diff -r 7c717ba55a0b -r fb9ae0727548 src/HOL/Transcendental.thy --- a/src/HOL/Transcendental.thy Wed Apr 02 18:35:02 2014 +0200 +++ b/src/HOL/Transcendental.thy Wed Apr 02 18:35:07 2014 +0200 @@ -994,7 +994,7 @@ "continuous F f \ continuous F (\x. exp (f x))" unfolding continuous_def by (rule tendsto_exp) -lemma continuous_on_exp [continuous_on_intros]: +lemma continuous_on_exp [continuous_intros]: "continuous_on s f \ continuous_on s (\x. exp (f x))" unfolding continuous_on_def by (auto intro: tendsto_exp) @@ -1303,7 +1303,7 @@ "continuous (at x within s) f \ 0 < f x \ continuous (at x within s) (\x. ln (f x))" unfolding continuous_within by (rule tendsto_ln) -lemma continuous_on_ln [continuous_on_intros]: +lemma continuous_on_ln [continuous_intros]: "continuous_on s f \ (\x\s. 0 < f x) \ continuous_on s (\x. ln (f x))" unfolding continuous_on_def by (auto intro: tendsto_ln) @@ -1747,7 +1747,7 @@ shows "isCont (\x. log (f x) (g x)) a" using assms unfolding continuous_at by (rule tendsto_log) -lemma continuous_on_log[continuous_on_intros]: +lemma continuous_on_log[continuous_intros]: assumes "continuous_on s f" "continuous_on s g" and "\x\s. 0 < f x" "\x\s. f x \ 1" "\x\s. 0 < g x" shows "continuous_on s (\x. log (f x) (g x))" @@ -2076,7 +2076,7 @@ shows "isCont (\x. (f x) powr g x) a" using assms unfolding continuous_at by (rule tendsto_powr) -lemma continuous_on_powr[continuous_on_intros]: +lemma continuous_on_powr[continuous_intros]: assumes "continuous_on s f" "continuous_on s g" and "\x\s. 0 < f x" shows "continuous_on s (\x. (f x) powr (g x))" using assms unfolding continuous_on_def by (fast intro: tendsto_powr) @@ -2215,7 +2215,7 @@ "continuous F f \ continuous F (\x. sin (f x))" unfolding continuous_def by (rule tendsto_sin) -lemma continuous_on_sin [continuous_on_intros]: +lemma continuous_on_sin [continuous_intros]: "continuous_on s f \ continuous_on s (\x. sin (f x))" unfolding continuous_on_def by (auto intro: tendsto_sin) @@ -2223,7 +2223,7 @@ "continuous F f \ continuous F (\x. cos (f x))" unfolding continuous_def by (rule tendsto_cos) -lemma continuous_on_cos [continuous_on_intros]: +lemma continuous_on_cos [continuous_intros]: "continuous_on s f \ continuous_on s (\x. cos (f x))" unfolding continuous_on_def by (auto intro: tendsto_cos) @@ -2884,7 +2884,7 @@ "continuous (at x within s) f \ cos (f x) \ 0 \ continuous (at x within s) (\x. tan (f x))" unfolding continuous_within by (rule tendsto_tan) -lemma continuous_on_tan [continuous_on_intros]: +lemma continuous_on_tan [continuous_intros]: "continuous_on s f \ (\x\s. cos (f x) \ 0) \ continuous_on s (\x. tan (f x))" unfolding continuous_on_def by (auto intro: tendsto_tan) @@ -3232,7 +3232,7 @@ lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin" proof - have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin" - by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arcsin_sin) + by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin) also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}" proof safe fix x :: real @@ -3244,7 +3244,7 @@ finally show ?thesis . qed -lemma continuous_on_arcsin [continuous_on_intros]: +lemma continuous_on_arcsin [continuous_intros]: "continuous_on s f \ (\x\s. -1 \ f x \ f x \ 1) \ continuous_on s (\x. arcsin (f x))" using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arcsin']] by (auto simp: comp_def subset_eq) @@ -3256,7 +3256,7 @@ lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos" proof - have "continuous_on (cos ` {0 .. pi}) arccos" - by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arccos_cos) + by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos) also have "cos ` {0 .. pi} = {-1 .. 1}" proof safe fix x :: real @@ -3268,7 +3268,7 @@ finally show ?thesis . qed -lemma continuous_on_arccos [continuous_on_intros]: +lemma continuous_on_arccos [continuous_intros]: "continuous_on s f \ (\x\s. -1 \ f x \ f x \ 1) \ continuous_on s (\x. arccos (f x))" using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arccos']] by (auto simp: comp_def subset_eq) @@ -3292,7 +3292,7 @@ lemma continuous_arctan [continuous_intros]: "continuous F f \ continuous F (\x. arctan (f x))" unfolding continuous_def by (rule tendsto_arctan) -lemma continuous_on_arctan [continuous_on_intros]: "continuous_on s f \ continuous_on s (\x. arctan (f x))" +lemma continuous_on_arctan [continuous_intros]: "continuous_on s f \ continuous_on s (\x. arctan (f x))" unfolding continuous_on_def by (auto intro: tendsto_arctan) lemma DERIV_arcsin: