src/HOL/Archimedean_Field.thy
 author eberlm Mon Nov 02 11:56:28 2015 +0100 (2015-11-02) changeset 61531 ab2e862263e7 parent 61378 3e04c9ca001a child 61649 268d88ec9087 permissions -rw-r--r--
Rounding function, uniform limits, cotangent, binomial identities
```     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 subsection \<open>Class of Archimedean fields\<close>
```
```    12
```
```    13 text \<open>Archimedean fields have no infinite elements.\<close>
```
```    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 \<open>Archimedean fields have no infinitesimal elements.\<close>
```
```    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 \<open>0 < x\<close> 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 \<open>0 < inverse x\<close> by (cases n) (simp_all del: of_nat_Suc)
```
```    63   then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
```
```    64     using \<open>0 < inverse x\<close> by (rule less_imp_inverse_less)
```
```    65   then have "inverse (of_nat (Suc m)) < x"
```
```    66     using \<open>0 < x\<close> 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 \<open>0 < x\<close>] 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 \<open>0 < x\<close> have "y < of_nat n * x" by (simp add: pos_divide_less_eq)
```
```    81   then show ?thesis ..
```
```    82 qed
```
```    83
```
```    84
```
```    85 subsection \<open>Existence and uniqueness of floor function\<close>
```
```    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 \<open>\<exists>n. P n\<close> have "P (Least P)" by (rule LeastI_ex)
```
```    92   with \<open>\<not> P 0\<close> 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 \<open>P (Least P)\<close> \<open>Least P = Suc n\<close> 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 \<open>Floor function\<close>
```
```   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 lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
```
```   148   using floor_correct [of x] floor_exists1 [of x] by auto
```
```   149
```
```   150 lemma floor_unique_iff:
```
```   151   fixes x :: "'a::floor_ceiling"
```
```   152   shows  "\<lfloor>x\<rfloor> = a \<longleftrightarrow> of_int a \<le> x \<and> x < of_int a + 1"
```
```   153 using floor_correct floor_unique by auto
```
```   154
```
```   155 lemma of_int_floor_le: "of_int (floor x) \<le> x"
```
```   156   using floor_correct ..
```
```   157
```
```   158 lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
```
```   159 proof
```
```   160   assume "z \<le> floor x"
```
```   161   then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
```
```   162   also have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
```
```   163   finally show "of_int z \<le> x" .
```
```   164 next
```
```   165   assume "of_int z \<le> x"
```
```   166   also have "x < of_int (floor x + 1)" using floor_correct ..
```
```   167   finally show "z \<le> floor x" by (simp del: of_int_add)
```
```   168 qed
```
```   169
```
```   170 lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
```
```   171   by (simp add: not_le [symmetric] le_floor_iff)
```
```   172
```
```   173 lemma less_floor_iff: "z < floor x \<longleftrightarrow> of_int z + 1 \<le> x"
```
```   174   using le_floor_iff [of "z + 1" x] by auto
```
```   175
```
```   176 lemma floor_le_iff: "floor x \<le> z \<longleftrightarrow> x < of_int z + 1"
```
```   177   by (simp add: not_less [symmetric] less_floor_iff)
```
```   178
```
```   179 lemma floor_split[arith_split]: "P (floor t) \<longleftrightarrow> (\<forall>i. of_int i \<le> t \<and> t < of_int i + 1 \<longrightarrow> P i)"
```
```   180   by (metis floor_correct floor_unique less_floor_iff not_le order_refl)
```
```   181
```
```   182 lemma floor_mono: assumes "x \<le> y" shows "floor x \<le> floor y"
```
```   183 proof -
```
```   184   have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
```
```   185   also note \<open>x \<le> y\<close>
```
```   186   finally show ?thesis by (simp add: le_floor_iff)
```
```   187 qed
```
```   188
```
```   189 lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
```
```   190   by (auto simp add: not_le [symmetric] floor_mono)
```
```   191
```
```   192 lemma floor_of_int [simp]: "floor (of_int z) = z"
```
```   193   by (rule floor_unique) simp_all
```
```   194
```
```   195 lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
```
```   196   using floor_of_int [of "of_nat n"] by simp
```
```   197
```
```   198 lemma le_floor_add: "floor x + floor y \<le> floor (x + y)"
```
```   199   by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le)
```
```   200
```
```   201 text \<open>Floor with numerals\<close>
```
```   202
```
```   203 lemma floor_zero [simp]: "floor 0 = 0"
```
```   204   using floor_of_int [of 0] by simp
```
```   205
```
```   206 lemma floor_one [simp]: "floor 1 = 1"
```
```   207   using floor_of_int [of 1] by simp
```
```   208
```
```   209 lemma floor_numeral [simp]: "floor (numeral v) = numeral v"
```
```   210   using floor_of_int [of "numeral v"] by simp
```
```   211
```
```   212 lemma floor_neg_numeral [simp]: "floor (- numeral v) = - numeral v"
```
```   213   using floor_of_int [of "- numeral v"] by simp
```
```   214
```
```   215 lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
```
```   216   by (simp add: le_floor_iff)
```
```   217
```
```   218 lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
```
```   219   by (simp add: le_floor_iff)
```
```   220
```
```   221 lemma numeral_le_floor [simp]:
```
```   222   "numeral v \<le> floor x \<longleftrightarrow> numeral v \<le> x"
```
```   223   by (simp add: le_floor_iff)
```
```   224
```
```   225 lemma neg_numeral_le_floor [simp]:
```
```   226   "- numeral v \<le> floor x \<longleftrightarrow> - numeral v \<le> x"
```
```   227   by (simp add: le_floor_iff)
```
```   228
```
```   229 lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
```
```   230   by (simp add: less_floor_iff)
```
```   231
```
```   232 lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
```
```   233   by (simp add: less_floor_iff)
```
```   234
```
```   235 lemma numeral_less_floor [simp]:
```
```   236   "numeral v < floor x \<longleftrightarrow> numeral v + 1 \<le> x"
```
```   237   by (simp add: less_floor_iff)
```
```   238
```
```   239 lemma neg_numeral_less_floor [simp]:
```
```   240   "- numeral v < floor x \<longleftrightarrow> - numeral v + 1 \<le> x"
```
```   241   by (simp add: less_floor_iff)
```
```   242
```
```   243 lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
```
```   244   by (simp add: floor_le_iff)
```
```   245
```
```   246 lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
```
```   247   by (simp add: floor_le_iff)
```
```   248
```
```   249 lemma floor_le_numeral [simp]:
```
```   250   "floor x \<le> numeral v \<longleftrightarrow> x < numeral v + 1"
```
```   251   by (simp add: floor_le_iff)
```
```   252
```
```   253 lemma floor_le_neg_numeral [simp]:
```
```   254   "floor x \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1"
```
```   255   by (simp add: floor_le_iff)
```
```   256
```
```   257 lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
```
```   258   by (simp add: floor_less_iff)
```
```   259
```
```   260 lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
```
```   261   by (simp add: floor_less_iff)
```
```   262
```
```   263 lemma floor_less_numeral [simp]:
```
```   264   "floor x < numeral v \<longleftrightarrow> x < numeral v"
```
```   265   by (simp add: floor_less_iff)
```
```   266
```
```   267 lemma floor_less_neg_numeral [simp]:
```
```   268   "floor x < - numeral v \<longleftrightarrow> x < - numeral v"
```
```   269   by (simp add: floor_less_iff)
```
```   270
```
```   271 text \<open>Addition and subtraction of integers\<close>
```
```   272
```
```   273 lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
```
```   274   using floor_correct [of x] by (simp add: floor_unique)
```
```   275
```
```   276 lemma floor_add_numeral [simp]:
```
```   277     "floor (x + numeral v) = floor x + numeral v"
```
```   278   using floor_add_of_int [of x "numeral v"] by simp
```
```   279
```
```   280 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
```
```   281   using floor_add_of_int [of x 1] by simp
```
```   282
```
```   283 lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
```
```   284   using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
```
```   285
```
```   286 lemma floor_uminus_of_int [simp]: "floor (- (of_int z)) = - z"
```
```   287   by (metis floor_diff_of_int [of 0] diff_0 floor_zero)
```
```   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_one [simp]: "floor (x - 1) = floor x - 1"
```
```   294   using floor_diff_of_int [of x 1] by simp
```
```   295
```
```   296 lemma le_mult_floor:
```
```   297   assumes "0 \<le> a" and "0 \<le> b"
```
```   298   shows "floor a * floor b \<le> floor (a * b)"
```
```   299 proof -
```
```   300   have "of_int (floor a) \<le> a"
```
```   301     and "of_int (floor b) \<le> b" by (auto intro: of_int_floor_le)
```
```   302   hence "of_int (floor a * floor b) \<le> a * b"
```
```   303     using assms by (auto intro!: mult_mono)
```
```   304   also have "a * b < of_int (floor (a * b) + 1)"
```
```   305     using floor_correct[of "a * b"] by auto
```
```   306   finally show ?thesis unfolding of_int_less_iff by simp
```
```   307 qed
```
```   308
```
```   309 lemma floor_divide_of_int_eq:
```
```   310   fixes k l :: int
```
```   311   shows "\<lfloor>of_int k / of_int l\<rfloor> = k div l"
```
```   312 proof (cases "l = 0")
```
```   313   case True then show ?thesis by simp
```
```   314 next
```
```   315   case False
```
```   316   have *: "\<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> = 0"
```
```   317   proof (cases "l > 0")
```
```   318     case True then show ?thesis
```
```   319       by (auto intro: floor_unique)
```
```   320   next
```
```   321     case False
```
```   322     obtain r where "r = - l" by blast
```
```   323     then have l: "l = - r" by simp
```
```   324     moreover with \<open>l \<noteq> 0\<close> False have "r > 0" by simp
```
```   325     ultimately show ?thesis using pos_mod_bound [of r]
```
```   326       by (auto simp add: zmod_zminus2_eq_if less_le field_simps intro: floor_unique)
```
```   327   qed
```
```   328   have "(of_int k :: 'a) = of_int (k div l * l + k mod l)"
```
```   329     by simp
```
```   330   also have "\<dots> = (of_int (k div l) + of_int (k mod l) / of_int l) * of_int l"
```
```   331     using False by (simp only: of_int_add) (simp add: field_simps)
```
```   332   finally have "(of_int k / of_int l :: 'a) = \<dots> / of_int l"
```
```   333     by simp
```
```   334   then have "(of_int k / of_int l :: 'a) = of_int (k div l) + of_int (k mod l) / of_int l"
```
```   335     using False by (simp only:) (simp add: field_simps)
```
```   336   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>"
```
```   337     by simp
```
```   338   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>"
```
```   339     by (simp add: ac_simps)
```
```   340   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"
```
```   341     by simp
```
```   342   with * show ?thesis by simp
```
```   343 qed
```
```   344
```
```   345 lemma floor_divide_of_nat_eq:
```
```   346   fixes m n :: nat
```
```   347   shows "\<lfloor>of_nat m / of_nat n\<rfloor> = of_nat (m div n)"
```
```   348 proof (cases "n = 0")
```
```   349   case True then show ?thesis by simp
```
```   350 next
```
```   351   case False
```
```   352   then have *: "\<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> = 0"
```
```   353     by (auto intro: floor_unique)
```
```   354   have "(of_nat m :: 'a) = of_nat (m div n * n + m mod n)"
```
```   355     by simp
```
```   356   also have "\<dots> = (of_nat (m div n) + of_nat (m mod n) / of_nat n) * of_nat n"
```
```   357     using False by (simp only: of_nat_add) (simp add: field_simps of_nat_mult)
```
```   358   finally have "(of_nat m / of_nat n :: 'a) = \<dots> / of_nat n"
```
```   359     by simp
```
```   360   then have "(of_nat m / of_nat n :: 'a) = of_nat (m div n) + of_nat (m mod n) / of_nat n"
```
```   361     using False by (simp only:) simp
```
```   362   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>"
```
```   363     by simp
```
```   364   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>"
```
```   365     by (simp add: ac_simps)
```
```   366   moreover have "(of_nat (m div n) :: 'a) = of_int (of_nat (m div n))"
```
```   367     by simp
```
```   368   ultimately have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> + of_nat (m div n)"
```
```   369     by (simp only: floor_add_of_int)
```
```   370   with * show ?thesis by simp
```
```   371 qed
```
```   372
```
```   373
```
```   374 subsection \<open>Ceiling function\<close>
```
```   375
```
```   376 definition
```
```   377   ceiling :: "'a::floor_ceiling \<Rightarrow> int" where
```
```   378   "ceiling x = - floor (- x)"
```
```   379
```
```   380 notation (xsymbols)
```
```   381   ceiling  ("\<lceil>_\<rceil>")
```
```   382
```
```   383 lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
```
```   384   unfolding ceiling_def using floor_correct [of "- x"] by simp
```
```   385
```
```   386 lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
```
```   387   unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
```
```   388
```
```   389 lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
```
```   390   using ceiling_correct ..
```
```   391
```
```   392 lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
```
```   393   unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
```
```   394
```
```   395 lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
```
```   396   by (simp add: not_le [symmetric] ceiling_le_iff)
```
```   397
```
```   398 lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
```
```   399   using ceiling_le_iff [of x "z - 1"] by simp
```
```   400
```
```   401 lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
```
```   402   by (simp add: not_less [symmetric] ceiling_less_iff)
```
```   403
```
```   404 lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
```
```   405   unfolding ceiling_def by (simp add: floor_mono)
```
```   406
```
```   407 lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
```
```   408   by (auto simp add: not_le [symmetric] ceiling_mono)
```
```   409
```
```   410 lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
```
```   411   by (rule ceiling_unique) simp_all
```
```   412
```
```   413 lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
```
```   414   using ceiling_of_int [of "of_nat n"] by simp
```
```   415
```
```   416 lemma ceiling_add_le: "ceiling (x + y) \<le> ceiling x + ceiling y"
```
```   417   by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling)
```
```   418
```
```   419 text \<open>Ceiling with numerals\<close>
```
```   420
```
```   421 lemma ceiling_zero [simp]: "ceiling 0 = 0"
```
```   422   using ceiling_of_int [of 0] by simp
```
```   423
```
```   424 lemma ceiling_one [simp]: "ceiling 1 = 1"
```
```   425   using ceiling_of_int [of 1] by simp
```
```   426
```
```   427 lemma ceiling_numeral [simp]: "ceiling (numeral v) = numeral v"
```
```   428   using ceiling_of_int [of "numeral v"] by simp
```
```   429
```
```   430 lemma ceiling_neg_numeral [simp]: "ceiling (- numeral v) = - numeral v"
```
```   431   using ceiling_of_int [of "- numeral v"] by simp
```
```   432
```
```   433 lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```   434   by (simp add: ceiling_le_iff)
```
```   435
```
```   436 lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
```
```   437   by (simp add: ceiling_le_iff)
```
```   438
```
```   439 lemma ceiling_le_numeral [simp]:
```
```   440   "ceiling x \<le> numeral v \<longleftrightarrow> x \<le> numeral v"
```
```   441   by (simp add: ceiling_le_iff)
```
```   442
```
```   443 lemma ceiling_le_neg_numeral [simp]:
```
```   444   "ceiling x \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v"
```
```   445   by (simp add: ceiling_le_iff)
```
```   446
```
```   447 lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
```
```   448   by (simp add: ceiling_less_iff)
```
```   449
```
```   450 lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
```
```   451   by (simp add: ceiling_less_iff)
```
```   452
```
```   453 lemma ceiling_less_numeral [simp]:
```
```   454   "ceiling x < numeral v \<longleftrightarrow> x \<le> numeral v - 1"
```
```   455   by (simp add: ceiling_less_iff)
```
```   456
```
```   457 lemma ceiling_less_neg_numeral [simp]:
```
```   458   "ceiling x < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1"
```
```   459   by (simp add: ceiling_less_iff)
```
```   460
```
```   461 lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
```
```   462   by (simp add: le_ceiling_iff)
```
```   463
```
```   464 lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
```
```   465   by (simp add: le_ceiling_iff)
```
```   466
```
```   467 lemma numeral_le_ceiling [simp]:
```
```   468   "numeral v \<le> ceiling x \<longleftrightarrow> numeral v - 1 < x"
```
```   469   by (simp add: le_ceiling_iff)
```
```   470
```
```   471 lemma neg_numeral_le_ceiling [simp]:
```
```   472   "- numeral v \<le> ceiling x \<longleftrightarrow> - numeral v - 1 < x"
```
```   473   by (simp add: le_ceiling_iff)
```
```   474
```
```   475 lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
```
```   476   by (simp add: less_ceiling_iff)
```
```   477
```
```   478 lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
```
```   479   by (simp add: less_ceiling_iff)
```
```   480
```
```   481 lemma numeral_less_ceiling [simp]:
```
```   482   "numeral v < ceiling x \<longleftrightarrow> numeral v < x"
```
```   483   by (simp add: less_ceiling_iff)
```
```   484
```
```   485 lemma neg_numeral_less_ceiling [simp]:
```
```   486   "- numeral v < ceiling x \<longleftrightarrow> - numeral v < x"
```
```   487   by (simp add: less_ceiling_iff)
```
```   488
```
```   489 lemma ceiling_altdef: "ceiling x = (if x = of_int (floor x) then floor x else floor x + 1)"
```
```   490   by (intro ceiling_unique, (simp, linarith?)+)
```
```   491
```
```   492 lemma floor_le_ceiling [simp]: "floor x \<le> ceiling x" by (simp add: ceiling_altdef)
```
```   493
```
```   494 text \<open>Addition and subtraction of integers\<close>
```
```   495
```
```   496 lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z"
```
```   497   using ceiling_correct [of x] by (simp add: ceiling_unique)
```
```   498
```
```   499 lemma ceiling_add_numeral [simp]:
```
```   500     "ceiling (x + numeral v) = ceiling x + numeral v"
```
```   501   using ceiling_add_of_int [of x "numeral v"] by simp
```
```   502
```
```   503 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
```
```   504   using ceiling_add_of_int [of x 1] by simp
```
```   505
```
```   506 lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
```
```   507   using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
```
```   508
```
```   509 lemma ceiling_diff_numeral [simp]:
```
```   510   "ceiling (x - numeral v) = ceiling x - numeral v"
```
```   511   using ceiling_diff_of_int [of x "numeral v"] by simp
```
```   512
```
```   513 lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
```
```   514   using ceiling_diff_of_int [of x 1] by simp
```
```   515
```
```   516 lemma ceiling_split[arith_split]: "P (ceiling t) \<longleftrightarrow> (\<forall>i. of_int i - 1 < t \<and> t \<le> of_int i \<longrightarrow> P i)"
```
```   517   by (auto simp add: ceiling_unique ceiling_correct)
```
```   518
```
```   519 lemma ceiling_diff_floor_le_1: "ceiling x - floor x \<le> 1"
```
```   520 proof -
```
```   521   have "of_int \<lceil>x\<rceil> - 1 < x"
```
```   522     using ceiling_correct[of x] by simp
```
```   523   also have "x < of_int \<lfloor>x\<rfloor> + 1"
```
```   524     using floor_correct[of x] by simp_all
```
```   525   finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)"
```
```   526     by simp
```
```   527   then show ?thesis
```
```   528     unfolding of_int_less_iff by simp
```
```   529 qed
```
```   530
```
```   531 subsection \<open>Negation\<close>
```
```   532
```
```   533 lemma floor_minus: "floor (- x) = - ceiling x"
```
```   534   unfolding ceiling_def by simp
```
```   535
```
```   536 lemma ceiling_minus: "ceiling (- x) = - floor x"
```
```   537   unfolding ceiling_def by simp
```
```   538
```
```   539 subsection \<open>Frac Function\<close>
```
```   540
```
```   541
```
```   542 definition frac :: "'a \<Rightarrow> 'a::floor_ceiling" where
```
```   543   "frac x \<equiv> x - of_int \<lfloor>x\<rfloor>"
```
```   544
```
```   545 lemma frac_lt_1: "frac x < 1"
```
```   546   by  (simp add: frac_def) linarith
```
```   547
```
```   548 lemma frac_eq_0_iff [simp]: "frac x = 0 \<longleftrightarrow> x \<in> \<int>"
```
```   549   by (simp add: frac_def) (metis Ints_cases Ints_of_int floor_of_int )
```
```   550
```
```   551 lemma frac_ge_0 [simp]: "frac x \<ge> 0"
```
```   552   unfolding frac_def
```
```   553   by linarith
```
```   554
```
```   555 lemma frac_gt_0_iff [simp]: "frac x > 0 \<longleftrightarrow> x \<notin> \<int>"
```
```   556   by (metis frac_eq_0_iff frac_ge_0 le_less less_irrefl)
```
```   557
```
```   558 lemma frac_of_int [simp]: "frac (of_int z) = 0"
```
```   559   by (simp add: frac_def)
```
```   560
```
```   561 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)"
```
```   562 proof -
```
```   563   {assume "x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>)"
```
```   564    then have "\<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>"
```
```   565      by (metis add.commute floor_unique le_floor_add le_floor_iff of_int_add)
```
```   566    }
```
```   567   moreover
```
```   568   { assume "\<not> x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>)"
```
```   569     then have "\<lfloor>x + y\<rfloor> = 1 + (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>)"
```
```   570       apply (simp add: floor_unique_iff)
```
```   571       apply (auto simp add: algebra_simps)
```
```   572       by linarith
```
```   573   }
```
```   574   ultimately show ?thesis
```
```   575     by (auto simp add: frac_def algebra_simps)
```
```   576 qed
```
```   577
```
```   578 lemma frac_add: "frac (x + y) = (if frac x + frac y < 1 then frac x + frac y
```
```   579                                  else (frac x + frac y) - 1)"
```
```   580   by (simp add: frac_def floor_add)
```
```   581
```
```   582 lemma frac_unique_iff:
```
```   583   fixes x :: "'a::floor_ceiling"
```
```   584   shows  "(frac x) = a \<longleftrightarrow> x - a \<in> \<int> \<and> 0 \<le> a \<and> a < 1"
```
```   585   apply (auto simp: Ints_def frac_def)
```
```   586   apply linarith
```
```   587   apply linarith
```
```   588   by (metis (no_types) add.commute add.left_neutral eq_diff_eq floor_add_of_int floor_unique of_int_0)
```
```   589
```
```   590 lemma frac_eq: "(frac x) = x \<longleftrightarrow> 0 \<le> x \<and> x < 1"
```
```   591   by (simp add: frac_unique_iff)
```
```   592
```
```   593 lemma frac_neg:
```
```   594   fixes x :: "'a::floor_ceiling"
```
```   595   shows  "frac (-x) = (if x \<in> \<int> then 0 else 1 - frac x)"
```
```   596   apply (auto simp add: frac_unique_iff)
```
```   597   apply (simp add: frac_def)
```
```   598   by (meson frac_lt_1 less_iff_diff_less_0 not_le not_less_iff_gr_or_eq)
```
```   599
```
```   600
```
```   601 subsection \<open>Rounding to the nearest integer\<close>
```
```   602
```
```   603 definition round where "round x = \<lfloor>x + 1/2\<rfloor>"
```
```   604
```
```   605 lemma of_int_round_ge:     "of_int (round x) \<ge> x - 1/2"
```
```   606   and of_int_round_le:     "of_int (round x) \<le> x + 1/2"
```
```   607   and of_int_round_abs_le: "\<bar>of_int (round x) - x\<bar> \<le> 1/2"
```
```   608   and of_int_round_gt:     "of_int (round x) > x - 1/2"
```
```   609 proof -
```
```   610   from floor_correct[of "x + 1/2"] have "x + 1/2 < of_int (round x) + 1" by (simp add: round_def)
```
```   611   from add_strict_right_mono[OF this, of "-1"] show A: "of_int (round x) > x - 1/2" by simp
```
```   612   thus "of_int (round x) \<ge> x - 1/2" by simp
```
```   613   from floor_correct[of "x + 1/2"] show "of_int (round x) \<le> x + 1/2" by (simp add: round_def)
```
```   614   with A show "\<bar>of_int (round x) - x\<bar> \<le> 1/2" by linarith
```
```   615 qed
```
```   616
```
```   617 lemma round_of_int [simp]: "round (of_int n) = n"
```
```   618   unfolding round_def by (subst floor_unique_iff) force
```
```   619
```
```   620 lemma round_0 [simp]: "round 0 = 0"
```
```   621   using round_of_int[of 0] by simp
```
```   622
```
```   623 lemma round_1 [simp]: "round 1 = 1"
```
```   624   using round_of_int[of 1] by simp
```
```   625
```
```   626 lemma round_numeral [simp]: "round (numeral n) = numeral n"
```
```   627   using round_of_int[of "numeral n"] by simp
```
```   628
```
```   629 lemma round_neg_numeral [simp]: "round (-numeral n) = -numeral n"
```
```   630   using round_of_int[of "-numeral n"] by simp
```
```   631
```
```   632 lemma round_of_nat [simp]: "round (of_nat n) = of_nat n"
```
```   633   using round_of_int[of "int n"] by simp
```
```   634
```
```   635 lemma round_mono: "x \<le> y \<Longrightarrow> round x \<le> round y"
```
```   636   unfolding round_def by (intro floor_mono) simp
```
```   637
```
```   638 lemma round_unique: "of_int y > x - 1/2 \<Longrightarrow> of_int y \<le> x + 1/2 \<Longrightarrow> round x = y"
```
```   639 unfolding round_def
```
```   640 proof (rule floor_unique)
```
```   641   assume "x - 1 / 2 < of_int y"
```
```   642   from add_strict_left_mono[OF this, of 1] show "x + 1 / 2 < of_int y + 1" by simp
```
```   643 qed
```
```   644
```
```   645 lemma round_altdef: "round x = (if frac x \<ge> 1/2 then ceiling x else floor x)"
```
```   646   by (cases "frac x \<ge> 1/2")
```
```   647      (rule round_unique, ((simp add: frac_def field_simps ceiling_altdef, linarith?)+))+
```
```   648
```
```   649 lemma floor_le_round: "\<lfloor>x\<rfloor> \<le> round x"
```
```   650   unfolding round_def by (intro floor_mono) simp
```
```   651
```
```   652 lemma ceiling_ge_round: "\<lceil>x\<rceil> \<ge> round x" unfolding round_altdef by simp
```
```   653
```
```   654 lemma round_diff_minimal:
```
```   655   fixes z :: "'a :: floor_ceiling"
```
```   656   shows "abs (z - of_int (round z)) \<le> abs (z - of_int m)"
```
```   657 proof (cases "of_int m \<ge> z")
```
```   658   case True
```
```   659   hence "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int (ceiling z) - z\<bar>"
```
```   660     unfolding round_altdef by (simp add: ceiling_altdef frac_def) linarith?
```
```   661   also have "of_int \<lceil>z\<rceil> - z \<ge> 0" by linarith
```
```   662   with True have "\<bar>of_int (ceiling z) - z\<bar> \<le> \<bar>z - of_int m\<bar>"
```
```   663     by (simp add: ceiling_le_iff)
```
```   664   finally show ?thesis .
```
```   665 next
```
```   666   case False
```
```   667   hence "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int (floor z) - z\<bar>"
```
```   668     unfolding round_altdef by (simp add: ceiling_altdef frac_def) linarith?
```
```   669   also have "z - of_int \<lfloor>z\<rfloor> \<ge> 0" by linarith
```
```   670   with False have "\<bar>of_int (floor z) - z\<bar> \<le> \<bar>z - of_int m\<bar>"
```
```   671     by (simp add: le_floor_iff)
```
```   672   finally show ?thesis .
```
```   673 qed
```
```   674
```
```   675 end
```