src/HOL/Log.thy
 author noschinl Mon Dec 19 14:41:08 2011 +0100 (2011-12-19) changeset 45930 2a882ef2cd73 parent 45916 758671e966a0 child 47593 69f0af2b7d54 permissions -rw-r--r--
1 (*  Title       : Log.thy
2     Author      : Jacques D. Fleuriot
4     Copyright   : 2000,2001 University of Edinburgh
5 *)
9 theory Log
10 imports Transcendental
11 begin
13 definition
14   powr  :: "[real,real] => real"     (infixr "powr" 80) where
15     --{*exponentation with real exponent*}
16   "x powr a = exp(a * ln x)"
18 definition
19   log :: "[real,real] => real" where
20     --{*logarithm of @{term x} to base @{term a}*}
21   "log a x = ln x / ln a"
25 lemma powr_one_eq_one [simp]: "1 powr a = 1"
28 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
31 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
33 declare powr_one_gt_zero_iff [THEN iffD2, simp]
35 lemma powr_mult:
36       "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
39 lemma powr_gt_zero [simp]: "0 < x powr a"
42 lemma powr_ge_pzero [simp]: "0 <= x powr y"
43 by (rule order_less_imp_le, rule powr_gt_zero)
45 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
48 lemma powr_divide:
49      "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
50 apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
52 done
54 lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
56   apply (subst exp_diff [THEN sym])
58 done
60 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
63 lemma powr_mult_base:
64   "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
65 using assms by (auto simp: powr_add)
67 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
70 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
71 by (simp add: powr_powr mult_commute)
73 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
74 by (simp add: powr_def exp_minus [symmetric])
76 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
77 by (simp add: divide_inverse powr_minus)
79 lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
82 lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
85 lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
86 by (blast intro: powr_less_cancel powr_less_mono)
88 lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
89 by (simp add: linorder_not_less [symmetric])
91 lemma log_ln: "ln x = log (exp(1)) x"
94 lemma DERIV_log: assumes "x > 0" shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
95 proof -
96   def lb \<equiv> "1 / ln b"
97   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
98     using `x > 0` by (auto intro!: DERIV_intros)
99   ultimately show ?thesis
101 qed
103 lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
105 lemma powr_log_cancel [simp]:
106      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
107 by (simp add: powr_def log_def)
109 lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
110 by (simp add: log_def powr_def)
112 lemma log_mult:
113      "[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]
114       ==> log a (x * y) = log a x + log a y"
115 by (simp add: log_def ln_mult divide_inverse left_distrib)
117 lemma log_eq_div_ln_mult_log:
118      "[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]
119       ==> log a x = (ln b/ln a) * log b x"
120 by (simp add: log_def divide_inverse)
122 text{*Base 10 logarithms*}
123 lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
126 lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
129 lemma log_one [simp]: "log a 1 = 0"
132 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
135 lemma log_inverse:
136      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
137 apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
138 apply (simp add: log_mult [symmetric])
139 done
141 lemma log_divide:
142      "[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
143 by (simp add: log_mult divide_inverse log_inverse)
145 lemma log_less_cancel_iff [simp]:
146      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
147 apply safe
148 apply (rule_tac [2] powr_less_cancel)
149 apply (drule_tac a = "log a x" in powr_less_mono, auto)
150 done
152 lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
153 proof (rule inj_onI, simp)
154   fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
155   show "x = y"
156   proof (cases rule: linorder_cases)
157     assume "x < y" hence "log b x < log b y"
158       using log_less_cancel_iff[OF `1 < b`] pos by simp
159     thus ?thesis using * by simp
160   next
161     assume "y < x" hence "log b y < log b x"
162       using log_less_cancel_iff[OF `1 < b`] pos by simp
163     thus ?thesis using * by simp
164   qed simp
165 qed
167 lemma log_le_cancel_iff [simp]:
168      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
169 by (simp add: linorder_not_less [symmetric])
172 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
173   apply (induct n, simp)
174   apply (subgoal_tac "real(Suc n) = real n + 1")
175   apply (erule ssubst)
176   apply (subst powr_add, simp, simp)
177 done
179 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0
180   else x powr (real n))"
181   apply (case_tac "x = 0", simp, simp)
182   apply (rule powr_realpow [THEN sym], simp)
183 done
185 lemma root_powr_inverse:
186   "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
187 by (auto simp: root_def powr_realpow[symmetric] powr_powr)
189 lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
190 by (unfold powr_def, simp)
192 lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
193   apply (case_tac "y = 0")
194   apply force
195   apply (auto simp add: log_def ln_powr field_simps)
196 done
198 lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
199   apply (subst powr_realpow [symmetric])
200   apply (auto simp add: log_powr)
201 done
203 lemma ln_bound: "1 <= x ==> ln x <= x"
204   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
205   apply simp
207 done
209 lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
210   apply (case_tac "x = 1", simp)
211   apply (case_tac "a = b", simp)
212   apply (rule order_less_imp_le)
213   apply (rule powr_less_mono, auto)
214 done
216 lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
217   apply (subst powr_zero_eq_one [THEN sym])
218   apply (rule powr_mono, assumption+)
219 done
221 lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
222     y powr a"
223   apply (unfold powr_def)
224   apply (rule exp_less_mono)
225   apply (rule mult_strict_left_mono)
226   apply (subst ln_less_cancel_iff, assumption)
227   apply (rule order_less_trans)
228   prefer 2
229   apply assumption+
230 done
232 lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
233     x powr a"
234   apply (unfold powr_def)
235   apply (rule exp_less_mono)
236   apply (rule mult_strict_left_mono_neg)
237   apply (subst ln_less_cancel_iff)
238   apply assumption
239   apply (rule order_less_trans)
240   prefer 2
241   apply assumption+
242 done
244 lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
245   apply (case_tac "a = 0", simp)
246   apply (case_tac "x = y", simp)
247   apply (rule order_less_imp_le)
248   apply (rule powr_less_mono2, auto)
249 done
251 lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
252   apply (rule mult_imp_le_div_pos)
253   apply (assumption)
254   apply (subst mult_commute)
255   apply (subst ln_powr [THEN sym])
256   apply auto
257   apply (rule ln_bound)
258   apply (erule ge_one_powr_ge_zero)
259   apply (erule order_less_imp_le)
260 done
262 lemma ln_powr_bound2:
263   assumes "1 < x" and "0 < a"
264   shows "(ln x) powr a <= (a powr a) * x"
265 proof -
266   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
267     apply (intro ln_powr_bound)
268     apply (erule order_less_imp_le)
269     apply (rule divide_pos_pos)
270     apply simp_all
271     done
272   also have "... = a * (x powr (1 / a))"
273     by simp
274   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
275     apply (intro powr_mono2)
276     apply (rule order_less_imp_le, rule assms)
277     apply (rule ln_gt_zero)
278     apply (rule assms)
279     apply assumption
280     done
281   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
282     apply (rule powr_mult)
283     apply (rule assms)
284     apply (rule powr_gt_zero)
285     done
286   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
287     by (rule powr_powr)
288   also have "... = x"
289     apply simp
290     apply (subgoal_tac "a ~= 0")
291     using assms apply auto
292     done
293   finally show ?thesis .
294 qed
296 lemma tendsto_powr [tendsto_intros]:
297   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
298   unfolding powr_def by (intro tendsto_intros)
300 (* FIXME: generalize by replacing d by with g x and g ---> d? *)
301 lemma tendsto_zero_powrI:
302   assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
303   assumes "0 < d"
304   shows "((\<lambda>x. f x powr d) ---> 0) F"
305 proof (rule tendstoI)
306   fix e :: real assume "0 < e"
307   def Z \<equiv> "e powr (1 / d)"
308   with `0 < e` have "0 < Z" by simp
309   with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
310     by (intro eventually_conj tendstoD)
311   moreover
312   from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
313     by (intro powr_less_mono2) (auto simp: dist_real_def)
314   with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
315     unfolding dist_real_def Z_def by (auto simp: powr_powr)
316   ultimately
317   show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
318 qed
320 lemma tendsto_neg_powr:
321   assumes "s < 0" and "real_tendsto_inf f F"
322   shows "((\<lambda>x. f x powr s) ---> 0) F"
323 proof (rule tendstoI)
324   fix e :: real assume "0 < e"
325   def Z \<equiv> "e powr (1 / s)"
326   from assms have "eventually (\<lambda>x. Z < f x) F" by (simp add: real_tendsto_inf_def)
327   moreover
328   from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
329     by (auto simp: Z_def intro!: powr_less_mono2_neg)
330   with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
331     by (simp add: powr_powr Z_def dist_real_def)
332   ultimately
333   show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
334 qed
336 end