haftmann@36349
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(* Title: HOL/Multivariate_Analysis/Derivative.thy
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Author: John Harrison
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Translation from HOL Light: Robert Himmelmann, TU Muenchen
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*)
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header {* Multivariate calculus in Euclidean space. *}
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theory Derivative
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imports Brouwer_Fixpoint Operator_Norm
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begin
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(* Because I do not want to type this all the time *)
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lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
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subsection {* Derivatives *}
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text {* The definition is slightly tricky since we make it work over
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nets of a particular form. This lets us prove theorems generally and use
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"at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}
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definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a filter \<Rightarrow> bool)"
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(infixl "(has'_derivative)" 12) where
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"(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
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(f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
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lemma derivative_linear[dest]:
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"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
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unfolding has_derivative_def by auto
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lemma netlimit_at_vector:
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(* TODO: move *)
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fixes a :: "'a::real_normed_vector"
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shows "netlimit (at a) = a"
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proof (cases "\<exists>x. x \<noteq> a")
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case True then obtain x where x: "x \<noteq> a" ..
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have "\<not> trivial_limit (at a)"
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unfolding trivial_limit_def eventually_at dist_norm
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apply clarsimp
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apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
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apply (simp add: norm_sgn sgn_zero_iff x)
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done
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thus ?thesis
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by (rule netlimit_within [of a UNIV, unfolded within_UNIV])
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qed simp
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lemma FDERIV_conv_has_derivative:
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shows "FDERIV f x :> f' = (f has_derivative f') (at x)"
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unfolding fderiv_def has_derivative_def netlimit_at_vector
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by (simp add: diff_diff_eq Lim_at_zero [where a=x]
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tendsto_norm_zero_iff [where 'b='b, symmetric])
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lemma DERIV_conv_has_derivative:
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"(DERIV f x :> f') = (f has_derivative op * f') (at x)"
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unfolding deriv_fderiv FDERIV_conv_has_derivative
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by (subst mult_commute, rule refl)
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text {* These are the only cases we'll care about, probably. *}
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lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
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bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
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unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)
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lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
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bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
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using has_derivative_within [of f f' x UNIV] by simp
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text {* More explicit epsilon-delta forms. *}
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lemma has_derivative_within':
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"(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
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(\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
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\<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
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unfolding has_derivative_within Lim_within dist_norm
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unfolding diff_0_right by (simp add: diff_diff_eq)
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lemma has_derivative_at':
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"(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
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(\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
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\<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
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using has_derivative_within' [of f f' x UNIV] by simp
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lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
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unfolding has_derivative_within' has_derivative_at' by meson
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lemma has_derivative_within_open:
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"a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
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by (simp only: at_within_interior interior_open)
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lemma has_derivative_right:
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fixes f :: "real \<Rightarrow> real" and y :: "real"
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shows "(f has_derivative (op * y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
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((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x <..} \<inter> I))"
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proof -
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have "((\<lambda>t. (f t - (f x + y * (t - x))) / \<bar>t - x\<bar>) ---> 0) (at x within ({x<..} \<inter> I)) \<longleftrightarrow>
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((\<lambda>t. (f t - f x) / (t - x) - y) ---> 0) (at x within ({x<..} \<inter> I))"
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by (intro Lim_cong_within) (auto simp add: diff_divide_distrib add_divide_distrib)
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also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f t - f x) / (t - x)) ---> y) (at x within ({x<..} \<inter> I))"
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by (simp add: Lim_null[symmetric])
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also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x<..} \<inter> I))"
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by (intro Lim_cong_within) (simp_all add: field_simps)
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finally show ?thesis
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by (simp add: bounded_linear_mult_right has_derivative_within)
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qed
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lemma bounded_linear_imp_linear: "bounded_linear f \<Longrightarrow> linear f" (* TODO: move elsewhere *)
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proof -
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assume "bounded_linear f"
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then interpret f: bounded_linear f .
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show "linear f"
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by (simp add: f.add f.scaleR linear_def)
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qed
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lemma derivative_is_linear:
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"(f has_derivative f') net \<Longrightarrow> linear f'"
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by (rule derivative_linear [THEN bounded_linear_imp_linear])
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subsubsection {* Combining theorems. *}
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lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
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unfolding has_derivative_def
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by (simp add: bounded_linear_ident tendsto_const)
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lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
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unfolding has_derivative_def
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by (simp add: bounded_linear_zero tendsto_const)
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lemma (in bounded_linear) has_derivative': "(f has_derivative f) net"
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unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
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unfolding diff by (simp add: tendsto_const)
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lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
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lemma (in bounded_linear) has_derivative:
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assumes "((\<lambda>x. g x) has_derivative (\<lambda>h. g' h)) net"
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shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>h. f (g' h))) net"
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using assms unfolding has_derivative_def
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apply safe
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apply (erule bounded_linear_compose [OF local.bounded_linear])
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apply (drule local.tendsto)
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apply (simp add: local.scaleR local.diff local.add local.zero)
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done
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lemmas scaleR_right_has_derivative =
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bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
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lemmas scaleR_left_has_derivative =
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bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
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lemmas inner_right_has_derivative =
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bounded_linear.has_derivative [OF bounded_linear_inner_right]
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lemmas inner_left_has_derivative =
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bounded_linear.has_derivative [OF bounded_linear_inner_left]
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lemmas mult_right_has_derivative =
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bounded_linear.has_derivative [OF bounded_linear_mult_right]
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lemmas mult_left_has_derivative =
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bounded_linear.has_derivative [OF bounded_linear_mult_left]
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lemmas euclidean_component_has_derivative =
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bounded_linear.has_derivative [OF bounded_linear_euclidean_component]
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lemma has_derivative_neg:
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assumes "(f has_derivative f') net"
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shows "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
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using scaleR_right_has_derivative [where r="-1", OF assms] by auto
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lemma has_derivative_add:
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assumes "(f has_derivative f') net" and "(g has_derivative g') net"
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shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net"
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proof-
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note as = assms[unfolded has_derivative_def]
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show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
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using tendsto_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
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by (auto simp add: algebra_simps)
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qed
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lemma has_derivative_add_const:
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"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
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by (drule has_derivative_add, rule has_derivative_const, auto)
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lemma has_derivative_sub:
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assumes "(f has_derivative f') net" and "(g has_derivative g') net"
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shows "((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
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huffman@44981
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unfolding diff_minus by (intro has_derivative_add has_derivative_neg assms)
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lemma has_derivative_setsum:
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assumes "finite s" and "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
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shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
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huffman@44981
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using assms by (induct, simp_all add: has_derivative_const has_derivative_add)
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text {* Somewhat different results for derivative of scalar multiplier. *}
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(** move **)
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lemma linear_vmul_component: (* TODO: delete *)
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assumes lf: "linear f"
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shows "linear (\<lambda>x. f x $$ k *\<^sub>R v)"
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using lf
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by (auto simp add: linear_def algebra_simps)
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huffman@44981
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lemmas has_derivative_intros =
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huffman@45011
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has_derivative_id has_derivative_const
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huffman@45011
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has_derivative_add has_derivative_sub has_derivative_neg
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huffman@45011
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has_derivative_add_const
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huffman@45145
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scaleR_left_has_derivative scaleR_right_has_derivative
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huffman@45145
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inner_left_has_derivative inner_right_has_derivative
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euclidean_component_has_derivative
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subsubsection {* Limit transformation for derivatives *}
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lemma has_derivative_transform_within:
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assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x within s)"
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hoelzl@33741
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shows "(g has_derivative f') (at x within s)"
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hoelzl@33741
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using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
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apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
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apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
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lemma has_derivative_transform_at:
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hoelzl@33741
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assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
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hoelzl@33741
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shows "(g has_derivative f') (at x)"
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huffman@45896
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using has_derivative_transform_within [of d x UNIV f g f'] assms by simp
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hoelzl@33741
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hoelzl@33741
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lemma has_derivative_transform_within_open:
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hoelzl@33741
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assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
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hoelzl@33741
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shows "(g has_derivative f') (at x)"
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hoelzl@33741
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using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
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hoelzl@33741
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apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
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hoelzl@33741
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apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
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subsection {* Differentiability *}
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huffman@36358
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no_notation Deriv.differentiable (infixl "differentiable" 60)
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huffman@36358
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huffman@44952
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definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" (infixr "differentiable" 30) where
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hoelzl@33741
|
235 |
"f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
|
hoelzl@33741
|
236 |
|
hoelzl@33741
|
237 |
definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
|
hoelzl@33741
|
238 |
"f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
|
hoelzl@33741
|
239 |
|
hoelzl@33741
|
240 |
lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
|
hoelzl@33741
|
241 |
unfolding differentiable_def by auto
|
hoelzl@33741
|
242 |
|
hoelzl@33741
|
243 |
lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
|
hoelzl@33741
|
244 |
unfolding differentiable_def using has_derivative_at_within by blast
|
hoelzl@33741
|
245 |
|
huffman@44981
|
246 |
lemma differentiable_within_open: (* TODO: delete *)
|
huffman@44981
|
247 |
assumes "a \<in> s" and "open s"
|
huffman@44981
|
248 |
shows "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
|
huffman@37729
|
249 |
using assms by (simp only: at_within_interior interior_open)
|
hoelzl@33741
|
250 |
|
huffman@44981
|
251 |
lemma differentiable_on_eq_differentiable_at:
|
huffman@44981
|
252 |
"open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
|
huffman@44981
|
253 |
unfolding differentiable_on_def
|
huffman@44981
|
254 |
by (auto simp add: at_within_interior interior_open)
|
hoelzl@33741
|
255 |
|
hoelzl@33741
|
256 |
lemma differentiable_transform_within:
|
huffman@44981
|
257 |
assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
|
huffman@44981
|
258 |
assumes "f differentiable (at x within s)"
|
hoelzl@33741
|
259 |
shows "g differentiable (at x within s)"
|
huffman@44981
|
260 |
using assms(4) unfolding differentiable_def
|
huffman@44981
|
261 |
by (auto intro!: has_derivative_transform_within[OF assms(1-3)])
|
hoelzl@33741
|
262 |
|
hoelzl@33741
|
263 |
lemma differentiable_transform_at:
|
hoelzl@33741
|
264 |
assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
|
hoelzl@33741
|
265 |
shows "g differentiable at x"
|
huffman@44981
|
266 |
using assms(3) unfolding differentiable_def
|
huffman@44981
|
267 |
using has_derivative_transform_at[OF assms(1-2)] by auto
|
hoelzl@33741
|
268 |
|
hoelzl@33741
|
269 |
subsection {* Frechet derivative and Jacobian matrix. *}
|
hoelzl@33741
|
270 |
|
hoelzl@33741
|
271 |
definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
|
hoelzl@33741
|
272 |
|
hoelzl@33741
|
273 |
lemma frechet_derivative_works:
|
hoelzl@33741
|
274 |
"f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
|
hoelzl@33741
|
275 |
unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
|
hoelzl@33741
|
276 |
|
huffman@37648
|
277 |
lemma linear_frechet_derivative:
|
hoelzl@33741
|
278 |
shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
|
huffman@44981
|
279 |
unfolding frechet_derivative_works has_derivative_def
|
huffman@44981
|
280 |
by (auto intro: bounded_linear_imp_linear)
|
hoelzl@33741
|
281 |
|
huffman@44982
|
282 |
subsection {* Differentiability implies continuity *}
|
hoelzl@33741
|
283 |
|
huffman@44981
|
284 |
lemma Lim_mul_norm_within:
|
huffman@44981
|
285 |
fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
|
hoelzl@33741
|
286 |
shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
|
huffman@44981
|
287 |
unfolding Lim_within apply(rule,rule)
|
huffman@44981
|
288 |
apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
|
huffman@44981
|
289 |
apply(rule_tac x="min d 1" in exI) apply rule defer
|
huffman@44981
|
290 |
apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
|
hoelzl@33741
|
291 |
by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
|
hoelzl@33741
|
292 |
|
huffman@44981
|
293 |
lemma differentiable_imp_continuous_within:
|
huffman@44981
|
294 |
assumes "f differentiable (at x within s)"
|
huffman@44981
|
295 |
shows "continuous (at x within s) f"
|
huffman@44981
|
296 |
proof-
|
huffman@44981
|
297 |
from assms guess f' unfolding differentiable_def has_derivative_within ..
|
huffman@44981
|
298 |
note f'=this
|
hoelzl@33741
|
299 |
then interpret bounded_linear f' by auto
|
hoelzl@33741
|
300 |
have *:"\<And>xa. x\<noteq>xa \<Longrightarrow> (f' \<circ> (\<lambda>y. y - x)) xa + norm (xa - x) *\<^sub>R ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) - ((f' \<circ> (\<lambda>y. y - x)) x + 0) = f xa - f x"
|
hoelzl@33741
|
301 |
using zero by auto
|
hoelzl@33741
|
302 |
have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
|
hoelzl@33741
|
303 |
apply(rule continuous_within_compose) apply(rule continuous_intros)+
|
hoelzl@33741
|
304 |
by(rule linear_continuous_within[OF f'[THEN conjunct1]])
|
huffman@44981
|
305 |
show ?thesis unfolding continuous_within
|
huffman@44981
|
306 |
using f'[THEN conjunct2, THEN Lim_mul_norm_within]
|
huffman@44983
|
307 |
apply- apply(drule tendsto_add)
|
huffman@44981
|
308 |
apply(rule **[unfolded continuous_within])
|
huffman@44981
|
309 |
unfolding Lim_within and dist_norm
|
huffman@44981
|
310 |
apply (rule, rule)
|
huffman@44981
|
311 |
apply (erule_tac x=e in allE)
|
huffman@44981
|
312 |
apply (erule impE | assumption)+
|
huffman@44981
|
313 |
apply (erule exE, rule_tac x=d in exI)
|
huffman@45011
|
314 |
by (auto simp add: zero *)
|
huffman@44981
|
315 |
qed
|
hoelzl@33741
|
316 |
|
huffman@44981
|
317 |
lemma differentiable_imp_continuous_at:
|
huffman@44981
|
318 |
"f differentiable at x \<Longrightarrow> continuous (at x) f"
|
hoelzl@33741
|
319 |
by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
|
hoelzl@33741
|
320 |
|
huffman@44981
|
321 |
lemma differentiable_imp_continuous_on:
|
huffman@44981
|
322 |
"f differentiable_on s \<Longrightarrow> continuous_on s f"
|
hoelzl@33741
|
323 |
unfolding differentiable_on_def continuous_on_eq_continuous_within
|
hoelzl@33741
|
324 |
using differentiable_imp_continuous_within by blast
|
hoelzl@33741
|
325 |
|
hoelzl@33741
|
326 |
lemma has_derivative_within_subset:
|
hoelzl@33741
|
327 |
"(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
|
hoelzl@33741
|
328 |
unfolding has_derivative_within using Lim_within_subset by blast
|
hoelzl@33741
|
329 |
|
hoelzl@33741
|
330 |
lemma differentiable_within_subset:
|
hoelzl@33741
|
331 |
"f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
|
hoelzl@33741
|
332 |
unfolding differentiable_def using has_derivative_within_subset by blast
|
hoelzl@33741
|
333 |
|
huffman@44981
|
334 |
lemma differentiable_on_subset:
|
huffman@44981
|
335 |
"f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
|
hoelzl@33741
|
336 |
unfolding differentiable_on_def using differentiable_within_subset by blast
|
hoelzl@33741
|
337 |
|
hoelzl@33741
|
338 |
lemma differentiable_on_empty: "f differentiable_on {}"
|
hoelzl@33741
|
339 |
unfolding differentiable_on_def by auto
|
hoelzl@33741
|
340 |
|
huffman@44982
|
341 |
text {* Several results are easier using a "multiplied-out" variant.
|
huffman@44982
|
342 |
(I got this idea from Dieudonne's proof of the chain rule). *}
|
hoelzl@33741
|
343 |
|
hoelzl@33741
|
344 |
lemma has_derivative_within_alt:
|
hoelzl@33741
|
345 |
"(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
|
hoelzl@33741
|
346 |
(\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm(f(y) - f(x) - f'(y - x)) \<le> e * norm(y - x))" (is "?lhs \<longleftrightarrow> ?rhs")
|
huffman@44981
|
347 |
proof
|
huffman@44981
|
348 |
assume ?lhs thus ?rhs
|
huffman@44981
|
349 |
unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
|
huffman@44981
|
350 |
unfolding Lim_within
|
huffman@44981
|
351 |
apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
|
huffman@44981
|
352 |
apply(erule exE,rule_tac x=d in exI)
|
huffman@44981
|
353 |
apply(erule conjE,rule,assumption,rule,rule)
|
huffman@44981
|
354 |
proof-
|
hoelzl@33741
|
355 |
fix x y e d assume as:"0 < e" "0 < d" "norm (y - x) < d" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
|
hoelzl@33741
|
356 |
dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
|
hoelzl@33741
|
357 |
then interpret bounded_linear f' by auto
|
hoelzl@33741
|
358 |
show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
|
huffman@44981
|
359 |
case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero)
|
huffman@44981
|
360 |
next
|
hoelzl@33741
|
361 |
case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
|
wenzelm@42829
|
362 |
unfolding dist_norm diff_0_right using as(3)
|
wenzelm@42829
|
363 |
using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded dist_norm]
|
wenzelm@42829
|
364 |
by (auto simp add: linear_0 linear_sub)
|
huffman@44981
|
365 |
thus ?thesis by(auto simp add:algebra_simps)
|
huffman@44981
|
366 |
qed
|
huffman@44981
|
367 |
qed
|
huffman@44981
|
368 |
next
|
huffman@44981
|
369 |
assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within
|
huffman@44981
|
370 |
apply-apply(erule conjE,rule,assumption)
|
huffman@44981
|
371 |
apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer
|
huffman@44981
|
372 |
apply(erule exE,rule_tac x=d in exI)
|
huffman@44981
|
373 |
apply(erule conjE,rule,assumption,rule,rule)
|
huffman@44981
|
374 |
unfolding dist_norm diff_0_right norm_scaleR
|
huffman@44981
|
375 |
apply(erule_tac x=xa in ballE,erule impE)
|
huffman@44981
|
376 |
proof-
|
hoelzl@33741
|
377 |
fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
|
hoelzl@33741
|
378 |
"norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
|
hoelzl@33741
|
379 |
thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
|
huffman@44981
|
380 |
apply(rule_tac le_less_trans[of _ "e/2"])
|
huffman@44981
|
381 |
by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps)
|
huffman@44981
|
382 |
qed auto
|
huffman@44981
|
383 |
qed
|
hoelzl@33741
|
384 |
|
hoelzl@33741
|
385 |
lemma has_derivative_at_alt:
|
himmelma@35172
|
386 |
"(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
|
hoelzl@33741
|
387 |
(\<forall>e>0. \<exists>d>0. \<forall>y. norm(y - x) < d \<longrightarrow> norm(f y - f x - f'(y - x)) \<le> e * norm(y - x))"
|
huffman@45896
|
388 |
using has_derivative_within_alt[where s=UNIV] by simp
|
hoelzl@33741
|
389 |
|
hoelzl@33741
|
390 |
subsection {* The chain rule. *}
|
hoelzl@33741
|
391 |
|
hoelzl@33741
|
392 |
lemma diff_chain_within:
|
huffman@44981
|
393 |
assumes "(f has_derivative f') (at x within s)"
|
huffman@44981
|
394 |
assumes "(g has_derivative g') (at (f x) within (f ` s))"
|
hoelzl@33741
|
395 |
shows "((g o f) has_derivative (g' o f'))(at x within s)"
|
huffman@44981
|
396 |
unfolding has_derivative_within_alt
|
huffman@44981
|
397 |
apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
|
hoelzl@33741
|
398 |
apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
|
huffman@44981
|
399 |
apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption)
|
huffman@44981
|
400 |
proof(rule,rule)
|
hoelzl@33741
|
401 |
note assms = assms[unfolded has_derivative_within_alt]
|
hoelzl@33741
|
402 |
fix e::real assume "0<e"
|
hoelzl@33741
|
403 |
guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
|
hoelzl@33741
|
404 |
guess B2 using bounded_linear.pos_bounded[OF assms(2)[THEN conjunct1, rule_format]] .. note B2 = this
|
hoelzl@33741
|
405 |
have *:"e / 2 / B2 > 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto
|
hoelzl@33741
|
406 |
guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this
|
hoelzl@33741
|
407 |
have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto
|
hoelzl@33741
|
408 |
guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this
|
hoelzl@33741
|
409 |
guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this
|
hoelzl@33741
|
410 |
|
hoelzl@33741
|
411 |
def d0 \<equiv> "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto
|
hoelzl@33741
|
412 |
def d \<equiv> "(min d0 (de / (B1 + 1))) / 2" have "de * 2 / (B1 + 1) > de / (B1 + 1)" apply(rule divide_strict_right_mono) using B1 de by auto
|
hoelzl@33741
|
413 |
hence d:"d>0" "d < d1" "d < d2" "d < (de / (B1 + 1))" unfolding d_def using d0 d1 d2 de B1 by(auto intro!: divide_pos_pos simp add:min_less_iff_disj not_less)
|
hoelzl@33741
|
414 |
|
hoelzl@33741
|
415 |
show "\<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" apply(rule_tac x=d in exI)
|
hoelzl@33741
|
416 |
proof(rule,rule `d>0`,rule,rule)
|
hoelzl@33741
|
417 |
fix y assume as:"y \<in> s" "norm (y - x) < d"
|
hoelzl@33741
|
418 |
hence 1:"norm (f y - f x - f' (y - x)) \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x))" using d1 d2 d by auto
|
hoelzl@33741
|
419 |
|
hoelzl@33741
|
420 |
have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
|
huffman@44981
|
421 |
using norm_triangle_sub[of "f y - f x" "f' (y - x)"]
|
huffman@44981
|
422 |
by(auto simp add:algebra_simps)
|
huffman@44981
|
423 |
also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)"
|
huffman@44981
|
424 |
apply(rule add_left_mono) using B1 by(auto simp add:algebra_simps)
|
huffman@44981
|
425 |
also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)"
|
huffman@44981
|
426 |
apply(rule add_right_mono) using d1 d2 d as by auto
|
hoelzl@33741
|
427 |
also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
|
hoelzl@33741
|
428 |
also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
|
hoelzl@33741
|
429 |
finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto
|
hoelzl@33741
|
430 |
|
huffman@44981
|
431 |
hence "norm (f y - f x) \<le> d * (1 + B1)" apply-
|
huffman@44981
|
432 |
apply(rule order_trans,assumption,rule mult_right_mono)
|
huffman@44981
|
433 |
using as B1 by auto
|
hoelzl@33741
|
434 |
also have "\<dots> < de" using d B1 by(auto simp add:field_simps)
|
hoelzl@33741
|
435 |
finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
|
huffman@44981
|
436 |
apply-apply(rule de[THEN conjunct2,rule_format])
|
huffman@44981
|
437 |
using `y\<in>s` using d as by auto
|
hoelzl@33741
|
438 |
also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto
|
huffman@44981
|
439 |
also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono)
|
huffman@44981
|
440 |
using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
|
hoelzl@33741
|
441 |
finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
|
hoelzl@33741
|
442 |
|
hoelzl@33741
|
443 |
interpret g': bounded_linear g' using assms(2) by auto
|
hoelzl@33741
|
444 |
interpret f': bounded_linear f' using assms(1) by auto
|
hoelzl@33741
|
445 |
have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
|
haftmann@36349
|
446 |
by(auto simp add:algebra_simps f'.diff g'.diff g'.add)
|
huffman@44981
|
447 |
also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2
|
huffman@44981
|
448 |
by (auto simp add: algebra_simps)
|
huffman@44981
|
449 |
also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))"
|
huffman@44981
|
450 |
apply (rule mult_left_mono) using as d1 d2 d B2 by auto
|
hoelzl@33741
|
451 |
also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
|
hoelzl@33741
|
452 |
finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
|
hoelzl@33741
|
453 |
|
huffman@44981
|
454 |
have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \<le> e * norm (y - x)"
|
huffman@44981
|
455 |
using 5 4 by auto
|
huffman@44981
|
456 |
thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)"
|
huffman@44981
|
457 |
unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub)
|
huffman@44981
|
458 |
by assumption
|
huffman@44981
|
459 |
qed
|
huffman@44981
|
460 |
qed
|
hoelzl@33741
|
461 |
|
hoelzl@33741
|
462 |
lemma diff_chain_at:
|
hoelzl@33741
|
463 |
"(f has_derivative f') (at x) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow> ((g o f) has_derivative (g' o f')) (at x)"
|
huffman@44981
|
464 |
using diff_chain_within[of f f' x UNIV g g']
|
huffman@44981
|
465 |
using has_derivative_within_subset[of g g' "f x" UNIV "range f"]
|
huffman@45896
|
466 |
by simp
|
hoelzl@33741
|
467 |
|
hoelzl@33741
|
468 |
subsection {* Composition rules stated just for differentiability. *}
|
hoelzl@33741
|
469 |
|
huffman@44981
|
470 |
lemma differentiable_const [intro]:
|
huffman@44981
|
471 |
"(\<lambda>z. c) differentiable (net::'a::real_normed_vector filter)"
|
hoelzl@33741
|
472 |
unfolding differentiable_def using has_derivative_const by auto
|
hoelzl@33741
|
473 |
|
huffman@44981
|
474 |
lemma differentiable_id [intro]:
|
huffman@44981
|
475 |
"(\<lambda>z. z) differentiable (net::'a::real_normed_vector filter)"
|
hoelzl@33741
|
476 |
unfolding differentiable_def using has_derivative_id by auto
|
hoelzl@33741
|
477 |
|
huffman@44981
|
478 |
lemma differentiable_cmul [intro]:
|
huffman@44981
|
479 |
"f differentiable net \<Longrightarrow>
|
huffman@44981
|
480 |
(\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)"
|
huffman@44981
|
481 |
unfolding differentiable_def
|
huffman@45145
|
482 |
apply(erule exE, drule scaleR_right_has_derivative) by auto
|
hoelzl@33741
|
483 |
|
huffman@44981
|
484 |
lemma differentiable_neg [intro]:
|
huffman@44981
|
485 |
"f differentiable net \<Longrightarrow>
|
huffman@44981
|
486 |
(\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector filter)"
|
huffman@44981
|
487 |
unfolding differentiable_def
|
huffman@44981
|
488 |
apply(erule exE, drule has_derivative_neg) by auto
|
hoelzl@33741
|
489 |
|
hoelzl@33741
|
490 |
lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
|
huffman@44952
|
491 |
\<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector filter)"
|
hoelzl@33741
|
492 |
unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
|
hoelzl@33741
|
493 |
apply(rule has_derivative_add) by auto
|
hoelzl@33741
|
494 |
|
hoelzl@33741
|
495 |
lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
|
huffman@44952
|
496 |
\<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector filter)"
|
huffman@44981
|
497 |
unfolding differentiable_def apply(erule exE)+
|
huffman@44981
|
498 |
apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
|
huffman@44981
|
499 |
apply(rule has_derivative_sub) by auto
|
hoelzl@33741
|
500 |
|
huffman@37648
|
501 |
lemma differentiable_setsum:
|
hoelzl@33741
|
502 |
assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
|
huffman@44981
|
503 |
shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net"
|
huffman@44981
|
504 |
proof-
|
hoelzl@33741
|
505 |
guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
|
huffman@44981
|
506 |
thus ?thesis unfolding differentiable_def apply-
|
huffman@44981
|
507 |
apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto
|
huffman@44981
|
508 |
qed
|
hoelzl@33741
|
509 |
|
huffman@37648
|
510 |
lemma differentiable_setsum_numseg:
|
hoelzl@33741
|
511 |
shows "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> (f i) differentiable net \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) {m::nat..n}) differentiable net"
|
hoelzl@33741
|
512 |
apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto
|
hoelzl@33741
|
513 |
|
hoelzl@33741
|
514 |
lemma differentiable_chain_at:
|
hoelzl@33741
|
515 |
"f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o f) differentiable (at x)"
|
hoelzl@33741
|
516 |
unfolding differentiable_def by(meson diff_chain_at)
|
hoelzl@33741
|
517 |
|
hoelzl@33741
|
518 |
lemma differentiable_chain_within:
|
hoelzl@33741
|
519 |
"f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s))
|
hoelzl@33741
|
520 |
\<Longrightarrow> (g o f) differentiable (at x within s)"
|
hoelzl@33741
|
521 |
unfolding differentiable_def by(meson diff_chain_within)
|
hoelzl@33741
|
522 |
|
huffman@37729
|
523 |
subsection {* Uniqueness of derivative *}
|
huffman@37729
|
524 |
|
huffman@37729
|
525 |
text {*
|
huffman@37729
|
526 |
The general result is a bit messy because we need approachability of the
|
huffman@37729
|
527 |
limit point from any direction. But OK for nontrivial intervals etc.
|
huffman@37729
|
528 |
*}
|
hoelzl@33741
|
529 |
|
huffman@44981
|
530 |
lemma frechet_derivative_unique_within:
|
huffman@44981
|
531 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
|
huffman@44981
|
532 |
assumes "(f has_derivative f') (at x within s)"
|
huffman@44981
|
533 |
assumes "(f has_derivative f'') (at x within s)"
|
huffman@44981
|
534 |
assumes "(\<forall>i<DIM('a). \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis i) \<in> s)"
|
huffman@44981
|
535 |
shows "f' = f''"
|
huffman@44981
|
536 |
proof-
|
hoelzl@33741
|
537 |
note as = assms(1,2)[unfolded has_derivative_def]
|
huffman@44981
|
538 |
then interpret f': bounded_linear f' by auto
|
huffman@44981
|
539 |
from as interpret f'': bounded_linear f'' by auto
|
huffman@44981
|
540 |
have "x islimpt s" unfolding islimpt_approachable
|
huffman@44981
|
541 |
proof(rule,rule)
|
huffman@44981
|
542 |
fix e::real assume "0<e" guess d
|
huffman@44981
|
543 |
using assms(3)[rule_format,OF DIM_positive `e>0`] ..
|
huffman@44981
|
544 |
thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
|
huffman@44981
|
545 |
apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI)
|
huffman@44981
|
546 |
unfolding dist_norm by auto
|
huffman@44981
|
547 |
qed
|
huffman@44981
|
548 |
hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within)
|
huffman@44981
|
549 |
unfolding trivial_limit_within by simp
|
huffman@44981
|
550 |
show ?thesis apply(rule linear_eq_stdbasis)
|
huffman@44981
|
551 |
unfolding linear_conv_bounded_linear
|
huffman@44981
|
552 |
apply(rule as(1,2)[THEN conjunct1])+
|
huffman@44981
|
553 |
proof(rule,rule,rule ccontr)
|
hoelzl@37489
|
554 |
fix i assume i:"i<DIM('a)" def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
|
huffman@44981
|
555 |
assume "f' (basis i) \<noteq> f'' (basis i)"
|
huffman@44981
|
556 |
hence "e>0" unfolding e_def by auto
|
huffman@44983
|
557 |
guess d using tendsto_diff [OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
|
hoelzl@37489
|
558 |
guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
|
hoelzl@33741
|
559 |
have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))"
|
hoelzl@33741
|
560 |
unfolding scaleR_right_distrib by auto
|
hoelzl@33741
|
561 |
also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"
|
huffman@44981
|
562 |
unfolding f'.scaleR f''.scaleR
|
huffman@44981
|
563 |
unfolding scaleR_right_distrib scaleR_minus_right by auto
|
huffman@44981
|
564 |
also have "\<dots> = e" unfolding e_def using c[THEN conjunct1]
|
huffman@44981
|
565 |
using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"]
|
huffman@44981
|
566 |
by (auto simp add: add.commute ab_diff_minus)
|
huffman@44981
|
567 |
finally show False using c
|
huffman@44981
|
568 |
using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"]
|
huffman@44981
|
569 |
unfolding dist_norm
|
huffman@44981
|
570 |
unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
|
huffman@44981
|
571 |
scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
|
huffman@44981
|
572 |
using i by auto
|
huffman@44981
|
573 |
qed
|
huffman@44981
|
574 |
qed
|
hoelzl@33741
|
575 |
|
huffman@37729
|
576 |
lemma frechet_derivative_unique_at:
|
hoelzl@33741
|
577 |
shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
|
huffman@37729
|
578 |
unfolding FDERIV_conv_has_derivative [symmetric]
|
huffman@37729
|
579 |
by (rule FDERIV_unique)
|
hoelzl@42700
|
580 |
|
huffman@44981
|
581 |
lemma continuous_isCont: "isCont f x = continuous (at x) f"
|
huffman@44981
|
582 |
unfolding isCont_def LIM_def
|
hoelzl@33741
|
583 |
unfolding continuous_at Lim_at unfolding dist_nz by auto
|
hoelzl@33741
|
584 |
|
huffman@44981
|
585 |
lemma frechet_derivative_unique_within_closed_interval:
|
huffman@44981
|
586 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
|
huffman@44981
|
587 |
assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" (is "x\<in>?I")
|
huffman@44981
|
588 |
assumes "(f has_derivative f' ) (at x within {a..b})"
|
huffman@44981
|
589 |
assumes "(f has_derivative f'') (at x within {a..b})"
|
huffman@44981
|
590 |
shows "f' = f''"
|
huffman@44981
|
591 |
apply(rule frechet_derivative_unique_within)
|
huffman@44981
|
592 |
apply(rule assms(3,4))+
|
huffman@44981
|
593 |
proof(rule,rule,rule,rule)
|
hoelzl@37489
|
594 |
fix e::real and i assume "e>0" and i:"i<DIM('a)"
|
huffman@44981
|
595 |
thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}"
|
huffman@44981
|
596 |
proof(cases "x$$i=a$$i")
|
huffman@44981
|
597 |
case True thus ?thesis
|
huffman@44981
|
598 |
apply(rule_tac x="(min (b$$i - a$$i) e) / 2" in exI)
|
hoelzl@33741
|
599 |
using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
|
huffman@45314
|
600 |
unfolding mem_interval euclidean_simps
|
huffman@44981
|
601 |
using i by (auto simp add: field_simps)
|
hoelzl@37489
|
602 |
next note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
|
hoelzl@37489
|
603 |
case False moreover have "a $$ i < x $$ i" using False * by auto
|
huffman@44981
|
604 |
moreover {
|
huffman@44981
|
605 |
have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> a$$i *2 + x$$i - a$$i"
|
huffman@44981
|
606 |
by auto
|
huffman@44981
|
607 |
also have "\<dots> = a$$i + x$$i" by auto
|
huffman@44981
|
608 |
also have "\<dots> \<le> 2 * x$$i" using * by auto
|
huffman@44981
|
609 |
finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> x $$ i * 2" by auto
|
huffman@44981
|
610 |
}
|
hoelzl@37489
|
611 |
moreover have "min (x $$ i - a $$ i) e \<ge> 0" using * and `e>0` by auto
|
hoelzl@37489
|
612 |
hence "x $$ i * 2 \<le> b $$ i * 2 + min (x $$ i - a $$ i) e" using * by auto
|
huffman@44981
|
613 |
ultimately show ?thesis
|
huffman@44981
|
614 |
apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI)
|
hoelzl@33741
|
615 |
using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
|
huffman@45314
|
616 |
unfolding mem_interval euclidean_simps
|
huffman@44981
|
617 |
using i by (auto simp add: field_simps)
|
huffman@44981
|
618 |
qed
|
huffman@44981
|
619 |
qed
|
hoelzl@33741
|
620 |
|
huffman@44981
|
621 |
lemma frechet_derivative_unique_within_open_interval:
|
huffman@44981
|
622 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
|
huffman@44981
|
623 |
assumes "x \<in> {a<..<b}"
|
huffman@44981
|
624 |
assumes "(f has_derivative f' ) (at x within {a<..<b})"
|
huffman@44981
|
625 |
assumes "(f has_derivative f'') (at x within {a<..<b})"
|
huffman@37650
|
626 |
shows "f' = f''"
|
huffman@37650
|
627 |
proof -
|
huffman@37650
|
628 |
from assms(1) have *: "at x within {a<..<b} = at x"
|
huffman@37650
|
629 |
by (simp add: at_within_interior interior_open open_interval)
|
huffman@37650
|
630 |
from assms(2,3) [unfolded *] show "f' = f''"
|
huffman@37650
|
631 |
by (rule frechet_derivative_unique_at)
|
huffman@37650
|
632 |
qed
|
hoelzl@33741
|
633 |
|
huffman@37729
|
634 |
lemma frechet_derivative_at:
|
hoelzl@33741
|
635 |
shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
|
hoelzl@33741
|
636 |
apply(rule frechet_derivative_unique_at[of f],assumption)
|
hoelzl@33741
|
637 |
unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
|
hoelzl@33741
|
638 |
|
huffman@44981
|
639 |
lemma frechet_derivative_within_closed_interval:
|
huffman@44981
|
640 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
|
huffman@44981
|
641 |
assumes "\<forall>i<DIM('a). a$$i < b$$i" and "x \<in> {a..b}"
|
huffman@44981
|
642 |
assumes "(f has_derivative f') (at x within {a.. b})"
|
hoelzl@33741
|
643 |
shows "frechet_derivative f (at x within {a.. b}) = f'"
|
hoelzl@33741
|
644 |
apply(rule frechet_derivative_unique_within_closed_interval[where f=f])
|
hoelzl@33741
|
645 |
apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
|
hoelzl@33741
|
646 |
unfolding differentiable_def using assms(3) by auto
|
hoelzl@33741
|
647 |
|
hoelzl@37489
|
648 |
subsection {* The traditional Rolle theorem in one dimension. *}
|
hoelzl@33741
|
649 |
|
hoelzl@37489
|
650 |
lemma linear_componentwise:
|
hoelzl@37489
|
651 |
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
|
hoelzl@37489
|
652 |
assumes lf: "linear f"
|
hoelzl@37489
|
653 |
shows "(f x) $$ j = (\<Sum>i<DIM('a). (x$$i) * (f (basis i)$$j))" (is "?lhs = ?rhs")
|
hoelzl@37489
|
654 |
proof -
|
hoelzl@37489
|
655 |
have fA: "finite {..<DIM('a)}" by simp
|
hoelzl@37489
|
656 |
have "?rhs = (\<Sum>i<DIM('a). x$$i *\<^sub>R f (basis i))$$j"
|
huffman@45314
|
657 |
by simp
|
hoelzl@37489
|
658 |
then show ?thesis
|
hoelzl@37489
|
659 |
unfolding linear_setsum_mul[OF lf fA, symmetric]
|
hoelzl@37489
|
660 |
unfolding euclidean_representation[symmetric] ..
|
hoelzl@37489
|
661 |
qed
|
hoelzl@37489
|
662 |
|
hoelzl@37489
|
663 |
text {* We do not introduce @{text jacobian}, which is defined on matrices, instead we use
|
hoelzl@37489
|
664 |
the unfolding of it. *}
|
hoelzl@37489
|
665 |
|
hoelzl@37489
|
666 |
lemma jacobian_works:
|
hoelzl@37489
|
667 |
"(f::('a::euclidean_space) \<Rightarrow> ('b::euclidean_space)) differentiable net \<longleftrightarrow>
|
hoelzl@37489
|
668 |
(f has_derivative (\<lambda>h. \<chi>\<chi> i.
|
hoelzl@37489
|
669 |
\<Sum>j<DIM('a). frechet_derivative f net (basis j) $$ i * h $$ j)) net"
|
hoelzl@37489
|
670 |
(is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net")
|
hoelzl@37489
|
671 |
proof
|
hoelzl@37489
|
672 |
assume *: ?differentiable
|
hoelzl@37489
|
673 |
{ fix h i
|
hoelzl@37489
|
674 |
have "?SUM h i = frechet_derivative f net h $$ i" using *
|
hoelzl@37489
|
675 |
by (auto intro!: setsum_cong
|
hoelzl@37489
|
676 |
simp: linear_componentwise[of _ h i] linear_frechet_derivative) }
|
hoelzl@37489
|
677 |
thus "(f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net"
|
hoelzl@37489
|
678 |
using * by (simp add: frechet_derivative_works)
|
hoelzl@37489
|
679 |
qed (auto intro!: differentiableI)
|
hoelzl@37489
|
680 |
|
hoelzl@37489
|
681 |
lemma differential_zero_maxmin_component:
|
hoelzl@37489
|
682 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
|
hoelzl@37489
|
683 |
assumes k: "k < DIM('b)"
|
hoelzl@37489
|
684 |
and ball: "0 < e" "((\<forall>y \<in> ball x e. (f y)$$k \<le> (f x)$$k) \<or> (\<forall>y\<in>ball x e. (f x)$$k \<le> (f y)$$k))"
|
hoelzl@37489
|
685 |
and diff: "f differentiable (at x)"
|
hoelzl@37489
|
686 |
shows "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j) $$ k) = (0::'a)" (is "?D k = 0")
|
hoelzl@37489
|
687 |
proof (rule ccontr)
|
hoelzl@37489
|
688 |
assume "?D k \<noteq> 0"
|
hoelzl@37489
|
689 |
then obtain j where j: "?D k $$ j \<noteq> 0" "j < DIM('a)"
|
hoelzl@37489
|
690 |
unfolding euclidean_lambda_beta euclidean_eq[of _ "0::'a"] by auto
|
hoelzl@37489
|
691 |
hence *: "\<bar>?D k $$ j\<bar> / 2 > 0" by auto
|
hoelzl@37489
|
692 |
note as = diff[unfolded jacobian_works has_derivative_at_alt]
|
hoelzl@37489
|
693 |
guess e' using as[THEN conjunct2, rule_format, OF *] .. note e' = this
|
hoelzl@37489
|
694 |
guess d using real_lbound_gt_zero[OF ball(1) e'[THEN conjunct1]] .. note d = this
|
hoelzl@37489
|
695 |
{ fix c assume "abs c \<le> d"
|
hoelzl@33741
|
696 |
hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
|
hoelzl@37489
|
697 |
let ?v = "(\<chi>\<chi> i. \<Sum>l<DIM('a). ?D i $$ l * (c *\<^sub>R basis j :: 'a) $$ l)"
|
hoelzl@37489
|
698 |
have if_dist: "\<And> P a b c. a * (if P then b else c) = (if P then a * b else a * c)" by auto
|
hoelzl@37489
|
699 |
have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le>
|
hoelzl@37489
|
700 |
norm (f (x + c *\<^sub>R basis j) - f x - ?v)" by (rule component_le_norm)
|
hoelzl@37489
|
701 |
also have "\<dots> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>"
|
nipkow@45761
|
702 |
using e'[THEN conjunct2, rule_format, OF *] and norm_basis[of j] by fastforce
|
hoelzl@37489
|
703 |
finally have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>" by simp
|
hoelzl@37489
|
704 |
hence "\<bar>f (x + c *\<^sub>R basis j) $$ k - f x $$ k - c * ?D k $$ j\<bar> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>"
|
hoelzl@37489
|
705 |
unfolding euclidean_simps euclidean_lambda_beta using j k
|
hoelzl@37489
|
706 |
by (simp add: if_dist setsum_cases field_simps) } note * = this
|
hoelzl@33741
|
707 |
have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
|
huffman@36587
|
708 |
unfolding mem_ball dist_norm using norm_basis[of j] d by auto
|
hoelzl@37489
|
709 |
hence **:"((f (x - d *\<^sub>R basis j))$$k \<le> (f x)$$k \<and> (f (x + d *\<^sub>R basis j))$$k \<le> (f x)$$k) \<or>
|
hoelzl@37489
|
710 |
((f (x - d *\<^sub>R basis j))$$k \<ge> (f x)$$k \<and> (f (x + d *\<^sub>R basis j))$$k \<ge> (f x)$$k)" using ball by auto
|
hoelzl@37489
|
711 |
have ***: "\<And>y y1 y2 d dx::real.
|
hoelzl@37489
|
712 |
(y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
|
hoelzl@37489
|
713 |
show False apply(rule ***[OF **, where dx="d * ?D k $$ j" and d="\<bar>?D k $$ j\<bar> / 2 * \<bar>d\<bar>"])
|
huffman@44981
|
714 |
using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
|
huffman@44981
|
715 |
unfolding mult_minus_left
|
huffman@45145
|
716 |
unfolding abs_mult diff_minus_eq_add scaleR_minus_left
|
huffman@44981
|
717 |
unfolding algebra_simps by (auto intro: mult_pos_pos)
|
haftmann@34906
|
718 |
qed
|
hoelzl@33741
|
719 |
|
huffman@44982
|
720 |
text {* In particular if we have a mapping into @{typ "real"}. *}
|
hoelzl@33741
|
721 |
|
hoelzl@37489
|
722 |
lemma differential_zero_maxmin:
|
huffman@37650
|
723 |
fixes f::"'a\<Colon>euclidean_space \<Rightarrow> real"
|
hoelzl@37489
|
724 |
assumes "x \<in> s" "open s"
|
hoelzl@37489
|
725 |
and deriv: "(f has_derivative f') (at x)"
|
hoelzl@37489
|
726 |
and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
|
hoelzl@37489
|
727 |
shows "f' = (\<lambda>v. 0)"
|
hoelzl@37489
|
728 |
proof -
|
huffman@44981
|
729 |
obtain e where e:"e>0" "ball x e \<subseteq> s"
|
huffman@44981
|
730 |
using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
|
hoelzl@37489
|
731 |
with differential_zero_maxmin_component[where 'b=real, of 0 e x f, simplified]
|
hoelzl@37489
|
732 |
have "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j)) = (0::'a)"
|
hoelzl@37489
|
733 |
unfolding differentiable_def using mono deriv by auto
|
hoelzl@37489
|
734 |
with frechet_derivative_at[OF deriv, symmetric]
|
hoelzl@37489
|
735 |
have "\<forall>i<DIM('a). f' (basis i) = 0"
|
hoelzl@37489
|
736 |
by (simp add: euclidean_eq[of _ "0::'a"])
|
hoelzl@37489
|
737 |
with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 0]
|
nipkow@39535
|
738 |
show ?thesis by (simp add: fun_eq_iff)
|
hoelzl@37489
|
739 |
qed
|
hoelzl@33741
|
740 |
|
hoelzl@33741
|
741 |
lemma rolle: fixes f::"real\<Rightarrow>real"
|
huffman@44981
|
742 |
assumes "a < b" and "f a = f b" and "continuous_on {a..b} f"
|
huffman@44981
|
743 |
assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
|
huffman@44981
|
744 |
shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)"
|
huffman@44981
|
745 |
proof-
|
huffman@44981
|
746 |
have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))"
|
huffman@44981
|
747 |
proof-
|
huffman@44981
|
748 |
have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto
|
huffman@44981
|
749 |
hence *:"{a .. b}\<noteq>{}" by auto
|
hoelzl@37489
|
750 |
guess d using continuous_attains_sup[OF compact_interval * assms(3)] .. note d=this
|
hoelzl@37489
|
751 |
guess c using continuous_attains_inf[OF compact_interval * assms(3)] .. note c=this
|
huffman@44981
|
752 |
show ?thesis
|
huffman@44981
|
753 |
proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
|
huffman@44981
|
754 |
case True thus ?thesis
|
huffman@44981
|
755 |
apply(erule_tac disjE) apply(rule_tac x=d in bexI)
|
huffman@44981
|
756 |
apply(rule_tac[3] x=c in bexI)
|
huffman@44981
|
757 |
using d c by auto
|
huffman@44981
|
758 |
next
|
huffman@44981
|
759 |
def e \<equiv> "(a + b) /2"
|
hoelzl@33741
|
760 |
case False hence "f d = f c" using d c assms(2) by auto
|
huffman@44981
|
761 |
hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d"
|
huffman@44981
|
762 |
using c d apply- apply(erule_tac x=x in ballE)+ by auto
|
huffman@44981
|
763 |
thus ?thesis
|
huffman@44981
|
764 |
apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto
|
huffman@44981
|
765 |
qed
|
huffman@44981
|
766 |
qed
|
hoelzl@33741
|
767 |
then guess x .. note x=this
|
huffman@44981
|
768 |
hence "f' x = (\<lambda>v. 0)"
|
huffman@44981
|
769 |
apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
|
hoelzl@37489
|
770 |
defer apply(rule open_interval)
|
hoelzl@37489
|
771 |
apply(rule assms(4)[unfolded has_derivative_at[THEN sym],THEN bspec[where x=x]],assumption)
|
hoelzl@37489
|
772 |
unfolding o_def apply(erule disjE,rule disjI2) by auto
|
hoelzl@37489
|
773 |
thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
|
huffman@44981
|
774 |
apply(drule_tac x=v in fun_cong) using x(1) by auto
|
huffman@44981
|
775 |
qed
|
hoelzl@33741
|
776 |
|
hoelzl@33741
|
777 |
subsection {* One-dimensional mean value theorem. *}
|
hoelzl@33741
|
778 |
|
hoelzl@33741
|
779 |
lemma mvt: fixes f::"real \<Rightarrow> real"
|
huffman@44981
|
780 |
assumes "a < b" and "continuous_on {a .. b} f"
|
huffman@44981
|
781 |
assumes "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
|
huffman@44981
|
782 |
shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))"
|
huffman@44981
|
783 |
proof-
|
hoelzl@33741
|
784 |
have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
|
huffman@44981
|
785 |
apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"])
|
huffman@44981
|
786 |
defer
|
huffman@45386
|
787 |
apply(rule continuous_on_intros assms(2))+
|
huffman@44981
|
788 |
proof
|
hoelzl@33741
|
789 |
fix x assume x:"x \<in> {a<..<b}"
|
hoelzl@33741
|
790 |
show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) has_derivative (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)"
|
huffman@45011
|
791 |
by (intro has_derivative_intros assms(3)[rule_format,OF x]
|
huffman@45145
|
792 |
mult_right_has_derivative)
|
hoelzl@33741
|
793 |
qed(insert assms(1), auto simp add:field_simps)
|
huffman@44981
|
794 |
then guess x ..
|
huffman@44981
|
795 |
thus ?thesis apply(rule_tac x=x in bexI)
|
huffman@44981
|
796 |
apply(drule fun_cong[of _ _ "b - a"]) by auto
|
huffman@44981
|
797 |
qed
|
hoelzl@33741
|
798 |
|
huffman@44981
|
799 |
lemma mvt_simple:
|
huffman@44981
|
800 |
fixes f::"real \<Rightarrow> real"
|
huffman@44981
|
801 |
assumes "a<b" and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
|
hoelzl@33741
|
802 |
shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
|
huffman@44981
|
803 |
apply(rule mvt)
|
huffman@44981
|
804 |
apply(rule assms(1), rule differentiable_imp_continuous_on)
|
huffman@44981
|
805 |
unfolding differentiable_on_def differentiable_def defer
|
huffman@44981
|
806 |
proof
|
hoelzl@37489
|
807 |
fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)"
|
hoelzl@37489
|
808 |
unfolding has_derivative_within_open[OF x open_interval,THEN sym]
|
huffman@44981
|
809 |
apply(rule has_derivative_within_subset)
|
huffman@44981
|
810 |
apply(rule assms(2)[rule_format])
|
huffman@44981
|
811 |
using x by auto
|
huffman@44981
|
812 |
qed(insert assms(2), auto)
|
hoelzl@33741
|
813 |
|
huffman@44981
|
814 |
lemma mvt_very_simple:
|
huffman@44981
|
815 |
fixes f::"real \<Rightarrow> real"
|
huffman@44981
|
816 |
assumes "a \<le> b" and "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
|
huffman@44981
|
817 |
shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
|
huffman@44981
|
818 |
proof (cases "a = b")
|
hoelzl@33741
|
819 |
interpret bounded_linear "f' b" using assms(2) assms(1) by auto
|
hoelzl@33741
|
820 |
case True thus ?thesis apply(rule_tac x=a in bexI)
|
hoelzl@33741
|
821 |
using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
|
hoelzl@33741
|
822 |
unfolding True using zero by auto next
|
huffman@44981
|
823 |
case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto
|
huffman@44981
|
824 |
qed
|
hoelzl@33741
|
825 |
|
huffman@44982
|
826 |
text {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
|
hoelzl@33741
|
827 |
|
huffman@44981
|
828 |
lemma mvt_general:
|
huffman@44981
|
829 |
fixes f::"real\<Rightarrow>'a::euclidean_space"
|
huffman@44981
|
830 |
assumes "a<b" and "continuous_on {a..b} f"
|
huffman@44981
|
831 |
assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
|
huffman@44981
|
832 |
shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))"
|
huffman@44981
|
833 |
proof-
|
hoelzl@33741
|
834 |
have "\<exists>x\<in>{a<..<b}. (op \<bullet> (f b - f a) \<circ> f) b - (op \<bullet> (f b - f a) \<circ> f) a = (f b - f a) \<bullet> f' x (b - a)"
|
huffman@44981
|
835 |
apply(rule mvt) apply(rule assms(1))
|
huffman@44981
|
836 |
apply(rule continuous_on_inner continuous_on_intros assms(2))+
|
huffman@45011
|
837 |
unfolding o_def apply(rule,rule has_derivative_intros)
|
huffman@44981
|
838 |
using assms(3) by auto
|
hoelzl@33741
|
839 |
then guess x .. note x=this
|
hoelzl@33741
|
840 |
show ?thesis proof(cases "f a = f b")
|
hoelzl@36839
|
841 |
case False
|
huffman@44981
|
842 |
have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2"
|
huffman@44981
|
843 |
by (simp add: power2_eq_square)
|
himmelma@35528
|
844 |
also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding power2_norm_eq_inner ..
|
huffman@44981
|
845 |
also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)"
|
huffman@44981
|
846 |
using x unfolding inner_simps by (auto simp add: inner_diff_left)
|
huffman@44981
|
847 |
also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))"
|
huffman@44981
|
848 |
by (rule norm_cauchy_schwarz)
|
huffman@44981
|
849 |
finally show ?thesis using False x(1)
|
huffman@44981
|
850 |
by (auto simp add: real_mult_left_cancel)
|
huffman@44981
|
851 |
next
|
huffman@44981
|
852 |
case True thus ?thesis using assms(1)
|
huffman@44981
|
853 |
apply (rule_tac x="(a + b) /2" in bexI) by auto
|
huffman@44981
|
854 |
qed
|
huffman@44981
|
855 |
qed
|
hoelzl@33741
|
856 |
|
huffman@44982
|
857 |
text {* Still more general bound theorem. *}
|
hoelzl@33741
|
858 |
|
huffman@44981
|
859 |
lemma differentiable_bound:
|
huffman@44981
|
860 |
fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
|
huffman@44981
|
861 |
assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)"
|
huffman@44981
|
862 |
assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
|
huffman@44981
|
863 |
shows "norm(f x - f y) \<le> B * norm(x - y)"
|
huffman@44981
|
864 |
proof-
|
hoelzl@33741
|
865 |
let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
|
hoelzl@33741
|
866 |
have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
|
huffman@44981
|
867 |
using assms(1)[unfolded convex_alt,rule_format,OF x y]
|
huffman@44981
|
868 |
unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib
|
huffman@44981
|
869 |
by (auto simp add: algebra_simps)
|
huffman@44981
|
870 |
hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply-
|
huffman@45386
|
871 |
apply(rule continuous_on_intros)+
|
huffman@44981
|
872 |
unfolding continuous_on_eq_continuous_within
|
huffman@44981
|
873 |
apply(rule,rule differentiable_imp_continuous_within)
|
hoelzl@33741
|
874 |
unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
|
huffman@44981
|
875 |
apply(rule has_derivative_within_subset)
|
huffman@44981
|
876 |
apply(rule assms(2)[rule_format]) by auto
|
huffman@44981
|
877 |
have 2:"\<forall>u\<in>{0<..<1}. ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)"
|
huffman@44981
|
878 |
proof rule
|
huffman@44981
|
879 |
case goal1
|
hoelzl@33741
|
880 |
let ?u = "x + u *\<^sub>R (y - x)"
|
hoelzl@33741
|
881 |
have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
|
hoelzl@33741
|
882 |
apply(rule diff_chain_within) apply(rule has_derivative_intros)+
|
huffman@44981
|
883 |
apply(rule has_derivative_within_subset)
|
huffman@44981
|
884 |
apply(rule assms(2)[rule_format]) using goal1 * by auto
|
huffman@44981
|
885 |
thus ?case
|
huffman@44981
|
886 |
unfolding has_derivative_within_open[OF goal1 open_interval] by auto
|
huffman@44981
|
887 |
qed
|
hoelzl@33741
|
888 |
guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
|
huffman@44981
|
889 |
have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y"
|
huffman@44981
|
890 |
proof-
|
huffman@44981
|
891 |
case goal1
|
hoelzl@33741
|
892 |
have "norm (f' x y) \<le> onorm (f' x) * norm y"
|
hoelzl@33741
|
893 |
using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
|
huffman@44981
|
894 |
also have "\<dots> \<le> B * norm y"
|
huffman@44981
|
895 |
apply(rule mult_right_mono)
|
huffman@44981
|
896 |
using assms(3)[rule_format,OF goal1]
|
huffman@44981
|
897 |
by(auto simp add:field_simps)
|
huffman@44981
|
898 |
finally show ?case by simp
|
huffman@44981
|
899 |
qed
|
hoelzl@33741
|
900 |
have "norm (f x - f y) = norm ((f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 1 - (f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 0)"
|
hoelzl@33741
|
901 |
by(auto simp add:norm_minus_commute)
|
hoelzl@33741
|
902 |
also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
|
hoelzl@33741
|
903 |
also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
|
huffman@44981
|
904 |
finally show ?thesis by(auto simp add:norm_minus_commute)
|
huffman@44981
|
905 |
qed
|
hoelzl@33741
|
906 |
|
huffman@44981
|
907 |
lemma differentiable_bound_real:
|
huffman@44981
|
908 |
fixes f::"real \<Rightarrow> real"
|
huffman@44981
|
909 |
assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
|
huffman@44981
|
910 |
assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
|
hoelzl@37489
|
911 |
shows "norm(f x - f y) \<le> B * norm(x - y)"
|
hoelzl@37489
|
912 |
using differentiable_bound[of s f f' B x y]
|
hoelzl@37489
|
913 |
unfolding Ball_def image_iff o_def using assms by auto
|
hoelzl@37489
|
914 |
|
huffman@44982
|
915 |
text {* In particular. *}
|
hoelzl@33741
|
916 |
|
huffman@44981
|
917 |
lemma has_derivative_zero_constant:
|
huffman@44981
|
918 |
fixes f::"real\<Rightarrow>real"
|
hoelzl@33741
|
919 |
assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
|
huffman@44981
|
920 |
shows "\<exists>c. \<forall>x\<in>s. f x = c"
|
huffman@44981
|
921 |
proof(cases "s={}")
|
hoelzl@33741
|
922 |
case False then obtain x where "x\<in>s" by auto
|
hoelzl@33741
|
923 |
have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
|
huffman@44981
|
924 |
thus ?case
|
huffman@44981
|
925 |
using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
|
huffman@44981
|
926 |
unfolding onorm_const by auto qed
|
huffman@44981
|
927 |
thus ?thesis apply(rule_tac x="f x" in exI) by auto
|
huffman@44981
|
928 |
qed auto
|
hoelzl@33741
|
929 |
|
hoelzl@33741
|
930 |
lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
|
huffman@44981
|
931 |
assumes "convex s" and "a \<in> s" and "f a = c"
|
huffman@44981
|
932 |
assumes "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" and "x\<in>s"
|
huffman@44981
|
933 |
shows "f x = c"
|
huffman@44981
|
934 |
using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
|
hoelzl@33741
|
935 |
|
hoelzl@33741
|
936 |
subsection {* Differentiability of inverse function (most basic form). *}
|
hoelzl@33741
|
937 |
|
huffman@44981
|
938 |
lemma has_derivative_inverse_basic:
|
huffman@44981
|
939 |
fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
|
huffman@44981
|
940 |
assumes "(f has_derivative f') (at (g y))"
|
huffman@44981
|
941 |
assumes "bounded_linear g'" and "g' \<circ> f' = id" and "continuous (at y) g"
|
huffman@44981
|
942 |
assumes "open t" and "y \<in> t" and "\<forall>z\<in>t. f(g z) = z"
|
huffman@44981
|
943 |
shows "(g has_derivative g') (at y)"
|
huffman@44981
|
944 |
proof-
|
huffman@44981
|
945 |
interpret f': bounded_linear f'
|
huffman@44981
|
946 |
using assms unfolding has_derivative_def by auto
|
hoelzl@33741
|
947 |
interpret g': bounded_linear g' using assms by auto
|
hoelzl@33741
|
948 |
guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
|
hoelzl@33741
|
949 |
(* have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
|
huffman@44981
|
950 |
have lem1:"\<forall>e>0. \<exists>d>0. \<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y - g'(z - y)) \<le> e * norm(g z - g y)"
|
huffman@44981
|
951 |
proof(rule,rule)
|
huffman@44981
|
952 |
case goal1
|
hoelzl@33741
|
953 |
have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
|
hoelzl@33741
|
954 |
guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
|
hoelzl@33741
|
955 |
guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
|
hoelzl@33741
|
956 |
guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
|
hoelzl@33741
|
957 |
guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
|
huffman@44981
|
958 |
thus ?case apply(rule_tac x=d in exI) apply rule defer
|
huffman@44981
|
959 |
proof(rule,rule)
|
huffman@44981
|
960 |
fix z assume as:"norm (z - y) < d" hence "z\<in>t"
|
huffman@44981
|
961 |
using d2 d unfolding dist_norm by auto
|
hoelzl@33741
|
962 |
have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
|
huffman@44981
|
963 |
unfolding g'.diff f'.diff
|
huffman@44981
|
964 |
unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
|
huffman@44981
|
965 |
unfolding assms(7)[rule_format,OF `z\<in>t`]
|
huffman@44981
|
966 |
apply(subst norm_minus_cancel[THEN sym]) by auto
|
huffman@44981
|
967 |
also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C"
|
huffman@44981
|
968 |
by (rule C [THEN conjunct2, rule_format])
|
huffman@44981
|
969 |
also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
|
huffman@44981
|
970 |
apply(rule mult_right_mono)
|
huffman@44981
|
971 |
apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]])
|
huffman@44981
|
972 |
apply(cases "z=y") defer
|
huffman@44981
|
973 |
apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format])
|
huffman@44981
|
974 |
using as d C d0 by auto
|
huffman@44981
|
975 |
also have "\<dots> \<le> e * norm (g z - g y)"
|
huffman@44981
|
976 |
using C by (auto simp add: field_simps)
|
huffman@44981
|
977 |
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
|
huffman@44981
|
978 |
by simp
|
huffman@44981
|
979 |
qed auto
|
huffman@44981
|
980 |
qed
|
huffman@44981
|
981 |
have *:"(0::real) < 1 / 2" by auto
|
huffman@44981
|
982 |
guess d using lem1[rule_format,OF *] .. note d=this
|
huffman@44981
|
983 |
def B\<equiv>"C*2"
|
hoelzl@33741
|
984 |
have "B>0" unfolding B_def using C by auto
|
huffman@44981
|
985 |
have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)"
|
huffman@44981
|
986 |
proof(rule,rule) case goal1
|
huffman@44981
|
987 |
have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
|
huffman@44981
|
988 |
by(rule norm_triangle_sub)
|
huffman@44981
|
989 |
also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)"
|
huffman@44981
|
990 |
apply(rule add_left_mono) using d and goal1 by auto
|
huffman@44981
|
991 |
also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
|
huffman@44981
|
992 |
apply(rule add_right_mono) using C by auto
|
huffman@44981
|
993 |
finally show ?case unfolding B_def by(auto simp add:field_simps)
|
huffman@44981
|
994 |
qed
|
huffman@44981
|
995 |
show ?thesis unfolding has_derivative_at_alt
|
huffman@44981
|
996 |
proof(rule,rule assms,rule,rule) case goal1
|
hoelzl@33741
|
997 |
hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
|
hoelzl@33741
|
998 |
guess d' using lem1[rule_format,OF *] .. note d'=this
|
hoelzl@33741
|
999 |
guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
|
huffman@44981
|
1000 |
show ?case
|
huffman@44981
|
1001 |
apply(rule_tac x=k in exI,rule) defer
|
huffman@44981
|
1002 |
proof(rule,rule)
|
huffman@44981
|
1003 |
fix z assume as:"norm(z - y) < k"
|
huffman@44981
|
1004 |
hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
|
huffman@44981
|
1005 |
using d' k by auto
|
huffman@44981
|
1006 |
also have "\<dots> \<le> e * norm(z - y)"
|
huffman@44981
|
1007 |
unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
|
huffman@44981
|
1008 |
using lem2[THEN spec[where x=z]] using k as using `e>0`
|
huffman@44981
|
1009 |
by (auto simp add: field_simps)
|
huffman@44981
|
1010 |
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
|
huffman@44981
|
1011 |
by simp qed(insert k, auto)
|
huffman@44981
|
1012 |
qed
|
huffman@44981
|
1013 |
qed
|
hoelzl@33741
|
1014 |
|
huffman@44982
|
1015 |
text {* Simply rewrite that based on the domain point x. *}
|
hoelzl@33741
|
1016 |
|
huffman@44981
|
1017 |
lemma has_derivative_inverse_basic_x:
|
huffman@44981
|
1018 |
fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
|
hoelzl@33741
|
1019 |
assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
|
hoelzl@33741
|
1020 |
"continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
|
hoelzl@33741
|
1021 |
shows "(g has_derivative g') (at (f(x)))"
|
hoelzl@33741
|
1022 |
apply(rule has_derivative_inverse_basic) using assms by auto
|
hoelzl@33741
|
1023 |
|
huffman@44982
|
1024 |
text {* This is the version in Dieudonne', assuming continuity of f and g. *}
|
hoelzl@33741
|
1025 |
|
huffman@44981
|
1026 |
lemma has_derivative_inverse_dieudonne:
|
huffman@44981
|
1027 |
fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
|
hoelzl@33741
|
1028 |
assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
|
hoelzl@33741
|
1029 |
(**) "x\<in>s" "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
|
hoelzl@33741
|
1030 |
shows "(g has_derivative g') (at (f x))"
|
hoelzl@33741
|
1031 |
apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
|
huffman@44981
|
1032 |
using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)]
|
huffman@44981
|
1033 |
continuous_on_eq_continuous_at[OF assms(2)] by auto
|
hoelzl@33741
|
1034 |
|
huffman@44982
|
1035 |
text {* Here's the simplest way of not assuming much about g. *}
|
hoelzl@33741
|
1036 |
|
huffman@44981
|
1037 |
lemma has_derivative_inverse:
|
huffman@44981
|
1038 |
fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
|
hoelzl@33741
|
1039 |
assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
|
hoelzl@33741
|
1040 |
"\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
|
huffman@44981
|
1041 |
shows "(g has_derivative g') (at (f x))"
|
huffman@44981
|
1042 |
proof-
|
hoelzl@33741
|
1043 |
{ fix y assume "y\<in>interior (f ` s)"
|
huffman@44981
|
1044 |
then obtain x where "x\<in>s" and *:"y = f x"
|
huffman@44981
|
1045 |
unfolding image_iff using interior_subset by auto
|
huffman@44981
|
1046 |
have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] ..
|
huffman@44981
|
1047 |
} note * = this
|
huffman@44981
|
1048 |
show ?thesis
|
huffman@44981
|
1049 |
apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
|
huffman@44981
|
1050 |
apply(rule continuous_on_interior[OF _ assms(3)])
|
huffman@45511
|
1051 |
apply(rule continuous_on_inv[OF assms(4,1)])
|
huffman@44981
|
1052 |
apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
|
huffman@44981
|
1053 |
by(rule, rule *, assumption)
|
huffman@44981
|
1054 |
qed
|
hoelzl@33741
|
1055 |
|
hoelzl@33741
|
1056 |
subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
|
hoelzl@33741
|
1057 |
|
huffman@44981
|
1058 |
lemma brouwer_surjective:
|
huffman@44981
|
1059 |
fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
|
hoelzl@33741
|
1060 |
assumes "compact t" "convex t" "t \<noteq> {}" "continuous_on t f"
|
hoelzl@33741
|
1061 |
"\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
|
huffman@44981
|
1062 |
shows "\<exists>y\<in>t. f y = x"
|
huffman@44981
|
1063 |
proof-
|
huffman@44981
|
1064 |
have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
|
huffman@44981
|
1065 |
by(auto simp add:algebra_simps)
|
huffman@44981
|
1066 |
show ?thesis
|
huffman@44981
|
1067 |
unfolding *
|
huffman@44981
|
1068 |
apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
|
huffman@44981
|
1069 |
apply(rule continuous_on_intros assms)+ using assms(4-6) by auto
|
huffman@44981
|
1070 |
qed
|
hoelzl@33741
|
1071 |
|
huffman@44981
|
1072 |
lemma brouwer_surjective_cball:
|
huffman@44981
|
1073 |
fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
|
hoelzl@33741
|
1074 |
assumes "0 < e" "continuous_on (cball a e) f"
|
hoelzl@33741
|
1075 |
"\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
|
huffman@44981
|
1076 |
shows "\<exists>y\<in>cball a e. f y = x"
|
huffman@44981
|
1077 |
apply(rule brouwer_surjective)
|
huffman@44981
|
1078 |
apply(rule compact_cball convex_cball)+
|
huffman@44981
|
1079 |
unfolding cball_eq_empty using assms by auto
|
hoelzl@33741
|
1080 |
|
hoelzl@33741
|
1081 |
text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
|
hoelzl@33741
|
1082 |
|
huffman@44981
|
1083 |
lemma sussmann_open_mapping:
|
huffman@44981
|
1084 |
fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
|
hoelzl@33741
|
1085 |
assumes "open s" "continuous_on s f" "x \<in> s"
|
hoelzl@33741
|
1086 |
"(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
|
hoelzl@37489
|
1087 |
"t \<subseteq> s" "x \<in> interior t"
|
huffman@44981
|
1088 |
shows "f x \<in> interior (f ` t)"
|
huffman@44981
|
1089 |
proof-
|
huffman@44981
|
1090 |
interpret f':bounded_linear f'
|
huffman@44981
|
1091 |
using assms unfolding has_derivative_def by auto
|
hoelzl@33741
|
1092 |
interpret g':bounded_linear g' using assms by auto
|
huffman@44981
|
1093 |
guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this
|
huffman@44981
|
1094 |
hence *:"1/(2*B)>0" by (auto intro!: divide_pos_pos)
|
hoelzl@33741
|
1095 |
guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
|
hoelzl@33741
|
1096 |
guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
|
huffman@44981
|
1097 |
have *:"0<e0/B" "0<e1/B"
|
huffman@44981
|
1098 |
apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
|
hoelzl@33741
|
1099 |
guess e using real_lbound_gt_zero[OF *] .. note e=this
|
hoelzl@33741
|
1100 |
have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
|
hoelzl@33741
|
1101 |
apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
|
huffman@44981
|
1102 |
prefer 3 apply(rule,rule)
|
huffman@44981
|
1103 |
proof-
|
huffman@44981
|
1104 |
show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
|
huffman@44981
|
1105 |
unfolding g'.diff
|
hoelzl@33741
|
1106 |
apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
|
hoelzl@33741
|
1107 |
apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
|
huffman@44981
|
1108 |
apply(rule continuous_on_subset[OF assms(2)])
|
huffman@44981
|
1109 |
apply(rule,unfold image_iff,erule bexE)
|
huffman@44981
|
1110 |
proof-
|
hoelzl@33741
|
1111 |
fix y z assume as:"y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
|
huffman@44981
|
1112 |
have "dist x z = norm (g' (f x) - g' y)"
|
huffman@44981
|
1113 |
unfolding as(2) and dist_norm by auto
|
huffman@44981
|
1114 |
also have "\<dots> \<le> norm (f x - y) * B"
|
huffman@44981
|
1115 |
unfolding g'.diff[THEN sym] using B by auto
|
huffman@44981
|
1116 |
also have "\<dots> \<le> e * B"
|
huffman@44981
|
1117 |
using as(1)[unfolded mem_cball dist_norm] using B by auto
|
hoelzl@33741
|
1118 |
also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
|
hoelzl@33741
|
1119 |
finally have "z\<in>cball x e1" unfolding mem_cball by force
|
huffman@44981
|
1120 |
thus "z \<in> s" using e1 assms(7) by auto
|
huffman@44981
|
1121 |
qed
|
huffman@44981
|
1122 |
next
|
hoelzl@33741
|
1123 |
fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
|
hoelzl@33741
|
1124 |
have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
|
huffman@44981
|
1125 |
also have "\<dots> \<le> e * B" apply(rule mult_right_mono)
|
huffman@44981
|
1126 |
using as(2)[unfolded mem_cball dist_norm] and B
|
huffman@44981
|
1127 |
unfolding norm_minus_commute by auto
|
hoelzl@33741
|
1128 |
also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
|
hoelzl@33741
|
1129 |
finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
|
huffman@44981
|
1130 |
have **:"f x + f' (x + g' (z - f x) - x) = z"
|
huffman@44981
|
1131 |
using assms(6)[unfolded o_def id_def,THEN cong] by auto
|
hoelzl@33741
|
1132 |
have "norm (f x - (y + (z - f (x + g' (z - f x))))) \<le> norm (f (x + g' (z - f x)) - z) + norm (f x - y)"
|
huffman@44981
|
1133 |
using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"]
|
huffman@44981
|
1134 |
by (auto simp add: algebra_simps)
|
huffman@44981
|
1135 |
also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)"
|
huffman@44981
|
1136 |
using e0[THEN conjunct2,rule_format,OF *]
|
huffman@44981
|
1137 |
unfolding algebra_simps ** by auto
|
huffman@44981
|
1138 |
also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2"
|
huffman@44981
|
1139 |
using as(1)[unfolded mem_cball dist_norm] by auto
|
huffman@44981
|
1140 |
also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2"
|
huffman@44981
|
1141 |
using * and B by (auto simp add: field_simps)
|
hoelzl@33741
|
1142 |
also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
|
huffman@44981
|
1143 |
also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono)
|
huffman@44981
|
1144 |
using as(2)[unfolded mem_cball dist_norm]
|
huffman@44981
|
1145 |
unfolding norm_minus_commute by auto
|
huffman@44981
|
1146 |
finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
|
huffman@44981
|
1147 |
unfolding mem_cball dist_norm by auto
|
hoelzl@33741
|
1148 |
qed(insert e, auto) note lem = this
|
hoelzl@33741
|
1149 |
show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
|
huffman@44981
|
1150 |
apply(rule,rule divide_pos_pos) prefer 3
|
huffman@44981
|
1151 |
proof
|
huffman@44981
|
1152 |
fix y assume "y \<in> ball (f x) (e/2)"
|
huffman@44981
|
1153 |
hence *:"y\<in>cball (f x) (e/2)" by auto
|
hoelzl@33741
|
1154 |
guess z using lem[rule_format,OF *] .. note z=this
|
huffman@44981
|
1155 |
hence "norm (g' (z - f x)) \<le> norm (z - f x) * B"
|
huffman@44981
|
1156 |
using B by (auto simp add: field_simps)
|
huffman@44981
|
1157 |
also have "\<dots> \<le> e * B"
|
huffman@44981
|
1158 |
apply (rule mult_right_mono) using z(1)
|
huffman@44981
|
1159 |
unfolding mem_cball dist_norm norm_minus_commute using B by auto
|
hoelzl@33741
|
1160 |
also have "\<dots> \<le> e1" using e B unfolding less_divide_eq by auto
|
huffman@44981
|
1161 |
finally have "x + g'(z - f x) \<in> t" apply-
|
huffman@44981
|
1162 |
apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
|
huffman@36587
|
1163 |
unfolding mem_cball dist_norm by auto
|
huffman@44981
|
1164 |
thus "y \<in> f ` t" using z by auto
|
huffman@44981
|
1165 |
qed(insert e, auto)
|
huffman@44981
|
1166 |
qed
|
hoelzl@33741
|
1167 |
|
hoelzl@33741
|
1168 |
text {* Hence the following eccentric variant of the inverse function theorem. *)
|
hoelzl@33741
|
1169 |
(* This has no continuity assumptions, but we do need the inverse function. *)
|
hoelzl@33741
|
1170 |
(* We could put f' o g = I but this happens to fit with the minimal linear *)
|
hoelzl@33741
|
1171 |
(* algebra theory I've set up so far. *}
|
hoelzl@33741
|
1172 |
|
hoelzl@37489
|
1173 |
(* move before left_inverse_linear in Euclidean_Space*)
|
hoelzl@37489
|
1174 |
|
huffman@44981
|
1175 |
lemma right_inverse_linear:
|
huffman@44981
|
1176 |
fixes f::"'a::euclidean_space => 'a"
|
hoelzl@37489
|
1177 |
assumes lf: "linear f" and gf: "f o g = id"
|
hoelzl@37489
|
1178 |
shows "linear g"
|
hoelzl@37489
|
1179 |
proof-
|
hoelzl@40950
|
1180 |
from gf have fi: "surj f" by (auto simp add: surj_def o_def id_def) metis
|
hoelzl@37489
|
1181 |
from linear_surjective_isomorphism[OF lf fi]
|
hoelzl@37489
|
1182 |
obtain h:: "'a => 'a" where
|
hoelzl@37489
|
1183 |
h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
|
hoelzl@37489
|
1184 |
have "h = g" apply (rule ext) using gf h(2,3)
|
hoelzl@40950
|
1185 |
by (simp add: o_def id_def fun_eq_iff) metis
|
hoelzl@37489
|
1186 |
with h(1) show ?thesis by blast
|
hoelzl@37489
|
1187 |
qed
|
hoelzl@37489
|
1188 |
|
huffman@44981
|
1189 |
lemma has_derivative_inverse_strong:
|
huffman@44981
|
1190 |
fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
|
huffman@44981
|
1191 |
assumes "open s" and "x \<in> s" and "continuous_on s f"
|
huffman@44981
|
1192 |
assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" and "f' o g' = id"
|
huffman@44981
|
1193 |
shows "(g has_derivative g') (at (f x))"
|
huffman@44981
|
1194 |
proof-
|
huffman@44981
|
1195 |
have linf:"bounded_linear f'"
|
huffman@44981
|
1196 |
using assms(5) unfolding has_derivative_def by auto
|
huffman@44981
|
1197 |
hence ling:"bounded_linear g'"
|
huffman@44981
|
1198 |
unfolding linear_conv_bounded_linear[THEN sym]
|
huffman@44981
|
1199 |
apply- apply(rule right_inverse_linear) using assms(6) by auto
|
huffman@44981
|
1200 |
moreover have "g' \<circ> f' = id" using assms(6) linf ling
|
huffman@44981
|
1201 |
unfolding linear_conv_bounded_linear[THEN sym]
|
hoelzl@33741
|
1202 |
using linear_inverse_left by auto
|
huffman@44981
|
1203 |
moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)"
|
huffman@44981
|
1204 |
apply(rule,rule,rule,rule sussmann_open_mapping )
|
hoelzl@33741
|
1205 |
apply(rule assms ling)+ by auto
|
huffman@44981
|
1206 |
have "continuous (at (f x)) g" unfolding continuous_at Lim_at
|
huffman@44981
|
1207 |
proof(rule,rule)
|
hoelzl@33741
|
1208 |
fix e::real assume "e>0"
|
huffman@44981
|
1209 |
hence "f x \<in> interior (f ` (ball x e \<inter> s))"
|
huffman@44981
|
1210 |
using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
|
hoelzl@33741
|
1211 |
by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
|
hoelzl@33741
|
1212 |
then guess d unfolding mem_interior .. note d=this
|
hoelzl@33741
|
1213 |
show "\<exists>d>0. \<forall>y. 0 < dist y (f x) \<and> dist y (f x) < d \<longrightarrow> dist (g y) (g (f x)) < e"
|
huffman@44981
|
1214 |
apply(rule_tac x=d in exI)
|
huffman@44981
|
1215 |
apply(rule,rule d[THEN conjunct1])
|
huffman@44981
|
1216 |
proof(rule,rule) case goal1
|
huffman@44981
|
1217 |
hence "g y \<in> g ` f ` (ball x e \<inter> s)"
|
huffman@44981
|
1218 |
using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
|
wenzelm@42829
|
1219 |
by(auto simp add:dist_commute)
|
hoelzl@33741
|
1220 |
hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
|
huffman@44981
|
1221 |
thus "dist (g y) (g (f x)) < e"
|
huffman@44981
|
1222 |
using assms(4)[rule_format,OF `x\<in>s`]
|
huffman@44981
|
1223 |
by (auto simp add: dist_commute)
|
huffman@44981
|
1224 |
qed
|
huffman@44981
|
1225 |
qed
|
huffman@44981
|
1226 |
moreover have "f x \<in> interior (f ` s)"
|
huffman@44981
|
1227 |
apply(rule sussmann_open_mapping)
|
huffman@44981
|
1228 |
apply(rule assms ling)+
|
huffman@44981
|
1229 |
using interior_open[OF assms(1)] and `x\<in>s` by auto
|
huffman@44981
|
1230 |
moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y"
|
huffman@44981
|
1231 |
proof- case goal1
|
huffman@44981
|
1232 |
hence "y\<in>f ` s" using interior_subset by auto
|
huffman@44981
|
1233 |
then guess z unfolding image_iff ..
|
huffman@44981
|
1234 |
thus ?case using assms(4) by auto
|
huffman@44981
|
1235 |
qed
|
huffman@44981
|
1236 |
ultimately show ?thesis
|
huffman@44981
|
1237 |
apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)])
|
huffman@44981
|
1238 |
using assms by auto
|
huffman@44981
|
1239 |
qed
|
hoelzl@33741
|
1240 |
|
huffman@44982
|
1241 |
text {* A rewrite based on the other domain. *}
|
hoelzl@33741
|
1242 |
|
huffman@44981
|
1243 |
lemma has_derivative_inverse_strong_x:
|
huffman@44981
|
1244 |
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
|
huffman@44981
|
1245 |
assumes "open s" and "g y \<in> s" and "continuous_on s f"
|
huffman@44981
|
1246 |
assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))"
|
huffman@44981
|
1247 |
assumes "f' o g' = id" and "f(g y) = y"
|
hoelzl@33741
|
1248 |
shows "(g has_derivative g') (at y)"
|
hoelzl@33741
|
1249 |
using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
|
hoelzl@33741
|
1250 |
|
huffman@44982
|
1251 |
text {* On a region. *}
|
hoelzl@33741
|
1252 |
|
huffman@44981
|
1253 |
lemma has_derivative_inverse_on:
|
huffman@44981
|
1254 |
fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
|
huffman@44981
|
1255 |
assumes "open s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
|
huffman@44981
|
1256 |
assumes "\<forall>x\<in>s. g(f x) = x" and "f'(x) o g'(x) = id" and "x\<in>s"
|
hoelzl@33741
|
1257 |
shows "(g has_derivative g'(x)) (at (f x))"
|
huffman@44981
|
1258 |
apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f])
|
huffman@44981
|
1259 |
apply(rule assms)+
|
hoelzl@33741
|
1260 |
unfolding continuous_on_eq_continuous_at[OF assms(1)]
|
huffman@44981
|
1261 |
apply(rule,rule differentiable_imp_continuous_at)
|
huffman@44981
|
1262 |
unfolding differentiable_def using assms by auto
|
hoelzl@33741
|
1263 |
|
huffman@44981
|
1264 |
text {* Invertible derivative continous at a point implies local
|
huffman@44981
|
1265 |
injectivity. It's only for this we need continuity of the derivative,
|
huffman@44981
|
1266 |
except of course if we want the fact that the inverse derivative is
|
huffman@44981
|
1267 |
also continuous. So if we know for some other reason that the inverse
|
huffman@44981
|
1268 |
function exists, it's OK. *}
|
hoelzl@33741
|
1269 |
|
huffman@44981
|
1270 |
lemma bounded_linear_sub:
|
huffman@44981
|
1271 |
"bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
|
huffman@44981
|
1272 |
using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g]
|
huffman@44981
|
1273 |
by (auto simp add: algebra_simps)
|
hoelzl@33741
|
1274 |
|
huffman@44981
|
1275 |
lemma has_derivative_locally_injective:
|
huffman@44981
|
1276 |
fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
|
hoelzl@33741
|
1277 |
assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
|
hoelzl@33741
|
1278 |
"\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
|
hoelzl@33741
|
1279 |
"\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
|
huffman@44981
|
1280 |
obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)"
|
huffman@44981
|
1281 |
proof-
|
hoelzl@33741
|
1282 |
interpret bounded_linear g' using assms by auto
|
hoelzl@33741
|
1283 |
note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
|
hoelzl@37489
|
1284 |
have "g' (f' a (\<chi>\<chi> i.1)) = (\<chi>\<chi> i.1)" "(\<chi>\<chi> i.1) \<noteq> (0::'n)" defer
|
hoelzl@37489
|
1285 |
apply(subst euclidean_eq) using f'g' by auto
|
huffman@44981
|
1286 |
hence *:"0 < onorm g'"
|
nipkow@45761
|
1287 |
unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastforce
|
hoelzl@33741
|
1288 |
def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
|
hoelzl@33741
|
1289 |
guess d1 using assms(6)[rule_format,OF *] .. note d1=this
|
hoelzl@33741
|
1290 |
from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
|
hoelzl@33741
|
1291 |
obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
|
huffman@44981
|
1292 |
guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] ..
|
huffman@44981
|
1293 |
note d2=this
|
huffman@44981
|
1294 |
guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] ..
|
huffman@44981
|
1295 |
note d = this
|
huffman@44981
|
1296 |
show ?thesis
|
huffman@44981
|
1297 |
proof
|
huffman@44981
|
1298 |
show "a\<in>ball a d" using d by auto
|
huffman@44981
|
1299 |
show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x"
|
huffman@44981
|
1300 |
proof (intro strip)
|
hoelzl@33741
|
1301 |
fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
|
huffman@44981
|
1302 |
def ph \<equiv> "\<lambda>w. w - g'(f w - f x)"
|
huffman@44981
|
1303 |
have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
|
huffman@44981
|
1304 |
unfolding ph_def o_def unfolding diff using f'g'
|
huffman@44981
|
1305 |
by (auto simp add: algebra_simps)
|
hoelzl@33741
|
1306 |
have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
|
wenzelm@42829
|
1307 |
apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
|
huffman@44981
|
1308 |
apply(rule_tac[!] ballI)
|
huffman@44981
|
1309 |
proof-
|
huffman@44981
|
1310 |
fix u assume u:"u \<in> ball a d"
|
huffman@44981
|
1311 |
hence "u\<in>s" using d d2 by auto
|
huffman@44981
|
1312 |
have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
|
huffman@44981
|
1313 |
unfolding o_def and diff using f'g' by auto
|
wenzelm@42829
|
1314 |
show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
|
huffman@44981
|
1315 |
unfolding ph' * apply(rule diff_chain_within) defer
|
huffman@45011
|
1316 |
apply(rule bounded_linear.has_derivative'[OF assms(3)])
|
huffman@44981
|
1317 |
apply(rule has_derivative_intros) defer
|
huffman@44981
|
1318 |
apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
|
huffman@44981
|
1319 |
apply(rule has_derivative_at_within)
|
huffman@44981
|
1320 |
using assms(5) and `u\<in>s` `a\<in>s`
|
huffman@45011
|
1321 |
by(auto intro!: has_derivative_intros bounded_linear.has_derivative' derivative_linear)
|
huffman@44981
|
1322 |
have **:"bounded_linear (\<lambda>x. f' u x - f' a x)"
|
huffman@44981
|
1323 |
"bounded_linear (\<lambda>x. f' a x - f' u x)"
|
huffman@44981
|
1324 |
apply(rule_tac[!] bounded_linear_sub)
|
huffman@44981
|
1325 |
apply(rule_tac[!] derivative_linear)
|
huffman@44981
|
1326 |
using assms(5) `u\<in>s` `a\<in>s` by auto
|
huffman@44981
|
1327 |
have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)"
|
huffman@44981
|
1328 |
unfolding * apply(rule onorm_compose)
|
huffman@44981
|
1329 |
unfolding linear_conv_bounded_linear by(rule assms(3) **)+
|
huffman@44981
|
1330 |
also have "\<dots> \<le> onorm g' * k"
|
huffman@44981
|
1331 |
apply(rule mult_left_mono)
|
huffman@44981
|
1332 |
using d1[THEN conjunct2,rule_format,of u]
|
huffman@44981
|
1333 |
using onorm_neg[OF **(1)[unfolded linear_linear]]
|
huffman@44981
|
1334 |
using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]]
|
huffman@44981
|
1335 |
by (auto simp add: algebra_simps)
|
wenzelm@42829
|
1336 |
also have "\<dots> \<le> 1/2" unfolding k_def by auto
|
huffman@44981
|
1337 |
finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption
|
huffman@44981
|
1338 |
qed
|
huffman@44981
|
1339 |
moreover have "norm (ph y - ph x) = norm (y - x)"
|
huffman@44981
|
1340 |
apply(rule arg_cong[where f=norm])
|
wenzelm@42829
|
1341 |
unfolding ph_def using diff unfolding as by auto
|
huffman@44981
|
1342 |
ultimately show "x = y" unfolding norm_minus_commute by auto
|
huffman@44981
|
1343 |
qed
|
huffman@44981
|
1344 |
qed auto
|
huffman@44981
|
1345 |
qed
|
hoelzl@33741
|
1346 |
|
hoelzl@33741
|
1347 |
subsection {* Uniformly convergent sequence of derivatives. *}
|
hoelzl@33741
|
1348 |
|
huffman@44981
|
1349 |
lemma has_derivative_sequence_lipschitz_lemma:
|
huffman@44981
|
1350 |
fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
|
huffman@44981
|
1351 |
assumes "convex s"
|
huffman@44981
|
1352 |
assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
|
huffman@44981
|
1353 |
assumes "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
|
huffman@44981
|
1354 |
shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
|
huffman@44981
|
1355 |
proof (default)+
|
hoelzl@33741
|
1356 |
fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
|
hoelzl@33741
|
1357 |
show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
|
huffman@44981
|
1358 |
apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)])
|
huffman@44981
|
1359 |
apply(rule_tac[!] ballI)
|
huffman@44981
|
1360 |
proof-
|
huffman@44981
|
1361 |
fix x assume "x\<in>s"
|
huffman@44981
|
1362 |
show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within s)"
|
hoelzl@33741
|
1363 |
by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
|
huffman@44981
|
1364 |
{ fix h
|
huffman@44981
|
1365 |
have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
|
huffman@44981
|
1366 |
using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"]
|
huffman@44981
|
1367 |
unfolding norm_minus_commute by (auto simp add: algebra_simps)
|
huffman@44981
|
1368 |
also have "\<dots> \<le> e * norm h+ e * norm h"
|
huffman@44981
|
1369 |
using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h]
|
huffman@44981
|
1370 |
using assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
|
wenzelm@42829
|
1371 |
by(auto simp add:field_simps)
|
hoelzl@33741
|
1372 |
finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
|
huffman@44981
|
1373 |
thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
|
huffman@44981
|
1374 |
apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
|
huffman@44981
|
1375 |
unfolding linear_conv_bounded_linear
|
huffman@44981
|
1376 |
using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear]
|
huffman@44981
|
1377 |
by auto
|
huffman@44981
|
1378 |
qed
|
huffman@44981
|
1379 |
qed
|
hoelzl@33741
|
1380 |
|
huffman@44981
|
1381 |
lemma has_derivative_sequence_lipschitz:
|
huffman@44981
|
1382 |
fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
|
huffman@44981
|
1383 |
assumes "convex s"
|
huffman@44981
|
1384 |
assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
|
huffman@44981
|
1385 |
assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
|
huffman@44981
|
1386 |
assumes "0 < e"
|
huffman@44981
|
1387 |
shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
|
huffman@44981
|
1388 |
proof(rule,rule)
|
hoelzl@33741
|
1389 |
case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
|
hoelzl@33741
|
1390 |
guess N using assms(3)[rule_format,OF *(2)] ..
|
huffman@44981
|
1391 |
thus ?case
|
huffman@44981
|
1392 |
apply(rule_tac x=N in exI)
|
huffman@44981
|
1393 |
apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
|
huffman@44981
|
1394 |
using assms by auto
|
huffman@44981
|
1395 |
qed
|
hoelzl@33741
|
1396 |
|
huffman@44981
|
1397 |
lemma has_derivative_sequence:
|
huffman@44981
|
1398 |
fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
|
huffman@44981
|
1399 |
assumes "convex s"
|
huffman@44981
|
1400 |
assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
|
huffman@44981
|
1401 |
assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
|
huffman@44981
|
1402 |
assumes "x0 \<in> s" and "((\<lambda>n. f n x0) ---> l) sequentially"
|
huffman@44981
|
1403 |
shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and>
|
huffman@44981
|
1404 |
(g has_derivative g'(x)) (at x within s)"
|
huffman@44981
|
1405 |
proof-
|
hoelzl@33741
|
1406 |
have lem1:"\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
|
huffman@44981
|
1407 |
apply(rule has_derivative_sequence_lipschitz[where e="42::nat"])
|
huffman@44981
|
1408 |
apply(rule assms)+ by auto
|
huffman@44981
|
1409 |
have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially"
|
huffman@44981
|
1410 |
apply(rule bchoice) unfolding convergent_eq_cauchy
|
huffman@44981
|
1411 |
proof
|
huffman@44981
|
1412 |
fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)"
|
huffman@44981
|
1413 |
proof(cases "x=x0")
|
huffman@44981
|
1414 |
case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto
|
huffman@44981
|
1415 |
next
|
huffman@44981
|
1416 |
case False show ?thesis unfolding Cauchy_def
|
huffman@44981
|
1417 |
proof(rule,rule)
|
huffman@44981
|
1418 |
fix e::real assume "e>0"
|
huffman@44981
|
1419 |
hence *:"e/2>0" "e/2/norm(x-x0)>0"
|
huffman@44981
|
1420 |
using False by (auto intro!: divide_pos_pos)
|
wenzelm@42829
|
1421 |
guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
|
wenzelm@42829
|
1422 |
guess N using lem1[rule_format,OF *(2)] .. note N = this
|
huffman@44981
|
1423 |
show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
|
huffman@44981
|
1424 |
apply(rule_tac x="max M N" in exI)
|
huffman@44981
|
1425 |
proof(default+)
|
wenzelm@42829
|
1426 |
fix m n assume as:"max M N \<le>m" "max M N\<le>n"
|
wenzelm@42829
|
1427 |
have "dist (f m x) (f n x) \<le> norm (f m x0 - f n x0) + norm (f m x - f n x - (f m x0 - f n x0))"
|
wenzelm@42829
|
1428 |
unfolding dist_norm by(rule norm_triangle_sub)
|
huffman@44981
|
1429 |
also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
|
huffman@44981
|
1430 |
using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False
|
huffman@44981
|
1431 |
by auto
|
huffman@44981
|
1432 |
also have "\<dots> < e / 2 + e / 2"
|
huffman@44981
|
1433 |
apply(rule add_strict_right_mono)
|
huffman@44981
|
1434 |
using as and M[rule_format] unfolding dist_norm by auto
|
huffman@44981
|
1435 |
finally show "dist (f m x) (f n x) < e" by auto
|
huffman@44981
|
1436 |
qed
|
huffman@44981
|
1437 |
qed
|
huffman@44981
|
1438 |
qed
|
huffman@44981
|
1439 |
qed
|
hoelzl@33741
|
1440 |
then guess g .. note g = this
|
huffman@44981
|
1441 |
have lem2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f n x - f n y) - (g x - g y)) \<le> e * norm(x - y)"
|
huffman@44981
|
1442 |
proof(rule,rule)
|
huffman@44981
|
1443 |
fix e::real assume *:"e>0"
|
huffman@44981
|
1444 |
guess N using lem1[rule_format,OF *] .. note N=this
|
huffman@44981
|
1445 |
show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
|
huffman@44981
|
1446 |
apply(rule_tac x=N in exI)
|
huffman@44981
|
1447 |
proof(default+)
|
hoelzl@33741
|
1448 |
fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
|
huffman@44981
|
1449 |
have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
|
huffman@44981
|
1450 |
unfolding eventually_sequentially
|
huffman@44981
|
1451 |
apply(rule_tac x=N in exI)
|
huffman@44981
|
1452 |
proof(rule,rule)
|
huffman@44981
|
1453 |
fix m assume "N\<le>m"
|
huffman@44981
|
1454 |
thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
|
huffman@44981
|
1455 |
using N[rule_format, of n m x y] and as
|
huffman@44981
|
1456 |
by (auto simp add: algebra_simps)
|
huffman@44981
|
1457 |
qed
|
huffman@44981
|
1458 |
thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
|
huffman@44981
|
1459 |
apply-
|
wenzelm@42829
|
1460 |
apply(rule Lim_norm_ubound[OF trivial_limit_sequentially, where f="\<lambda>m. (f n x - f n y) - (f m x - f m y)"])
|
huffman@44983
|
1461 |
apply(rule tendsto_intros g[rule_format] as)+ by assumption
|
huffman@44981
|
1462 |
qed
|
huffman@44981
|
1463 |
qed
|
hoelzl@33741
|
1464 |
show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
|
huffman@44981
|
1465 |
apply(rule,rule,rule g[rule_format],assumption)
|
huffman@44981
|
1466 |
proof fix x assume "x\<in>s"
|
huffman@44981
|
1467 |
have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
|
huffman@45778
|
1468 |
unfolding LIMSEQ_def
|
huffman@44981
|
1469 |
proof(rule,rule,rule)
|
huffman@44981
|
1470 |
fix u and e::real assume "e>0"
|
huffman@44981
|
1471 |
show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e"
|
huffman@44981
|
1472 |
proof(cases "u=0")
|
wenzelm@42829
|
1473 |
case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
|
wenzelm@42829
|
1474 |
show ?thesis apply(rule_tac x=N in exI) unfolding True
|
huffman@44981
|
1475 |
using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto
|
huffman@44981
|
1476 |
next
|
huffman@44981
|
1477 |
case False hence *:"e / 2 / norm u > 0"
|
huffman@44981
|
1478 |
using `e>0` by (auto intro!: divide_pos_pos)
|
wenzelm@42829
|
1479 |
guess N using assms(3)[rule_format,OF *] .. note N=this
|
huffman@44981
|
1480 |
show ?thesis apply(rule_tac x=N in exI)
|
huffman@44981
|
1481 |
proof(rule,rule) case goal1
|
huffman@44981
|
1482 |
show ?case unfolding dist_norm
|
huffman@44981
|
1483 |
using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
|
huffman@44981
|
1484 |
by (auto simp add:field_simps)
|
huffman@44981
|
1485 |
qed
|
huffman@44981
|
1486 |
qed
|
huffman@44981
|
1487 |
qed
|
huffman@44981
|
1488 |
show "bounded_linear (g' x)"
|
huffman@44981
|
1489 |
unfolding linear_linear linear_def
|
huffman@44981
|
1490 |
apply(rule,rule,rule) defer
|
huffman@44981
|
1491 |
proof(rule,rule)
|
hoelzl@37489
|
1492 |
fix x' y z::"'m" and c::real
|
hoelzl@33741
|
1493 |
note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
|
huffman@44981
|
1494 |
show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
|
huffman@44981
|
1495 |
apply(rule tendsto_unique[OF trivial_limit_sequentially])
|
wenzelm@42829
|
1496 |
apply(rule lem3[rule_format])
|
hoelzl@33741
|
1497 |
unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
|
huffman@44983
|
1498 |
apply (intro tendsto_intros) by(rule lem3[rule_format])
|
huffman@44981
|
1499 |
show "g' x (y + z) = g' x y + g' x z"
|
huffman@44981
|
1500 |
apply(rule tendsto_unique[OF trivial_limit_sequentially])
|
huffman@44981
|
1501 |
apply(rule lem3[rule_format])
|
huffman@44981
|
1502 |
unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
|
huffman@44983
|
1503 |
apply(rule tendsto_add) by(rule lem3[rule_format])+
|
huffman@44981
|
1504 |
qed
|
huffman@44981
|
1505 |
show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
|
huffman@44981
|
1506 |
proof(rule,rule) case goal1
|
huffman@44981
|
1507 |
have *:"e/3>0" using goal1 by auto
|
huffman@44981
|
1508 |
guess N1 using assms(3)[rule_format,OF *] .. note N1=this
|
hoelzl@33741
|
1509 |
guess N2 using lem2[rule_format,OF *] .. note N2=this
|
hoelzl@33741
|
1510 |
guess d1 using assms(2)[unfolded has_derivative_within_alt, rule_format,OF `x\<in>s`, of "max N1 N2",THEN conjunct2,rule_format,OF *] .. note d1=this
|
huffman@44981
|
1511 |
show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1])
|
huffman@44981
|
1512 |
proof(rule,rule)
|
huffman@44981
|
1513 |
fix y assume as:"y \<in> s" "norm (y - x) < d1"
|
huffman@44981
|
1514 |
let ?N ="max N1 N2"
|
huffman@44981
|
1515 |
have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)"
|
huffman@44981
|
1516 |
apply(subst norm_minus_cancel[THEN sym])
|
huffman@44981
|
1517 |
using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto
|
huffman@44981
|
1518 |
moreover
|
huffman@44981
|
1519 |
have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
|
huffman@44981
|
1520 |
using d1 and as by auto
|
huffman@44981
|
1521 |
ultimately
|
wenzelm@42829
|
1522 |
have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
|
huffman@44981
|
1523 |
using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
|
huffman@44981
|
1524 |
by (auto simp add:algebra_simps)
|
huffman@44981
|
1525 |
moreover
|
huffman@44981
|
1526 |
have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
|
huffman@44981
|
1527 |
using N1 `x\<in>s` by auto
|
wenzelm@42829
|
1528 |
ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
|
huffman@44981
|
1529 |
using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"]
|
huffman@44981
|
1530 |
by(auto simp add:algebra_simps)
|
huffman@44981
|
1531 |
qed
|
huffman@44981
|
1532 |
qed
|
huffman@44981
|
1533 |
qed
|
huffman@44981
|
1534 |
qed
|
hoelzl@33741
|
1535 |
|
huffman@44982
|
1536 |
text {* Can choose to line up antiderivatives if we want. *}
|
hoelzl@33741
|
1537 |
|
huffman@44981
|
1538 |
lemma has_antiderivative_sequence:
|
huffman@44981
|
1539 |
fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
|
huffman@44981
|
1540 |
assumes "convex s"
|
huffman@44981
|
1541 |
assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
|
huffman@44981
|
1542 |
assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm h"
|
huffman@44981
|
1543 |
shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
|
huffman@44981
|
1544 |
proof(cases "s={}")
|
huffman@44981
|
1545 |
case False then obtain a where "a\<in>s" by auto
|
huffman@44981
|
1546 |
have *:"\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x" by auto
|
huffman@44981
|
1547 |
show ?thesis
|
huffman@44981
|
1548 |
apply(rule *)
|
huffman@44981
|
1549 |
apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
|
huffman@44981
|
1550 |
apply(rule,rule)
|
huffman@44981
|
1551 |
apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
|
huffman@45314
|
1552 |
apply(rule `a\<in>s`) by auto
|
huffman@44981
|
1553 |
qed auto
|
hoelzl@33741
|
1554 |
|
huffman@44981
|
1555 |
lemma has_antiderivative_limit:
|
huffman@44981
|
1556 |
fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
|
huffman@44981
|
1557 |
assumes "convex s"
|
huffman@44981
|
1558 |
assumes "\<forall>e>0. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> e * norm(h))"
|
huffman@44981
|
1559 |
shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
|
huffman@44981
|
1560 |
proof-
|
hoelzl@33741
|
1561 |
have *:"\<forall>n. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm(h))"
|
huffman@44981
|
1562 |
apply(rule) using assms(2)
|
huffman@44981
|
1563 |
apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
|
huffman@44981
|
1564 |
guess f using *[THEN choice] .. note * = this
|
huffman@44981
|
1565 |
guess f' using *[THEN choice] .. note f=this
|
huffman@44981
|
1566 |
show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer
|
huffman@44981
|
1567 |
proof(rule,rule)
|
hoelzl@33741
|
1568 |
fix e::real assume "0<e" guess N using reals_Archimedean[OF `e>0`] .. note N=this
|
huffman@44981
|
1569 |
show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
|
huffman@44981
|
1570 |
apply(rule_tac x=N in exI)
|
huffman@44981
|
1571 |
proof(default+)
|
huffman@44981
|
1572 |
case goal1
|
hoelzl@33741
|
1573 |
have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
|
wenzelm@42829
|
1574 |
using goal1(1) by(auto simp add:field_simps)
|
huffman@44981
|
1575 |
show ?case
|
huffman@44981
|
1576 |
using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
|
huffman@44981
|
1577 |
apply(rule order_trans) using N * apply(cases "h=0") by auto
|
huffman@44981
|
1578 |
qed
|
huffman@44981
|
1579 |
qed(insert f,auto)
|
huffman@44981
|
1580 |
qed
|
hoelzl@33741
|
1581 |
|
hoelzl@33741
|
1582 |
subsection {* Differentiation of a series. *}
|
hoelzl@33741
|
1583 |
|
hoelzl@33741
|
1584 |
definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
|
hoelzl@33741
|
1585 |
(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
|
hoelzl@33741
|
1586 |
|
huffman@44981
|
1587 |
lemma has_derivative_series:
|
huffman@44981
|
1588 |
fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
|
huffman@44981
|
1589 |
assumes "convex s"
|
huffman@44981
|
1590 |
assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
|
huffman@44981
|
1591 |
assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(setsum (\<lambda>i. f' i x h) (k \<inter> {0..n}) - g' x h) \<le> e * norm(h)"
|
huffman@44981
|
1592 |
assumes "x\<in>s" and "((\<lambda>n. f n x) sums_seq l) k"
|
hoelzl@33741
|
1593 |
shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) sums_seq (g x)) k \<and> (g has_derivative g'(x)) (at x within s)"
|
huffman@44981
|
1594 |
unfolding sums_seq_def
|
huffman@44981
|
1595 |
apply(rule has_derivative_sequence[OF assms(1) _ assms(3)])
|
huffman@44981
|
1596 |
apply(rule,rule)
|
huffman@44981
|
1597 |
apply(rule has_derivative_setsum) defer
|
huffman@44981
|
1598 |
apply(rule,rule assms(2)[rule_format],assumption)
|
hoelzl@33741
|
1599 |
using assms(4-5) unfolding sums_seq_def by auto
|
hoelzl@33741
|
1600 |
|
hoelzl@33741
|
1601 |
subsection {* Derivative with composed bilinear function. *}
|
hoelzl@33741
|
1602 |
|
huffman@37650
|
1603 |
lemma has_derivative_bilinear_within:
|
huffman@44981
|
1604 |
assumes "(f has_derivative f') (at x within s)"
|
huffman@44981
|
1605 |
assumes "(g has_derivative g') (at x within s)"
|
huffman@44981
|
1606 |
assumes "bounded_bilinear h"
|
huffman@44981
|
1607 |
shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)"
|
huffman@44981
|
1608 |
proof-
|
huffman@44981
|
1609 |
have "(g ---> g x) (at x within s)"
|
huffman@44981
|
1610 |
apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
|
huffman@44981
|
1611 |
using assms(2) unfolding differentiable_def by auto
|
huffman@44981
|
1612 |
moreover
|
huffman@44981
|
1613 |
interpret f':bounded_linear f'
|
huffman@44981
|
1614 |
using assms unfolding has_derivative_def by auto
|
huffman@44981
|
1615 |
interpret g':bounded_linear g'
|
huffman@44981
|
1616 |
using assms unfolding has_derivative_def by auto
|
huffman@44981
|
1617 |
interpret h:bounded_bilinear h
|
huffman@44981
|
1618 |
using assms by auto
|
huffman@44981
|
1619 |
have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)"
|
huffman@44981
|
1620 |
unfolding f'.zero[THEN sym]
|
huffman@44983
|
1621 |
using bounded_linear.tendsto [of f' "\<lambda>y. y - x" 0 "at x within s"]
|
huffman@44983
|
1622 |
using tendsto_diff [OF Lim_within_id tendsto_const, of x x s]
|
hoelzl@33741
|
1623 |
unfolding id_def using assms(1) unfolding has_derivative_def by auto
|
hoelzl@33741
|
1624 |
hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
|
huffman@44983
|
1625 |
using tendsto_add[OF tendsto_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"]
|
huffman@44981
|
1626 |
by auto
|
huffman@44981
|
1627 |
ultimately
|
hoelzl@33741
|
1628 |
have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
|
hoelzl@33741
|
1629 |
+ h ((1/ (norm (x' - x))) *\<^sub>R (f x' - (f x + f' (x' - x)))) (g x')) ---> h (f x) 0 + h 0 (g x)) (at x within s)"
|
huffman@44983
|
1630 |
apply-apply(rule tendsto_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)])
|
huffman@44981
|
1631 |
using assms(1-2) unfolding has_derivative_within by auto
|
hoelzl@33741
|
1632 |
guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
|
hoelzl@33741
|
1633 |
guess C using f'.pos_bounded .. note C=this
|
hoelzl@33741
|
1634 |
guess D using g'.pos_bounded .. note D=this
|
hoelzl@33741
|
1635 |
have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
|
huffman@44981
|
1636 |
have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)"
|
huffman@44981
|
1637 |
unfolding Lim_within
|
huffman@44981
|
1638 |
proof(rule,rule) case goal1
|
hoelzl@33741
|
1639 |
hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
|
huffman@44981
|
1640 |
thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule)
|
huffman@44981
|
1641 |
proof(rule,rule,erule conjE)
|
hoelzl@33741
|
1642 |
fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
|
hoelzl@33741
|
1643 |
have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
|
huffman@44981
|
1644 |
also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B"
|
huffman@44981
|
1645 |
apply(rule mult_right_mono)
|
huffman@44981
|
1646 |
apply(rule mult_mono) using B C D
|
huffman@44981
|
1647 |
by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
|
huffman@44981
|
1648 |
also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)"
|
huffman@44981
|
1649 |
by (auto simp add: field_simps)
|
huffman@44981
|
1650 |
also have "\<dots> < e * norm (y - x)"
|
huffman@44981
|
1651 |
apply(rule mult_strict_right_mono)
|
huffman@44981
|
1652 |
using as(3)[unfolded dist_norm] and as(2)
|
huffman@44981
|
1653 |
unfolding pos_less_divide_eq[OF bcd] by (auto simp add: field_simps)
|
hoelzl@33741
|
1654 |
finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
|
huffman@44981
|
1655 |
unfolding dist_norm apply-apply(cases "y = x")
|
huffman@44981
|
1656 |
by(auto simp add: field_simps)
|
huffman@44981
|
1657 |
qed
|
huffman@44981
|
1658 |
qed
|
huffman@37650
|
1659 |
have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))"
|
huffman@37650
|
1660 |
apply (rule bounded_linear_add)
|
huffman@37650
|
1661 |
apply (rule bounded_linear_compose [OF h.bounded_linear_right `bounded_linear g'`])
|
huffman@37650
|
1662 |
apply (rule bounded_linear_compose [OF h.bounded_linear_left `bounded_linear f'`])
|
huffman@37650
|
1663 |
done
|
huffman@44983
|
1664 |
thus ?thesis using tendsto_add[OF * **] unfolding has_derivative_within
|
hoelzl@37489
|
1665 |
unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
|
hoelzl@33741
|
1666 |
h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
|
huffman@44981
|
1667 |
scaleR_right_diff_distrib h.zero_right h.zero_left
|
huffman@44981
|
1668 |
by(auto simp add:field_simps)
|
huffman@44981
|
1669 |
qed
|
hoelzl@33741
|
1670 |
|
huffman@37650
|
1671 |
lemma has_derivative_bilinear_at:
|
huffman@44981
|
1672 |
assumes "(f has_derivative f') (at x)"
|
huffman@44981
|
1673 |
assumes "(g has_derivative g') (at x)"
|
huffman@44981
|
1674 |
assumes "bounded_bilinear h"
|
hoelzl@33741
|
1675 |
shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
|
huffman@45896
|
1676 |
using has_derivative_bilinear_within[of f f' x UNIV g g' h] assms by simp
|
hoelzl@33741
|
1677 |
|
hoelzl@37489
|
1678 |
subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
|
hoelzl@33741
|
1679 |
|
huffman@44952
|
1680 |
definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real filter \<Rightarrow> bool)"
|
hoelzl@33741
|
1681 |
(infixl "has'_vector'_derivative" 12) where
|
hoelzl@33741
|
1682 |
"(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
|
hoelzl@33741
|
1683 |
|
hoelzl@33741
|
1684 |
definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
|
hoelzl@33741
|
1685 |
|
huffman@44981
|
1686 |
lemma vector_derivative_works:
|
huffman@44981
|
1687 |
fixes f::"real \<Rightarrow> 'a::real_normed_vector"
|
hoelzl@33741
|
1688 |
shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
|
huffman@44981
|
1689 |
proof
|
huffman@44981
|
1690 |
assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
|
hoelzl@33741
|
1691 |
then interpret bounded_linear f' by auto
|
wenzelm@47770
|
1692 |
show ?r unfolding vector_derivative_def has_vector_derivative_def
|
hoelzl@33741
|
1693 |
apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
|
hoelzl@33741
|
1694 |
using f' unfolding scaleR[THEN sym] by auto
|
huffman@44981
|
1695 |
next
|
huffman@44981
|
1696 |
assume ?r thus ?l
|
huffman@44981
|
1697 |
unfolding vector_derivative_def has_vector_derivative_def differentiable_def
|
huffman@44981
|
1698 |
by auto
|
huffman@44981
|
1699 |
qed
|
hoelzl@33741
|
1700 |
|
huffman@37729
|
1701 |
lemma vector_derivative_unique_at:
|
huffman@37729
|
1702 |
assumes "(f has_vector_derivative f') (at x)"
|
huffman@37729
|
1703 |
assumes "(f has_vector_derivative f'') (at x)"
|
huffman@37729
|
1704 |
shows "f' = f''"
|
huffman@37729
|
1705 |
proof-
|
huffman@37729
|
1706 |
have "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
|
huffman@37729
|
1707 |
using assms [unfolded has_vector_derivative_def]
|
huffman@37729
|
1708 |
by (rule frechet_derivative_unique_at)
|
nipkow@39535
|
1709 |
thus ?thesis unfolding fun_eq_iff by auto
|
huffman@37729
|
1710 |
qed
|
hoelzl@33741
|
1711 |
|
huffman@44981
|
1712 |
lemma vector_derivative_unique_within_closed_interval:
|
huffman@44981
|
1713 |
fixes f::"real \<Rightarrow> 'n::ordered_euclidean_space"
|
huffman@44981
|
1714 |
assumes "a < b" and "x \<in> {a..b}"
|
huffman@44981
|
1715 |
assumes "(f has_vector_derivative f') (at x within {a..b})"
|
huffman@44981
|
1716 |
assumes "(f has_vector_derivative f'') (at x within {a..b})"
|
huffman@44981
|
1717 |
shows "f' = f''"
|
huffman@44981
|
1718 |
proof-
|
hoelzl@37489
|
1719 |
have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
|
hoelzl@37489
|
1720 |
apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
|
huffman@44981
|
1721 |
using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2)
|
huffman@44981
|
1722 |
by auto
|
huffman@44981
|
1723 |
show ?thesis
|
huffman@44981
|
1724 |
proof(rule ccontr)
|
huffman@44981
|
1725 |
assume "f' \<noteq> f''"
|
huffman@44981
|
1726 |
moreover
|
huffman@44981
|
1727 |
hence "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1"
|
huffman@44981
|
1728 |
using * by (auto simp: fun_eq_iff)
|
huffman@44981
|
1729 |
ultimately show False unfolding o_def by auto
|
huffman@44981
|
1730 |
qed
|
huffman@44981
|
1731 |
qed
|
hoelzl@33741
|
1732 |
|
huffman@37729
|
1733 |
lemma vector_derivative_at:
|
huffman@37729
|
1734 |
shows "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
|
hoelzl@33741
|
1735 |
apply(rule vector_derivative_unique_at) defer apply assumption
|
hoelzl@33741
|
1736 |
unfolding vector_derivative_works[THEN sym] differentiable_def
|
hoelzl@33741
|
1737 |
unfolding has_vector_derivative_def by auto
|
hoelzl@33741
|
1738 |
|
huffman@44981
|
1739 |
lemma vector_derivative_within_closed_interval:
|
huffman@44981
|
1740 |
fixes f::"real \<Rightarrow> 'a::ordered_euclidean_space"
|
huffman@44981
|
1741 |
assumes "a < b" and "x \<in> {a..b}"
|
huffman@44981
|
1742 |
assumes "(f has_vector_derivative f') (at x within {a..b})"
|
hoelzl@33741
|
1743 |
shows "vector_derivative f (at x within {a..b}) = f'"
|
hoelzl@33741
|
1744 |
apply(rule vector_derivative_unique_within_closed_interval)
|
hoelzl@33741
|
1745 |
using vector_derivative_works[unfolded differentiable_def]
|
hoelzl@33741
|
1746 |
using assms by(auto simp add:has_vector_derivative_def)
|
hoelzl@33741
|
1747 |
|
himmelma@34968
|
1748 |
lemma has_vector_derivative_within_subset:
|
hoelzl@33741
|
1749 |
"(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
|
hoelzl@33741
|
1750 |
unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
|
hoelzl@33741
|
1751 |
|
himmelma@34968
|
1752 |
lemma has_vector_derivative_const:
|
hoelzl@33741
|
1753 |
"((\<lambda>x. c) has_vector_derivative 0) net"
|
hoelzl@33741
|
1754 |
unfolding has_vector_derivative_def using has_derivative_const by auto
|
hoelzl@33741
|
1755 |
|
hoelzl@33741
|
1756 |
lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
|
hoelzl@33741
|
1757 |
unfolding has_vector_derivative_def using has_derivative_id by auto
|
hoelzl@33741
|
1758 |
|
huffman@44981
|
1759 |
lemma has_vector_derivative_cmul:
|
huffman@44981
|
1760 |
"(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
|
huffman@45011
|
1761 |
unfolding has_vector_derivative_def
|
huffman@45145
|
1762 |
apply (drule scaleR_right_has_derivative)
|
huffman@44981
|
1763 |
by (auto simp add: algebra_simps)
|
hoelzl@33741
|
1764 |
|
huffman@44981
|
1765 |
lemma has_vector_derivative_cmul_eq:
|
huffman@44981
|
1766 |
assumes "c \<noteq> 0"
|
hoelzl@33741
|
1767 |
shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
|
hoelzl@33741
|
1768 |
apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
|
hoelzl@33741
|
1769 |
apply(rule has_vector_derivative_cmul) using assms by auto
|
hoelzl@33741
|
1770 |
|
hoelzl@33741
|
1771 |
lemma has_vector_derivative_neg:
|
huffman@44981
|
1772 |
"(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
|
hoelzl@33741
|
1773 |
unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
|
hoelzl@33741
|
1774 |
|
hoelzl@33741
|
1775 |
lemma has_vector_derivative_add:
|
huffman@44981
|
1776 |
assumes "(f has_vector_derivative f') net"
|
huffman@44981
|
1777 |
assumes "(g has_vector_derivative g') net"
|
hoelzl@33741
|
1778 |
shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
|
hoelzl@33741
|
1779 |
using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
|
hoelzl@33741
|
1780 |
unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
|
hoelzl@33741
|
1781 |
|
hoelzl@33741
|
1782 |
lemma has_vector_derivative_sub:
|
huffman@44981
|
1783 |
assumes "(f has_vector_derivative f') net"
|
huffman@44981
|
1784 |
assumes "(g has_vector_derivative g') net"
|
hoelzl@33741
|
1785 |
shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
|
hoelzl@33741
|
1786 |
using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
|
hoelzl@33741
|
1787 |
unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
|
hoelzl@33741
|
1788 |
|
huffman@37650
|
1789 |
lemma has_vector_derivative_bilinear_within:
|
huffman@44981
|
1790 |
assumes "(f has_vector_derivative f') (at x within s)"
|
huffman@44981
|
1791 |
assumes "(g has_vector_derivative g') (at x within s)"
|
huffman@44981
|
1792 |
assumes "bounded_bilinear h"
|
huffman@44981
|
1793 |
shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)"
|
huffman@44981
|
1794 |
proof-
|
hoelzl@33741
|
1795 |
interpret bounded_bilinear h using assms by auto
|
hoelzl@37489
|
1796 |
show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
|
hoelzl@37489
|
1797 |
unfolding o_def has_vector_derivative_def
|
huffman@44981
|
1798 |
using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib
|
huffman@44981
|
1799 |
by auto
|
huffman@44981
|
1800 |
qed
|
hoelzl@33741
|
1801 |
|
huffman@37650
|
1802 |
lemma has_vector_derivative_bilinear_at:
|
huffman@44981
|
1803 |
assumes "(f has_vector_derivative f') (at x)"
|
huffman@44981
|
1804 |
assumes "(g has_vector_derivative g') (at x)"
|
huffman@44981
|
1805 |
assumes "bounded_bilinear h"
|
hoelzl@33741
|
1806 |
shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
|
huffman@45896
|
1807 |
using has_vector_derivative_bilinear_within[where s=UNIV] assms by simp
|
hoelzl@33741
|
1808 |
|
huffman@44981
|
1809 |
lemma has_vector_derivative_at_within:
|
huffman@44981
|
1810 |
"(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
|
huffman@44981
|
1811 |
unfolding has_vector_derivative_def
|
huffman@45896
|
1812 |
by (rule has_derivative_at_within)
|
hoelzl@33741
|
1813 |
|
hoelzl@33741
|
1814 |
lemma has_vector_derivative_transform_within:
|
huffman@44981
|
1815 |
assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
|
huffman@44981
|
1816 |
assumes "(f has_vector_derivative f') (at x within s)"
|
hoelzl@33741
|
1817 |
shows "(g has_vector_derivative f') (at x within s)"
|
huffman@44981
|
1818 |
using assms unfolding has_vector_derivative_def
|
huffman@44981
|
1819 |
by (rule has_derivative_transform_within)
|
hoelzl@33741
|
1820 |
|
hoelzl@33741
|
1821 |
lemma has_vector_derivative_transform_at:
|
huffman@44981
|
1822 |
assumes "0 < d" and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
|
huffman@44981
|
1823 |
assumes "(f has_vector_derivative f') (at x)"
|
hoelzl@33741
|
1824 |
shows "(g has_vector_derivative f') (at x)"
|
huffman@44981
|
1825 |
using assms unfolding has_vector_derivative_def
|
huffman@44981
|
1826 |
by (rule has_derivative_transform_at)
|
hoelzl@33741
|
1827 |
|
hoelzl@33741
|
1828 |
lemma has_vector_derivative_transform_within_open:
|
huffman@44981
|
1829 |
assumes "open s" and "x \<in> s" and "\<forall>y\<in>s. f y = g y"
|
huffman@44981
|
1830 |
assumes "(f has_vector_derivative f') (at x)"
|
hoelzl@33741
|
1831 |
shows "(g has_vector_derivative f') (at x)"
|
huffman@44981
|
1832 |
using assms unfolding has_vector_derivative_def
|
huffman@44981
|
1833 |
by (rule has_derivative_transform_within_open)
|
hoelzl@33741
|
1834 |
|
hoelzl@33741
|
1835 |
lemma vector_diff_chain_at:
|
huffman@44981
|
1836 |
assumes "(f has_vector_derivative f') (at x)"
|
huffman@44981
|
1837 |
assumes "(g has_vector_derivative g') (at (f x))"
|
hoelzl@33741
|
1838 |
shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
|
huffman@44981
|
1839 |
using assms(2) unfolding has_vector_derivative_def apply-
|
huffman@44981
|
1840 |
apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
|
huffman@45145
|
1841 |
unfolding o_def real_scaleR_def scaleR_scaleR .
|
hoelzl@33741
|
1842 |
|
hoelzl@33741
|
1843 |
lemma vector_diff_chain_within:
|
huffman@44981
|
1844 |
assumes "(f has_vector_derivative f') (at x within s)"
|
huffman@44981
|
1845 |
assumes "(g has_vector_derivative g') (at (f x) within f ` s)"
|
hoelzl@33741
|
1846 |
shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
|
huffman@44981
|
1847 |
using assms(2) unfolding has_vector_derivative_def apply-
|
huffman@44981
|
1848 |
apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
|
huffman@45145
|
1849 |
unfolding o_def real_scaleR_def scaleR_scaleR .
|
hoelzl@33741
|
1850 |
|
hoelzl@33741
|
1851 |
end
|