src/HOL/Hyperreal/Log.thy
 author nipkow Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) changeset 15140 322485b816ac parent 15131 c69542757a4d child 16819 00d8f9300d13 permissions -rw-r--r--
import -> imports
1 (*  Title       : Log.thy
2     Author      : Jacques D. Fleuriot
3     Copyright   : 2000,2001 University of Edinburgh
4 *)
8 theory Log
9 imports Transcendental
10 begin
12 constdefs
14   powr  :: "[real,real] => real"     (infixr "powr" 80)
15     --{*exponentation with real exponent*}
16     "x powr a == exp(a * ln x)"
18   log :: "[real,real] => real"
19     --{*logarithm of @{term x} to base @{term a}*}
20     "log a x == ln x / ln a"
24 lemma powr_one_eq_one [simp]: "1 powr a = 1"
27 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
30 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
32 declare powr_one_gt_zero_iff [THEN iffD2, simp]
34 lemma powr_mult:
35       "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
38 lemma powr_gt_zero [simp]: "0 < x powr a"
41 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
44 lemma powr_divide:
45      "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
46 apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
48 done
50 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
53 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
56 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
57 by (simp add: powr_powr real_mult_commute)
59 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
60 by (simp add: powr_def exp_minus [symmetric])
62 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
63 by (simp add: divide_inverse powr_minus)
65 lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
68 lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
71 lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
72 by (blast intro: powr_less_cancel powr_less_mono)
74 lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
75 by (simp add: linorder_not_less [symmetric])
77 lemma log_ln: "ln x = log (exp(1)) x"
80 lemma powr_log_cancel [simp]:
81      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
82 by (simp add: powr_def log_def)
84 lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
85 by (simp add: log_def powr_def)
87 lemma log_mult:
88      "[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]
89       ==> log a (x * y) = log a x + log a y"
90 by (simp add: log_def ln_mult divide_inverse left_distrib)
92 lemma log_eq_div_ln_mult_log:
93      "[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]
94       ==> log a x = (ln b/ln a) * log b x"
95 by (simp add: log_def divide_inverse)
97 text{*Base 10 logarithms*}
98 lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
101 lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
104 lemma log_one [simp]: "log a 1 = 0"
107 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
110 lemma log_inverse:
111      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
112 apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
113 apply (simp add: log_mult [symmetric])
114 done
116 lemma log_divide:
117      "[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
118 by (simp add: log_mult divide_inverse log_inverse)
120 lemma log_less_cancel_iff [simp]:
121      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
122 apply safe
123 apply (rule_tac [2] powr_less_cancel)
124 apply (drule_tac a = "log a x" in powr_less_mono, auto)
125 done
127 lemma log_le_cancel_iff [simp]:
128      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
129 by (simp add: linorder_not_less [symmetric])
132 subsection{*Further Results Courtesy Jeremy Avigad*}
134 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
135   apply (induct n, simp)
136   apply (subgoal_tac "real(Suc n) = real n + 1")
137   apply (erule ssubst)
138   apply (subst powr_add, simp, simp)
139 done
141 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0
142   else x powr (real n))"
143   apply (case_tac "x = 0", simp, simp)
144   apply (rule powr_realpow [THEN sym], simp)
145 done
147 lemma ln_pwr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
148 by (unfold powr_def, simp)
150 lemma ln_bound: "1 <= x ==> ln x <= x"
151   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
152   apply simp
154 done
156 lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
157   apply (case_tac "x = 1", simp)
158   apply (case_tac "a = b", simp)
159   apply (rule order_less_imp_le)
160   apply (rule powr_less_mono, auto)
161 done
163 lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
164   apply (subst powr_zero_eq_one [THEN sym])
165   apply (rule powr_mono, assumption+)
166 done
168 lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
169     y powr a"
170   apply (unfold powr_def)
171   apply (rule exp_less_mono)
172   apply (rule mult_strict_left_mono)
173   apply (subst ln_less_cancel_iff, assumption)
174   apply (rule order_less_trans)
175   prefer 2
176   apply assumption+
177 done
179 lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a";
180   apply (case_tac "a = 0", simp)
181   apply (case_tac "x = y", simp)
182   apply (rule order_less_imp_le)
183   apply (rule powr_less_mono2, auto)
184 done
188 ML
189 {*
190 val powr_one_eq_one = thm "powr_one_eq_one";
191 val powr_zero_eq_one = thm "powr_zero_eq_one";
192 val powr_one_gt_zero_iff = thm "powr_one_gt_zero_iff";
193 val powr_mult = thm "powr_mult";
194 val powr_gt_zero = thm "powr_gt_zero";
195 val powr_not_zero = thm "powr_not_zero";
196 val powr_divide = thm "powr_divide";
198 val powr_powr = thm "powr_powr";
199 val powr_powr_swap = thm "powr_powr_swap";
200 val powr_minus = thm "powr_minus";
201 val powr_minus_divide = thm "powr_minus_divide";
202 val powr_less_mono = thm "powr_less_mono";
203 val powr_less_cancel = thm "powr_less_cancel";
204 val powr_less_cancel_iff = thm "powr_less_cancel_iff";
205 val powr_le_cancel_iff = thm "powr_le_cancel_iff";
206 val log_ln = thm "log_ln";
207 val powr_log_cancel = thm "powr_log_cancel";
208 val log_powr_cancel = thm "log_powr_cancel";
209 val log_mult = thm "log_mult";
210 val log_eq_div_ln_mult_log = thm "log_eq_div_ln_mult_log";
211 val log_base_10_eq1 = thm "log_base_10_eq1";
212 val log_base_10_eq2 = thm "log_base_10_eq2";
213 val log_one = thm "log_one";
214 val log_eq_one = thm "log_eq_one";
215 val log_inverse = thm "log_inverse";
216 val log_divide = thm "log_divide";
217 val log_less_cancel_iff = thm "log_less_cancel_iff";
218 val log_le_cancel_iff = thm "log_le_cancel_iff";
219 *}
221 end