiling_le_iff)
```
```   546
```
```   547 lemma ceiling_le_one [simp]: "\<lceil>x\<rceil> \<le> 1 \<longleftrightarrow> x \<le> 1"
```
```   548   by (simp add: ceiling_le_iff)
```
```   549
```
```   550 lemma ceiling_le_numeral [simp]: "\<lceil>x\<rceil> \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
```
```   551   by (simp add: ceiling_le_iff)
```
```   552
```
```   553 lemma ceiling_le_neg_numeral [simp]: "\<lceil>x\<rceil> \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v"
```
```   554   by (simp add: ceiling_le_iff)
```
```   555
```
```   556 lemma ceiling_less_zero [simp]: "\<lceil>x\<rceil> < 0 \<longleftrightarrow> x \<le> -1"
```
```   557   by (simp add: ceiling_less_iff)
```
```   558
```
```   559 lemma ceiling_less_one [simp]: "\<lceil>x\<rceil> < 1 \<longleftrightarrow> x \<le> 0"
```
```   560   by (simp add: ceiling_less_iff)
```
```   561
```
```   562 lemma ceiling_less_numeral [simp]: "\<lceil>x\<rceil> < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
```
```   563   by (simp add: ceiling_less_iff)
```
```   564
```
```   565 lemma ceiling_less_neg_numeral [simp]: "\<lceil>x\<rceil> < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1"
```
```   566   by (simp add: ceiling_less_iff)
```
```   567
```
```   568 lemma zero_le_ceiling [simp]: "0 \<le> \<lceil>x\<rceil> \<longleftrightarrow> -1 < x"
```
```   569   by (simp add: le_ceiling_iff)
```
```   570
```
```   571 lemma one_le_ceiling [simp]: "1 \<le> \<lceil>x\<rceil> \<longleftrightarrow> 0 < x"
```
```   572   by (simp add: le_ceiling_iff)
```
```   573
```
```   574 lemma numeral_le_ceiling [simp]: "numeral v \<le> \<lceil>x\<rceil> \<longleftrightarrow> numeral v - 1 < x"
```
```   575   by (simp add: le_ceiling_iff)
```
```   576
```
```   577 lemma neg_numeral_le_ceiling [simp]: "- numeral v \<le> \<lceil>x\<rceil> \<longleftrightarrow> - numeral v - 1 < x"
```
```   578   by (simp add: le_ceiling_iff)
```
```   579
```
```   580 lemma zero_less_ceiling [simp]: "0 < \<lceil>x\<rceil> \<longleftrightarrow> 0 < x"
```
```   581   by (simp add: less_ceiling_iff)
```
```   582
```
```   583 lemma one_less_ceiling [simp]: "1 < \<lceil>x\<rceil> \<longleftrightarrow> 1 < x"
```
```   584   by (simp add: less_ceiling_iff)
```
```   585
```
```   586 lemma numeral_less_ceiling [simp]: "numeral v < \<lceil>x\<rceil> \<longleftrightarrow> numeral v < x"
```
```   587   by (simp add: less_ceiling_iff)
```
```   588
```
```   589 lemma neg_numeral_less_ceiling [simp]: "- numeral v < \<lceil>x\<rceil> \<longleftrightarrow> - numeral v < x"
```
```   590   by (simp add: less_ceiling_iff)
```
```   591
```
```   592 lemma ceiling_altdef: "\<lceil>x\<rceil> = (if x = of_int \<lfloor>x\<rfloor> then \<lfloor>x\<rfloor> else \<lfloor>x\<rfloor> + 1)"
```
```   593   by (intro ceiling_unique; simp, linarith?)
```
```   594
```
```   595 lemma floor_le_ceiling [simp]: "\<lfloor>x\<rfloor> \<le> \<lceil>x\<rceil>"
```
```   596   by (simp add: ceiling_altdef)
```
```   597
```
```   598
```
```   599 subsubsection \<open>Addition and subtraction of integers.\<close>
```
```   600
```
```   601 lemma ceiling_add_of_int [simp]: "\<lceil>x + of_int z\<rceil> = \<lceil>x\<rceil> + z"
```
```   602   using ceiling_correct [of x] by (simp add: ceiling_def)
```
```   603
```
```   604 lemma ceiling_add_numeral [simp]: "\<lceil>x + numeral v\<rceil> = \<lceil>x\<rceil> + numeral v"
```
```   605   using ceiling_add_of_int [of x "numeral v"] by simp
```
```   606
```
```   607 lemma ceiling_add_one [simp]: "\<lceil>x + 1\<rceil> = \<lceil>x\<rceil> + 1"
```
```   608   using ceiling_add_of_int [of x 1] by simp
```
```   609
```
```   610 lemma ceiling_diff_of_int [simp]: "\<lceil>x - of_int z\<rceil> = \<lceil>x\<rceil> - z"
```
```   611   using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
```
```   612
```
```   613 lemma ceiling_diff_numeral [simp]: "\<lceil>x - numeral v\<rceil> = \<lceil>x\<rceil> - numeral v"
```
```   614   using ceiling_diff_of_int [of x "numeral v"] by simp
```
```   615
```
```   616 lemma ceiling_diff_one [simp]: "\<lceil>x - 1\<rceil> = \<lceil>x\<rceil> - 1"
```
```   617   using ceiling_diff_of_int [of x 1] by simp
```
```   618
```
```   619 lemma ceiling_split[arith_split]: "P \<lceil>t\<rceil> \<longleftrightarrow> (\<forall>i. of_int i - 1 < t \<and> t \<le> of_int i \<longrightarrow> P i)"
```
```   620   by (auto simp add: ceiling_unique ceiling_correct)
```
```   621
```
```   622 lemma ceiling_diff_floor_le_1: "\<lceil>x\<rceil> - \<lfloor>x\<rfloor> \<le> 1"
```
```   623 proof -
```
```   624   have "of_int \<lceil>x\<rceil> - 1 < x"
```
```   625     using ceiling_correct[of x] by simp
```
```   626   also have "x < of_int \<lfloor>x\<rfloor> + 1"
```
```   627     using floor_correct[of x] by simp_all
```
```   628   finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
```
```   629     by simp
```
```   630   then show ?thesis
```
```   631     unfolding of_int_less_iff by simp
```
```   632 qed
```
```   633
```
```   634 lemma nat_approx_posE:
```
```   635   fixes e:: "'a::{archimedean_field,floor_ceiling}"
```
```   636   assumes "0 < e"
```
```   637   obtains n :: nat where "1 / of_nat(Suc n) < e"
```
```   638 proof
```
```   639   have "(1::'a) / of_nat (Suc (nat \<lceil>1/e\<rceil>)) < 1 / of_int (\<lceil>1/e\<rceil>)"
```
```   640   proof (rule divide_strict_left_mono)
```
```   641     show "(of_int \<lceil>1 / e\<rceil>::'a) < of_nat (Suc (nat \<lceil>1 / e\<rceil>))"
```
```   642       using assms by (simp add: field_simps)
```
```   643     show "(0::'a) < of_nat (Suc (nat \<lceil>1 / e\<rceil>)) * of_int \<lceil>1 / e\<rceil>"
```
```   644       using assms by (auto simp: zero_less_mult_iff pos_add_strict)
```
```   645   qed auto
```
```   646   also have "1 / of_int (\<lceil>1/e\<rceil>) \<le> 1 / (1/e)"
```
```   647     by (rule divide_left_mono) (auto simp: \<open>0 < e\<close> ceiling_correct)
```
```   648   also have "\<dots> = e" by simp
```
```   649   finally show  "1 / of_nat (Suc (nat \<lceil>1 / e\<rceil>)) < e"
```
```   650     by metis
```
```   651 qed
```
```   652
```
```   653 subsection \<open>Negation\<close>
```
```   654
```
```   655 lemma floor_minus: "\<lfloor>- x\<rfloor> = - \<lceil>x\<rceil>"
```
```   656   unfolding ceiling_def by simp
```
```   657
```
```   658 lemma ceiling_minus: "\<lceil>- x\<rceil> = - \<lfloor>x\<rfloor>"
```
```   659   unfolding ceiling_def by simp
```
```   660
```
```   661
```
```   662 subsection \<open>Natural numbers\<close>
```
```   663
```
```   664 lemma of_nat_floor: "r\<ge>0 \<Longrightarrow> of_nat (nat \<lfloor>r\<rfloor>) \<le> r"
```
```   665   by simp
```
```   666
```
```   667 lemma of_nat_ceiling: "of_nat (nat \<lceil>r\<rceil>) \<ge> r"
```
```   668   by (cases "r\<ge>0") auto
```
```   669
```
```   670
```
```   671 subsection \<open>Frac Function\<close>
```
```   672
```
```   673 definition frac :: "'a \<Rightarrow> 'a::floor_ceiling"
```
```   674   where "frac x \<equiv> x - of_int \<lfloor>x\<rfloor>"
```
```   675
```
```   676 lemma frac_lt_1: "frac x < 1"
```
```   677   by (simp add: frac_def) linarith
```
```   678
```
```   679 lemma frac_eq_0_iff [simp]: "frac x = 0 \<longleftrightarrow> x \<in> \<int>"
```
```   680   by (simp add: frac_def) (metis Ints_cases Ints_of_int floor_of_int )
```
```   681
```
```   682 lemma frac_ge_0 [simp]: "frac x \<ge> 0"
```
```   683   unfolding frac_def by linarith
```
```   684
```
```   685 lemma frac_gt_0_iff [simp]: "frac x > 0 \<longleftrightarrow> x \<notin> \<int>"
```
```   686   by (metis frac_eq_0_iff frac_ge_0 le_less less_irrefl)
```
```   687
```
```   688 lemma frac_of_int [simp]: "frac (of_int z) = 0"
```
```   689   by (simp add: frac_def)
```
```   690
```
```   691 lemma floor_add: "\<lfloor>x + y\<rfloor> = (if frac x + frac y < 1 then \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> else (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>) + 1)"
```
```   692 proof -
```
```   693   have "x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>) \<Longrightarrow> \<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>"
```
```   694     by (metis add.commute floor_unique le_floor_add le_floor_iff of_int_add)
```
```   695   moreover
```
```   696   have "\<not> x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>) \<Longrightarrow> \<lfloor>x + y\<rfloor> = 1 + (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>)"
```
```   697     apply (simp add: floor_eq_iff)
```
```   698     apply (auto simp add: algebra_simps)
```
```   699     apply linarith
```
```   700     done
```
```   701   ultimately show ?thesis by (auto simp add: frac_def algebra_simps)
```
```   702 qed
```
```   703
```
```   704 lemma floor_add2[simp]: "x \<in> \<int> \<or> y \<in> \<int> \<Longrightarrow> \<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>"
```
```   705 by (metis add.commute add.left_neutral frac_lt_1 floor_add frac_eq_0_iff)
```
```   706
```
```   707 lemma frac_add:
```
```   708   "frac (x + y) = (if frac x + frac y < 1 then frac x + frac y else (frac x + frac y) - 1)"
```
```   709   by (simp add: frac_def floor_add)
```
```   710
```
```   711 lemma frac_unique_iff: "frac x = a \<longleftrightarrow> x - a \<in> \<int> \<and> 0 \<le> a \<and> a < 1"
```
```   712   for x :: "'a::floor_ceiling"
```
```   713   apply (auto simp: Ints_def frac_def algebra_simps floor_unique)
```
```   714    apply linarith+
```
```   715   done
```
```   716
```
```   717 lemma frac_eq: "frac x = x \<longleftrightarrow> 0 \<le> x \<and> x < 1"
```
```   718   by (simp add: frac_unique_iff)
```
```   719
```
```   720 lemma frac_neg: "frac (- x) = (if x \<in> \<int> then 0 else 1 - frac x)"
```
```   721   for x :: "'a::floor_ceiling"
```
```   722   apply (auto simp add: frac_unique_iff)
```
```   723    apply (simp add: frac_def)
```
```   724   apply (meson frac_lt_1 less_iff_diff_less_0 not_le not_less_iff_gr_or_eq)
```
```   725   done
```
```   726
```
```   727
```
```   728 subsection \<open>Rounding to the nearest integer\<close>
```
```   729
```
```   730 definition round :: "'a::floor_ceiling \<Rightarrow> int"
```
```   731   where "round x = \<lfloor>x + 1/2\<rfloor>"
```
```   732
```
```   733 lemma of_int_round_ge: "of_int (round x) \<ge> x - 1/2"
```
```   734   and of_int_round_le: "of_int (round x) \<le> x + 1/2"
```
```   735   and of_int_round_abs_le: "\<bar>of_int (round x) - x\<bar> \<le> 1/2"
```
```   736   and of_int_round_gt: "of_int (round x) > x - 1/2"
```
```   737 proof -
```
```   738   from floor_correct[of "x + 1/2"] have "x + 1/2 < of_int (round x) + 1"
```
```   739     by (simp add: round_def)
```
```   740   from add_strict_right_mono[OF this, of "-1"] show A: "of_int (round x) > x - 1/2"
```
```   741     by simp
```
```   742   then show "of_int (round x) \<ge> x - 1/2"
```
```   743     by simp
```
```   744   from floor_correct[of "x + 1/2"] show "of_int (round x) \<le> x + 1/2"
```
```   745     by (simp add: round_def)
```
```   746   with A show "\<bar>of_int (round x) - x\<bar> \<le> 1/2"
```
```   747     by linarith
```
```   748 qed
```
```   749
```
```   750 lemma round_of_int [simp]: "round (of_int n) = n"
```
```   751   unfolding round_def by (subst floor_eq_iff) force
```
```   752
```
```   753 lemma round_0 [simp]: "round 0 = 0"
```
```   754   using round_of_int[of 0] by simp
```
```   755
```
```   756 lemma round_1 [simp]: "round 1 = 1"
```
```   757   using round_of_int[of 1] by simp
```
```   758
```
```   759 lemma round_numeral [simp]: "round (numeral n) = numeral n"
```
```   760   using round_of_int[of "numeral n"] by simp
```
```   761
```
```   762 lemma round_neg_numeral [simp]: "round (-numeral n) = -numeral n"
```
```   763   using round_of_int[of "-numeral n"] by simp
```
```   764
```
```   765 lemma round_of_nat [simp]: "round (of_nat n) = of_nat n"
```
```   766   using round_of_int[of "int n"] by simp
```
```   767
```
```   768 lemma round_mono: "x \<le> y \<Longrightarrow> round x \<le> round y"
```
```   769   unfolding round_def by (intro floor_mono) simp
```
```   770
```
```   771 lemma round_unique: "of_int y > x - 1/2 \<Longrightarrow> of_int y \<le> x + 1/2 \<Longrightarrow> round x = y"
```
```   772   unfolding round_def
```
```   773 proof (rule floor_unique)
```
```   774   assume "x - 1 / 2 < of_int y"
```
```   775   from add_strict_left_mono[OF this, of 1] show "x + 1 / 2 < of_int y + 1"
```
```   776     by simp
```
```   777 qed
```
```   778
```
```   779 lemma round_unique': "\<bar>x - of_int n\<bar> < 1/2 \<Longrightarrow> round x = n"
```
```   780   by (subst (asm) abs_less_iff, rule round_unique) (simp_all add: field_simps)
```
```   781
```
```   782 lemma round_altdef: "round x = (if frac x \<ge> 1/2 then \<lceil>x\<rceil> else \<lfloor>x\<rfloor>)"
```
```   783   by (cases "frac x \<ge> 1/2")
```
```   784     (rule round_unique, ((simp add: frac_def field_simps ceiling_altdef; linarith)+)[2])+
```
```   785
```
```   786 lemma floor_le_round: "\<lfloor>x\<rfloor> \<le> round x"
```
```   787   unfolding round_def by (intro floor_mono) simp
```
```   788
```
```   789 lemma ceiling_ge_round: "\<lceil>x\<rceil> \<ge> round x"
```
```   790   unfolding round_altdef by simp
```
```   791
```
```   792 lemma round_diff_minimal: "\<bar>z - of_int (round z)\<bar> \<le> \<bar>z - of_int m\<bar>"
```
```   793   for z :: "'a::floor_ceiling"
```
```   794 proof (cases "of_int m \<ge> z")
```
```   795   case True
```
```   796   then have "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int \<lceil>z\<rceil> - z\<bar>"
```
```   797     unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith
```
```   798   also have "of_int \<lceil>z\<rceil> - z \<ge> 0"
```
```   799     by linarith
```
```   800   with True have "\<bar>of_int \<lceil>z\<rceil> - z\<bar> \<le> \<bar>z - of_int m\<bar>"
```
```   801     by (simp add: ceiling_le_iff)
```
```   802   finally show ?thesis .
```
```   803 next
```
```   804   case False
```
```   805   then have "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int \<lfloor>z\<rfloor> - z\<bar>"
```
```   806     unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith
```
```   807   also have "z - of_int \<lfloor>z\<rfloor> \<ge> 0"
```
```   808     by linarith
```
```   809   with False have "\<bar>of_int \<lfloor>z\<rfloor> - z\<bar> \<le> \<bar>z - of_int m\<bar>"
```
```   810     by (simp add: le_floor_iff)
```
```   811   finally show ?thesis .
```
```   812 qed
```
```   813
```
```   814 end
```