--- a/src/HOL/Complex_Main.thy Tue Mar 26 12:20:58 2013 +0100
+++ b/src/HOL/Complex_Main.thy Tue Mar 26 12:20:59 2013 +0100
@@ -3,9 +3,9 @@
theory Complex_Main
imports
Main
+ Real
Complex
- Log
- Ln
+ Transcendental
Taylor
Deriv
begin
--- a/src/HOL/Ln.thy Tue Mar 26 12:20:58 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,229 +0,0 @@
-(* Title: HOL/Ln.thy
- Author: Jeremy Avigad
-*)
-
-header {* Properties of ln *}
-
-theory Ln
-imports Transcendental
-begin
-
-lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==>
- x - x^2 <= ln (1 + x)"
-proof -
- assume a: "0 <= x" and b: "x <= 1"
- have "exp (x - x^2) = exp x / exp (x^2)"
- by (rule exp_diff)
- also have "... <= (1 + x + x^2) / exp (x ^2)"
- apply (rule divide_right_mono)
- apply (rule exp_bound)
- apply (rule a, rule b)
- apply simp
- done
- also have "... <= (1 + x + x^2) / (1 + x^2)"
- apply (rule divide_left_mono)
- apply (simp add: exp_ge_add_one_self_aux)
- apply (simp add: a)
- apply (simp add: mult_pos_pos add_pos_nonneg)
- done
- also from a have "... <= 1 + x"
- by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
- finally have "exp (x - x^2) <= 1 + x" .
- also have "... = exp (ln (1 + x))"
- proof -
- from a have "0 < 1 + x" by auto
- thus ?thesis
- by (auto simp only: exp_ln_iff [THEN sym])
- qed
- finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" .
- thus ?thesis by (auto simp only: exp_le_cancel_iff)
-qed
-
-lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
-proof -
- assume a: "x < 1"
- have "ln(1 - x) = - ln(1 / (1 - x))"
- proof -
- have "ln(1 - x) = - (- ln (1 - x))"
- by auto
- also have "- ln(1 - x) = ln 1 - ln(1 - x)"
- by simp
- also have "... = ln(1 / (1 - x))"
- apply (rule ln_div [THEN sym])
- by (insert a, auto)
- finally show ?thesis .
- qed
- also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
- finally show ?thesis .
-qed
-
-lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
- - x - 2 * x^2 <= ln (1 - x)"
-proof -
- assume a: "0 <= x" and b: "x <= (1 / 2)"
- from b have c: "x < 1"
- by auto
- then have "ln (1 - x) = - ln (1 + x / (1 - x))"
- by (rule aux5)
- also have "- (x / (1 - x)) <= ..."
- proof -
- have "ln (1 + x / (1 - x)) <= x / (1 - x)"
- apply (rule ln_add_one_self_le_self)
- apply (rule divide_nonneg_pos)
- by (insert a c, auto)
- thus ?thesis
- by auto
- qed
- also have "- (x / (1 - x)) = -x / (1 - x)"
- by auto
- finally have d: "- x / (1 - x) <= ln (1 - x)" .
- have "0 < 1 - x" using a b by simp
- hence e: "-x - 2 * x^2 <= - x / (1 - x)"
- using mult_right_le_one_le[of "x*x" "2*x"] a b
- by (simp add:field_simps power2_eq_square)
- from e d show "- x - 2 * x^2 <= ln (1 - x)"
- by (rule order_trans)
-qed
-
-lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
- apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
- apply (subst ln_le_cancel_iff)
- apply auto
-done
-
-lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
- "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
-proof -
- assume x: "0 <= x"
- assume x1: "x <= 1"
- from x have "ln (1 + x) <= x"
- by (rule ln_add_one_self_le_self)
- then have "ln (1 + x) - x <= 0"
- by simp
- then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
- by (rule abs_of_nonpos)
- also have "... = x - ln (1 + x)"
- by simp
- also have "... <= x^2"
- proof -
- from x x1 have "x - x^2 <= ln (1 + x)"
- by (intro ln_one_plus_pos_lower_bound)
- thus ?thesis
- by simp
- qed
- finally show ?thesis .
-qed
-
-lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
- "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
-proof -
- assume a: "-(1 / 2) <= x"
- assume b: "x <= 0"
- have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
- apply (subst abs_of_nonpos)
- apply simp
- apply (rule ln_add_one_self_le_self2)
- using a apply auto
- done
- also have "... <= 2 * x^2"
- apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
- apply (simp add: algebra_simps)
- apply (rule ln_one_minus_pos_lower_bound)
- using a b apply auto
- done
- finally show ?thesis .
-qed
-
-lemma abs_ln_one_plus_x_minus_x_bound:
- "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2"
- apply (case_tac "0 <= x")
- apply (rule order_trans)
- apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
- apply auto
- apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
- apply auto
-done
-
-lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
-proof -
- assume x: "exp 1 <= x" "x <= y"
- moreover have "0 < exp (1::real)" by simp
- ultimately have a: "0 < x" and b: "0 < y"
- by (fast intro: less_le_trans order_trans)+
- have "x * ln y - x * ln x = x * (ln y - ln x)"
- by (simp add: algebra_simps)
- also have "... = x * ln(y / x)"
- by (simp only: ln_div a b)
- also have "y / x = (x + (y - x)) / x"
- by simp
- also have "... = 1 + (y - x) / x"
- using x a by (simp add: field_simps)
- also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
- apply (rule mult_left_mono)
- apply (rule ln_add_one_self_le_self)
- apply (rule divide_nonneg_pos)
- using x a apply simp_all
- done
- also have "... = y - x" using a by simp
- also have "... = (y - x) * ln (exp 1)" by simp
- also have "... <= (y - x) * ln x"
- apply (rule mult_left_mono)
- apply (subst ln_le_cancel_iff)
- apply fact
- apply (rule a)
- apply (rule x)
- using x apply simp
- done
- also have "... = y * ln x - x * ln x"
- by (rule left_diff_distrib)
- finally have "x * ln y <= y * ln x"
- by arith
- then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
- also have "... = y * (ln x / x)" by simp
- finally show ?thesis using b by (simp add: field_simps)
-qed
-
-lemma ln_le_minus_one:
- "0 < x \<Longrightarrow> ln x \<le> x - 1"
- using exp_ge_add_one_self[of "ln x"] by simp
-
-lemma ln_eq_minus_one:
- assumes "0 < x" "ln x = x - 1" shows "x = 1"
-proof -
- let "?l y" = "ln y - y + 1"
- have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
- by (auto intro!: DERIV_intros)
-
- show ?thesis
- proof (cases rule: linorder_cases)
- assume "x < 1"
- from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
- from `x < a` have "?l x < ?l a"
- proof (rule DERIV_pos_imp_increasing, safe)
- fix y assume "x \<le> y" "y \<le> a"
- with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
- by (auto simp: field_simps)
- with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
- by auto
- qed
- also have "\<dots> \<le> 0"
- using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
- finally show "x = 1" using assms by auto
- next
- assume "1 < x"
- from dense[OF `1 < x`] obtain a where "1 < a" "a < x" by blast
- from `a < x` have "?l x < ?l a"
- proof (rule DERIV_neg_imp_decreasing, safe)
- fix y assume "a \<le> y" "y \<le> x"
- with `1 < a` have "1 / y - 1 < 0" "0 < y"
- by (auto simp: field_simps)
- with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
- by blast
- qed
- also have "\<dots> \<le> 0"
- using ln_le_minus_one `1 < a` by (auto simp: field_simps)
- finally show "x = 1" using assms by auto
- qed simp
-qed
-
-end
--- a/src/HOL/Log.thy Tue Mar 26 12:20:58 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,421 +0,0 @@
-(* 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 tendsto_log [tendsto_intros]:
- "\<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"
- unfolding log_def by (intro tendsto_intros) auto
-
-lemma continuous_log:
- 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))"
- shows "continuous F (\<lambda>x. log (f x) (g x))"
- using assms unfolding continuous_def by (rule tendsto_log)
-
-lemma continuous_at_within_log[continuous_intros]:
- 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"
- shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
- using assms unfolding continuous_within by (rule tendsto_log)
-
-lemma isCont_log[continuous_intros, simp]:
- assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
- shows "isCont (\<lambda>x. log (f x) (g x)) a"
- using assms unfolding continuous_at by (rule tendsto_log)
-
-lemma continuous_on_log[continuous_on_intros]:
- 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"
- shows "continuous_on s (\<lambda>x. log (f x) (g x))"
- using assms unfolding continuous_on_def by (fast intro: tendsto_log)
-
-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 (rule real_root_pos_unique) (auto simp: 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)
-
-lemma continuous_powr:
- assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))"
- shows "continuous F (\<lambda>x. (f x) powr (g x))"
- using assms unfolding continuous_def by (rule tendsto_powr)
-
-lemma continuous_at_within_powr[continuous_intros]:
- assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a"
- shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
- using assms unfolding continuous_within by (rule tendsto_powr)
-
-lemma isCont_powr[continuous_intros, simp]:
- assumes "isCont f a" "isCont g a" "0 < f a"
- shows "isCont (\<lambda>x. (f x) powr g x) a"
- using assms unfolding continuous_at by (rule tendsto_powr)
-
-lemma continuous_on_powr[continuous_on_intros]:
- assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x"
- shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
- using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
-
-(* 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
--- a/src/HOL/Transcendental.thy Tue Mar 26 12:20:58 2013 +0100
+++ b/src/HOL/Transcendental.thy Tue Mar 26 12:20:59 2013 +0100
@@ -1,6 +1,8 @@
(* Title: HOL/Transcendental.thy
Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
Author: Lawrence C Paulson
+ Author: Jeremy Avigad
+
*)
header{*Power Series, Transcendental Functions etc.*}
@@ -871,6 +873,8 @@
apply (simp del: of_real_add)
done
+declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
+
lemma isCont_exp: "isCont exp x"
by (rule DERIV_exp [THEN DERIV_isCont])
@@ -1200,6 +1204,8 @@
lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
+declare DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
+
lemma ln_series: assumes "0 < x" and "x < 2"
shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
proof -
@@ -1337,6 +1343,223 @@
apply auto
done
+lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==> x - x^2 <= ln (1 + x)"
+proof -
+ assume a: "0 <= x" and b: "x <= 1"
+ have "exp (x - x^2) = exp x / exp (x^2)"
+ by (rule exp_diff)
+ also have "... <= (1 + x + x^2) / exp (x ^2)"
+ apply (rule divide_right_mono)
+ apply (rule exp_bound)
+ apply (rule a, rule b)
+ apply simp
+ done
+ also have "... <= (1 + x + x^2) / (1 + x^2)"
+ apply (rule divide_left_mono)
+ apply (simp add: exp_ge_add_one_self_aux)
+ apply (simp add: a)
+ apply (simp add: mult_pos_pos add_pos_nonneg)
+ done
+ also from a have "... <= 1 + x"
+ by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
+ finally have "exp (x - x^2) <= 1 + x" .
+ also have "... = exp (ln (1 + x))"
+ proof -
+ from a have "0 < 1 + x" by auto
+ thus ?thesis
+ by (auto simp only: exp_ln_iff [THEN sym])
+ qed
+ finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" .
+ thus ?thesis by (auto simp only: exp_le_cancel_iff)
+qed
+
+lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
+proof -
+ assume a: "x < 1"
+ have "ln(1 - x) = - ln(1 / (1 - x))"
+ proof -
+ have "ln(1 - x) = - (- ln (1 - x))"
+ by auto
+ also have "- ln(1 - x) = ln 1 - ln(1 - x)"
+ by simp
+ also have "... = ln(1 / (1 - x))"
+ apply (rule ln_div [THEN sym])
+ by (insert a, auto)
+ finally show ?thesis .
+ qed
+ also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
+ finally show ?thesis .
+qed
+
+lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
+ - x - 2 * x^2 <= ln (1 - x)"
+proof -
+ assume a: "0 <= x" and b: "x <= (1 / 2)"
+ from b have c: "x < 1"
+ by auto
+ then have "ln (1 - x) = - ln (1 + x / (1 - x))"
+ by (rule aux5)
+ also have "- (x / (1 - x)) <= ..."
+ proof -
+ have "ln (1 + x / (1 - x)) <= x / (1 - x)"
+ apply (rule ln_add_one_self_le_self)
+ apply (rule divide_nonneg_pos)
+ by (insert a c, auto)
+ thus ?thesis
+ by auto
+ qed
+ also have "- (x / (1 - x)) = -x / (1 - x)"
+ by auto
+ finally have d: "- x / (1 - x) <= ln (1 - x)" .
+ have "0 < 1 - x" using a b by simp
+ hence e: "-x - 2 * x^2 <= - x / (1 - x)"
+ using mult_right_le_one_le[of "x*x" "2*x"] a b
+ by (simp add:field_simps power2_eq_square)
+ from e d show "- x - 2 * x^2 <= ln (1 - x)"
+ by (rule order_trans)
+qed
+
+lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
+ apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
+ apply (subst ln_le_cancel_iff)
+ apply auto
+done
+
+lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
+ "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
+proof -
+ assume x: "0 <= x"
+ assume x1: "x <= 1"
+ from x have "ln (1 + x) <= x"
+ by (rule ln_add_one_self_le_self)
+ then have "ln (1 + x) - x <= 0"
+ by simp
+ then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
+ by (rule abs_of_nonpos)
+ also have "... = x - ln (1 + x)"
+ by simp
+ also have "... <= x^2"
+ proof -
+ from x x1 have "x - x^2 <= ln (1 + x)"
+ by (intro ln_one_plus_pos_lower_bound)
+ thus ?thesis
+ by simp
+ qed
+ finally show ?thesis .
+qed
+
+lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
+ "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
+proof -
+ assume a: "-(1 / 2) <= x"
+ assume b: "x <= 0"
+ have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
+ apply (subst abs_of_nonpos)
+ apply simp
+ apply (rule ln_add_one_self_le_self2)
+ using a apply auto
+ done
+ also have "... <= 2 * x^2"
+ apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
+ apply (simp add: algebra_simps)
+ apply (rule ln_one_minus_pos_lower_bound)
+ using a b apply auto
+ done
+ finally show ?thesis .
+qed
+
+lemma abs_ln_one_plus_x_minus_x_bound:
+ "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2"
+ apply (case_tac "0 <= x")
+ apply (rule order_trans)
+ apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
+ apply auto
+ apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
+ apply auto
+done
+
+lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
+proof -
+ assume x: "exp 1 <= x" "x <= y"
+ moreover have "0 < exp (1::real)" by simp
+ ultimately have a: "0 < x" and b: "0 < y"
+ by (fast intro: less_le_trans order_trans)+
+ have "x * ln y - x * ln x = x * (ln y - ln x)"
+ by (simp add: algebra_simps)
+ also have "... = x * ln(y / x)"
+ by (simp only: ln_div a b)
+ also have "y / x = (x + (y - x)) / x"
+ by simp
+ also have "... = 1 + (y - x) / x"
+ using x a by (simp add: field_simps)
+ also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
+ apply (rule mult_left_mono)
+ apply (rule ln_add_one_self_le_self)
+ apply (rule divide_nonneg_pos)
+ using x a apply simp_all
+ done
+ also have "... = y - x" using a by simp
+ also have "... = (y - x) * ln (exp 1)" by simp
+ also have "... <= (y - x) * ln x"
+ apply (rule mult_left_mono)
+ apply (subst ln_le_cancel_iff)
+ apply fact
+ apply (rule a)
+ apply (rule x)
+ using x apply simp
+ done
+ also have "... = y * ln x - x * ln x"
+ by (rule left_diff_distrib)
+ finally have "x * ln y <= y * ln x"
+ by arith
+ then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
+ also have "... = y * (ln x / x)" by simp
+ finally show ?thesis using b by (simp add: field_simps)
+qed
+
+lemma ln_le_minus_one:
+ "0 < x \<Longrightarrow> ln x \<le> x - 1"
+ using exp_ge_add_one_self[of "ln x"] by simp
+
+lemma ln_eq_minus_one:
+ assumes "0 < x" "ln x = x - 1" shows "x = 1"
+proof -
+ let "?l y" = "ln y - y + 1"
+ have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
+ by (auto intro!: DERIV_intros)
+
+ show ?thesis
+ proof (cases rule: linorder_cases)
+ assume "x < 1"
+ from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
+ from `x < a` have "?l x < ?l a"
+ proof (rule DERIV_pos_imp_increasing, safe)
+ fix y assume "x \<le> y" "y \<le> a"
+ with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
+ by (auto simp: field_simps)
+ with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
+ by auto
+ qed
+ also have "\<dots> \<le> 0"
+ using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
+ finally show "x = 1" using assms by auto
+ next
+ assume "1 < x"
+ from dense[OF `1 < x`] obtain a where "1 < a" "a < x" by blast
+ from `a < x` have "?l x < ?l a"
+ proof (rule DERIV_neg_imp_decreasing, safe)
+ fix y assume "a \<le> y" "y \<le> x"
+ with `1 < a` have "1 / y - 1 < 0" "0 < y"
+ by (auto simp: field_simps)
+ with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
+ by blast
+ qed
+ also have "\<dots> \<le> 0"
+ using ln_le_minus_one `1 < a` by (auto simp: field_simps)
+ finally show "x = 1" using assms by auto
+ qed simp
+qed
+
lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
unfolding tendsto_Zfun_iff
proof (rule ZfunI, simp add: eventually_at_bot_dense)
@@ -1383,6 +1606,415 @@
qed (rule exp_at_top)
qed
+
+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 tendsto_log [tendsto_intros]:
+ "\<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"
+ unfolding log_def by (intro tendsto_intros) auto
+
+lemma continuous_log:
+ 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))"
+ shows "continuous F (\<lambda>x. log (f x) (g x))"
+ using assms unfolding continuous_def by (rule tendsto_log)
+
+lemma continuous_at_within_log[continuous_intros]:
+ 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"
+ shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
+ using assms unfolding continuous_within by (rule tendsto_log)
+
+lemma isCont_log[continuous_intros, simp]:
+ assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
+ shows "isCont (\<lambda>x. log (f x) (g x)) a"
+ using assms unfolding continuous_at by (rule tendsto_log)
+
+lemma continuous_on_log[continuous_on_intros]:
+ 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"
+ shows "continuous_on s (\<lambda>x. log (f x) (g x))"
+ using assms unfolding continuous_on_def by (fast intro: tendsto_log)
+
+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 (rule real_root_pos_unique) (auto simp: 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)
+
+lemma continuous_powr:
+ assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))"
+ shows "continuous F (\<lambda>x. (f x) powr (g x))"
+ using assms unfolding continuous_def by (rule tendsto_powr)
+
+lemma continuous_at_within_powr[continuous_intros]:
+ assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a"
+ shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
+ using assms unfolding continuous_within by (rule tendsto_powr)
+
+lemma isCont_powr[continuous_intros, simp]:
+ assumes "isCont f a" "isCont g a" "0 < f a"
+ shows "isCont (\<lambda>x. (f x) powr g x) a"
+ using assms unfolding continuous_at by (rule tendsto_powr)
+
+lemma continuous_on_powr[continuous_on_intros]:
+ assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x"
+ shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
+ using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
+
+(* 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
+
subsection {* Sine and Cosine *}
definition sin_coeff :: "nat \<Rightarrow> real" where
@@ -1444,6 +2076,8 @@
summable_minus summable_sin summable_cos)
done
+declare DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
+
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
unfolding cos_def sin_def
apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
@@ -1451,6 +2085,8 @@
summable_minus summable_sin summable_cos suminf_minus)
done
+declare DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
+
lemma isCont_sin: "isCont sin x"
by (rule DERIV_sin [THEN DERIV_isCont])
@@ -1487,12 +2123,6 @@
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
unfolding continuous_on_def by (auto intro: tendsto_cos)
-declare
- DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
- DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
- DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
- DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
-
subsection {* Properties of Sine and Cosine *}
lemma sin_zero [simp]: "sin 0 = 0"