src/HOL/Archimedean_Field.thy
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     1 (* Title:      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 = ordered_field + number_ring +
       
    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 definition
       
   145   floor :: "'a::archimedean_field \<Rightarrow> int" where
       
   146   [code del]: "floor x = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
       
   147 
       
   148 notation (xsymbols)
       
   149   floor  ("\<lfloor>_\<rfloor>")
       
   150 
       
   151 notation (HTML output)
       
   152   floor  ("\<lfloor>_\<rfloor>")
       
   153 
       
   154 lemma floor_correct: "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
       
   155   unfolding floor_def using floor_exists1 by (rule theI')
       
   156 
       
   157 lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
       
   158   using floor_correct [of x] floor_exists1 [of x] by auto
       
   159 
       
   160 lemma of_int_floor_le: "of_int (floor x) \<le> x"
       
   161   using floor_correct ..
       
   162 
       
   163 lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
       
   164 proof
       
   165   assume "z \<le> floor x"
       
   166   then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
       
   167   also have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
       
   168   finally show "of_int z \<le> x" .
       
   169 next
       
   170   assume "of_int z \<le> x"
       
   171   also have "x < of_int (floor x + 1)" using floor_correct ..
       
   172   finally show "z \<le> floor x" by (simp del: of_int_add)
       
   173 qed
       
   174 
       
   175 lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
       
   176   by (simp add: not_le [symmetric] le_floor_iff)
       
   177 
       
   178 lemma less_floor_iff: "z < floor x \<longleftrightarrow> of_int z + 1 \<le> x"
       
   179   using le_floor_iff [of "z + 1" x] by auto
       
   180 
       
   181 lemma floor_le_iff: "floor x \<le> z \<longleftrightarrow> x < of_int z + 1"
       
   182   by (simp add: not_less [symmetric] less_floor_iff)
       
   183 
       
   184 lemma floor_mono: assumes "x \<le> y" shows "floor x \<le> floor y"
       
   185 proof -
       
   186   have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
       
   187   also note `x \<le> y`
       
   188   finally show ?thesis by (simp add: le_floor_iff)
       
   189 qed
       
   190 
       
   191 lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
       
   192   by (auto simp add: not_le [symmetric] floor_mono)
       
   193 
       
   194 lemma floor_of_int [simp]: "floor (of_int z) = z"
       
   195   by (rule floor_unique) simp_all
       
   196 
       
   197 lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
       
   198   using floor_of_int [of "of_nat n"] by simp
       
   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_number_of [simp]: "floor (number_of v) = number_of v"
       
   209   using floor_of_int [of "number_of v"] by simp
       
   210 
       
   211 lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
       
   212   by (simp add: le_floor_iff)
       
   213 
       
   214 lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
       
   215   by (simp add: le_floor_iff)
       
   216 
       
   217 lemma number_of_le_floor [simp]: "number_of v \<le> floor x \<longleftrightarrow> number_of v \<le> x"
       
   218   by (simp add: le_floor_iff)
       
   219 
       
   220 lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
       
   221   by (simp add: less_floor_iff)
       
   222 
       
   223 lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
       
   224   by (simp add: less_floor_iff)
       
   225 
       
   226 lemma number_of_less_floor [simp]:
       
   227   "number_of v < floor x \<longleftrightarrow> number_of v + 1 \<le> x"
       
   228   by (simp add: less_floor_iff)
       
   229 
       
   230 lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
       
   231   by (simp add: floor_le_iff)
       
   232 
       
   233 lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
       
   234   by (simp add: floor_le_iff)
       
   235 
       
   236 lemma floor_le_number_of [simp]:
       
   237   "floor x \<le> number_of v \<longleftrightarrow> x < number_of v + 1"
       
   238   by (simp add: floor_le_iff)
       
   239 
       
   240 lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
       
   241   by (simp add: floor_less_iff)
       
   242 
       
   243 lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
       
   244   by (simp add: floor_less_iff)
       
   245 
       
   246 lemma floor_less_number_of [simp]:
       
   247   "floor x < number_of v \<longleftrightarrow> x < number_of v"
       
   248   by (simp add: floor_less_iff)
       
   249 
       
   250 text {* Addition and subtraction of integers *}
       
   251 
       
   252 lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
       
   253   using floor_correct [of x] by (simp add: floor_unique)
       
   254 
       
   255 lemma floor_add_number_of [simp]:
       
   256     "floor (x + number_of v) = floor x + number_of v"
       
   257   using floor_add_of_int [of x "number_of v"] by simp
       
   258 
       
   259 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
       
   260   using floor_add_of_int [of x 1] by simp
       
   261 
       
   262 lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
       
   263   using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
       
   264 
       
   265 lemma floor_diff_number_of [simp]:
       
   266   "floor (x - number_of v) = floor x - number_of v"
       
   267   using floor_diff_of_int [of x "number_of v"] by simp
       
   268 
       
   269 lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
       
   270   using floor_diff_of_int [of x 1] by simp
       
   271 
       
   272 
       
   273 subsection {* Ceiling function *}
       
   274 
       
   275 definition
       
   276   ceiling :: "'a::archimedean_field \<Rightarrow> int" where
       
   277   [code del]: "ceiling x = - floor (- x)"
       
   278 
       
   279 notation (xsymbols)
       
   280   ceiling  ("\<lceil>_\<rceil>")
       
   281 
       
   282 notation (HTML output)
       
   283   ceiling  ("\<lceil>_\<rceil>")
       
   284 
       
   285 lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
       
   286   unfolding ceiling_def using floor_correct [of "- x"] by simp
       
   287 
       
   288 lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
       
   289   unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
       
   290 
       
   291 lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
       
   292   using ceiling_correct ..
       
   293 
       
   294 lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
       
   295   unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
       
   296 
       
   297 lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
       
   298   by (simp add: not_le [symmetric] ceiling_le_iff)
       
   299 
       
   300 lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
       
   301   using ceiling_le_iff [of x "z - 1"] by simp
       
   302 
       
   303 lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
       
   304   by (simp add: not_less [symmetric] ceiling_less_iff)
       
   305 
       
   306 lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
       
   307   unfolding ceiling_def by (simp add: floor_mono)
       
   308 
       
   309 lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
       
   310   by (auto simp add: not_le [symmetric] ceiling_mono)
       
   311 
       
   312 lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
       
   313   by (rule ceiling_unique) simp_all
       
   314 
       
   315 lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
       
   316   using ceiling_of_int [of "of_nat n"] by simp
       
   317 
       
   318 text {* Ceiling with numerals *}
       
   319 
       
   320 lemma ceiling_zero [simp]: "ceiling 0 = 0"
       
   321   using ceiling_of_int [of 0] by simp
       
   322 
       
   323 lemma ceiling_one [simp]: "ceiling 1 = 1"
       
   324   using ceiling_of_int [of 1] by simp
       
   325 
       
   326 lemma ceiling_number_of [simp]: "ceiling (number_of v) = number_of v"
       
   327   using ceiling_of_int [of "number_of v"] by simp
       
   328 
       
   329 lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
       
   330   by (simp add: ceiling_le_iff)
       
   331 
       
   332 lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
       
   333   by (simp add: ceiling_le_iff)
       
   334 
       
   335 lemma ceiling_le_number_of [simp]:
       
   336   "ceiling x \<le> number_of v \<longleftrightarrow> x \<le> number_of v"
       
   337   by (simp add: ceiling_le_iff)
       
   338 
       
   339 lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
       
   340   by (simp add: ceiling_less_iff)
       
   341 
       
   342 lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
       
   343   by (simp add: ceiling_less_iff)
       
   344 
       
   345 lemma ceiling_less_number_of [simp]:
       
   346   "ceiling x < number_of v \<longleftrightarrow> x \<le> number_of v - 1"
       
   347   by (simp add: ceiling_less_iff)
       
   348 
       
   349 lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
       
   350   by (simp add: le_ceiling_iff)
       
   351 
       
   352 lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
       
   353   by (simp add: le_ceiling_iff)
       
   354 
       
   355 lemma number_of_le_ceiling [simp]:
       
   356   "number_of v \<le> ceiling x\<longleftrightarrow> number_of v - 1 < x"
       
   357   by (simp add: le_ceiling_iff)
       
   358 
       
   359 lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
       
   360   by (simp add: less_ceiling_iff)
       
   361 
       
   362 lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
       
   363   by (simp add: less_ceiling_iff)
       
   364 
       
   365 lemma number_of_less_ceiling [simp]:
       
   366   "number_of v < ceiling x \<longleftrightarrow> number_of v < x"
       
   367   by (simp add: less_ceiling_iff)
       
   368 
       
   369 text {* Addition and subtraction of integers *}
       
   370 
       
   371 lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z"
       
   372   using ceiling_correct [of x] by (simp add: ceiling_unique)
       
   373 
       
   374 lemma ceiling_add_number_of [simp]:
       
   375     "ceiling (x + number_of v) = ceiling x + number_of v"
       
   376   using ceiling_add_of_int [of x "number_of v"] by simp
       
   377 
       
   378 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
       
   379   using ceiling_add_of_int [of x 1] by simp
       
   380 
       
   381 lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
       
   382   using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
       
   383 
       
   384 lemma ceiling_diff_number_of [simp]:
       
   385   "ceiling (x - number_of v) = ceiling x - number_of v"
       
   386   using ceiling_diff_of_int [of x "number_of v"] by simp
       
   387 
       
   388 lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
       
   389   using ceiling_diff_of_int [of x 1] by simp
       
   390 
       
   391 
       
   392 subsection {* Negation *}
       
   393 
       
   394 lemma floor_minus [simp]: "floor (- x) = - ceiling x"
       
   395   unfolding ceiling_def by simp
       
   396 
       
   397 lemma ceiling_minus [simp]: "ceiling (- x) = - floor x"
       
   398   unfolding ceiling_def by simp
       
   399 
       
   400 end