avigad@16959
|
1 |
(* Title: Ln.thy
|
avigad@16959
|
2 |
Author: Jeremy Avigad
|
nipkow@16963
|
3 |
ID: $Id$
|
avigad@16959
|
4 |
*)
|
avigad@16959
|
5 |
|
avigad@16959
|
6 |
header {* Properties of ln *}
|
avigad@16959
|
7 |
|
avigad@16959
|
8 |
theory Ln
|
avigad@16959
|
9 |
imports Transcendental
|
avigad@16959
|
10 |
begin
|
avigad@16959
|
11 |
|
avigad@16959
|
12 |
lemma exp_first_two_terms: "exp x = 1 + x + suminf (%n.
|
avigad@16959
|
13 |
inverse(real (fact (n+2))) * (x ^ (n+2)))"
|
avigad@16959
|
14 |
proof -
|
avigad@16959
|
15 |
have "exp x = suminf (%n. inverse(real (fact n)) * (x ^ n))"
|
wenzelm@19765
|
16 |
by (simp add: exp_def)
|
avigad@16959
|
17 |
also from summable_exp have "... = (SUM n : {0..<2}.
|
avigad@16959
|
18 |
inverse(real (fact n)) * (x ^ n)) + suminf (%n.
|
avigad@16959
|
19 |
inverse(real (fact (n+2))) * (x ^ (n+2)))" (is "_ = ?a + _")
|
avigad@16959
|
20 |
by (rule suminf_split_initial_segment)
|
avigad@16959
|
21 |
also have "?a = 1 + x"
|
avigad@16959
|
22 |
by (simp add: numerals)
|
avigad@16959
|
23 |
finally show ?thesis .
|
avigad@16959
|
24 |
qed
|
avigad@16959
|
25 |
|
avigad@16959
|
26 |
lemma exp_tail_after_first_two_terms_summable:
|
avigad@16959
|
27 |
"summable (%n. inverse(real (fact (n+2))) * (x ^ (n+2)))"
|
avigad@16959
|
28 |
proof -
|
avigad@16959
|
29 |
note summable_exp
|
avigad@16959
|
30 |
thus ?thesis
|
avigad@16959
|
31 |
by (frule summable_ignore_initial_segment)
|
avigad@16959
|
32 |
qed
|
avigad@16959
|
33 |
|
avigad@16959
|
34 |
lemma aux1: assumes a: "0 <= x" and b: "x <= 1"
|
avigad@16959
|
35 |
shows "inverse (real (fact (n + 2))) * x ^ (n + 2) <= (x^2/2) * ((1/2)^n)"
|
avigad@16959
|
36 |
proof (induct n)
|
avigad@16959
|
37 |
show "inverse (real (fact (0 + 2))) * x ^ (0 + 2) <=
|
avigad@16959
|
38 |
x ^ 2 / 2 * (1 / 2) ^ 0"
|
avigad@16959
|
39 |
apply (simp add: power2_eq_square)
|
avigad@16959
|
40 |
apply (subgoal_tac "real (Suc (Suc 0)) = 2")
|
avigad@16959
|
41 |
apply (erule ssubst)
|
avigad@16959
|
42 |
apply simp
|
avigad@16959
|
43 |
apply simp
|
avigad@16959
|
44 |
done
|
avigad@16959
|
45 |
next
|
avigad@16959
|
46 |
fix n
|
avigad@16959
|
47 |
assume c: "inverse (real (fact (n + 2))) * x ^ (n + 2)
|
avigad@16959
|
48 |
<= x ^ 2 / 2 * (1 / 2) ^ n"
|
avigad@16959
|
49 |
show "inverse (real (fact (Suc n + 2))) * x ^ (Suc n + 2)
|
avigad@16959
|
50 |
<= x ^ 2 / 2 * (1 / 2) ^ Suc n"
|
avigad@16959
|
51 |
proof -
|
avigad@16959
|
52 |
have "inverse(real (fact (Suc n + 2))) <=
|
avigad@16959
|
53 |
(1 / 2) *inverse (real (fact (n+2)))"
|
avigad@16959
|
54 |
proof -
|
avigad@16959
|
55 |
have "Suc n + 2 = Suc (n + 2)" by simp
|
avigad@16959
|
56 |
then have "fact (Suc n + 2) = Suc (n + 2) * fact (n + 2)"
|
avigad@16959
|
57 |
by simp
|
avigad@16959
|
58 |
then have "real(fact (Suc n + 2)) = real(Suc (n + 2) * fact (n + 2))"
|
avigad@16959
|
59 |
apply (rule subst)
|
avigad@16959
|
60 |
apply (rule refl)
|
avigad@16959
|
61 |
done
|
avigad@16959
|
62 |
also have "... = real(Suc (n + 2)) * real(fact (n + 2))"
|
avigad@16959
|
63 |
by (rule real_of_nat_mult)
|
avigad@16959
|
64 |
finally have "real (fact (Suc n + 2)) =
|
avigad@16959
|
65 |
real (Suc (n + 2)) * real (fact (n + 2))" .
|
avigad@16959
|
66 |
then have "inverse(real (fact (Suc n + 2))) =
|
avigad@16959
|
67 |
inverse(real (Suc (n + 2))) * inverse(real (fact (n + 2)))"
|
avigad@16959
|
68 |
apply (rule ssubst)
|
avigad@16959
|
69 |
apply (rule inverse_mult_distrib)
|
avigad@16959
|
70 |
done
|
avigad@16959
|
71 |
also have "... <= (1/2) * inverse(real (fact (n + 2)))"
|
avigad@16959
|
72 |
apply (rule mult_right_mono)
|
avigad@16959
|
73 |
apply (subst inverse_eq_divide)
|
avigad@16959
|
74 |
apply simp
|
avigad@16959
|
75 |
apply (rule inv_real_of_nat_fact_ge_zero)
|
avigad@16959
|
76 |
done
|
avigad@16959
|
77 |
finally show ?thesis .
|
avigad@16959
|
78 |
qed
|
avigad@16959
|
79 |
moreover have "x ^ (Suc n + 2) <= x ^ (n + 2)"
|
avigad@16959
|
80 |
apply (simp add: mult_compare_simps)
|
avigad@16959
|
81 |
apply (simp add: prems)
|
avigad@16959
|
82 |
apply (subgoal_tac "0 <= x * (x * x^n)")
|
avigad@16959
|
83 |
apply force
|
avigad@16959
|
84 |
apply (rule mult_nonneg_nonneg, rule a)+
|
avigad@16959
|
85 |
apply (rule zero_le_power, rule a)
|
avigad@16959
|
86 |
done
|
avigad@16959
|
87 |
ultimately have "inverse (real (fact (Suc n + 2))) * x ^ (Suc n + 2) <=
|
avigad@16959
|
88 |
(1 / 2 * inverse (real (fact (n + 2)))) * x ^ (n + 2)"
|
avigad@16959
|
89 |
apply (rule mult_mono)
|
avigad@16959
|
90 |
apply (rule mult_nonneg_nonneg)
|
avigad@16959
|
91 |
apply simp
|
avigad@16959
|
92 |
apply (subst inverse_nonnegative_iff_nonnegative)
|
avigad@16959
|
93 |
apply (rule real_of_nat_fact_ge_zero)
|
avigad@16959
|
94 |
apply (rule zero_le_power)
|
avigad@16959
|
95 |
apply assumption
|
avigad@16959
|
96 |
done
|
avigad@16959
|
97 |
also have "... = 1 / 2 * (inverse (real (fact (n + 2))) * x ^ (n + 2))"
|
avigad@16959
|
98 |
by simp
|
avigad@16959
|
99 |
also have "... <= 1 / 2 * (x ^ 2 / 2 * (1 / 2) ^ n)"
|
avigad@16959
|
100 |
apply (rule mult_left_mono)
|
avigad@16959
|
101 |
apply (rule prems)
|
avigad@16959
|
102 |
apply simp
|
avigad@16959
|
103 |
done
|
avigad@16959
|
104 |
also have "... = x ^ 2 / 2 * (1 / 2 * (1 / 2) ^ n)"
|
avigad@16959
|
105 |
by auto
|
avigad@16959
|
106 |
also have "(1::real) / 2 * (1 / 2) ^ n = (1 / 2) ^ (Suc n)"
|
avigad@16959
|
107 |
by (rule realpow_Suc [THEN sym])
|
avigad@16959
|
108 |
finally show ?thesis .
|
avigad@16959
|
109 |
qed
|
avigad@16959
|
110 |
qed
|
avigad@16959
|
111 |
|
huffman@20692
|
112 |
lemma aux2: "(%n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2"
|
avigad@16959
|
113 |
proof -
|
huffman@20692
|
114 |
have "(%n. (1 / 2::real)^n) sums (1 / (1 - (1/2)))"
|
avigad@16959
|
115 |
apply (rule geometric_sums)
|
huffman@22998
|
116 |
by (simp add: abs_less_iff)
|
avigad@16959
|
117 |
also have "(1::real) / (1 - 1/2) = 2"
|
avigad@16959
|
118 |
by simp
|
huffman@20692
|
119 |
finally have "(%n. (1 / 2::real)^n) sums 2" .
|
avigad@16959
|
120 |
then have "(%n. x ^ 2 / 2 * (1 / 2) ^ n) sums (x^2 / 2 * 2)"
|
avigad@16959
|
121 |
by (rule sums_mult)
|
avigad@16959
|
122 |
also have "x^2 / 2 * 2 = x^2"
|
avigad@16959
|
123 |
by simp
|
avigad@16959
|
124 |
finally show ?thesis .
|
avigad@16959
|
125 |
qed
|
avigad@16959
|
126 |
|
avigad@16959
|
127 |
lemma exp_bound: "0 <= x ==> x <= 1 ==> exp x <= 1 + x + x^2"
|
avigad@16959
|
128 |
proof -
|
avigad@16959
|
129 |
assume a: "0 <= x"
|
avigad@16959
|
130 |
assume b: "x <= 1"
|
avigad@16959
|
131 |
have c: "exp x = 1 + x + suminf (%n. inverse(real (fact (n+2))) *
|
avigad@16959
|
132 |
(x ^ (n+2)))"
|
avigad@16959
|
133 |
by (rule exp_first_two_terms)
|
avigad@16959
|
134 |
moreover have "suminf (%n. inverse(real (fact (n+2))) * (x ^ (n+2))) <= x^2"
|
avigad@16959
|
135 |
proof -
|
avigad@16959
|
136 |
have "suminf (%n. inverse(real (fact (n+2))) * (x ^ (n+2))) <=
|
avigad@16959
|
137 |
suminf (%n. (x^2/2) * ((1/2)^n))"
|
avigad@16959
|
138 |
apply (rule summable_le)
|
avigad@16959
|
139 |
apply (auto simp only: aux1 prems)
|
avigad@16959
|
140 |
apply (rule exp_tail_after_first_two_terms_summable)
|
avigad@16959
|
141 |
by (rule sums_summable, rule aux2)
|
avigad@16959
|
142 |
also have "... = x^2"
|
avigad@16959
|
143 |
by (rule sums_unique [THEN sym], rule aux2)
|
avigad@16959
|
144 |
finally show ?thesis .
|
avigad@16959
|
145 |
qed
|
avigad@16959
|
146 |
ultimately show ?thesis
|
avigad@16959
|
147 |
by auto
|
avigad@16959
|
148 |
qed
|
avigad@16959
|
149 |
|
avigad@16959
|
150 |
lemma aux3: "(0::real) <= x ==> (1 + x + x^2)/(1 + x^2) <= 1 + x"
|
avigad@16959
|
151 |
apply (subst pos_divide_le_eq)
|
avigad@16959
|
152 |
apply (simp add: zero_compare_simps)
|
avigad@16959
|
153 |
apply (simp add: ring_eq_simps zero_compare_simps)
|
avigad@16959
|
154 |
done
|
avigad@16959
|
155 |
|
avigad@16959
|
156 |
lemma aux4: "0 <= x ==> x <= 1 ==> exp (x - x^2) <= 1 + x"
|
avigad@16959
|
157 |
proof -
|
avigad@16959
|
158 |
assume a: "0 <= x" and b: "x <= 1"
|
avigad@16959
|
159 |
have "exp (x - x^2) = exp x / exp (x^2)"
|
avigad@16959
|
160 |
by (rule exp_diff)
|
avigad@16959
|
161 |
also have "... <= (1 + x + x^2) / exp (x ^2)"
|
avigad@16959
|
162 |
apply (rule divide_right_mono)
|
avigad@16959
|
163 |
apply (rule exp_bound)
|
avigad@16959
|
164 |
apply (rule a, rule b)
|
avigad@16959
|
165 |
apply simp
|
avigad@16959
|
166 |
done
|
avigad@16959
|
167 |
also have "... <= (1 + x + x^2) / (1 + x^2)"
|
avigad@16959
|
168 |
apply (rule divide_left_mono)
|
avigad@17013
|
169 |
apply (auto simp add: exp_ge_add_one_self_aux)
|
avigad@16959
|
170 |
apply (rule add_nonneg_nonneg)
|
avigad@16959
|
171 |
apply (insert prems, auto)
|
avigad@16959
|
172 |
apply (rule mult_pos_pos)
|
avigad@16959
|
173 |
apply auto
|
avigad@16959
|
174 |
apply (rule add_pos_nonneg)
|
avigad@16959
|
175 |
apply auto
|
avigad@16959
|
176 |
done
|
avigad@16959
|
177 |
also from a have "... <= 1 + x"
|
avigad@16959
|
178 |
by (rule aux3)
|
avigad@16959
|
179 |
finally show ?thesis .
|
avigad@16959
|
180 |
qed
|
avigad@16959
|
181 |
|
avigad@16959
|
182 |
lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==>
|
avigad@16959
|
183 |
x - x^2 <= ln (1 + x)"
|
avigad@16959
|
184 |
proof -
|
avigad@16959
|
185 |
assume a: "0 <= x" and b: "x <= 1"
|
avigad@16959
|
186 |
then have "exp (x - x^2) <= 1 + x"
|
avigad@16959
|
187 |
by (rule aux4)
|
avigad@16959
|
188 |
also have "... = exp (ln (1 + x))"
|
avigad@16959
|
189 |
proof -
|
avigad@16959
|
190 |
from a have "0 < 1 + x" by auto
|
avigad@16959
|
191 |
thus ?thesis
|
avigad@16959
|
192 |
by (auto simp only: exp_ln_iff [THEN sym])
|
avigad@16959
|
193 |
qed
|
avigad@16959
|
194 |
finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" .
|
avigad@16959
|
195 |
thus ?thesis by (auto simp only: exp_le_cancel_iff)
|
avigad@16959
|
196 |
qed
|
avigad@16959
|
197 |
|
avigad@16959
|
198 |
lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
|
avigad@16959
|
199 |
proof -
|
avigad@16959
|
200 |
assume a: "0 <= (x::real)" and b: "x < 1"
|
avigad@16959
|
201 |
have "(1 - x) * (1 + x + x^2) = (1 - x^3)"
|
avigad@16959
|
202 |
by (simp add: ring_eq_simps power2_eq_square power3_eq_cube)
|
avigad@16959
|
203 |
also have "... <= 1"
|
avigad@16959
|
204 |
by (auto intro: zero_le_power simp add: a)
|
avigad@16959
|
205 |
finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
|
avigad@16959
|
206 |
moreover have "0 < 1 + x + x^2"
|
avigad@16959
|
207 |
apply (rule add_pos_nonneg)
|
avigad@16959
|
208 |
apply (insert a, auto)
|
avigad@16959
|
209 |
done
|
avigad@16959
|
210 |
ultimately have "1 - x <= 1 / (1 + x + x^2)"
|
avigad@16959
|
211 |
by (elim mult_imp_le_div_pos)
|
avigad@16959
|
212 |
also have "... <= 1 / exp x"
|
avigad@16959
|
213 |
apply (rule divide_left_mono)
|
avigad@16959
|
214 |
apply (rule exp_bound, rule a)
|
avigad@16959
|
215 |
apply (insert prems, auto)
|
avigad@16959
|
216 |
apply (rule mult_pos_pos)
|
avigad@16959
|
217 |
apply (rule add_pos_nonneg)
|
avigad@16959
|
218 |
apply auto
|
avigad@16959
|
219 |
done
|
avigad@16959
|
220 |
also have "... = exp (-x)"
|
avigad@16959
|
221 |
by (auto simp add: exp_minus real_divide_def)
|
avigad@16959
|
222 |
finally have "1 - x <= exp (- x)" .
|
avigad@16959
|
223 |
also have "1 - x = exp (ln (1 - x))"
|
avigad@16959
|
224 |
proof -
|
avigad@16959
|
225 |
have "0 < 1 - x"
|
avigad@16959
|
226 |
by (insert b, auto)
|
avigad@16959
|
227 |
thus ?thesis
|
avigad@16959
|
228 |
by (auto simp only: exp_ln_iff [THEN sym])
|
avigad@16959
|
229 |
qed
|
avigad@16959
|
230 |
finally have "exp (ln (1 - x)) <= exp (- x)" .
|
avigad@16959
|
231 |
thus ?thesis by (auto simp only: exp_le_cancel_iff)
|
avigad@16959
|
232 |
qed
|
avigad@16959
|
233 |
|
avigad@16959
|
234 |
lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
|
avigad@16959
|
235 |
proof -
|
avigad@16959
|
236 |
assume a: "x < 1"
|
avigad@16959
|
237 |
have "ln(1 - x) = - ln(1 / (1 - x))"
|
avigad@16959
|
238 |
proof -
|
avigad@16959
|
239 |
have "ln(1 - x) = - (- ln (1 - x))"
|
avigad@16959
|
240 |
by auto
|
avigad@16959
|
241 |
also have "- ln(1 - x) = ln 1 - ln(1 - x)"
|
avigad@16959
|
242 |
by simp
|
avigad@16959
|
243 |
also have "... = ln(1 / (1 - x))"
|
avigad@16959
|
244 |
apply (rule ln_div [THEN sym])
|
avigad@16959
|
245 |
by (insert a, auto)
|
avigad@16959
|
246 |
finally show ?thesis .
|
avigad@16959
|
247 |
qed
|
avigad@16959
|
248 |
also have " 1 / (1 - x) = 1 + x / (1 - x)"
|
avigad@16959
|
249 |
proof -
|
avigad@16959
|
250 |
have "1 / (1 - x) = (1 - x + x) / (1 - x)"
|
avigad@16959
|
251 |
by auto
|
avigad@16959
|
252 |
also have "... = (1 - x) / (1 - x) + x / (1 - x)"
|
avigad@16959
|
253 |
by (rule add_divide_distrib)
|
avigad@16959
|
254 |
also have "... = 1 + x / (1-x)"
|
avigad@16959
|
255 |
apply (subst add_right_cancel)
|
avigad@16959
|
256 |
apply (insert a, simp)
|
avigad@16959
|
257 |
done
|
avigad@16959
|
258 |
finally show ?thesis .
|
avigad@16959
|
259 |
qed
|
avigad@16959
|
260 |
finally show ?thesis .
|
avigad@16959
|
261 |
qed
|
avigad@16959
|
262 |
|
avigad@16959
|
263 |
lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
|
avigad@16959
|
264 |
- x - 2 * x^2 <= ln (1 - x)"
|
avigad@16959
|
265 |
proof -
|
avigad@16959
|
266 |
assume a: "0 <= x" and b: "x <= (1 / 2)"
|
avigad@16959
|
267 |
from b have c: "x < 1"
|
avigad@16959
|
268 |
by auto
|
avigad@16959
|
269 |
then have "ln (1 - x) = - ln (1 + x / (1 - x))"
|
avigad@16959
|
270 |
by (rule aux5)
|
avigad@16959
|
271 |
also have "- (x / (1 - x)) <= ..."
|
avigad@16959
|
272 |
proof -
|
avigad@16959
|
273 |
have "ln (1 + x / (1 - x)) <= x / (1 - x)"
|
avigad@16959
|
274 |
apply (rule ln_add_one_self_le_self)
|
avigad@16959
|
275 |
apply (rule divide_nonneg_pos)
|
avigad@16959
|
276 |
by (insert a c, auto)
|
avigad@16959
|
277 |
thus ?thesis
|
avigad@16959
|
278 |
by auto
|
avigad@16959
|
279 |
qed
|
avigad@16959
|
280 |
also have "- (x / (1 - x)) = -x / (1 - x)"
|
avigad@16959
|
281 |
by auto
|
avigad@16959
|
282 |
finally have d: "- x / (1 - x) <= ln (1 - x)" .
|
avigad@16959
|
283 |
have e: "-x - 2 * x^2 <= - x / (1 - x)"
|
avigad@16959
|
284 |
apply (rule mult_imp_le_div_pos)
|
avigad@16959
|
285 |
apply (insert prems, force)
|
avigad@16959
|
286 |
apply (auto simp add: ring_eq_simps power2_eq_square)
|
avigad@16959
|
287 |
apply (subgoal_tac "- (x * x) + x * (x * (x * 2)) = x^2 * (2 * x - 1)")
|
avigad@16959
|
288 |
apply (erule ssubst)
|
avigad@16959
|
289 |
apply (rule mult_nonneg_nonpos)
|
avigad@16959
|
290 |
apply auto
|
avigad@16959
|
291 |
apply (auto simp add: ring_eq_simps power2_eq_square)
|
avigad@16959
|
292 |
done
|
avigad@16959
|
293 |
from e d show "- x - 2 * x^2 <= ln (1 - x)"
|
avigad@16959
|
294 |
by (rule order_trans)
|
avigad@16959
|
295 |
qed
|
avigad@16959
|
296 |
|
avigad@17013
|
297 |
lemma exp_ge_add_one_self [simp]: "1 + x <= exp x"
|
avigad@16959
|
298 |
apply (case_tac "0 <= x")
|
avigad@17013
|
299 |
apply (erule exp_ge_add_one_self_aux)
|
avigad@16959
|
300 |
apply (case_tac "x <= -1")
|
avigad@16959
|
301 |
apply (subgoal_tac "1 + x <= 0")
|
avigad@16959
|
302 |
apply (erule order_trans)
|
avigad@16959
|
303 |
apply simp
|
avigad@16959
|
304 |
apply simp
|
avigad@16959
|
305 |
apply (subgoal_tac "1 + x = exp(ln (1 + x))")
|
avigad@16959
|
306 |
apply (erule ssubst)
|
avigad@16959
|
307 |
apply (subst exp_le_cancel_iff)
|
avigad@16959
|
308 |
apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
|
avigad@16959
|
309 |
apply simp
|
avigad@16959
|
310 |
apply (rule ln_one_minus_pos_upper_bound)
|
avigad@16959
|
311 |
apply auto
|
avigad@16959
|
312 |
done
|
avigad@16959
|
313 |
|
avigad@16959
|
314 |
lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
|
avigad@16959
|
315 |
apply (subgoal_tac "x = ln (exp x)")
|
avigad@16959
|
316 |
apply (erule ssubst)back
|
avigad@16959
|
317 |
apply (subst ln_le_cancel_iff)
|
avigad@16959
|
318 |
apply auto
|
avigad@16959
|
319 |
done
|
avigad@16959
|
320 |
|
avigad@16959
|
321 |
lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
|
avigad@16959
|
322 |
"0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
|
avigad@16959
|
323 |
proof -
|
avigad@16959
|
324 |
assume "0 <= x"
|
avigad@16959
|
325 |
assume "x <= 1"
|
avigad@16959
|
326 |
have "ln (1 + x) <= x"
|
avigad@16959
|
327 |
by (rule ln_add_one_self_le_self)
|
avigad@16959
|
328 |
then have "ln (1 + x) - x <= 0"
|
avigad@16959
|
329 |
by simp
|
avigad@16959
|
330 |
then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
|
avigad@16959
|
331 |
by (rule abs_of_nonpos)
|
avigad@16959
|
332 |
also have "... = x - ln (1 + x)"
|
avigad@16959
|
333 |
by simp
|
avigad@16959
|
334 |
also have "... <= x^2"
|
avigad@16959
|
335 |
proof -
|
avigad@16959
|
336 |
from prems have "x - x^2 <= ln (1 + x)"
|
avigad@16959
|
337 |
by (intro ln_one_plus_pos_lower_bound)
|
avigad@16959
|
338 |
thus ?thesis
|
avigad@16959
|
339 |
by simp
|
avigad@16959
|
340 |
qed
|
avigad@16959
|
341 |
finally show ?thesis .
|
avigad@16959
|
342 |
qed
|
avigad@16959
|
343 |
|
avigad@16959
|
344 |
lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
|
avigad@16959
|
345 |
"-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
|
avigad@16959
|
346 |
proof -
|
avigad@16959
|
347 |
assume "-(1 / 2) <= x"
|
avigad@16959
|
348 |
assume "x <= 0"
|
avigad@16959
|
349 |
have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
|
avigad@16959
|
350 |
apply (subst abs_of_nonpos)
|
avigad@16959
|
351 |
apply simp
|
avigad@16959
|
352 |
apply (rule ln_add_one_self_le_self2)
|
avigad@16959
|
353 |
apply (insert prems, auto)
|
avigad@16959
|
354 |
done
|
avigad@16959
|
355 |
also have "... <= 2 * x^2"
|
avigad@16959
|
356 |
apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
|
avigad@16959
|
357 |
apply (simp add: compare_rls)
|
avigad@16959
|
358 |
apply (rule ln_one_minus_pos_lower_bound)
|
avigad@16959
|
359 |
apply (insert prems, auto)
|
avigad@16959
|
360 |
done
|
avigad@16959
|
361 |
finally show ?thesis .
|
avigad@16959
|
362 |
qed
|
avigad@16959
|
363 |
|
avigad@16959
|
364 |
lemma abs_ln_one_plus_x_minus_x_bound:
|
avigad@16959
|
365 |
"abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2"
|
avigad@16959
|
366 |
apply (case_tac "0 <= x")
|
avigad@16959
|
367 |
apply (rule order_trans)
|
avigad@16959
|
368 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
|
avigad@16959
|
369 |
apply auto
|
avigad@16959
|
370 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
|
avigad@16959
|
371 |
apply auto
|
avigad@16959
|
372 |
done
|
avigad@16959
|
373 |
|
avigad@16959
|
374 |
lemma DERIV_ln: "0 < x ==> DERIV ln x :> 1 / x"
|
avigad@16959
|
375 |
apply (unfold deriv_def, unfold LIM_def, clarsimp)
|
avigad@16959
|
376 |
apply (rule exI)
|
avigad@16959
|
377 |
apply (rule conjI)
|
avigad@16959
|
378 |
prefer 2
|
avigad@16959
|
379 |
apply clarsimp
|
huffman@20563
|
380 |
apply (subgoal_tac "(ln (x + xa) - ln x) / xa - (1 / x) =
|
avigad@16959
|
381 |
(ln (1 + xa / x) - xa / x) / xa")
|
avigad@16959
|
382 |
apply (erule ssubst)
|
avigad@16959
|
383 |
apply (subst abs_divide)
|
avigad@16959
|
384 |
apply (rule mult_imp_div_pos_less)
|
avigad@16959
|
385 |
apply force
|
avigad@16959
|
386 |
apply (rule order_le_less_trans)
|
avigad@16959
|
387 |
apply (rule abs_ln_one_plus_x_minus_x_bound)
|
avigad@16959
|
388 |
apply (subst abs_divide)
|
avigad@16959
|
389 |
apply (subst abs_of_pos, assumption)
|
avigad@16959
|
390 |
apply (erule mult_imp_div_pos_le)
|
avigad@16959
|
391 |
apply (subgoal_tac "abs xa < min (x / 2) (r * x^2 / 2)")
|
avigad@16959
|
392 |
apply force
|
avigad@16959
|
393 |
apply assumption
|
webertj@20432
|
394 |
apply (simp add: power2_eq_square mult_compare_simps)
|
avigad@16959
|
395 |
apply (rule mult_imp_div_pos_less)
|
avigad@16959
|
396 |
apply (rule mult_pos_pos, assumption, assumption)
|
avigad@16959
|
397 |
apply (subgoal_tac "xa * xa = abs xa * abs xa")
|
avigad@16959
|
398 |
apply (erule ssubst)
|
avigad@16959
|
399 |
apply (subgoal_tac "abs xa * (abs xa * 2) < abs xa * (r * (x * x))")
|
avigad@16959
|
400 |
apply (simp only: mult_ac)
|
avigad@16959
|
401 |
apply (rule mult_strict_left_mono)
|
avigad@16959
|
402 |
apply (erule conjE, assumption)
|
avigad@16959
|
403 |
apply force
|
avigad@16959
|
404 |
apply simp
|
avigad@16959
|
405 |
apply (subst ln_div [THEN sym])
|
avigad@16959
|
406 |
apply arith
|
avigad@16959
|
407 |
apply (auto simp add: ring_eq_simps add_frac_eq frac_eq_eq
|
avigad@16959
|
408 |
add_divide_distrib power2_eq_square)
|
avigad@16959
|
409 |
apply (rule mult_pos_pos, assumption)+
|
avigad@16959
|
410 |
apply assumption
|
avigad@16959
|
411 |
done
|
avigad@16959
|
412 |
|
avigad@16959
|
413 |
lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
|
avigad@16959
|
414 |
proof -
|
avigad@16959
|
415 |
assume "exp 1 <= x" and "x <= y"
|
avigad@16959
|
416 |
have a: "0 < x" and b: "0 < y"
|
avigad@16959
|
417 |
apply (insert prems)
|
avigad@16959
|
418 |
apply (subgoal_tac "0 < exp 1")
|
avigad@16959
|
419 |
apply arith
|
avigad@16959
|
420 |
apply auto
|
avigad@16959
|
421 |
apply (subgoal_tac "0 < exp 1")
|
avigad@16959
|
422 |
apply arith
|
avigad@16959
|
423 |
apply auto
|
avigad@16959
|
424 |
done
|
avigad@16959
|
425 |
have "x * ln y - x * ln x = x * (ln y - ln x)"
|
avigad@16959
|
426 |
by (simp add: ring_eq_simps)
|
avigad@16959
|
427 |
also have "... = x * ln(y / x)"
|
avigad@16959
|
428 |
apply (subst ln_div)
|
avigad@16959
|
429 |
apply (rule b, rule a, rule refl)
|
avigad@16959
|
430 |
done
|
avigad@16959
|
431 |
also have "y / x = (x + (y - x)) / x"
|
avigad@16959
|
432 |
by simp
|
avigad@16959
|
433 |
also have "... = 1 + (y - x) / x"
|
avigad@16959
|
434 |
apply (simp only: add_divide_distrib)
|
avigad@16959
|
435 |
apply (simp add: prems)
|
avigad@16959
|
436 |
apply (insert a, arith)
|
avigad@16959
|
437 |
done
|
avigad@16959
|
438 |
also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
|
avigad@16959
|
439 |
apply (rule mult_left_mono)
|
avigad@16959
|
440 |
apply (rule ln_add_one_self_le_self)
|
avigad@16959
|
441 |
apply (rule divide_nonneg_pos)
|
avigad@16959
|
442 |
apply (insert prems a, simp_all)
|
avigad@16959
|
443 |
done
|
avigad@16959
|
444 |
also have "... = y - x"
|
avigad@16959
|
445 |
by (insert a, simp)
|
avigad@16959
|
446 |
also have "... = (y - x) * ln (exp 1)"
|
avigad@16959
|
447 |
by simp
|
avigad@16959
|
448 |
also have "... <= (y - x) * ln x"
|
avigad@16959
|
449 |
apply (rule mult_left_mono)
|
avigad@16959
|
450 |
apply (subst ln_le_cancel_iff)
|
avigad@16959
|
451 |
apply force
|
avigad@16959
|
452 |
apply (rule a)
|
avigad@16959
|
453 |
apply (rule prems)
|
avigad@16959
|
454 |
apply (insert prems, simp)
|
avigad@16959
|
455 |
done
|
avigad@16959
|
456 |
also have "... = y * ln x - x * ln x"
|
avigad@16959
|
457 |
by (rule left_diff_distrib)
|
avigad@16959
|
458 |
finally have "x * ln y <= y * ln x"
|
avigad@16959
|
459 |
by arith
|
avigad@16959
|
460 |
then have "ln y <= (y * ln x) / x"
|
avigad@16959
|
461 |
apply (subst pos_le_divide_eq)
|
avigad@16959
|
462 |
apply (rule a)
|
avigad@16959
|
463 |
apply (simp add: mult_ac)
|
avigad@16959
|
464 |
done
|
avigad@16959
|
465 |
also have "... = y * (ln x / x)"
|
avigad@16959
|
466 |
by simp
|
avigad@16959
|
467 |
finally show ?thesis
|
avigad@16959
|
468 |
apply (subst pos_divide_le_eq)
|
avigad@16959
|
469 |
apply (rule b)
|
avigad@16959
|
470 |
apply (simp add: mult_ac)
|
avigad@16959
|
471 |
done
|
avigad@16959
|
472 |
qed
|
avigad@16959
|
473 |
|
avigad@16959
|
474 |
end
|