src/HOL/Archimedean_Field.thy
 author wenzelm Thu May 24 17:25:53 2012 +0200 (2012-05-24) changeset 47988 e4b69e10b990 parent 47592 a6b76247534d child 54281 b01057e72233 permissions -rw-r--r--
tuned proofs;
```     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   then have
```
```   133     "of_int y \<le> x" "x < of_int (y + 1)"
```
```   134     "of_int z \<le> x" "x < of_int (z + 1)"
```
```   135     by simp_all
```
```   136   from le_less_trans [OF `of_int y \<le> x` `x < of_int (z + 1)`]
```
```   137        le_less_trans [OF `of_int z \<le> x` `x < of_int (y + 1)`]
```
```   138   show "y = z" by (simp del: of_int_add)
```
```   139 qed
```
```   140
```
```   141
```
```   142 subsection {* Floor function *}
```
```   143
```
```   144 class floor_ceiling = archimedean_field +
```
```   145   fixes floor :: "'a \<Rightarrow> int"
```
```   146   assumes floor_correct: "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
```
```   147
```
```   148 notation (xsymbols)
```
```   149   floor  ("\<lfloor>_\<rfloor>")
```
```   150
```
```   151 notation (HTML output)
```
```   152   floor  ("\<lfloor>_\<rfloor>")
```
```   153
```
```   154 lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
```
```   155   using floor_correct [of x] floor_exists1 [of x] by auto
```
```   156
```
```   157 lemma of_int_floor_le: "of_int (floor x) \<le> x"
```
```   158   using floor_correct ..
```
```   159
```
```   160 lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
```
```   161 proof
```
```   162   assume "z \<le> floor x"
```
```   163   then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
```
```   164   also have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
```
```   165   finally show "of_int z \<le> x" .
```
```   166 next
```
```   167   assume "of_int z \<le> x"
```
```   168   also have "x < of_int (floor x + 1)" using floor_correct ..
```
```   169   finally show "z \<le> floor x" by (simp del: of_int_add)
```
```   170 qed
```
```   171
```
```   172 lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
```
```   173   by (simp add: not_le [symmetric] le_floor_iff)
```
```   174
```
```   175 lemma less_floor_iff: "z < floor x \<longleftrightarrow> of_int z + 1 \<le> x"
```
```   176   using le_floor_iff [of "z + 1" x] by auto
```
```   177
```
```   178 lemma floor_le_iff: "floor x \<le> z \<longleftrightarrow> x < of_int z + 1"
```
```   179   by (simp add: not_less [symmetric] less_floor_iff)
```
```   180
```
```   181 lemma floor_mono: assumes "x \<le> y" shows "floor x \<le> floor y"
```
```   182 proof -
```
```   183   have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
```
```   184   also note `x \<le> y`
```
```   185   finally show ?thesis by (simp add: le_floor_iff)
```
```   186 qed
```
```   187
```
```   188 lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
```
```   189   by (auto simp add: not_le [symmetric] floor_mono)
```
```   190
```
```   191 lemma floor_of_int [simp]: "floor (of_int z) = z"
```
```   192   by (rule floor_unique) simp_all
```
```   193
```
```   194 lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
```
```   195   using floor_of_int [of "of_nat n"] by simp
```
```   196
```
```   197 lemma le_floor_add: "floor x + floor y \<le> floor (x + y)"
```
```   198   by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
```
```   199
```
```   200 text {* Floor with numerals *}
```
```   201
```
```   202 lemma floor_zero [simp]: "floor 0 = 0"
```
```   203   using floor_of_int [of 0] by simp
```
```   204
```
```   205 lemma floor_one [simp]: "floor 1 = 1"
```
```   206   using floor_of_int [of 1] by simp
```
```   207
```
```   208 lemma floor_numeral [simp]: "floor (numeral v) = numeral v"
```
```   209   using floor_of_int [of "numeral v"] by simp
```
```   210
```
```   211 lemma floor_neg_numeral [simp]: "floor (neg_numeral v) = neg_numeral v"
```
```   212   using floor_of_int [of "neg_numeral v"] by simp
```
```   213
```
```   214 lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
```
```   215   by (simp add: le_floor_iff)
```
```   216
```
```   217 lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
```
```   218   by (simp add: le_floor_iff)
```
```   219
```
```   220 lemma 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 neg_numeral_le_floor [simp]:
```
```   225   "neg_numeral v \<le> floor x \<longleftrightarrow> neg_numeral v \<le> x"
```
```   226   by (simp add: le_floor_iff)
```
```   227
```
```   228 lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
```
```   229   by (simp add: less_floor_iff)
```
```   230
```
```   231 lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
```
```   232   by (simp add: less_floor_iff)
```
```   233
```
```   234 lemma 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 neg_numeral_less_floor [simp]:
```
```   239   "neg_numeral v < floor x \<longleftrightarrow> neg_numeral v + 1 \<le> x"
```
```   240   by (simp add: less_floor_iff)
```
```   241
```
```   242 lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
```
```   243   by (simp add: floor_le_iff)
```
```   244
```
```   245 lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
```
```   246   by (simp add: floor_le_iff)
```
```   247
```
```   248 lemma floor_le_numeral [simp]:
```
```   249   "floor x \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
```
```   250   by (simp add: floor_le_iff)
```
```   251
```
```   252 lemma floor_le_neg_numeral [simp]:
```
```   253   "floor x \<le> neg_numeral v \<longleftrightarrow> x < neg_numeral v + 1"
```
```   254   by (simp add: floor_le_iff)
```
```   255
```
```   256 lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
```
```   257   by (simp add: floor_less_iff)
```
```   258
```
```   259 lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
```
```   260   by (simp add: floor_less_iff)
```
```   261
```
```   262 lemma floor_less_numeral [simp]:
```
```   263   "floor x < numeral v \<longleftrightarrow> x < numeral v"
```
```   264   by (simp add: floor_less_iff)
```
```   265
```
```   266 lemma floor_less_neg_numeral [simp]:
```
```   267   "floor x < neg_numeral v \<longleftrightarrow> x < neg_numeral v"
```
```   268   by (simp add: floor_less_iff)
```
```   269
```
```   270 text {* Addition and subtraction of integers *}
```
```   271
```
```   272 lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
```
```   273   using floor_correct [of x] by (simp add: floor_unique)
```
```   274
```
```   275 lemma floor_add_numeral [simp]:
```
```   276     "floor (x + numeral v) = floor x + numeral v"
```
```   277   using floor_add_of_int [of x "numeral v"] by simp
```
```   278
```
```   279 lemma floor_add_neg_numeral [simp]:
```
```   280     "floor (x + neg_numeral v) = floor x + neg_numeral v"
```
```   281   using floor_add_of_int [of x "neg_numeral v"] by simp
```
```   282
```
```   283 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
```
```   284   using floor_add_of_int [of x 1] by simp
```
```   285
```
```   286 lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
```
```   287   using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
```
```   288
```
```   289 lemma floor_diff_numeral [simp]:
```
```   290   "floor (x - numeral v) = floor x - numeral v"
```
```   291   using floor_diff_of_int [of x "numeral v"] by simp
```
```   292
```
```   293 lemma floor_diff_neg_numeral [simp]:
```
```   294   "floor (x - neg_numeral v) = floor x - neg_numeral v"
```
```   295   using floor_diff_of_int [of x "neg_numeral v"] by simp
```
```   296
```
```   297 lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
```
```   298   using floor_diff_of_int [of x 1] by simp
```
```   299
```
```   300
```
```   301 subsection {* Ceiling function *}
```
```   302
```
```   303 definition
```
```   304   ceiling :: "'a::floor_ceiling \<Rightarrow> int" where
```
```   305   "ceiling x = - floor (- x)"
```
```   306
```
```   307 notation (xsymbols)
```
```   308   ceiling  ("\<lceil>_\<rceil>")
```
```   309
```
```   310 notation (HTML output)
```
```   311   ceiling  ("\<lceil>_\<rceil>")
```
```   312
```
```   313 lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
```
```   314   unfolding ceiling_def using floor_correct [of "- x"] by simp
```
```   315
```
```   316 lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
```
```   317   unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
```
```   318
```
```   319 lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
```
```   320   using ceiling_correct ..
```
```   321
```
```   322 lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
```
```   323   unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
```
```   324
```
```   325 lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
```
```   326   by (simp add: not_le [symmetric] ceiling_le_iff)
```
```   327
```
```   328 lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
```
```   329   using ceiling_le_iff [of x "z - 1"] by simp
```
```   330
```
```   331 lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
```
```   332   by (simp add: not_less [symmetric] ceiling_less_iff)
```
```   333
```
```   334 lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
```
```   335   unfolding ceiling_def by (simp add: floor_mono)
```
```   336
```
```   337 lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
```
```   338   by (auto simp add: not_le [symmetric] ceiling_mono)
```
```   339
```
```   340 lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
```
```   341   by (rule ceiling_unique) simp_all
```
```   342
```
```   343 lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
```
```   344   using ceiling_of_int [of "of_nat n"] by simp
```
```   345
```
```   346 lemma ceiling_add_le: "ceiling (x + y) \<le> ceiling x + ceiling y"
```
```   347   by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
```
```   348
```
```   349 text {* Ceiling with numerals *}
```
```   350
```
```   351 lemma ceiling_zero [simp]: "ceiling 0 = 0"
```
```   352   using ceiling_of_int [of 0] by simp
```
```   353
```
```   354 lemma ceiling_one [simp]: "ceiling 1 = 1"
```
```   355   using ceiling_of_int [of 1] by simp
```
```   356
```
```   357 lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v"
```
```   358   using ceiling_of_int [of "numeral v"] by simp
```
```   359
```
```   360 lemma ceiling_neg_numeral [simp]: "ceiling (neg_numeral v) = neg_numeral v"
```
```   361   using ceiling_of_int [of "neg_numeral v"] by simp
```
```   362
```
```   363 lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```   364   by (simp add: ceiling_le_iff)
```
```   365
```
```   366 lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
```
```   367   by (simp add: ceiling_le_iff)
```
```   368
```
```   369 lemma ceiling_le_numeral [simp]:
```
```   370   "ceiling x \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
```
```   371   by (simp add: ceiling_le_iff)
```
```   372
```
```   373 lemma ceiling_le_neg_numeral [simp]:
```
```   374   "ceiling x \<le> neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v"
```
```   375   by (simp add: ceiling_le_iff)
```
```   376
```
```   377 lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
```
```   378   by (simp add: ceiling_less_iff)
```
```   379
```
```   380 lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
```
```   381   by (simp add: ceiling_less_iff)
```
```   382
```
```   383 lemma ceiling_less_numeral [simp]:
```
```   384   "ceiling x < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
```
```   385   by (simp add: ceiling_less_iff)
```
```   386
```
```   387 lemma ceiling_less_neg_numeral [simp]:
```
```   388   "ceiling x < neg_numeral v \<longleftrightarrow> x \<le> neg_numeral v - 1"
```
```   389   by (simp add: ceiling_less_iff)
```
```   390
```
```   391 lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
```
```   392   by (simp add: le_ceiling_iff)
```
```   393
```
```   394 lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
```
```   395   by (simp add: le_ceiling_iff)
```
```   396
```
```   397 lemma numeral_le_ceiling [simp]:
```
```   398   "numeral v \<le> ceiling x \<longleftrightarrow> numeral v - 1 < x"
```
```   399   by (simp add: le_ceiling_iff)
```
```   400
```
```   401 lemma neg_numeral_le_ceiling [simp]:
```
```   402   "neg_numeral v \<le> ceiling x \<longleftrightarrow> neg_numeral v - 1 < x"
```
```   403   by (simp add: le_ceiling_iff)
```
```   404
```
```   405 lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
```
```   406   by (simp add: less_ceiling_iff)
```
```   407
```
```   408 lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
```
```   409   by (simp add: less_ceiling_iff)
```
```   410
```
```   411 lemma numeral_less_ceiling [simp]:
```
```   412   "numeral v < ceiling x \<longleftrightarrow> numeral v < x"
```
```   413   by (simp add: less_ceiling_iff)
```
```   414
```
```   415 lemma neg_numeral_less_ceiling [simp]:
```
```   416   "neg_numeral v < ceiling x \<longleftrightarrow> neg_numeral v < x"
```
```   417   by (simp add: less_ceiling_iff)
```
```   418
```
```   419 text {* Addition and subtraction of integers *}
```
```   420
```
```   421 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)
```
```   423
```
```   424 lemma ceiling_add_numeral [simp]:
```
```   425     "ceiling (x + numeral v) = ceiling x + numeral v"
```
```   426   using ceiling_add_of_int [of x "numeral v"] by simp
```
```   427
```
```   428 lemma ceiling_add_neg_numeral [simp]:
```
```   429     "ceiling (x + neg_numeral v) = ceiling x + neg_numeral v"
```
```   430   using ceiling_add_of_int [of x "neg_numeral v"] by simp
```
```   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
```
```   435 lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
```
```   436   using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
```
```   437
```
```   438 lemma ceiling_diff_numeral [simp]:
```
```   439   "ceiling (x - numeral v) = ceiling x - numeral v"
```
```   440   using ceiling_diff_of_int [of x "numeral v"] by simp
```
```   441
```
```   442 lemma ceiling_diff_neg_numeral [simp]:
```
```   443   "ceiling (x - neg_numeral v) = ceiling x - neg_numeral v"
```
```   444   using ceiling_diff_of_int [of x "neg_numeral v"] by simp
```
```   445
```
```   446 lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
```
```   447   using ceiling_diff_of_int [of x 1] by simp
```
```   448
```
```   449 lemma ceiling_diff_floor_le_1: "ceiling x - floor x \<le> 1"
```
```   450 proof -
```
```   451   have "of_int \<lceil>x\<rceil> - 1 < x"
```
```   452     using ceiling_correct[of x] by simp
```
```   453   also have "x < of_int \<lfloor>x\<rfloor> + 1"
```
```   454     using floor_correct[of x] by simp_all
```
```   455   finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
```
```   456     by simp
```
```   457   then show ?thesis
```
```   458     unfolding of_int_less_iff by simp
```
```   459 qed
```
```   460
```
```   461 subsection {* Negation *}
```
```   462
```
```   463 lemma floor_minus: "floor (- x) = - ceiling x"
```
```   464   unfolding ceiling_def by simp
```
```   465
```
```   466 lemma ceiling_minus: "ceiling (- x) = - floor x"
```
```   467   unfolding ceiling_def by simp
```
```   468
```
```   469 end
```