new theory of Archimedean fields
authorhuffman
Wed Feb 25 11:26:01 2009 -0800 (2009-02-25)
changeset 30096c5497842ee35
parent 30095 c6e184561159
child 30097 57df8626c23b
new theory of Archimedean fields
src/HOL/Archimedean_Field.thy
src/HOL/IsaMakefile
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Archimedean_Field.thy	Wed Feb 25 11:26:01 2009 -0800
     1.3 @@ -0,0 +1,400 @@
     1.4 +(* Title:      Archimedean_Field.thy
     1.5 +   Author:     Brian Huffman
     1.6 +*)
     1.7 +
     1.8 +header {* Archimedean Fields, Floor and Ceiling Functions *}
     1.9 +
    1.10 +theory Archimedean_Field
    1.11 +imports Main
    1.12 +begin
    1.13 +
    1.14 +subsection {* Class of Archimedean fields *}
    1.15 +
    1.16 +text {* Archimedean fields have no infinite elements. *}
    1.17 +
    1.18 +class archimedean_field = ordered_field + number_ring +
    1.19 +  assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
    1.20 +
    1.21 +lemma ex_less_of_int:
    1.22 +  fixes x :: "'a::archimedean_field" shows "\<exists>z. x < of_int z"
    1.23 +proof -
    1.24 +  from ex_le_of_int obtain z where "x \<le> of_int z" ..
    1.25 +  then have "x < of_int (z + 1)" by simp
    1.26 +  then show ?thesis ..
    1.27 +qed
    1.28 +
    1.29 +lemma ex_of_int_less:
    1.30 +  fixes x :: "'a::archimedean_field" shows "\<exists>z. of_int z < x"
    1.31 +proof -
    1.32 +  from ex_less_of_int obtain z where "- x < of_int z" ..
    1.33 +  then have "of_int (- z) < x" by simp
    1.34 +  then show ?thesis ..
    1.35 +qed
    1.36 +
    1.37 +lemma ex_less_of_nat:
    1.38 +  fixes x :: "'a::archimedean_field" shows "\<exists>n. x < of_nat n"
    1.39 +proof -
    1.40 +  obtain z where "x < of_int z" using ex_less_of_int ..
    1.41 +  also have "\<dots> \<le> of_int (int (nat z))" by simp
    1.42 +  also have "\<dots> = of_nat (nat z)" by (simp only: of_int_of_nat_eq)
    1.43 +  finally show ?thesis ..
    1.44 +qed
    1.45 +
    1.46 +lemma ex_le_of_nat:
    1.47 +  fixes x :: "'a::archimedean_field" shows "\<exists>n. x \<le> of_nat n"
    1.48 +proof -
    1.49 +  obtain n where "x < of_nat n" using ex_less_of_nat ..
    1.50 +  then have "x \<le> of_nat n" by simp
    1.51 +  then show ?thesis ..
    1.52 +qed
    1.53 +
    1.54 +text {* Archimedean fields have no infinitesimal elements. *}
    1.55 +
    1.56 +lemma ex_inverse_of_nat_Suc_less:
    1.57 +  fixes x :: "'a::archimedean_field"
    1.58 +  assumes "0 < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
    1.59 +proof -
    1.60 +  from `0 < x` have "0 < inverse x"
    1.61 +    by (rule positive_imp_inverse_positive)
    1.62 +  obtain n where "inverse x < of_nat n"
    1.63 +    using ex_less_of_nat ..
    1.64 +  then obtain m where "inverse x < of_nat (Suc m)"
    1.65 +    using `0 < inverse x` by (cases n) (simp_all del: of_nat_Suc)
    1.66 +  then have "inverse (of_nat (Suc m)) < inverse (inverse x)"
    1.67 +    using `0 < inverse x` by (rule less_imp_inverse_less)
    1.68 +  then have "inverse (of_nat (Suc m)) < x"
    1.69 +    using `0 < x` by (simp add: nonzero_inverse_inverse_eq)
    1.70 +  then show ?thesis ..
    1.71 +qed
    1.72 +
    1.73 +lemma ex_inverse_of_nat_less:
    1.74 +  fixes x :: "'a::archimedean_field"
    1.75 +  assumes "0 < x" shows "\<exists>n>0. inverse (of_nat n) < x"
    1.76 +  using ex_inverse_of_nat_Suc_less [OF `0 < x`] by auto
    1.77 +
    1.78 +lemma ex_less_of_nat_mult:
    1.79 +  fixes x :: "'a::archimedean_field"
    1.80 +  assumes "0 < x" shows "\<exists>n. y < of_nat n * x"
    1.81 +proof -
    1.82 +  obtain n where "y / x < of_nat n" using ex_less_of_nat ..
    1.83 +  with `0 < x` have "y < of_nat n * x" by (simp add: pos_divide_less_eq)
    1.84 +  then show ?thesis ..
    1.85 +qed
    1.86 +
    1.87 +
    1.88 +subsection {* Existence and uniqueness of floor function *}
    1.89 +
    1.90 +lemma exists_least_lemma:
    1.91 +  assumes "\<not> P 0" and "\<exists>n. P n"
    1.92 +  shows "\<exists>n. \<not> P n \<and> P (Suc n)"
    1.93 +proof -
    1.94 +  from `\<exists>n. P n` have "P (Least P)" by (rule LeastI_ex)
    1.95 +  with `\<not> P 0` obtain n where "Least P = Suc n"
    1.96 +    by (cases "Least P") auto
    1.97 +  then have "n < Least P" by simp
    1.98 +  then have "\<not> P n" by (rule not_less_Least)
    1.99 +  then have "\<not> P n \<and> P (Suc n)"
   1.100 +    using `P (Least P)` `Least P = Suc n` by simp
   1.101 +  then show ?thesis ..
   1.102 +qed
   1.103 +
   1.104 +lemma floor_exists:
   1.105 +  fixes x :: "'a::archimedean_field"
   1.106 +  shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
   1.107 +proof (cases)
   1.108 +  assume "0 \<le> x"
   1.109 +  then have "\<not> x < of_nat 0" by simp
   1.110 +  then have "\<exists>n. \<not> x < of_nat n \<and> x < of_nat (Suc n)"
   1.111 +    using ex_less_of_nat by (rule exists_least_lemma)
   1.112 +  then obtain n where "\<not> x < of_nat n \<and> x < of_nat (Suc n)" ..
   1.113 +  then have "of_int (int n) \<le> x \<and> x < of_int (int n + 1)" by simp
   1.114 +  then show ?thesis ..
   1.115 +next
   1.116 +  assume "\<not> 0 \<le> x"
   1.117 +  then have "\<not> - x \<le> of_nat 0" by simp
   1.118 +  then have "\<exists>n. \<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)"
   1.119 +    using ex_le_of_nat by (rule exists_least_lemma)
   1.120 +  then obtain n where "\<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" ..
   1.121 +  then have "of_int (- int n - 1) \<le> x \<and> x < of_int (- int n - 1 + 1)" by simp
   1.122 +  then show ?thesis ..
   1.123 +qed
   1.124 +
   1.125 +lemma floor_exists1:
   1.126 +  fixes x :: "'a::archimedean_field"
   1.127 +  shows "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)"
   1.128 +proof (rule ex_ex1I)
   1.129 +  show "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
   1.130 +    by (rule floor_exists)
   1.131 +next
   1.132 +  fix y z assume
   1.133 +    "of_int y \<le> x \<and> x < of_int (y + 1)"
   1.134 +    "of_int z \<le> x \<and> x < of_int (z + 1)"
   1.135 +  then have
   1.136 +    "of_int y \<le> x" "x < of_int (y + 1)"
   1.137 +    "of_int z \<le> x" "x < of_int (z + 1)"
   1.138 +    by simp_all
   1.139 +  from le_less_trans [OF `of_int y \<le> x` `x < of_int (z + 1)`]
   1.140 +       le_less_trans [OF `of_int z \<le> x` `x < of_int (y + 1)`]
   1.141 +  show "y = z" by (simp del: of_int_add)
   1.142 +qed
   1.143 +
   1.144 +
   1.145 +subsection {* Floor function *}
   1.146 +
   1.147 +definition
   1.148 +  floor :: "'a::archimedean_field \<Rightarrow> int" where
   1.149 +  [code del]: "floor x = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
   1.150 +
   1.151 +notation (xsymbols)
   1.152 +  floor  ("\<lfloor>_\<rfloor>")
   1.153 +
   1.154 +notation (HTML output)
   1.155 +  floor  ("\<lfloor>_\<rfloor>")
   1.156 +
   1.157 +lemma floor_correct: "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
   1.158 +  unfolding floor_def using floor_exists1 by (rule theI')
   1.159 +
   1.160 +lemma floor_unique: "\<lbrakk>of_int z \<le> x; x < of_int z + 1\<rbrakk> \<Longrightarrow> floor x = z"
   1.161 +  using floor_correct [of x] floor_exists1 [of x] by auto
   1.162 +
   1.163 +lemma of_int_floor_le: "of_int (floor x) \<le> x"
   1.164 +  using floor_correct ..
   1.165 +
   1.166 +lemma le_floor_iff: "z \<le> floor x \<longleftrightarrow> of_int z \<le> x"
   1.167 +proof
   1.168 +  assume "z \<le> floor x"
   1.169 +  then have "(of_int z :: 'a) \<le> of_int (floor x)" by simp
   1.170 +  also have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
   1.171 +  finally show "of_int z \<le> x" .
   1.172 +next
   1.173 +  assume "of_int z \<le> x"
   1.174 +  also have "x < of_int (floor x + 1)" using floor_correct ..
   1.175 +  finally show "z \<le> floor x" by (simp del: of_int_add)
   1.176 +qed
   1.177 +
   1.178 +lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
   1.179 +  by (simp add: not_le [symmetric] le_floor_iff)
   1.180 +
   1.181 +lemma less_floor_iff: "z < floor x \<longleftrightarrow> of_int z + 1 \<le> x"
   1.182 +  using le_floor_iff [of "z + 1" x] by auto
   1.183 +
   1.184 +lemma floor_le_iff: "floor x \<le> z \<longleftrightarrow> x < of_int z + 1"
   1.185 +  by (simp add: not_less [symmetric] less_floor_iff)
   1.186 +
   1.187 +lemma floor_mono: assumes "x \<le> y" shows "floor x \<le> floor y"
   1.188 +proof -
   1.189 +  have "of_int (floor x) \<le> x" by (rule of_int_floor_le)
   1.190 +  also note `x \<le> y`
   1.191 +  finally show ?thesis by (simp add: le_floor_iff)
   1.192 +qed
   1.193 +
   1.194 +lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
   1.195 +  by (auto simp add: not_le [symmetric] floor_mono)
   1.196 +
   1.197 +lemma floor_of_int [simp]: "floor (of_int z) = z"
   1.198 +  by (rule floor_unique) simp_all
   1.199 +
   1.200 +lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
   1.201 +  using floor_of_int [of "of_nat n"] by simp
   1.202 +
   1.203 +text {* Floor with numerals *}
   1.204 +
   1.205 +lemma floor_zero [simp]: "floor 0 = 0"
   1.206 +  using floor_of_int [of 0] by simp
   1.207 +
   1.208 +lemma floor_one [simp]: "floor 1 = 1"
   1.209 +  using floor_of_int [of 1] by simp
   1.210 +
   1.211 +lemma floor_number_of [simp]: "floor (number_of v) = number_of v"
   1.212 +  using floor_of_int [of "number_of v"] by simp
   1.213 +
   1.214 +lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
   1.215 +  by (simp add: le_floor_iff)
   1.216 +
   1.217 +lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
   1.218 +  by (simp add: le_floor_iff)
   1.219 +
   1.220 +lemma number_of_le_floor [simp]: "number_of v \<le> floor x \<longleftrightarrow> number_of v \<le> x"
   1.221 +  by (simp add: le_floor_iff)
   1.222 +
   1.223 +lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
   1.224 +  by (simp add: less_floor_iff)
   1.225 +
   1.226 +lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
   1.227 +  by (simp add: less_floor_iff)
   1.228 +
   1.229 +lemma number_of_less_floor [simp]:
   1.230 +  "number_of v < floor x \<longleftrightarrow> number_of v + 1 \<le> x"
   1.231 +  by (simp add: less_floor_iff)
   1.232 +
   1.233 +lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
   1.234 +  by (simp add: floor_le_iff)
   1.235 +
   1.236 +lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
   1.237 +  by (simp add: floor_le_iff)
   1.238 +
   1.239 +lemma floor_le_number_of [simp]:
   1.240 +  "floor x \<le> number_of v \<longleftrightarrow> x < number_of v + 1"
   1.241 +  by (simp add: floor_le_iff)
   1.242 +
   1.243 +lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
   1.244 +  by (simp add: floor_less_iff)
   1.245 +
   1.246 +lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
   1.247 +  by (simp add: floor_less_iff)
   1.248 +
   1.249 +lemma floor_less_number_of [simp]:
   1.250 +  "floor x < number_of v \<longleftrightarrow> x < number_of v"
   1.251 +  by (simp add: floor_less_iff)
   1.252 +
   1.253 +text {* Addition and subtraction of integers *}
   1.254 +
   1.255 +lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z"
   1.256 +  using floor_correct [of x] by (simp add: floor_unique)
   1.257 +
   1.258 +lemma floor_add_number_of [simp]:
   1.259 +    "floor (x + number_of v) = floor x + number_of v"
   1.260 +  using floor_add_of_int [of x "number_of v"] by simp
   1.261 +
   1.262 +lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
   1.263 +  using floor_add_of_int [of x 1] by simp
   1.264 +
   1.265 +lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
   1.266 +  using floor_add_of_int [of x "- z"] by (simp add: algebra_simps)
   1.267 +
   1.268 +lemma floor_diff_number_of [simp]:
   1.269 +  "floor (x - number_of v) = floor x - number_of v"
   1.270 +  using floor_diff_of_int [of x "number_of v"] by simp
   1.271 +
   1.272 +lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
   1.273 +  using floor_diff_of_int [of x 1] by simp
   1.274 +
   1.275 +
   1.276 +subsection {* Ceiling function *}
   1.277 +
   1.278 +definition
   1.279 +  ceiling :: "'a::archimedean_field \<Rightarrow> int" where
   1.280 +  [code del]: "ceiling x = - floor (- x)"
   1.281 +
   1.282 +notation (xsymbols)
   1.283 +  ceiling  ("\<lceil>_\<rceil>")
   1.284 +
   1.285 +notation (HTML output)
   1.286 +  ceiling  ("\<lceil>_\<rceil>")
   1.287 +
   1.288 +lemma ceiling_correct: "of_int (ceiling x) - 1 < x \<and> x \<le> of_int (ceiling x)"
   1.289 +  unfolding ceiling_def using floor_correct [of "- x"] by simp
   1.290 +
   1.291 +lemma ceiling_unique: "\<lbrakk>of_int z - 1 < x; x \<le> of_int z\<rbrakk> \<Longrightarrow> ceiling x = z"
   1.292 +  unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp
   1.293 +
   1.294 +lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
   1.295 +  using ceiling_correct ..
   1.296 +
   1.297 +lemma ceiling_le_iff: "ceiling x \<le> z \<longleftrightarrow> x \<le> of_int z"
   1.298 +  unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto
   1.299 +
   1.300 +lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
   1.301 +  by (simp add: not_le [symmetric] ceiling_le_iff)
   1.302 +
   1.303 +lemma ceiling_less_iff: "ceiling x < z \<longleftrightarrow> x \<le> of_int z - 1"
   1.304 +  using ceiling_le_iff [of x "z - 1"] by simp
   1.305 +
   1.306 +lemma le_ceiling_iff: "z \<le> ceiling x \<longleftrightarrow> of_int z - 1 < x"
   1.307 +  by (simp add: not_less [symmetric] ceiling_less_iff)
   1.308 +
   1.309 +lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
   1.310 +  unfolding ceiling_def by (simp add: floor_mono)
   1.311 +
   1.312 +lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
   1.313 +  by (auto simp add: not_le [symmetric] ceiling_mono)
   1.314 +
   1.315 +lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
   1.316 +  by (rule ceiling_unique) simp_all
   1.317 +
   1.318 +lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
   1.319 +  using ceiling_of_int [of "of_nat n"] by simp
   1.320 +
   1.321 +text {* Ceiling with numerals *}
   1.322 +
   1.323 +lemma ceiling_zero [simp]: "ceiling 0 = 0"
   1.324 +  using ceiling_of_int [of 0] by simp
   1.325 +
   1.326 +lemma ceiling_one [simp]: "ceiling 1 = 1"
   1.327 +  using ceiling_of_int [of 1] by simp
   1.328 +
   1.329 +lemma ceiling_number_of [simp]: "ceiling (number_of v) = number_of v"
   1.330 +  using ceiling_of_int [of "number_of v"] by simp
   1.331 +
   1.332 +lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
   1.333 +  by (simp add: ceiling_le_iff)
   1.334 +
   1.335 +lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
   1.336 +  by (simp add: ceiling_le_iff)
   1.337 +
   1.338 +lemma ceiling_le_number_of [simp]:
   1.339 +  "ceiling x \<le> number_of v \<longleftrightarrow> x \<le> number_of v"
   1.340 +  by (simp add: ceiling_le_iff)
   1.341 +
   1.342 +lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
   1.343 +  by (simp add: ceiling_less_iff)
   1.344 +
   1.345 +lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
   1.346 +  by (simp add: ceiling_less_iff)
   1.347 +
   1.348 +lemma ceiling_less_number_of [simp]:
   1.349 +  "ceiling x < number_of v \<longleftrightarrow> x \<le> number_of v - 1"
   1.350 +  by (simp add: ceiling_less_iff)
   1.351 +
   1.352 +lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
   1.353 +  by (simp add: le_ceiling_iff)
   1.354 +
   1.355 +lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
   1.356 +  by (simp add: le_ceiling_iff)
   1.357 +
   1.358 +lemma number_of_le_ceiling [simp]:
   1.359 +  "number_of v \<le> ceiling x\<longleftrightarrow> number_of v - 1 < x"
   1.360 +  by (simp add: le_ceiling_iff)
   1.361 +
   1.362 +lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
   1.363 +  by (simp add: less_ceiling_iff)
   1.364 +
   1.365 +lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
   1.366 +  by (simp add: less_ceiling_iff)
   1.367 +
   1.368 +lemma number_of_less_ceiling [simp]:
   1.369 +  "number_of v < ceiling x \<longleftrightarrow> number_of v < x"
   1.370 +  by (simp add: less_ceiling_iff)
   1.371 +
   1.372 +text {* Addition and subtraction of integers *}
   1.373 +
   1.374 +lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z"
   1.375 +  using ceiling_correct [of x] by (simp add: ceiling_unique)
   1.376 +
   1.377 +lemma ceiling_add_number_of [simp]:
   1.378 +    "ceiling (x + number_of v) = ceiling x + number_of v"
   1.379 +  using ceiling_add_of_int [of x "number_of v"] by simp
   1.380 +
   1.381 +lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
   1.382 +  using ceiling_add_of_int [of x 1] by simp
   1.383 +
   1.384 +lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"
   1.385 +  using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps)
   1.386 +
   1.387 +lemma ceiling_diff_number_of [simp]:
   1.388 +  "ceiling (x - number_of v) = ceiling x - number_of v"
   1.389 +  using ceiling_diff_of_int [of x "number_of v"] by simp
   1.390 +
   1.391 +lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1"
   1.392 +  using ceiling_diff_of_int [of x 1] by simp
   1.393 +
   1.394 +
   1.395 +subsection {* Negation *}
   1.396 +
   1.397 +lemma floor_minus [simp]: "floor (- x) = - ceiling x"
   1.398 +  unfolding ceiling_def by simp
   1.399 +
   1.400 +lemma ceiling_minus [simp]: "ceiling (- x) = - floor x"
   1.401 +  unfolding ceiling_def by simp
   1.402 +
   1.403 +end
     2.1 --- a/src/HOL/IsaMakefile	Wed Feb 25 09:09:50 2009 -0800
     2.2 +++ b/src/HOL/IsaMakefile	Wed Feb 25 11:26:01 2009 -0800
     2.3 @@ -267,6 +267,7 @@
     2.4  	@$(ISABELLE_TOOL) usedir -b -f main.ML -g true $(OUT)/Pure HOL-Main
     2.5  
     2.6  $(OUT)/HOL: ROOT.ML $(MAIN_DEPENDENCIES) \
     2.7 +  Archimedean_Field.thy \
     2.8    Complex_Main.thy \
     2.9    Complex.thy \
    2.10    Deriv.thy \