src/HOL/Hyperreal/Log.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 15085 5693a977a767 child 15140 322485b816ac permissions -rw-r--r--
```     1 (*  Title       : Log.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 2000,2001 University of Edinburgh
```
```     4 *)
```
```     5
```
```     6 header{*Logarithms: Standard Version*}
```
```     7
```
```     8 theory Log
```
```     9 import Transcendental
```
```    10 begin
```
```    11
```
```    12 constdefs
```
```    13
```
```    14   powr  :: "[real,real] => real"     (infixr "powr" 80)
```
```    15     --{*exponentation with real exponent*}
```
```    16     "x powr a == exp(a * ln x)"
```
```    17
```
```    18   log :: "[real,real] => real"
```
```    19     --{*logarithm of @{term x} to base @{term a}*}
```
```    20     "log a x == ln x / ln a"
```
```    21
```
```    22
```
```    23
```
```    24 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```    25 by (simp add: powr_def)
```
```    26
```
```    27 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
```
```    28 by (simp add: powr_def)
```
```    29
```
```    30 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
```
```    31 by (simp add: powr_def)
```
```    32 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```    33
```
```    34 lemma powr_mult:
```
```    35       "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
```
```    36 by (simp add: powr_def exp_add [symmetric] ln_mult right_distrib)
```
```    37
```
```    38 lemma powr_gt_zero [simp]: "0 < x powr a"
```
```    39 by (simp add: powr_def)
```
```    40
```
```    41 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
```
```    42 by (simp add: powr_def)
```
```    43
```
```    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)
```
```    47 apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```    48 done
```
```    49
```
```    50 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
```
```    51 by (simp add: powr_def exp_add [symmetric] left_distrib)
```
```    52
```
```    53 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
```
```    54 by (simp add: powr_def)
```
```    55
```
```    56 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
```
```    57 by (simp add: powr_powr real_mult_commute)
```
```    58
```
```    59 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
```
```    60 by (simp add: powr_def exp_minus [symmetric])
```
```    61
```
```    62 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
```
```    63 by (simp add: divide_inverse powr_minus)
```
```    64
```
```    65 lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
```
```    66 by (simp add: powr_def)
```
```    67
```
```    68 lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
```
```    69 by (simp add: powr_def)
```
```    70
```
```    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)
```
```    73
```
```    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])
```
```    76
```
```    77 lemma log_ln: "ln x = log (exp(1)) x"
```
```    78 by (simp add: log_def)
```
```    79
```
```    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)
```
```    83
```
```    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)
```
```    86
```
```    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)
```
```    91
```
```    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)
```
```    96
```
```    97 text{*Base 10 logarithms*}
```
```    98 lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```    99 by (simp add: log_def)
```
```   100
```
```   101 lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
```
```   102 by (simp add: log_def)
```
```   103
```
```   104 lemma log_one [simp]: "log a 1 = 0"
```
```   105 by (simp add: log_def)
```
```   106
```
```   107 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```   108 by (simp add: log_def)
```
```   109
```
```   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
```
```   115
```
```   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)
```
```   119
```
```   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
```
```   126
```
```   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])
```
```   130
```
```   131
```
```   132 subsection{*Further Results Courtesy Jeremy Avigad*}
```
```   133
```
```   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
```
```   140
```
```   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
```
```   146
```
```   147 lemma ln_pwr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
```
```   148 by (unfold powr_def, simp)
```
```   149
```
```   150 lemma ln_bound: "1 <= x ==> ln x <= x"
```
```   151   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```   152   apply simp
```
```   153   apply (rule ln_add_one_self_le_self, simp)
```
```   154 done
```
```   155
```
```   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
```
```   162
```
```   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
```
```   167
```
```   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
```
```   178
```
```   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
```
```   185
```
```   186
```
```   187
```
```   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";
```
```   197 val powr_add = thm "powr_add";
```
```   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 *}
```
```   220
```
```   221 end
```