src/HOL/Log.thy
changeset 28952 15a4b2cf8c34
parent 21404 eb85850d3eb7
child 31336 e17f13cd1280
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Log.thy	Wed Dec 03 15:58:44 2008 +0100
     1.3 @@ -0,0 +1,276 @@
     1.4 +(*  Title       : Log.thy
     1.5 +    Author      : Jacques D. Fleuriot
     1.6 +                  Additional contributions by Jeremy Avigad
     1.7 +    Copyright   : 2000,2001 University of Edinburgh
     1.8 +*)
     1.9 +
    1.10 +header{*Logarithms: Standard Version*}
    1.11 +
    1.12 +theory Log
    1.13 +imports Transcendental
    1.14 +begin
    1.15 +
    1.16 +definition
    1.17 +  powr  :: "[real,real] => real"     (infixr "powr" 80) where
    1.18 +    --{*exponentation with real exponent*}
    1.19 +  "x powr a = exp(a * ln x)"
    1.20 +
    1.21 +definition
    1.22 +  log :: "[real,real] => real" where
    1.23 +    --{*logarithm of @{term x} to base @{term a}*}
    1.24 +  "log a x = ln x / ln a"
    1.25 +
    1.26 +
    1.27 +
    1.28 +lemma powr_one_eq_one [simp]: "1 powr a = 1"
    1.29 +by (simp add: powr_def)
    1.30 +
    1.31 +lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
    1.32 +by (simp add: powr_def)
    1.33 +
    1.34 +lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
    1.35 +by (simp add: powr_def)
    1.36 +declare powr_one_gt_zero_iff [THEN iffD2, simp]
    1.37 +
    1.38 +lemma powr_mult: 
    1.39 +      "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
    1.40 +by (simp add: powr_def exp_add [symmetric] ln_mult right_distrib)
    1.41 +
    1.42 +lemma powr_gt_zero [simp]: "0 < x powr a"
    1.43 +by (simp add: powr_def)
    1.44 +
    1.45 +lemma powr_ge_pzero [simp]: "0 <= x powr y"
    1.46 +by (rule order_less_imp_le, rule powr_gt_zero)
    1.47 +
    1.48 +lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
    1.49 +by (simp add: powr_def)
    1.50 +
    1.51 +lemma powr_divide:
    1.52 +     "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
    1.53 +apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
    1.54 +apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
    1.55 +done
    1.56 +
    1.57 +lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
    1.58 +  apply (simp add: powr_def)
    1.59 +  apply (subst exp_diff [THEN sym])
    1.60 +  apply (simp add: left_diff_distrib)
    1.61 +done
    1.62 +
    1.63 +lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
    1.64 +by (simp add: powr_def exp_add [symmetric] left_distrib)
    1.65 +
    1.66 +lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
    1.67 +by (simp add: powr_def)
    1.68 +
    1.69 +lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
    1.70 +by (simp add: powr_powr real_mult_commute)
    1.71 +
    1.72 +lemma powr_minus: "x powr (-a) = inverse (x powr a)"
    1.73 +by (simp add: powr_def exp_minus [symmetric])
    1.74 +
    1.75 +lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
    1.76 +by (simp add: divide_inverse powr_minus)
    1.77 +
    1.78 +lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
    1.79 +by (simp add: powr_def)
    1.80 +
    1.81 +lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
    1.82 +by (simp add: powr_def)
    1.83 +
    1.84 +lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
    1.85 +by (blast intro: powr_less_cancel powr_less_mono)
    1.86 +
    1.87 +lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
    1.88 +by (simp add: linorder_not_less [symmetric])
    1.89 +
    1.90 +lemma log_ln: "ln x = log (exp(1)) x"
    1.91 +by (simp add: log_def)
    1.92 +
    1.93 +lemma powr_log_cancel [simp]:
    1.94 +     "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
    1.95 +by (simp add: powr_def log_def)
    1.96 +
    1.97 +lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
    1.98 +by (simp add: log_def powr_def)
    1.99 +
   1.100 +lemma log_mult: 
   1.101 +     "[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]  
   1.102 +      ==> log a (x * y) = log a x + log a y"
   1.103 +by (simp add: log_def ln_mult divide_inverse left_distrib)
   1.104 +
   1.105 +lemma log_eq_div_ln_mult_log: 
   1.106 +     "[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]  
   1.107 +      ==> log a x = (ln b/ln a) * log b x"
   1.108 +by (simp add: log_def divide_inverse)
   1.109 +
   1.110 +text{*Base 10 logarithms*}
   1.111 +lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
   1.112 +by (simp add: log_def)
   1.113 +
   1.114 +lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
   1.115 +by (simp add: log_def)
   1.116 +
   1.117 +lemma log_one [simp]: "log a 1 = 0"
   1.118 +by (simp add: log_def)
   1.119 +
   1.120 +lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
   1.121 +by (simp add: log_def)
   1.122 +
   1.123 +lemma log_inverse:
   1.124 +     "[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
   1.125 +apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
   1.126 +apply (simp add: log_mult [symmetric])
   1.127 +done
   1.128 +
   1.129 +lemma log_divide:
   1.130 +     "[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
   1.131 +by (simp add: log_mult divide_inverse log_inverse)
   1.132 +
   1.133 +lemma log_less_cancel_iff [simp]:
   1.134 +     "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
   1.135 +apply safe
   1.136 +apply (rule_tac [2] powr_less_cancel)
   1.137 +apply (drule_tac a = "log a x" in powr_less_mono, auto)
   1.138 +done
   1.139 +
   1.140 +lemma log_le_cancel_iff [simp]:
   1.141 +     "[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
   1.142 +by (simp add: linorder_not_less [symmetric])
   1.143 +
   1.144 +
   1.145 +lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
   1.146 +  apply (induct n, simp)
   1.147 +  apply (subgoal_tac "real(Suc n) = real n + 1")
   1.148 +  apply (erule ssubst)
   1.149 +  apply (subst powr_add, simp, simp)
   1.150 +done
   1.151 +
   1.152 +lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0
   1.153 +  else x powr (real n))"
   1.154 +  apply (case_tac "x = 0", simp, simp)
   1.155 +  apply (rule powr_realpow [THEN sym], simp)
   1.156 +done
   1.157 +
   1.158 +lemma ln_pwr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
   1.159 +by (unfold powr_def, simp)
   1.160 +
   1.161 +lemma ln_bound: "1 <= x ==> ln x <= x"
   1.162 +  apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
   1.163 +  apply simp
   1.164 +  apply (rule ln_add_one_self_le_self, simp)
   1.165 +done
   1.166 +
   1.167 +lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
   1.168 +  apply (case_tac "x = 1", simp)
   1.169 +  apply (case_tac "a = b", simp)
   1.170 +  apply (rule order_less_imp_le)
   1.171 +  apply (rule powr_less_mono, auto)
   1.172 +done
   1.173 +
   1.174 +lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
   1.175 +  apply (subst powr_zero_eq_one [THEN sym])
   1.176 +  apply (rule powr_mono, assumption+)
   1.177 +done
   1.178 +
   1.179 +lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
   1.180 +    y powr a"
   1.181 +  apply (unfold powr_def)
   1.182 +  apply (rule exp_less_mono)
   1.183 +  apply (rule mult_strict_left_mono)
   1.184 +  apply (subst ln_less_cancel_iff, assumption)
   1.185 +  apply (rule order_less_trans)
   1.186 +  prefer 2
   1.187 +  apply assumption+
   1.188 +done
   1.189 +
   1.190 +lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
   1.191 +    x powr a"
   1.192 +  apply (unfold powr_def)
   1.193 +  apply (rule exp_less_mono)
   1.194 +  apply (rule mult_strict_left_mono_neg)
   1.195 +  apply (subst ln_less_cancel_iff)
   1.196 +  apply assumption
   1.197 +  apply (rule order_less_trans)
   1.198 +  prefer 2
   1.199 +  apply assumption+
   1.200 +done
   1.201 +
   1.202 +lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
   1.203 +  apply (case_tac "a = 0", simp)
   1.204 +  apply (case_tac "x = y", simp)
   1.205 +  apply (rule order_less_imp_le)
   1.206 +  apply (rule powr_less_mono2, auto)
   1.207 +done
   1.208 +
   1.209 +lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
   1.210 +  apply (rule mult_imp_le_div_pos)
   1.211 +  apply (assumption)
   1.212 +  apply (subst mult_commute)
   1.213 +  apply (subst ln_pwr [THEN sym])
   1.214 +  apply auto
   1.215 +  apply (rule ln_bound)
   1.216 +  apply (erule ge_one_powr_ge_zero)
   1.217 +  apply (erule order_less_imp_le)
   1.218 +done
   1.219 +
   1.220 +lemma ln_powr_bound2: "1 < x ==> 0 < a ==> (ln x) powr a <= (a powr a) * x"
   1.221 +proof -
   1.222 +  assume "1 < x" and "0 < a"
   1.223 +  then have "ln x <= (x powr (1 / a)) / (1 / a)"
   1.224 +    apply (intro ln_powr_bound)
   1.225 +    apply (erule order_less_imp_le)
   1.226 +    apply (rule divide_pos_pos)
   1.227 +    apply simp_all
   1.228 +    done
   1.229 +  also have "... = a * (x powr (1 / a))"
   1.230 +    by simp
   1.231 +  finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
   1.232 +    apply (intro powr_mono2)
   1.233 +    apply (rule order_less_imp_le, rule prems)
   1.234 +    apply (rule ln_gt_zero)
   1.235 +    apply (rule prems)
   1.236 +    apply assumption
   1.237 +    done
   1.238 +  also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
   1.239 +    apply (rule powr_mult)
   1.240 +    apply (rule prems)
   1.241 +    apply (rule powr_gt_zero)
   1.242 +    done
   1.243 +  also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
   1.244 +    by (rule powr_powr)
   1.245 +  also have "... = x"
   1.246 +    apply simp
   1.247 +    apply (subgoal_tac "a ~= 0")
   1.248 +    apply (insert prems, auto)
   1.249 +    done
   1.250 +  finally show ?thesis .
   1.251 +qed
   1.252 +
   1.253 +lemma LIMSEQ_neg_powr: "0 < s ==> (%x. (real x) powr - s) ----> 0"
   1.254 +  apply (unfold LIMSEQ_def)
   1.255 +  apply clarsimp
   1.256 +  apply (rule_tac x = "natfloor(r powr (1 / - s)) + 1" in exI)
   1.257 +  apply clarify
   1.258 +  proof -
   1.259 +    fix r fix n
   1.260 +    assume "0 < s" and "0 < r" and "natfloor (r powr (1 / - s)) + 1 <= n"
   1.261 +    have "r powr (1 / - s) < real(natfloor(r powr (1 / - s))) + 1"
   1.262 +      by (rule real_natfloor_add_one_gt)
   1.263 +    also have "... = real(natfloor(r powr (1 / -s)) + 1)"
   1.264 +      by simp
   1.265 +    also have "... <= real n"
   1.266 +      apply (subst real_of_nat_le_iff)
   1.267 +      apply (rule prems)
   1.268 +      done
   1.269 +    finally have "r powr (1 / - s) < real n".
   1.270 +    then have "real n powr (- s) < (r powr (1 / - s)) powr - s" 
   1.271 +      apply (intro powr_less_mono2_neg)
   1.272 +      apply (auto simp add: prems)
   1.273 +      done
   1.274 +    also have "... = r"
   1.275 +      by (simp add: powr_powr prems less_imp_neq [THEN not_sym])
   1.276 +    finally show "real n powr - s < r" .
   1.277 +  qed
   1.278 +
   1.279 +end