src/HOL/Library/Discrete.thy
 author Manuel Eberl Mon Mar 26 16:14:16 2018 +0200 (19 months ago) changeset 67951 655aa11359dc parent 67399 eab6ce8368fa child 69064 5840724b1d71 permissions -rw-r--r--
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
1 (* Author: Florian Haftmann, TU Muenchen *)
3 section \<open>Common discrete functions\<close>
5 theory Discrete
6 imports Complex_Main
7 begin
9 subsection \<open>Discrete logarithm\<close>
11 context
12 begin
14 qualified fun log :: "nat \<Rightarrow> nat"
15   where [simp del]: "log n = (if n < 2 then 0 else Suc (log (n div 2)))"
17 lemma log_induct [consumes 1, case_names one double]:
18   fixes n :: nat
19   assumes "n > 0"
20   assumes one: "P 1"
21   assumes double: "\<And>n. n \<ge> 2 \<Longrightarrow> P (n div 2) \<Longrightarrow> P n"
22   shows "P n"
23 using \<open>n > 0\<close> proof (induct n rule: log.induct)
24   fix n
25   assume "\<not> n < 2 \<Longrightarrow>
26           0 < n div 2 \<Longrightarrow> P (n div 2)"
27   then have *: "n \<ge> 2 \<Longrightarrow> P (n div 2)" by simp
28   assume "n > 0"
29   show "P n"
30   proof (cases "n = 1")
31     case True
32     with one show ?thesis by simp
33   next
34     case False
35     with \<open>n > 0\<close> have "n \<ge> 2" by auto
36     with * have "P (n div 2)" .
37     with \<open>n \<ge> 2\<close> show ?thesis by (rule double)
38   qed
39 qed
41 lemma log_zero [simp]: "log 0 = 0"
42   by (simp add: log.simps)
44 lemma log_one [simp]: "log 1 = 0"
45   by (simp add: log.simps)
47 lemma log_Suc_zero [simp]: "log (Suc 0) = 0"
48   using log_one by simp
50 lemma log_rec: "n \<ge> 2 \<Longrightarrow> log n = Suc (log (n div 2))"
51   by (simp add: log.simps)
53 lemma log_twice [simp]: "n \<noteq> 0 \<Longrightarrow> log (2 * n) = Suc (log n)"
54   by (simp add: log_rec)
56 lemma log_half [simp]: "log (n div 2) = log n - 1"
57 proof (cases "n < 2")
58   case True
59   then have "n = 0 \<or> n = 1" by arith
60   then show ?thesis by (auto simp del: One_nat_def)
61 next
62   case False
63   then show ?thesis by (simp add: log_rec)
64 qed
66 lemma log_exp [simp]: "log (2 ^ n) = n"
67   by (induct n) simp_all
69 lemma log_mono: "mono log"
70 proof
71   fix m n :: nat
72   assume "m \<le> n"
73   then show "log m \<le> log n"
74   proof (induct m arbitrary: n rule: log.induct)
75     case (1 m)
76     then have mn2: "m div 2 \<le> n div 2" by arith
77     show "log m \<le> log n"
78     proof (cases "m \<ge> 2")
79       case False
80       then have "m = 0 \<or> m = 1" by arith
81       then show ?thesis by (auto simp del: One_nat_def)
82     next
83       case True then have "\<not> m < 2" by simp
84       with mn2 have "n \<ge> 2" by arith
85       from True have m2_0: "m div 2 \<noteq> 0" by arith
86       with mn2 have n2_0: "n div 2 \<noteq> 0" by arith
87       from \<open>\<not> m < 2\<close> "1.hyps" mn2 have "log (m div 2) \<le> log (n div 2)" by blast
88       with m2_0 n2_0 have "log (2 * (m div 2)) \<le> log (2 * (n div 2))" by simp
89       with m2_0 n2_0 \<open>m \<ge> 2\<close> \<open>n \<ge> 2\<close> show ?thesis by (simp only: log_rec [of m] log_rec [of n]) simp
90     qed
91   qed
92 qed
94 lemma log_exp2_le:
95   assumes "n > 0"
96   shows "2 ^ log n \<le> n"
97   using assms
98 proof (induct n rule: log_induct)
99   case one
100   then show ?case by simp
101 next
102   case (double n)
103   with log_mono have "log n \<ge> Suc 0"
104     by (simp add: log.simps)
105   assume "2 ^ log (n div 2) \<le> n div 2"
106   with \<open>n \<ge> 2\<close> have "2 ^ (log n - Suc 0) \<le> n div 2" by simp
107   then have "2 ^ (log n - Suc 0) * 2 ^ 1 \<le> n div 2 * 2" by simp
108   with \<open>log n \<ge> Suc 0\<close> have "2 ^ log n \<le> n div 2 * 2"
109     unfolding power_add [symmetric] by simp
110   also have "n div 2 * 2 \<le> n" by (cases "even n") simp_all
111   finally show ?case .
112 qed
114 lemma log_exp2_gt: "2 * 2 ^ log n > n"
115 proof (cases "n > 0")
116   case True
117   thus ?thesis
118   proof (induct n rule: log_induct)
119     case (double n)
120     thus ?case
121       by (cases "even n") (auto elim!: evenE oddE simp: field_simps log.simps)
122   qed simp_all
123 qed simp_all
125 lemma log_exp2_ge: "2 * 2 ^ log n \<ge> n"
126   using log_exp2_gt[of n] by simp
128 lemma log_le_iff: "m \<le> n \<Longrightarrow> log m \<le> log n"
129   by (rule monoD [OF log_mono])
131 lemma log_eqI:
132   assumes "n > 0" "2^k \<le> n" "n < 2 * 2^k"
133   shows   "log n = k"
134 proof (rule antisym)
135   from \<open>n > 0\<close> have "2 ^ log n \<le> n" by (rule log_exp2_le)
136   also have "\<dots> < 2 ^ Suc k" using assms by simp
137   finally have "log n < Suc k" by (subst (asm) power_strict_increasing_iff) simp_all
138   thus "log n \<le> k" by simp
139 next
140   have "2^k \<le> n" by fact
141   also have "\<dots> < 2^(Suc (log n))" by (simp add: log_exp2_gt)
142   finally have "k < Suc (log n)" by (subst (asm) power_strict_increasing_iff) simp_all
143   thus "k \<le> log n" by simp
144 qed
146 lemma log_altdef: "log n = (if n = 0 then 0 else nat \<lfloor>Transcendental.log 2 (real_of_nat n)\<rfloor>)"
147 proof (cases "n = 0")
148   case False
149   have "\<lfloor>Transcendental.log 2 (real_of_nat n)\<rfloor> = int (log n)"
150   proof (rule floor_unique)
151     from False have "2 powr (real (log n)) \<le> real n"
152       by (simp add: powr_realpow log_exp2_le)
153     hence "Transcendental.log 2 (2 powr (real (log n))) \<le> Transcendental.log 2 (real n)"
154       using False by (subst Transcendental.log_le_cancel_iff) simp_all
155     also have "Transcendental.log 2 (2 powr (real (log n))) = real (log n)" by simp
156     finally show "real_of_int (int (log n)) \<le> Transcendental.log 2 (real n)" by simp
157   next
158     have "real n < real (2 * 2 ^ log n)"
159       by (subst of_nat_less_iff) (rule log_exp2_gt)
160     also have "\<dots> = 2 powr (real (log n) + 1)"
162     finally have "Transcendental.log 2 (real n) < Transcendental.log 2 \<dots>"
163       using False by (subst Transcendental.log_less_cancel_iff) simp_all
164     also have "\<dots> = real (log n) + 1" by simp
165     finally show "Transcendental.log 2 (real n) < real_of_int (int (log n)) + 1" by simp
166   qed
167   thus ?thesis by simp
168 qed simp_all
171 subsection \<open>Discrete square root\<close>
173 qualified definition sqrt :: "nat \<Rightarrow> nat"
174   where "sqrt n = Max {m. m\<^sup>2 \<le> n}"
176 lemma sqrt_aux:
177   fixes n :: nat
178   shows "finite {m. m\<^sup>2 \<le> n}" and "{m. m\<^sup>2 \<le> n} \<noteq> {}"
179 proof -
180   { fix m
181     assume "m\<^sup>2 \<le> n"
182     then have "m \<le> n"
183       by (cases m) (simp_all add: power2_eq_square)
184   } note ** = this
185   then have "{m. m\<^sup>2 \<le> n} \<subseteq> {m. m \<le> n}" by auto
186   then show "finite {m. m\<^sup>2 \<le> n}" by (rule finite_subset) rule
187   have "0\<^sup>2 \<le> n" by simp
188   then show *: "{m. m\<^sup>2 \<le> n} \<noteq> {}" by blast
189 qed
191 lemma sqrt_unique:
192   assumes "m^2 \<le> n" "n < (Suc m)^2"
193   shows   "Discrete.sqrt n = m"
194 proof -
195   have "m' \<le> m" if "m'^2 \<le> n" for m'
196   proof -
197     note that
198     also note assms(2)
199     finally have "m' < Suc m" by (rule power_less_imp_less_base) simp_all
200     thus "m' \<le> m" by simp
201   qed
202   with \<open>m^2 \<le> n\<close> sqrt_aux[of n] show ?thesis unfolding Discrete.sqrt_def
203     by (intro antisym Max.boundedI Max.coboundedI) simp_all
204 qed
207 lemma sqrt_code[code]: "sqrt n = Max (Set.filter (\<lambda>m. m\<^sup>2 \<le> n) {0..n})"
208 proof -
209   from power2_nat_le_imp_le [of _ n] have "{m. m \<le> n \<and> m\<^sup>2 \<le> n} = {m. m\<^sup>2 \<le> n}" by auto
210   then show ?thesis by (simp add: sqrt_def Set.filter_def)
211 qed
213 lemma sqrt_inverse_power2 [simp]: "sqrt (n\<^sup>2) = n"
214 proof -
215   have "{m. m \<le> n} \<noteq> {}" by auto
216   then have "Max {m. m \<le> n} \<le> n" by auto
217   then show ?thesis
218     by (auto simp add: sqrt_def power2_nat_le_eq_le intro: antisym)
219 qed
221 lemma sqrt_zero [simp]: "sqrt 0 = 0"
222   using sqrt_inverse_power2 [of 0] by simp
224 lemma sqrt_one [simp]: "sqrt 1 = 1"
225   using sqrt_inverse_power2 [of 1] by simp
227 lemma mono_sqrt: "mono sqrt"
228 proof
229   fix m n :: nat
230   have *: "0 * 0 \<le> m" by simp
231   assume "m \<le> n"
232   then show "sqrt m \<le> sqrt n"
233     by (auto intro!: Max_mono \<open>0 * 0 \<le> m\<close> finite_less_ub simp add: power2_eq_square sqrt_def)
234 qed
236 lemma mono_sqrt': "m \<le> n \<Longrightarrow> Discrete.sqrt m \<le> Discrete.sqrt n"
237   using mono_sqrt unfolding mono_def by auto
239 lemma sqrt_greater_zero_iff [simp]: "sqrt n > 0 \<longleftrightarrow> n > 0"
240 proof -
241   have *: "0 < Max {m. m\<^sup>2 \<le> n} \<longleftrightarrow> (\<exists>a\<in>{m. m\<^sup>2 \<le> n}. 0 < a)"
242     by (rule Max_gr_iff) (fact sqrt_aux)+
243   show ?thesis
244   proof
245     assume "0 < sqrt n"
246     then have "0 < Max {m. m\<^sup>2 \<le> n}" by (simp add: sqrt_def)
247     with * show "0 < n" by (auto dest: power2_nat_le_imp_le)
248   next
249     assume "0 < n"
250     then have "1\<^sup>2 \<le> n \<and> 0 < (1::nat)" by simp
251     then have "\<exists>q. q\<^sup>2 \<le> n \<and> 0 < q" ..
252     with * have "0 < Max {m. m\<^sup>2 \<le> n}" by blast
253     then show "0 < sqrt n" by  (simp add: sqrt_def)
254   qed
255 qed
257 lemma sqrt_power2_le [simp]: "(sqrt n)\<^sup>2 \<le> n" (* FIXME tune proof *)
258 proof (cases "n > 0")
259   case False then show ?thesis by simp
260 next
261   case True then have "sqrt n > 0" by simp
262   then have "mono (times (Max {m. m\<^sup>2 \<le> n}))" by (auto intro: mono_times_nat simp add: sqrt_def)
263   then have *: "Max {m. m\<^sup>2 \<le> n} * Max {m. m\<^sup>2 \<le> n} = Max (times (Max {m. m\<^sup>2 \<le> n}) ` {m. m\<^sup>2 \<le> n})"
264     using sqrt_aux [of n] by (rule mono_Max_commute)
265   have "\<And>a. a * a \<le> n \<Longrightarrow> Max {m. m * m \<le> n} * a \<le> n"
266   proof -
267     fix q
268     assume "q * q \<le> n"
269     show "Max {m. m * m \<le> n} * q \<le> n"
270     proof (cases "q > 0")
271       case False then show ?thesis by simp
272     next
273       case True then have "mono (times q)" by (rule mono_times_nat)
274       then have "q * Max {m. m * m \<le> n} = Max (times q ` {m. m * m \<le> n})"
275         using sqrt_aux [of n] by (auto simp add: power2_eq_square intro: mono_Max_commute)
276       then have "Max {m. m * m \<le> n} * q = Max (times q ` {m. m * m \<le> n})" by (simp add: ac_simps)
277       moreover have "finite (( * ) q ` {m. m * m \<le> n})"
278         by (metis (mono_tags) finite_imageI finite_less_ub le_square)
279       moreover have "\<exists>x. x * x \<le> n"
280         by (metis \<open>q * q \<le> n\<close>)
281       ultimately show ?thesis
282         by simp (metis \<open>q * q \<le> n\<close> le_cases mult_le_mono1 mult_le_mono2 order_trans)
283     qed
284   qed
285   then have "Max (( * ) (Max {m. m * m \<le> n}) ` {m. m * m \<le> n}) \<le> n"
286     apply (subst Max_le_iff)
287       apply (metis (mono_tags) finite_imageI finite_less_ub le_square)
288      apply auto
289     apply (metis le0 mult_0_right)
290     done
291   with * show ?thesis by (simp add: sqrt_def power2_eq_square)
292 qed
294 lemma sqrt_le: "sqrt n \<le> n"
295   using sqrt_aux [of n] by (auto simp add: sqrt_def intro: power2_nat_le_imp_le)
297 text \<open>Additional facts about the discrete square root, thanks to Julian Biendarra, Manuel Eberl\<close>
299 lemma Suc_sqrt_power2_gt: "n < (Suc (Discrete.sqrt n))^2"
300   using Max_ge[OF Discrete.sqrt_aux(1), of "Discrete.sqrt n + 1" n]
301   by (cases "n < (Suc (Discrete.sqrt n))^2") (simp_all add: Discrete.sqrt_def)
303 lemma le_sqrt_iff: "x \<le> Discrete.sqrt y \<longleftrightarrow> x^2 \<le> y"
304 proof -
305   have "x \<le> Discrete.sqrt y \<longleftrightarrow> (\<exists>z. z\<^sup>2 \<le> y \<and> x \<le> z)"
306     using Max_ge_iff[OF Discrete.sqrt_aux, of x y] by (simp add: Discrete.sqrt_def)
307   also have "\<dots> \<longleftrightarrow> x^2 \<le> y"
308   proof safe
309     fix z assume "x \<le> z" "z ^ 2 \<le> y"
310     thus "x^2 \<le> y" by (intro le_trans[of "x^2" "z^2" y]) (simp_all add: power2_nat_le_eq_le)
311   qed auto
312   finally show ?thesis .
313 qed
315 lemma le_sqrtI: "x^2 \<le> y \<Longrightarrow> x \<le> Discrete.sqrt y"
316   by (simp add: le_sqrt_iff)
318 lemma sqrt_le_iff: "Discrete.sqrt y \<le> x \<longleftrightarrow> (\<forall>z. z^2 \<le> y \<longrightarrow> z \<le> x)"
319   using Max.bounded_iff[OF Discrete.sqrt_aux] by (simp add: Discrete.sqrt_def)
321 lemma sqrt_leI:
322   "(\<And>z. z^2 \<le> y \<Longrightarrow> z \<le> x) \<Longrightarrow> Discrete.sqrt y \<le> x"
323   by (simp add: sqrt_le_iff)
325 lemma sqrt_Suc:
326   "Discrete.sqrt (Suc n) = (if \<exists>m. Suc n = m^2 then Suc (Discrete.sqrt n) else Discrete.sqrt n)"
327 proof cases
328   assume "\<exists> m. Suc n = m^2"
329   then obtain m where m_def: "Suc n = m^2" by blast
330   then have lhs: "Discrete.sqrt (Suc n) = m" by simp
331   from m_def sqrt_power2_le[of n]
332     have "(Discrete.sqrt n)^2 < m^2" by linarith
333   with power2_less_imp_less have lt_m: "Discrete.sqrt n < m" by blast
334   from m_def Suc_sqrt_power2_gt[of "n"]
335     have "m^2 \<le> (Suc(Discrete.sqrt n))^2" by simp
336   with power2_nat_le_eq_le have "m \<le> Suc (Discrete.sqrt n)" by blast
337   with lt_m have "m = Suc (Discrete.sqrt n)" by simp
338   with lhs m_def show ?thesis by fastforce
339 next
340   assume asm: "\<not> (\<exists> m. Suc n = m^2)"
341   hence "Suc n \<noteq> (Discrete.sqrt (Suc n))^2" by simp
342   with sqrt_power2_le[of "Suc n"]
343     have "Discrete.sqrt (Suc n) \<le> Discrete.sqrt n" by (intro le_sqrtI) linarith
344   moreover have "Discrete.sqrt (Suc n) \<ge> Discrete.sqrt n"
345     by (intro monoD[OF mono_sqrt]) simp_all
346   ultimately show ?thesis using asm by simp
347 qed
349 end
351 end