src/HOL/Archimedean_Field.thy
 author haftmann Fri Jun 11 17:14:02 2010 +0200 (2010-06-11) changeset 37407 61dd8c145da7 parent 35028 108662d50512 child 37765 26bdfb7b680b permissions -rw-r--r--
declare lex_prod_def [code del]
1 (* Title:      Archimedean_Field.thy
2    Author:     Brian Huffman
3 *)
5 header {* Archimedean Fields, Floor and Ceiling Functions *}
7 theory Archimedean_Field
8 imports Main
9 begin
11 subsection {* Class of Archimedean fields *}
13 text {* Archimedean fields have no infinite elements. *}
15 class archimedean_field = linordered_field + number_ring +
16   assumes ex_le_of_int: "\<exists>z. x \<le> of_int z"
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
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
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
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
51 text {* Archimedean fields have no infinitesimal elements. *}
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
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
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
85 subsection {* Existence and uniqueness of floor function *}
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
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
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
142 subsection {* Floor function *}
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))"
148 notation (xsymbols)
149   floor  ("\<lfloor>_\<rfloor>")
151 notation (HTML output)
152   floor  ("\<lfloor>_\<rfloor>")
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')
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
160 lemma of_int_floor_le: "of_int (floor x) \<le> x"
161   using floor_correct ..
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
175 lemma floor_less_iff: "floor x < z \<longleftrightarrow> x < of_int z"
176   by (simp add: not_le [symmetric] le_floor_iff)
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
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)
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
191 lemma floor_less_cancel: "floor x < floor y \<Longrightarrow> x < y"
192   by (auto simp add: not_le [symmetric] floor_mono)
194 lemma floor_of_int [simp]: "floor (of_int z) = z"
195   by (rule floor_unique) simp_all
197 lemma floor_of_nat [simp]: "floor (of_nat n) = int n"
198   using floor_of_int [of "of_nat n"] by simp
200 text {* Floor with numerals *}
202 lemma floor_zero [simp]: "floor 0 = 0"
203   using floor_of_int [of 0] by simp
205 lemma floor_one [simp]: "floor 1 = 1"
206   using floor_of_int [of 1] by simp
208 lemma floor_number_of [simp]: "floor (number_of v) = number_of v"
209   using floor_of_int [of "number_of v"] by simp
211 lemma zero_le_floor [simp]: "0 \<le> floor x \<longleftrightarrow> 0 \<le> x"
214 lemma one_le_floor [simp]: "1 \<le> floor x \<longleftrightarrow> 1 \<le> x"
217 lemma number_of_le_floor [simp]: "number_of v \<le> floor x \<longleftrightarrow> number_of v \<le> x"
220 lemma zero_less_floor [simp]: "0 < floor x \<longleftrightarrow> 1 \<le> x"
223 lemma one_less_floor [simp]: "1 < floor x \<longleftrightarrow> 2 \<le> x"
226 lemma number_of_less_floor [simp]:
227   "number_of v < floor x \<longleftrightarrow> number_of v + 1 \<le> x"
230 lemma floor_le_zero [simp]: "floor x \<le> 0 \<longleftrightarrow> x < 1"
233 lemma floor_le_one [simp]: "floor x \<le> 1 \<longleftrightarrow> x < 2"
236 lemma floor_le_number_of [simp]:
237   "floor x \<le> number_of v \<longleftrightarrow> x < number_of v + 1"
240 lemma floor_less_zero [simp]: "floor x < 0 \<longleftrightarrow> x < 0"
243 lemma floor_less_one [simp]: "floor x < 1 \<longleftrightarrow> x < 1"
246 lemma floor_less_number_of [simp]:
247   "floor x < number_of v \<longleftrightarrow> x < number_of v"
250 text {* Addition and subtraction of integers *}
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)
256     "floor (x + number_of v) = floor x + number_of v"
257   using floor_add_of_int [of x "number_of v"] by simp
259 lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1"
260   using floor_add_of_int [of x 1] by simp
262 lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z"
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
269 lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1"
270   using floor_diff_of_int [of x 1] by simp
273 subsection {* Ceiling function *}
275 definition
276   ceiling :: "'a::archimedean_field \<Rightarrow> int" where
277   [code del]: "ceiling x = - floor (- x)"
279 notation (xsymbols)
280   ceiling  ("\<lceil>_\<rceil>")
282 notation (HTML output)
283   ceiling  ("\<lceil>_\<rceil>")
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
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
291 lemma le_of_int_ceiling: "x \<le> of_int (ceiling x)"
292   using ceiling_correct ..
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
297 lemma less_ceiling_iff: "z < ceiling x \<longleftrightarrow> of_int z < x"
298   by (simp add: not_le [symmetric] ceiling_le_iff)
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
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)
306 lemma ceiling_mono: "x \<ge> y \<Longrightarrow> ceiling x \<ge> ceiling y"
307   unfolding ceiling_def by (simp add: floor_mono)
309 lemma ceiling_less_cancel: "ceiling x < ceiling y \<Longrightarrow> x < y"
310   by (auto simp add: not_le [symmetric] ceiling_mono)
312 lemma ceiling_of_int [simp]: "ceiling (of_int z) = z"
313   by (rule ceiling_unique) simp_all
315 lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n"
316   using ceiling_of_int [of "of_nat n"] by simp
318 text {* Ceiling with numerals *}
320 lemma ceiling_zero [simp]: "ceiling 0 = 0"
321   using ceiling_of_int [of 0] by simp
323 lemma ceiling_one [simp]: "ceiling 1 = 1"
324   using ceiling_of_int [of 1] by simp
326 lemma ceiling_number_of [simp]: "ceiling (number_of v) = number_of v"
327   using ceiling_of_int [of "number_of v"] by simp
329 lemma ceiling_le_zero [simp]: "ceiling x \<le> 0 \<longleftrightarrow> x \<le> 0"
332 lemma ceiling_le_one [simp]: "ceiling x \<le> 1 \<longleftrightarrow> x \<le> 1"
335 lemma ceiling_le_number_of [simp]:
336   "ceiling x \<le> number_of v \<longleftrightarrow> x \<le> number_of v"
339 lemma ceiling_less_zero [simp]: "ceiling x < 0 \<longleftrightarrow> x \<le> -1"
342 lemma ceiling_less_one [simp]: "ceiling x < 1 \<longleftrightarrow> x \<le> 0"
345 lemma ceiling_less_number_of [simp]:
346   "ceiling x < number_of v \<longleftrightarrow> x \<le> number_of v - 1"
349 lemma zero_le_ceiling [simp]: "0 \<le> ceiling x \<longleftrightarrow> -1 < x"
352 lemma one_le_ceiling [simp]: "1 \<le> ceiling x \<longleftrightarrow> 0 < x"
355 lemma number_of_le_ceiling [simp]:
356   "number_of v \<le> ceiling x\<longleftrightarrow> number_of v - 1 < x"
359 lemma zero_less_ceiling [simp]: "0 < ceiling x \<longleftrightarrow> 0 < x"
362 lemma one_less_ceiling [simp]: "1 < ceiling x \<longleftrightarrow> 1 < x"
365 lemma number_of_less_ceiling [simp]:
366   "number_of v < ceiling x \<longleftrightarrow> number_of v < x"
369 text {* Addition and subtraction of integers *}
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)
375     "ceiling (x + number_of v) = ceiling x + number_of v"
376   using ceiling_add_of_int [of x "number_of v"] by simp
378 lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1"
379   using ceiling_add_of_int [of x 1] by simp
381 lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z"