src/HOL/Log.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51478 270b21f3ae0a parent 50346 a75c6429c3c3 child 51483 dc39d69774bb permissions -rw-r--r--
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
```     1 (*  Title       : Log.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3                   Additional contributions by Jeremy Avigad
```
```     4     Copyright   : 2000,2001 University of Edinburgh
```
```     5 *)
```
```     6
```
```     7 header{*Logarithms: Standard Version*}
```
```     8
```
```     9 theory Log
```
```    10 imports Transcendental
```
```    11 begin
```
```    12
```
```    13 definition
```
```    14   powr  :: "[real,real] => real"     (infixr "powr" 80) where
```
```    15     --{*exponentation with real exponent*}
```
```    16   "x powr a = exp(a * ln x)"
```
```    17
```
```    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"
```
```    22
```
```    23
```
```    24 lemma tendsto_log [tendsto_intros]:
```
```    25   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) ---> log a b) F"
```
```    26   unfolding log_def by (intro tendsto_intros) auto
```
```    27
```
```    28 lemma continuous_log:
```
```    29   assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))" and "f (Lim F (\<lambda>x. x)) \<noteq> 1" and "0 < g (Lim F (\<lambda>x. x))"
```
```    30   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```    31   using assms unfolding continuous_def by (rule tendsto_log)
```
```    32
```
```    33 lemma continuous_at_within_log[continuous_intros]:
```
```    34   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a" and "f a \<noteq> 1" and "0 < g a"
```
```    35   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```    36   using assms unfolding continuous_within by (rule tendsto_log)
```
```    37
```
```    38 lemma isCont_log[continuous_intros, simp]:
```
```    39   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```    40   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```    41   using assms unfolding continuous_at by (rule tendsto_log)
```
```    42
```
```    43 lemma continuous_on_log[continuous_on_intros]:
```
```    44   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```    45   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```    46   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```    47
```
```    48 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```    49 by (simp add: powr_def)
```
```    50
```
```    51 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
```
```    52 by (simp add: powr_def)
```
```    53
```
```    54 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
```
```    55 by (simp add: powr_def)
```
```    56 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```    57
```
```    58 lemma powr_mult:
```
```    59       "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
```
```    60 by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```    61
```
```    62 lemma powr_gt_zero [simp]: "0 < x powr a"
```
```    63 by (simp add: powr_def)
```
```    64
```
```    65 lemma powr_ge_pzero [simp]: "0 <= x powr y"
```
```    66 by (rule order_less_imp_le, rule powr_gt_zero)
```
```    67
```
```    68 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
```
```    69 by (simp add: powr_def)
```
```    70
```
```    71 lemma powr_divide:
```
```    72      "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
```
```    73 apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```    74 apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```    75 done
```
```    76
```
```    77 lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
```
```    78   apply (simp add: powr_def)
```
```    79   apply (subst exp_diff [THEN sym])
```
```    80   apply (simp add: left_diff_distrib)
```
```    81 done
```
```    82
```
```    83 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
```
```    84 by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```    85
```
```    86 lemma powr_mult_base:
```
```    87   "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```    88 using assms by (auto simp: powr_add)
```
```    89
```
```    90 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
```
```    91 by (simp add: powr_def)
```
```    92
```
```    93 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
```
```    94 by (simp add: powr_powr mult_commute)
```
```    95
```
```    96 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
```
```    97 by (simp add: powr_def exp_minus [symmetric])
```
```    98
```
```    99 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
```
```   100 by (simp add: divide_inverse powr_minus)
```
```   101
```
```   102 lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
```
```   103 by (simp add: powr_def)
```
```   104
```
```   105 lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
```
```   106 by (simp add: powr_def)
```
```   107
```
```   108 lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
```
```   109 by (blast intro: powr_less_cancel powr_less_mono)
```
```   110
```
```   111 lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
```
```   112 by (simp add: linorder_not_less [symmetric])
```
```   113
```
```   114 lemma log_ln: "ln x = log (exp(1)) x"
```
```   115 by (simp add: log_def)
```
```   116
```
```   117 lemma DERIV_log: assumes "x > 0" shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```   118 proof -
```
```   119   def lb \<equiv> "1 / ln b"
```
```   120   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```   121     using `x > 0` by (auto intro!: DERIV_intros)
```
```   122   ultimately show ?thesis
```
```   123     by (simp add: log_def)
```
```   124 qed
```
```   125
```
```   126 lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```   127
```
```   128 lemma powr_log_cancel [simp]:
```
```   129      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
```
```   130 by (simp add: powr_def log_def)
```
```   131
```
```   132 lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
```
```   133 by (simp add: log_def powr_def)
```
```   134
```
```   135 lemma log_mult:
```
```   136      "[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]
```
```   137       ==> log a (x * y) = log a x + log a y"
```
```   138 by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```   139
```
```   140 lemma log_eq_div_ln_mult_log:
```
```   141      "[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]
```
```   142       ==> log a x = (ln b/ln a) * log b x"
```
```   143 by (simp add: log_def divide_inverse)
```
```   144
```
```   145 text{*Base 10 logarithms*}
```
```   146 lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```   147 by (simp add: log_def)
```
```   148
```
```   149 lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
```
```   150 by (simp add: log_def)
```
```   151
```
```   152 lemma log_one [simp]: "log a 1 = 0"
```
```   153 by (simp add: log_def)
```
```   154
```
```   155 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```   156 by (simp add: log_def)
```
```   157
```
```   158 lemma log_inverse:
```
```   159      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
```
```   160 apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```   161 apply (simp add: log_mult [symmetric])
```
```   162 done
```
```   163
```
```   164 lemma log_divide:
```
```   165      "[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
```
```   166 by (simp add: log_mult divide_inverse log_inverse)
```
```   167
```
```   168 lemma log_less_cancel_iff [simp]:
```
```   169      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
```
```   170 apply safe
```
```   171 apply (rule_tac [2] powr_less_cancel)
```
```   172 apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```   173 done
```
```   174
```
```   175 lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
```
```   176 proof (rule inj_onI, simp)
```
```   177   fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```   178   show "x = y"
```
```   179   proof (cases rule: linorder_cases)
```
```   180     assume "x < y" hence "log b x < log b y"
```
```   181       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```   182     thus ?thesis using * by simp
```
```   183   next
```
```   184     assume "y < x" hence "log b y < log b x"
```
```   185       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```   186     thus ?thesis using * by simp
```
```   187   qed simp
```
```   188 qed
```
```   189
```
```   190 lemma log_le_cancel_iff [simp]:
```
```   191      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
```
```   192 by (simp add: linorder_not_less [symmetric])
```
```   193
```
```   194 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```   195   using log_less_cancel_iff[of a 1 x] by simp
```
```   196
```
```   197 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```   198   using log_le_cancel_iff[of a 1 x] by simp
```
```   199
```
```   200 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```   201   using log_less_cancel_iff[of a x 1] by simp
```
```   202
```
```   203 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```   204   using log_le_cancel_iff[of a x 1] by simp
```
```   205
```
```   206 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```   207   using log_less_cancel_iff[of a a x] by simp
```
```   208
```
```   209 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```   210   using log_le_cancel_iff[of a a x] by simp
```
```   211
```
```   212 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```   213   using log_less_cancel_iff[of a x a] by simp
```
```   214
```
```   215 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```   216   using log_le_cancel_iff[of a x a] by simp
```
```   217
```
```   218 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```   219   apply (induct n, simp)
```
```   220   apply (subgoal_tac "real(Suc n) = real n + 1")
```
```   221   apply (erule ssubst)
```
```   222   apply (subst powr_add, simp, simp)
```
```   223 done
```
```   224
```
```   225 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```   226   apply (case_tac "x = 0", simp, simp)
```
```   227   apply (rule powr_realpow [THEN sym], simp)
```
```   228 done
```
```   229
```
```   230 lemma powr_int:
```
```   231   assumes "x > 0"
```
```   232   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```   233 proof cases
```
```   234   assume "i < 0"
```
```   235   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```   236   show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
```
```   237 qed (simp add: assms powr_realpow[symmetric])
```
```   238
```
```   239 lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x^numeral n"
```
```   240   using powr_realpow[of x "numeral n"] by simp
```
```   241
```
```   242 lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr neg_numeral n = 1 / x^numeral n"
```
```   243   using powr_int[of x "neg_numeral n"] by simp
```
```   244
```
```   245 lemma root_powr_inverse:
```
```   246   "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```   247 by (auto simp: root_def powr_realpow[symmetric] powr_powr)
```
```   248
```
```   249 lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
```
```   250 by (unfold powr_def, simp)
```
```   251
```
```   252 lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
```
```   253   apply (case_tac "y = 0")
```
```   254   apply force
```
```   255   apply (auto simp add: log_def ln_powr field_simps)
```
```   256 done
```
```   257
```
```   258 lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
```
```   259   apply (subst powr_realpow [symmetric])
```
```   260   apply (auto simp add: log_powr)
```
```   261 done
```
```   262
```
```   263 lemma ln_bound: "1 <= x ==> ln x <= x"
```
```   264   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```   265   apply simp
```
```   266   apply (rule ln_add_one_self_le_self, simp)
```
```   267 done
```
```   268
```
```   269 lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```   270   apply (case_tac "x = 1", simp)
```
```   271   apply (case_tac "a = b", simp)
```
```   272   apply (rule order_less_imp_le)
```
```   273   apply (rule powr_less_mono, auto)
```
```   274 done
```
```   275
```
```   276 lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```   277   apply (subst powr_zero_eq_one [THEN sym])
```
```   278   apply (rule powr_mono, assumption+)
```
```   279 done
```
```   280
```
```   281 lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
```
```   282     y powr a"
```
```   283   apply (unfold powr_def)
```
```   284   apply (rule exp_less_mono)
```
```   285   apply (rule mult_strict_left_mono)
```
```   286   apply (subst ln_less_cancel_iff, assumption)
```
```   287   apply (rule order_less_trans)
```
```   288   prefer 2
```
```   289   apply assumption+
```
```   290 done
```
```   291
```
```   292 lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
```
```   293     x powr a"
```
```   294   apply (unfold powr_def)
```
```   295   apply (rule exp_less_mono)
```
```   296   apply (rule mult_strict_left_mono_neg)
```
```   297   apply (subst ln_less_cancel_iff)
```
```   298   apply assumption
```
```   299   apply (rule order_less_trans)
```
```   300   prefer 2
```
```   301   apply assumption+
```
```   302 done
```
```   303
```
```   304 lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
```
```   305   apply (case_tac "a = 0", simp)
```
```   306   apply (case_tac "x = y", simp)
```
```   307   apply (rule order_less_imp_le)
```
```   308   apply (rule powr_less_mono2, auto)
```
```   309 done
```
```   310
```
```   311 lemma powr_inj:
```
```   312   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```   313   unfolding powr_def exp_inj_iff by simp
```
```   314
```
```   315 lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```   316   apply (rule mult_imp_le_div_pos)
```
```   317   apply (assumption)
```
```   318   apply (subst mult_commute)
```
```   319   apply (subst ln_powr [THEN sym])
```
```   320   apply auto
```
```   321   apply (rule ln_bound)
```
```   322   apply (erule ge_one_powr_ge_zero)
```
```   323   apply (erule order_less_imp_le)
```
```   324 done
```
```   325
```
```   326 lemma ln_powr_bound2:
```
```   327   assumes "1 < x" and "0 < a"
```
```   328   shows "(ln x) powr a <= (a powr a) * x"
```
```   329 proof -
```
```   330   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```   331     apply (intro ln_powr_bound)
```
```   332     apply (erule order_less_imp_le)
```
```   333     apply (rule divide_pos_pos)
```
```   334     apply simp_all
```
```   335     done
```
```   336   also have "... = a * (x powr (1 / a))"
```
```   337     by simp
```
```   338   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```   339     apply (intro powr_mono2)
```
```   340     apply (rule order_less_imp_le, rule assms)
```
```   341     apply (rule ln_gt_zero)
```
```   342     apply (rule assms)
```
```   343     apply assumption
```
```   344     done
```
```   345   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```   346     apply (rule powr_mult)
```
```   347     apply (rule assms)
```
```   348     apply (rule powr_gt_zero)
```
```   349     done
```
```   350   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```   351     by (rule powr_powr)
```
```   352   also have "... = x"
```
```   353     apply simp
```
```   354     apply (subgoal_tac "a ~= 0")
```
```   355     using assms apply auto
```
```   356     done
```
```   357   finally show ?thesis .
```
```   358 qed
```
```   359
```
```   360 lemma tendsto_powr [tendsto_intros]:
```
```   361   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```   362   unfolding powr_def by (intro tendsto_intros)
```
```   363
```
```   364 lemma continuous_powr:
```
```   365   assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))"
```
```   366   shows "continuous F (\<lambda>x. (f x) powr (g x))"
```
```   367   using assms unfolding continuous_def by (rule tendsto_powr)
```
```   368
```
```   369 lemma continuous_at_within_powr[continuous_intros]:
```
```   370   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a"
```
```   371   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
```
```   372   using assms unfolding continuous_within by (rule tendsto_powr)
```
```   373
```
```   374 lemma isCont_powr[continuous_intros, simp]:
```
```   375   assumes "isCont f a" "isCont g a" "0 < f a"
```
```   376   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```   377   using assms unfolding continuous_at by (rule tendsto_powr)
```
```   378
```
```   379 lemma continuous_on_powr[continuous_on_intros]:
```
```   380   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x"
```
```   381   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```   382   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```   383
```
```   384 (* FIXME: generalize by replacing d by with g x and g ---> d? *)
```
```   385 lemma tendsto_zero_powrI:
```
```   386   assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
```
```   387   assumes "0 < d"
```
```   388   shows "((\<lambda>x. f x powr d) ---> 0) F"
```
```   389 proof (rule tendstoI)
```
```   390   fix e :: real assume "0 < e"
```
```   391   def Z \<equiv> "e powr (1 / d)"
```
```   392   with `0 < e` have "0 < Z" by simp
```
```   393   with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
```
```   394     by (intro eventually_conj tendstoD)
```
```   395   moreover
```
```   396   from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
```
```   397     by (intro powr_less_mono2) (auto simp: dist_real_def)
```
```   398   with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
```
```   399     unfolding dist_real_def Z_def by (auto simp: powr_powr)
```
```   400   ultimately
```
```   401   show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
```
```   402 qed
```
```   403
```
```   404 lemma tendsto_neg_powr:
```
```   405   assumes "s < 0" and "LIM x F. f x :> at_top"
```
```   406   shows "((\<lambda>x. f x powr s) ---> 0) F"
```
```   407 proof (rule tendstoI)
```
```   408   fix e :: real assume "0 < e"
```
```   409   def Z \<equiv> "e powr (1 / s)"
```
```   410   from assms have "eventually (\<lambda>x. Z < f x) F"
```
```   411     by (simp add: filterlim_at_top_dense)
```
```   412   moreover
```
```   413   from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
```
```   414     by (auto simp: Z_def intro!: powr_less_mono2_neg)
```
```   415   with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
```
```   416     by (simp add: powr_powr Z_def dist_real_def)
```
```   417   ultimately
```
```   418   show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
```
```   419 qed
```
```   420
```
```   421 end
```