src/HOL/Archimedean_Field.thy
 author blanchet Wed Feb 12 08:35:57 2014 +0100 (2014-02-12) changeset 55415 05f5fdb8d093 parent 54489 03ff4d1e6784 child 58040 9a867afaab5a permissions -rw-r--r--
renamed 'nat_{case,rec}' to '{case,rec}_nat'
```     1 (*  Title:      HOL/Archimedean_Field.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Archimedean Fields, Floor and Ceiling Functions *}
```
```     6
```
```     7 theory Archimedean_Field
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 subsection {* Class of Archimedean fields *}
```
```    12
```
```    13 text {* Archimedean fields have no infinite elements. *}
```
```    14
```
```    15 class archimedean_field = linordered_field +
```
```    16   assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
```
```    17
```
```    18 lemma ex_less_of_int:
```
```    19   fixes x :: "'a::archimedean_field" shows "\<exists>z. x < of_int z"
```
```    20 proof -
```
```    21   from ex_le_of_int obtain z where "x \<le> of_int z" ..
```
```    22   then have "x < of_int (z + 1)" by simp
```
```    23   then show ?thesis ..
```
```    24 qed
```
```    25
```
```    26 lemma ex_of_int_less:
```
```    27   fixes x :: "'a::archimedean_field" shows "\<exists>z. of_int z < x"
```
```    28 proof -
```
```    29   from ex_less_of_int obtain z where "- x < of_int z" ..
```
```    30   then have "of_int (- z) < x" by simp
```
```    31   then show ?thesis ..
```
```    32 qed
```
```    33
```
```    34 lemma ex_less_of_nat:
```
```    35   fixes x :: "'a::archimedean_field" shows "\<exists>n. x < of_nat n"
```
```    36 proof -
```
```    37   obtain z where "x < of_int z" using ex_less_of_int ..
```
```    38   also have "\<dots> \<le> of_int (int (nat z))" by simp
```
```    39   also have "\<dots> = of_nat (nat z)" by (simp only: of_int_of_nat_eq)
```
```    40   finally show ?thesis ..
```
```    41 qed
```
```    42
```
```    43 lemma ex_le_of_nat:
```
```    44   fixes x :: "'a::archimedean_field" shows "\<exists>n. x \<le> of_nat n"
```
```    45 proof -
```
```    46   obtain n where "x < of_nat n" using ex_less_of_nat ..
```
```    47   then have "x \<le> of_nat n" by simp
```
```    48   then show ?thesis ..
```
```    49 qed
```
```    50
```
```    51 text {* Archimedean fields have no infinitesimal elements. *}
```
```    52
```
```    53 lemma ex_inverse_of_nat_Suc_less:
```
```    54   fixes x :: "'a::archimedean_field"
```
```    55   assumes "0 < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
```
```    56 proof -
```
```    57   from `0 < x` have "0 < inverse x"
```
```    58     by (rule positive_imp_inverse_positive)
```
```    59   obtain n where "inverse x < of_nat n"
```
```    60     using ex_less_of_nat ..
```
```    61   then obtain m where "inverse x < of_nat (Suc m)"
```
```    62     using `0 < inverse x` by (cases n) (simp_all del: of_nat_Suc)
```
```    63   then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
```
```    64     using `0 < inverse x` by (rule less_imp_inverse_less)
```
```    65   then have "inverse (of_nat (Suc m)) < x"
```
```    66     using `0 < x` by (simp add: nonzero_inverse_inverse_eq)
```
```    67   then show ?thesis ..
```
```    68 qed
```
```    69
```
```    70 lemma ex_inverse_of_nat_less:
```
```    71   fixes x :: "'a::archimedean_field"
```
```    72   assumes "0 < x" shows "\<exists>n>0. inverse (of_nat n) < x"
```
```    73   using ex_inverse_of_nat_Suc_less [OF `0 < x`] by auto
```
```    74
```
```    75 lemma ex_less_of_nat_mult:
```
```    76   fixes x :: "'a::archimedean_field"
```
```    77   assumes "0 < x" shows "\<exists>n. y < of_nat n * x"
```
```    78 proof -
```
```    79   obtain n where "y / x < of_nat n" using ex_less_of_nat ..
```
```    80   with `0 < x` have "y < of_nat n * x" by (simp add: pos_divide_less_eq)
```
```    81   then show ?thesis ..
```
```    82 qed
```
```    83
```
```    84
```
```    85 subsection {* Existence and uniqueness of floor function *}
```
```    86
```
```    87 lemma exists_least_lemma:
```
```    88   assumes "\<not> P 0" and "\<exists>n. P n"
```
```    89   shows "\<exists>n. \<not> P n \<and> P (Suc n)"
```
```    90 proof -
```
```    91   from `\<exists>n. P n` have "P (Least P)" by (rule LeastI_ex)
```
```    92   with `\<not> P 0` obtain n where "Least P = Suc n"
```
```    93     by (cases "Least P") auto
```
```    94   then have "n < Least P" by simp
```
```    95   then have "\<not> P n" by (rule not_less_Least)
```
```    96   then have "\<not> P n \<and> P (Suc n)"
```
```    97     using `P (Least P)` `Least P = Suc n` by simp
```
```    98   then show ?thesis ..
```
```    99 qed
```
```   100
```
```   101 lemma floor_exists:
```
```   102   fixes x :: "'a::archimedean_field"
```
```   103   shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   104 proof (cases)
```
```   105   assume "0 \<le> x"
```
```   106   then have "\<not> x < of_nat 0" by simp
```
```   107   then have "\<exists>n. \<not> x < of_nat n \<and> x < of_nat (Suc n)"
```
```   108     using ex_less_of_nat by (rule exists_least_lemma)
```
```   109   then obtain n where "\<not> x < of_nat n \<and> x < of_nat (Suc n)" ..
```
```   110   then have "of_int (int n) \<le> x \<and> x < of_int (int n + 1)" by simp
```
```   111   then show ?thesis ..
```
```   112 next
```
```   113   assume "\<not> 0 \<le> x"
```
```   114   then have "\<not> - x \<le> of_nat 0" by simp
```
```   115   then have "\<exists>n. \<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)"
```
```   116     using ex_le_of_nat by (rule exists_least_lemma)
```
```   117   then obtain n where "\<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" ..
```
```   118   then have "of_int (- int n - 1) \<le> x \<and> x < of_int (- int n - 1 + 1)" by simp
```
```   119   then show ?thesis ..
```
```   120 qed
```
```   121
```
```   122 lemma floor_exists1:
```
```   123   fixes x :: "'a::archimedean_field"
```
```   124   shows "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   125 proof (rule ex_ex1I)
```
```   126   show "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   127     by (rule floor_exists)
```
```   128 next
```
```   129   fix y z assume
```
```   130     "of_int y \<le> x \<and> x < of_int (y + 1)"
```
```   131     "of_int z \<le> x \<and> x < of_int (z + 1)"
```
```   132   with le_less_trans [of "of_int y" "x" "of_int (z + 1)"]
```
```   133        le_less_trans [of "of_int z" "x" "of_int (y + 1)"]
```
```   134   show "y = z" by (simp del: of_int_add)
```
```   135 qed
```
```   136
```
```   137
```
```   138 subsection {* Floor function *}
```
```   139
```
```   140 class floor_ceiling = archimedean_field +
```
```   141   fixes floor :: "'a \<Rightarrow> int"
```
```   142   assumes floor_correct: "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
```
```   143
```
```   144 notation (xsymbols)
```
```   145   floor  ("\<lfloor>_\<rfloor>")
```
```   146
```
```   147 notation (HTML output)
```
```   148   floor  ("\<lfloor>_\<rfloor>")
```
```   149
```
```   150 lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
```
```   151   using floor_correct [of x] floor_exists1 [of x] by auto
```
```   152
```
```   153 lemma of_int_floor_le: "of_int (floor x) \<le> x"
```
```   154   using floor_correct ..
```
```   155
```
```   156 lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
```
```   157 proof
```
```   158   assume "z \<le> floor x"
```
```   159   then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
```
```   160   also have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
```
```   161   finally show "of_int z \<le> x" .
```
```   162 next
```
```   163   assume "of_int z \<le> x"
```
```   164   also have "x < of_int (floor x + 1)" using floor_correct ..
```
```   165   finally show "z \<le> floor x" by (simp del: of_int_add)
```
```   166 qed
```
```   167
```
```   168 lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
```
```   169   by (simp add: not_le [symmetric] le_floor_iff)
```
```   170
```
```   171 lemma less_floor_iff: "z < floor x \<longleftrightarrow> of_int z + 1 \<le> x"
```
```   172   using le_floor_iff [of "z + 1" x] by auto
```
```   173
```
```   174 lemma floor_le_iff: "floor x \<le> z \<longleftrightarrow> x < of_int z + 1"
```
```   175   by (simp add: not_less [symmetric] less_floor_iff)
```
```   176
```
```   177 lemma floor_mono: assumes "x \<le> y" shows "floor x \<le> floor y"
```
```   178 proof -
```
```   179   have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
```
```   180   also note `x \<le> y`
```
```   181   finally show ?thesis by (simp add: le_floor_iff)
```
```   182 qed
```
```   183
```
```   184 lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
```
```   185   by (auto simp add: not_le [symmetric] floor_mono)
```
```   186
```
```   187 lemma floor_of_int [simp]: "floor (of_int z) = z"
```
```   188   by (rule floor_unique) simp_all
```
```   189
```
```   190 lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
```
```   191   using floor_of_int [of "of_nat n"] by simp
```
```   192
```
```   193 lemma le_floor_add: "floor x + floor y \<le> floor (x + y)"
```
```   194   by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
```
```   195
```
```   196 text {* Floor with numerals *}
```
```   197
```
```   198 lemma floor_zero [simp]: "floor 0 = 0"
```
```   199   using floor_of_int [of 0] by simp
```
```   200
```
```   201 lemma floor_one [simp]: "floor 1 = 1"
```
```   202   using floor_of_int [of 1] by simp
```
```   203
```
```   204 lemma floor_numeral [simp]: "floor (numeral v) = numeral v"
```
```   205   using floor_of_int [of "numeral v"] by simp
```
```   206
```
```   207 lemma floor_neg_numeral [simp]: "floor (- numeral v) = - numeral v"
```
```   208   using floor_of_int [of "- numeral v"] by simp
```
```   209
```
```   210 lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
```
```   211   by (simp add: le_floor_iff)
```
```   212
```
```   213 lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
```
```   214   by (simp add: le_floor_iff)
```
```   215
```
```   216 lemma numeral_le_floor [simp]:
```
```   217   "numeral v \<le> floor x \<longleftrightarrow> numeral v \<le> x"
```
```   218   by (simp add: le_floor_iff)
```
```   219
```
```   220 lemma neg_numeral_le_floor [simp]:
```
```   221   "- numeral v \<le> floor x \<longleftrightarrow> - numeral v \<le> x"
```
```   222   by (simp add: le_floor_iff)
```
```   223
```
```   224 lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
```
```   225   by (simp add: less_floor_iff)
```
```   226
```
```   227 lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
```
```   228   by (simp add: less_floor_iff)
```
```   229
```
```   230 lemma numeral_less_floor [simp]:
```
```   231   "numeral v < floor x \<longleftrightarrow> numeral v + 1 \<le> x"
```
```   232   by (simp add: less_floor_iff)
```
```   233
```
```   234 lemma neg_numeral_less_floor [simp]:
```
```   235   "- numeral v < floor x \<longleftrightarrow> - numeral v + 1 \<le> x"
```
```   236   by (simp add: less_floor_iff)
```
```   237
```
```   238 lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
```
```   239   by (simp add: floor_le_iff)
```
```   240
```
```   241 lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
```
```   242   by (simp add: floor_le_iff)
```
```   243
```
```   244 lemma floor_le_numeral [simp]:
```
```   245   "floor x \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
```
```   246   by (simp add: floor_le_iff)
```
```   247
```
```   248 lemma floor_le_neg_numeral [simp]:
```
```   249   "floor x \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1"
```
```   250   by (simp add: floor_le_iff)
```
```   251
```
```   252 lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
```
```   253   by (simp add: floor_less_iff)
```
```   254
```
```   255 lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
```
```   256   by (simp add: floor_less_iff)
```
```   257
```
```   258 lemma floor_less_numeral [simp]:
```
```   259   "floor x < numeral v \<longleftrightarrow> x < numeral v"
```
```   260   by (simp add: floor_less_iff)
```
```   261
```
```   262 lemma floor_less_neg_numeral [simp]:
```
```   263   "floor x < - numeral v \<longleftrightarrow> x < - numeral v"
```
```   264   by (simp add: floor_less_iff)
```
```   265
```
```   266 text {* Addition and subtraction of integers *}
```
```   267
```
```   268 lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
```
```   269   using floor_correct [of x] by (simp add: floor_unique)
```
```   270
```
```   271 lemma floor_add_numeral [simp]:
```
```   272     "floor (x + numeral v) = floor x + numeral v"
```
```   273   using floor_add_of_int [of x "numeral v"] by simp
```
```   274
```
```   275 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
```
```   276   using floor_add_of_int [of x 1] by simp
```
```   277
```
```   278 lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
```
```   279   using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
```
```   280
```
```   281 lemma floor_diff_numeral [simp]:
```
```   282   "floor (x - numeral v) = floor x - numeral v"
```
```   283   using floor_diff_of_int [of x "numeral v"] by simp
```
```   284
```
```   285 lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
```
```   286   using floor_diff_of_int [of x 1] by simp
```
```   287
```
```   288
```
```   289 subsection {* Ceiling function *}
```
```   290
```
```   291 definition
```
```   292   ceiling :: "'a::floor_ceiling \<Rightarrow> int" where
```
```   293   "ceiling x = - floor (- x)"
```
```   294
```
```   295 notation (xsymbols)
```
```   296   ceiling  ("\<lceil>_\<rceil>")
```
```   297
```
```   298 notation (HTML output)
```
```   299   ceiling  ("\<lceil>_\<rceil>")
```
```   300
```
```   301 lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
```
```   302   unfolding ceiling_def using floor_correct [of "- x"] by simp
```
```   303
```
```   304 lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
```
```   305   unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
```
```   306
```
```   307 lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
```
```   308   using ceiling_correct ..
```
```   309
```
```   310 lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
```
```   311   unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
```
```   312
```
```   313 lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
```
```   314   by (simp add: not_le [symmetric] ceiling_le_iff)
```
```   315
```
```   316 lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
```
```   317   using ceiling_le_iff [of x "z - 1"] by simp
```
```   318
```
```   319 lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
```
```   320   by (simp add: not_less [symmetric] ceiling_less_iff)
```
```   321
```
```   322 lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
```
```   323   unfolding ceiling_def by (simp add: floor_mono)
```
```   324
```
```   325 lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
```
```   326   by (auto simp add: not_le [symmetric] ceiling_mono)
```
```   327
```
```   328 lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
```
```   329   by (rule ceiling_unique) simp_all
```
```   330
```
```   331 lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
```
```   332   using ceiling_of_int [of "of_nat n"] by simp
```
```   333
```
```   334 lemma ceiling_add_le: "ceiling (x + y) \<le> ceiling x + ceiling y"
```
```   335   by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
```
```   336
```
```   337 text {* Ceiling with numerals *}
```
```   338
```
```   339 lemma ceiling_zero [simp]: "ceiling 0 = 0"
```
```   340   using ceiling_of_int [of 0] by simp
```
```   341
```
```   342 lemma ceiling_one [simp]: "ceiling 1 = 1"
```
```   343   using ceiling_of_int [of 1] by simp
```
```   344
```
```   345 lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v"
```
```   346   using ceiling_of_int [of "numeral v"] by simp
```
```   347
```
```   348 lemma ceiling_neg_numeral [simp]: "ceiling (- numeral v) = - numeral v"
```
```   349   using ceiling_of_int [of "- numeral v"] by simp
```
```   350
```
```   351 lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```   352   by (simp add: ceiling_le_iff)
```
```   353
```
```   354 lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
```
```   355   by (simp add: ceiling_le_iff)
```
```   356
```
```   357 lemma ceiling_le_numeral [simp]:
```
```   358   "ceiling x \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
```
```   359   by (simp add: ceiling_le_iff)
```
```   360
```
```   361 lemma ceiling_le_neg_numeral [simp]:
```
```   362   "ceiling x \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v"
```
```   363   by (simp add: ceiling_le_iff)
```
```   364
```
```   365 lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
```
```   366   by (simp add: ceiling_less_iff)
```
```   367
```
```   368 lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
```
```   369   by (simp add: ceiling_less_iff)
```
```   370
```
```   371 lemma ceiling_less_numeral [simp]:
```
```   372   "ceiling x < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
```
```   373   by (simp add: ceiling_less_iff)
```
```   374
```
```   375 lemma ceiling_less_neg_numeral [simp]:
```
```   376   "ceiling x < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1"
```
```   377   by (simp add: ceiling_less_iff)
```
```   378
```
```   379 lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
```
```   380   by (simp add: le_ceiling_iff)
```
```   381
```
```   382 lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
```
```   383   by (simp add: le_ceiling_iff)
```
```   384
```
```   385 lemma numeral_le_ceiling [simp]:
```
```   386   "numeral v \<le> ceiling x \<longleftrightarrow> numeral v - 1 < x"
```
```   387   by (simp add: le_ceiling_iff)
```
```   388
```
```   389 lemma neg_numeral_le_ceiling [simp]:
```
```   390   "- numeral v \<le> ceiling x \<longleftrightarrow> - numeral v - 1 < x"
```
```   391   by (simp add: le_ceiling_iff)
```
```   392
```
```   393 lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
```
```   394   by (simp add: less_ceiling_iff)
```
```   395
```
```   396 lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
```
```   397   by (simp add: less_ceiling_iff)
```
```   398
```
```   399 lemma numeral_less_ceiling [simp]:
```
```   400   "numeral v < ceiling x \<longleftrightarrow> numeral v < x"
```
```   401   by (simp add: less_ceiling_iff)
```
```   402
```
```   403 lemma neg_numeral_less_ceiling [simp]:
```
```   404   "- numeral v < ceiling x \<longleftrightarrow> - numeral v < x"
```
```   405   by (simp add: less_ceiling_iff)
```
```   406
```
```   407 text {* Addition and subtraction of integers *}
```
```   408
```
```   409 lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z"
```
```   410   using ceiling_correct [of x] by (simp add: ceiling_unique)
```
```   411
```
```   412 lemma ceiling_add_numeral [simp]:
```
```   413     "ceiling (x + numeral v) = ceiling x + numeral v"
```
```   414   using ceiling_add_of_int [of x "numeral v"] by simp
```
```   415
```
```   416 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
```
```   417   using ceiling_add_of_int [of x 1] by simp
```
```   418
```
```   419 lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
```
```   420   using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
```
```   421
```
```   422 lemma ceiling_diff_numeral [simp]:
```
```   423   "ceiling (x - numeral v) = ceiling x - numeral v"
```
```   424   using ceiling_diff_of_int [of x "numeral v"] by simp
```
```   425
```
```   426 lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
```
```   427   using ceiling_diff_of_int [of x 1] by simp
```
```   428
```
```   429 lemma ceiling_diff_floor_le_1: "ceiling x - floor x \<le> 1"
```
```   430 proof -
```
```   431   have "of_int \<lceil>x\<rceil> - 1 < x"
```
```   432     using ceiling_correct[of x] by simp
```
```   433   also have "x < of_int \<lfloor>x\<rfloor> + 1"
```
```   434     using floor_correct[of x] by simp_all
```
```   435   finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
```
```   436     by simp
```
```   437   then show ?thesis
```
```   438     unfolding of_int_less_iff by simp
```
```   439 qed
```
```   440
```
```   441 subsection {* Negation *}
```
```   442
```
```   443 lemma floor_minus: "floor (- x) = - ceiling x"
```
```   444   unfolding ceiling_def by simp
```
```   445
```
```   446 lemma ceiling_minus: "ceiling (- x) = - floor x"
```
```   447   unfolding ceiling_def by simp
```
```   448
```
```   449 end
```