add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
(* Title : Log.thy
Author : Jacques D. Fleuriot
Additional contributions by Jeremy Avigad
Copyright : 2000,2001 University of Edinburgh
*)
header{*Logarithms: Standard Version*}
theory Log
imports Transcendental
begin
definition
powr :: "[real,real] => real" (infixr "powr" 80) where
--{*exponentation with real exponent*}
"x powr a = exp(a * ln x)"
definition
log :: "[real,real] => real" where
--{*logarithm of @{term x} to base @{term a}*}
"log a x = ln x / ln a"
lemma powr_one_eq_one [simp]: "1 powr a = 1"
by (simp add: powr_def)
lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
by (simp add: powr_def)
lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
by (simp add: powr_def)
declare powr_one_gt_zero_iff [THEN iffD2, simp]
lemma powr_mult:
"[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
lemma powr_gt_zero [simp]: "0 < x powr a"
by (simp add: powr_def)
lemma powr_ge_pzero [simp]: "0 <= x powr y"
by (rule order_less_imp_le, rule powr_gt_zero)
lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
by (simp add: powr_def)
lemma powr_divide:
"[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
done
lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
apply (simp add: powr_def)
apply (subst exp_diff [THEN sym])
apply (simp add: left_diff_distrib)
done
lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
by (simp add: powr_def exp_add [symmetric] distrib_right)
lemma powr_mult_base:
"0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
using assms by (auto simp: powr_add)
lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
by (simp add: powr_def)
lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
by (simp add: powr_powr mult_commute)
lemma powr_minus: "x powr (-a) = inverse (x powr a)"
by (simp add: powr_def exp_minus [symmetric])
lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
by (simp add: divide_inverse powr_minus)
lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
by (simp add: powr_def)
lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
by (simp add: powr_def)
lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
by (blast intro: powr_less_cancel powr_less_mono)
lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
by (simp add: linorder_not_less [symmetric])
lemma log_ln: "ln x = log (exp(1)) x"
by (simp add: log_def)
lemma DERIV_log: assumes "x > 0" shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
proof -
def lb \<equiv> "1 / ln b"
moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
using `x > 0` by (auto intro!: DERIV_intros)
ultimately show ?thesis
by (simp add: log_def)
qed
lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
lemma powr_log_cancel [simp]:
"[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
by (simp add: powr_def log_def)
lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
by (simp add: log_def powr_def)
lemma log_mult:
"[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]
==> log a (x * y) = log a x + log a y"
by (simp add: log_def ln_mult divide_inverse distrib_right)
lemma log_eq_div_ln_mult_log:
"[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]
==> log a x = (ln b/ln a) * log b x"
by (simp add: log_def divide_inverse)
text{*Base 10 logarithms*}
lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
by (simp add: log_def)
lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
by (simp add: log_def)
lemma log_one [simp]: "log a 1 = 0"
by (simp add: log_def)
lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
by (simp add: log_def)
lemma log_inverse:
"[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
apply (simp add: log_mult [symmetric])
done
lemma log_divide:
"[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
by (simp add: log_mult divide_inverse log_inverse)
lemma log_less_cancel_iff [simp]:
"[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
apply safe
apply (rule_tac [2] powr_less_cancel)
apply (drule_tac a = "log a x" in powr_less_mono, auto)
done
lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
proof (rule inj_onI, simp)
fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
show "x = y"
proof (cases rule: linorder_cases)
assume "x < y" hence "log b x < log b y"
using log_less_cancel_iff[OF `1 < b`] pos by simp
thus ?thesis using * by simp
next
assume "y < x" hence "log b y < log b x"
using log_less_cancel_iff[OF `1 < b`] pos by simp
thus ?thesis using * by simp
qed simp
qed
lemma log_le_cancel_iff [simp]:
"[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
by (simp add: linorder_not_less [symmetric])
lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
using log_less_cancel_iff[of a 1 x] by simp
lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
using log_le_cancel_iff[of a 1 x] by simp
lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
using log_less_cancel_iff[of a x 1] by simp
lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
using log_le_cancel_iff[of a x 1] by simp
lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
using log_less_cancel_iff[of a a x] by simp
lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
using log_le_cancel_iff[of a a x] by simp
lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
using log_less_cancel_iff[of a x a] by simp
lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
using log_le_cancel_iff[of a x a] by simp
lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
apply (induct n, simp)
apply (subgoal_tac "real(Suc n) = real n + 1")
apply (erule ssubst)
apply (subst powr_add, simp, simp)
done
lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
apply (case_tac "x = 0", simp, simp)
apply (rule powr_realpow [THEN sym], simp)
done
lemma powr_int:
assumes "x > 0"
shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
proof cases
assume "i < 0"
have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
qed (simp add: assms powr_realpow[symmetric])
lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x^numeral n"
using powr_realpow[of x "numeral n"] by simp
lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr neg_numeral n = 1 / x^numeral n"
using powr_int[of x "neg_numeral n"] by simp
lemma root_powr_inverse:
"0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
by (auto simp: root_def powr_realpow[symmetric] powr_powr)
lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
by (unfold powr_def, simp)
lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
apply (case_tac "y = 0")
apply force
apply (auto simp add: log_def ln_powr field_simps)
done
lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
apply (subst powr_realpow [symmetric])
apply (auto simp add: log_powr)
done
lemma ln_bound: "1 <= x ==> ln x <= x"
apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
apply simp
apply (rule ln_add_one_self_le_self, simp)
done
lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
apply (case_tac "x = 1", simp)
apply (case_tac "a = b", simp)
apply (rule order_less_imp_le)
apply (rule powr_less_mono, auto)
done
lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
apply (subst powr_zero_eq_one [THEN sym])
apply (rule powr_mono, assumption+)
done
lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
y powr a"
apply (unfold powr_def)
apply (rule exp_less_mono)
apply (rule mult_strict_left_mono)
apply (subst ln_less_cancel_iff, assumption)
apply (rule order_less_trans)
prefer 2
apply assumption+
done
lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
x powr a"
apply (unfold powr_def)
apply (rule exp_less_mono)
apply (rule mult_strict_left_mono_neg)
apply (subst ln_less_cancel_iff)
apply assumption
apply (rule order_less_trans)
prefer 2
apply assumption+
done
lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
apply (case_tac "a = 0", simp)
apply (case_tac "x = y", simp)
apply (rule order_less_imp_le)
apply (rule powr_less_mono2, auto)
done
lemma powr_inj:
"0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
unfolding powr_def exp_inj_iff by simp
lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
apply (rule mult_imp_le_div_pos)
apply (assumption)
apply (subst mult_commute)
apply (subst ln_powr [THEN sym])
apply auto
apply (rule ln_bound)
apply (erule ge_one_powr_ge_zero)
apply (erule order_less_imp_le)
done
lemma ln_powr_bound2:
assumes "1 < x" and "0 < a"
shows "(ln x) powr a <= (a powr a) * x"
proof -
from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
apply (intro ln_powr_bound)
apply (erule order_less_imp_le)
apply (rule divide_pos_pos)
apply simp_all
done
also have "... = a * (x powr (1 / a))"
by simp
finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
apply (intro powr_mono2)
apply (rule order_less_imp_le, rule assms)
apply (rule ln_gt_zero)
apply (rule assms)
apply assumption
done
also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
apply (rule powr_mult)
apply (rule assms)
apply (rule powr_gt_zero)
done
also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
by (rule powr_powr)
also have "... = x"
apply simp
apply (subgoal_tac "a ~= 0")
using assms apply auto
done
finally show ?thesis .
qed
lemma tendsto_powr [tendsto_intros]:
"\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
unfolding powr_def by (intro tendsto_intros)
(* FIXME: generalize by replacing d by with g x and g ---> d? *)
lemma tendsto_zero_powrI:
assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
assumes "0 < d"
shows "((\<lambda>x. f x powr d) ---> 0) F"
proof (rule tendstoI)
fix e :: real assume "0 < e"
def Z \<equiv> "e powr (1 / d)"
with `0 < e` have "0 < Z" by simp
with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
by (intro eventually_conj tendstoD)
moreover
from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
by (intro powr_less_mono2) (auto simp: dist_real_def)
with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
unfolding dist_real_def Z_def by (auto simp: powr_powr)
ultimately
show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
qed
lemma tendsto_neg_powr:
assumes "s < 0" and "LIM x F. f x :> at_top"
shows "((\<lambda>x. f x powr s) ---> 0) F"
proof (rule tendstoI)
fix e :: real assume "0 < e"
def Z \<equiv> "e powr (1 / s)"
from assms have "eventually (\<lambda>x. Z < f x) F"
by (simp add: filterlim_at_top_dense)
moreover
from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
by (auto simp: Z_def intro!: powr_less_mono2_neg)
with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
by (simp add: powr_powr Z_def dist_real_def)
ultimately
show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
qed
end