--- a/src/HOL/Multivariate_Analysis/Derivative.thy Sat Sep 21 20:58:32 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy Sat Sep 21 22:33:42 2013 +0200
@@ -1,19 +1,21 @@
-(* Title: HOL/Multivariate_Analysis/Derivative.thy
- Author: John Harrison
- Translation from HOL Light: Robert Himmelmann, TU Muenchen
+(* Title: HOL/Multivariate_Analysis/Derivative.thy
+ Author: John Harrison
+ Author: Robert Himmelmann, TU Muenchen (translation from HOL Light)
*)
-header {* Multivariate calculus in Euclidean space. *}
+header {* Multivariate calculus in Euclidean space *}
theory Derivative
imports Brouwer_Fixpoint Operator_Norm
begin
-lemma bounded_linear_imp_linear: "bounded_linear f \<Longrightarrow> linear f" (* TODO: move elsewhere *)
+lemma bounded_linear_imp_linear: (* TODO: move elsewhere *)
+ assumes "bounded_linear f"
+ shows "linear f"
proof -
- assume "bounded_linear f"
- then interpret f: bounded_linear f .
- show "linear f"
+ interpret f: bounded_linear f
+ using assms .
+ show ?thesis
by (simp add: f.add f.scaleR linear_iff)
qed
@@ -28,25 +30,23 @@
apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
apply (simp add: norm_sgn sgn_zero_iff x)
done
- thus ?thesis
+ then show ?thesis
by (rule netlimit_within [of a UNIV])
qed simp
(* Because I do not want to type this all the time *)
-lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
+lemmas linear_linear = linear_conv_bounded_linear[symmetric]
-lemma derivative_linear[dest]:
- "(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
+lemma derivative_linear[dest]: "(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
unfolding has_derivative_def by auto
-lemma derivative_is_linear:
- "(f has_derivative f') net \<Longrightarrow> linear f'"
+lemma derivative_is_linear: "(f has_derivative f') net \<Longrightarrow> linear f'"
by (rule derivative_linear [THEN bounded_linear_imp_linear])
-lemma DERIV_conv_has_derivative:
- "(DERIV f x :> f') = (f has_derivative op * f') (at x)"
+lemma DERIV_conv_has_derivative: "(DERIV f x :> f') \<longleftrightarrow> (f has_derivative op * f') (at x)"
using deriv_fderiv[of f x UNIV f'] by (subst (asm) mult_commute)
+
subsection {* Derivatives *}
subsubsection {* Combining theorems. *}
@@ -68,56 +68,66 @@
"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
by (intro FDERIV_eq_intros) auto
+
subsection {* Derivative with composed bilinear function. *}
lemma has_derivative_bilinear_within:
assumes "(f has_derivative f') (at x within s)"
- assumes "(g has_derivative g') (at x within s)"
- assumes "bounded_bilinear h"
+ and "(g has_derivative g') (at x within s)"
+ and "bounded_bilinear h"
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)"
using bounded_bilinear.FDERIV[OF assms(3,1,2)] .
lemma has_derivative_bilinear_at:
assumes "(f has_derivative f') (at x)"
- assumes "(g has_derivative g') (at x)"
- assumes "bounded_bilinear h"
+ and "(g has_derivative g') (at x)"
+ and "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
using has_derivative_bilinear_within[of f f' x UNIV g g' h] assms by simp
text {* These are the only cases we'll care about, probably. *}
lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
- bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
- unfolding has_derivative_def Lim by (auto simp add: netlimit_within inverse_eq_divide field_simps)
+ bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
+ unfolding has_derivative_def Lim
+ by (auto simp add: netlimit_within inverse_eq_divide field_simps)
lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
- bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
- using has_derivative_within [of f f' x UNIV] by simp
+ bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
+ using has_derivative_within [of f f' x UNIV]
+ by simp
text {* More explicit epsilon-delta forms. *}
lemma has_derivative_within':
- "(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
- (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
- \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
+ "(f has_derivative f')(at x within s) \<longleftrightarrow>
+ bounded_linear f' \<and>
+ (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
+ norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
unfolding has_derivative_within Lim_within dist_norm
- unfolding diff_0_right by (simp add: diff_diff_eq)
+ unfolding diff_0_right
+ by (simp add: diff_diff_eq)
lemma has_derivative_at':
- "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
- (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
- \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
- using has_derivative_within' [of f f' x UNIV] by simp
+ "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
+ (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
+ norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
+ using has_derivative_within' [of f f' x UNIV]
+ by simp
-lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
- unfolding has_derivative_within' has_derivative_at' by blast
+lemma has_derivative_at_within:
+ "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
+ unfolding has_derivative_within' has_derivative_at'
+ by blast
lemma has_derivative_within_open:
- "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
+ "a \<in> s \<Longrightarrow> open s \<Longrightarrow>
+ (f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a)"
by (simp only: at_within_interior interior_open)
lemma has_derivative_right:
- fixes f :: "real \<Rightarrow> real" and y :: "real"
+ fixes f :: "real \<Rightarrow> real"
+ and y :: "real"
shows "(f has_derivative (op * y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x <..} \<inter> I))"
proof -
@@ -132,91 +142,154 @@
by (simp add: bounded_linear_mult_right has_derivative_within)
qed
+
subsubsection {* Limit transformation for derivatives *}
lemma has_derivative_transform_within:
- 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)"
+ assumes "0 < d"
+ and "x \<in> s"
+ and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+ and "(f has_derivative f') (at x within s)"
shows "(g has_derivative f') (at x within s)"
- using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
- apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
- apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
+ using assms(4)
+ unfolding has_derivative_within
+ apply -
+ apply (erule conjE)
+ apply rule
+ apply assumption
+ apply (rule Lim_transform_within[OF assms(1)])
+ defer
+ apply assumption
+ apply rule
+ apply rule
+ apply (drule assms(3)[rule_format])
+ using assms(3)[rule_format, OF assms(2)]
+ apply auto
+ done
lemma has_derivative_transform_at:
- assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
+ assumes "0 < d"
+ and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
+ and "(f has_derivative f') (at x)"
shows "(g has_derivative f') (at x)"
- using has_derivative_transform_within [of d x UNIV f g f'] assms by simp
+ using has_derivative_transform_within [of d x UNIV f g f'] assms
+ by simp
lemma has_derivative_transform_within_open:
- assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
+ assumes "open s"
+ and "x \<in> s"
+ and "\<forall>y\<in>s. f y = g y"
+ and "(f has_derivative f') (at x)"
shows "(g has_derivative f') (at x)"
- using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
- apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
- apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
+ using assms(4)
+ unfolding has_derivative_at
+ apply -
+ apply (erule conjE)
+ apply rule
+ apply assumption
+ apply (rule Lim_transform_within_open[OF assms(1,2)])
+ defer
+ apply assumption
+ apply rule
+ apply rule
+ apply (drule assms(3)[rule_format])
+ using assms(3)[rule_format, OF assms(2)]
+ apply auto
+ done
subsection {* Differentiability *}
no_notation Deriv.differentiable (infixl "differentiable" 60)
-abbreviation differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" (infixr "differentiable" 30) where
- "f differentiable net \<equiv> isDiff net f"
+abbreviation
+ differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
+ (infixr "differentiable" 30)
+ where "f differentiable net \<equiv> isDiff net f"
-definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
- "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
+definition
+ differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool"
+ (infixr "differentiable'_on" 30)
+ where "f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable (at x within s))"
lemmas differentiable_def = isDiff_def
lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
- unfolding differentiable_def by auto
+ unfolding differentiable_def
+ by auto
lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
- unfolding differentiable_def using has_derivative_at_within by blast
+ unfolding differentiable_def
+ using has_derivative_at_within
+ by blast
lemma differentiable_within_open: (* TODO: delete *)
- assumes "a \<in> s" and "open s"
- shows "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
- using assms by (simp only: at_within_interior interior_open)
+ assumes "a \<in> s"
+ and "open s"
+ shows "f differentiable (at a within s) \<longleftrightarrow> f differentiable (at a)"
+ using assms
+ by (simp only: at_within_interior interior_open)
lemma differentiable_on_eq_differentiable_at:
- "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
+ "open s \<Longrightarrow> f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x)"
unfolding differentiable_on_def
by (metis at_within_interior interior_open)
lemma differentiable_transform_within:
- assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+ assumes "0 < d"
+ and "x \<in> s"
+ and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
assumes "f differentiable (at x within s)"
shows "g differentiable (at x within s)"
- using assms(4) unfolding differentiable_def
+ using assms(4)
+ unfolding differentiable_def
by (auto intro!: has_derivative_transform_within[OF assms(1-3)])
lemma differentiable_transform_at:
- assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
+ assumes "0 < d"
+ and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
+ and "f differentiable at x"
shows "g differentiable at x"
- using assms(3) unfolding differentiable_def
- using has_derivative_transform_at[OF assms(1-2)] by auto
+ using assms(3)
+ unfolding differentiable_def
+ using has_derivative_transform_at[OF assms(1-2)]
+ by auto
-subsection {* Frechet derivative and Jacobian matrix. *}
+
+subsection {* Frechet derivative and Jacobian matrix *}
definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
lemma frechet_derivative_works:
- "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
- unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
+ "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
+ unfolding frechet_derivative_def differentiable_def
+ unfolding some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
-lemma linear_frechet_derivative:
- shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
+lemma linear_frechet_derivative: "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
unfolding frechet_derivative_works has_derivative_def
by (auto intro: bounded_linear_imp_linear)
+
subsection {* Differentiability implies continuity *}
lemma Lim_mul_norm_within:
- fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
- unfolding Lim_within apply(rule,rule)
- apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
- apply(rule_tac x="min d 1" in exI) apply rule defer
- apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
- by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
+ fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
+ shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f x) ---> 0) (at a within s)"
+ unfolding Lim_within
+ apply rule
+ apply rule
+ apply (erule_tac x=e in allE)
+ apply (erule impE)
+ apply assumption
+ apply (erule exE)
+ apply (erule conjE)
+ apply (rule_tac x="min d 1" in exI)
+ apply rule
+ defer
+ apply rule
+ apply (erule_tac x=x in ballE)
+ unfolding dist_norm diff_0_right
+ apply (auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
+ done
lemma differentiable_imp_continuous_within:
"f differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
@@ -228,91 +301,162 @@
using differentiable_imp_continuous_within by blast
lemma has_derivative_within_subset:
- "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
- unfolding has_derivative_within using tendsto_within_subset by blast
+ "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
+ (f has_derivative f') (at x within t)"
+ unfolding has_derivative_within
+ using tendsto_within_subset
+ by blast
lemma differentiable_within_subset:
- "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
- unfolding differentiable_def using has_derivative_within_subset by blast
+ "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow>
+ f differentiable (at x within s)"
+ unfolding differentiable_def
+ using has_derivative_within_subset
+ by blast
lemma differentiable_on_subset:
"f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
- unfolding differentiable_on_def using differentiable_within_subset by blast
+ unfolding differentiable_on_def
+ using differentiable_within_subset
+ by blast
lemma differentiable_on_empty: "f differentiable_on {}"
- unfolding differentiable_on_def by auto
+ unfolding differentiable_on_def
+ by auto
text {* Several results are easier using a "multiplied-out" variant.
(I got this idea from Dieudonne's proof of the chain rule). *}
lemma has_derivative_within_alt:
- "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
- (\<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")
+ "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
+ (\<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")
proof
- assume ?lhs thus ?rhs
- unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
+ assume ?lhs
+ then show ?rhs
+ unfolding has_derivative_within
+ apply -
+ apply (erule conjE)
+ apply rule
+ apply assumption
unfolding Lim_within
- apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
- apply(erule exE,rule_tac x=d in exI)
- apply(erule conjE,rule,assumption,rule,rule)
+ apply rule
+ apply (erule_tac x=e in allE)
+ apply rule
+ apply (erule impE)
+ apply assumption
+ apply (erule exE)
+ apply (rule_tac x=d in exI)
+ apply (erule conjE)
+ apply rule
+ apply assumption
+ apply rule
+ apply rule
proof-
- 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>
- dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
- then interpret bounded_linear f' by auto
- show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
- case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero)
+ 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>
+ dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e"
+ "y \<in> s"
+ "bounded_linear f'"
+ then interpret bounded_linear f'
+ by auto
+ show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)"
+ proof (cases "y = x")
+ case True
+ then show ?thesis
+ using `bounded_linear f'` by (auto simp add: zero)
next
- case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
- unfolding dist_norm diff_0_right using as(3)
+ case False
+ then have "norm (f y - (f x + f' (y - x))) < e * norm (y - x)"
+ using as(4)[rule_format, OF `y \<in> s`]
+ unfolding dist_norm diff_0_right
+ using as(3)
using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded dist_norm]
by (auto simp add: linear_0 linear_sub)
- thus ?thesis by(auto simp add:algebra_simps)
+ then show ?thesis
+ by (auto simp add: algebra_simps)
qed
qed
next
- assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within
- apply-apply(erule conjE,rule,assumption)
- apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer
- apply(erule exE,rule_tac x=d in exI)
- apply(erule conjE,rule,assumption,rule,rule)
+ assume ?rhs
+ then show ?lhs
+ unfolding has_derivative_within Lim_within
+ apply -
+ apply (erule conjE)
+ apply rule
+ apply assumption
+ apply rule
+ apply (erule_tac x="e/2" in allE)
+ apply rule
+ apply (erule impE)
+ defer
+ apply (erule exE)
+ apply (rule_tac x=d in exI)
+ apply (erule conjE)
+ apply rule
+ apply assumption
+ apply rule
+ apply rule
unfolding dist_norm diff_0_right norm_scaleR
- apply(erule_tac x=xa in ballE,erule impE)
- proof-
- fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
- "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
- thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
- apply(rule_tac le_less_trans[of _ "e/2"])
- by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps)
+ apply (erule_tac x=xa in ballE)
+ apply (erule impE)
+ proof -
+ fix e d y
+ assume "bounded_linear f'"
+ and "0 < e"
+ and "0 < d"
+ and "y \<in> s"
+ and "0 < norm (y - x) \<and> norm (y - x) < d"
+ and "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
+ then show "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
+ apply (rule_tac le_less_trans[of _ "e/2"])
+ apply (auto intro!: mult_imp_div_pos_le simp add: algebra_simps)
+ done
qed auto
qed
lemma has_derivative_at_alt:
- "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
- (\<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))"
- using has_derivative_within_alt[where s=UNIV] by simp
+ "(f has_derivative f') (at x) \<longleftrightarrow>
+ bounded_linear f' \<and>
+ (\<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))"
+ using has_derivative_within_alt[where s=UNIV]
+ by simp
-subsection {* The chain rule. *}
+
+subsection {* The chain rule *}
lemma diff_chain_within[FDERIV_intros]:
assumes "(f has_derivative f') (at x within s)"
- assumes "(g has_derivative g') (at (f x) within (f ` s))"
- shows "((g o f) has_derivative (g' o f'))(at x within s)"
- using FDERIV_in_compose[OF assms] by (simp add: comp_def)
+ and "(g has_derivative g') (at (f x) within (f ` s))"
+ shows "((g \<circ> f) has_derivative (g' \<circ> f'))(at x within s)"
+ using FDERIV_in_compose[OF assms]
+ by (simp add: comp_def)
lemma diff_chain_at[FDERIV_intros]:
- "(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)"
- using FDERIV_compose[of f f' x UNIV g g'] by (simp add: comp_def)
+ "(f has_derivative f') (at x) \<Longrightarrow>
+ (g has_derivative g') (at (f x)) \<Longrightarrow> ((g \<circ> f) has_derivative (g' \<circ> f')) (at x)"
+ using FDERIV_compose[of f f' x UNIV g g']
+ by (simp add: comp_def)
-subsection {* Composition rules stated just for differentiability. *}
+subsection {* Composition rules stated just for differentiability *}
lemma differentiable_chain_at:
- "f differentiable (at x) \<Longrightarrow> g differentiable (at (f x)) \<Longrightarrow> (g o f) differentiable (at x)"
- unfolding differentiable_def by(meson diff_chain_at)
+ "f differentiable (at x) \<Longrightarrow>
+ g differentiable (at (f x)) \<Longrightarrow> (g \<circ> f) differentiable (at x)"
+ unfolding differentiable_def
+ by (meson diff_chain_at)
lemma differentiable_chain_within:
- "f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s)) \<Longrightarrow> (g o f) differentiable (at x within s)"
- unfolding differentiable_def by(meson diff_chain_within)
+ "f differentiable (at x within s) \<Longrightarrow>
+ g differentiable (at(f x) within (f ` s)) \<Longrightarrow> (g \<circ> f) differentiable (at x within s)"
+ unfolding differentiable_def
+ by (meson diff_chain_within)
+
subsection {* Uniqueness of derivative *}
@@ -324,95 +468,127 @@
lemma frechet_derivative_unique_within:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "(f has_derivative f') (at x within s)"
- assumes "(f has_derivative f'') (at x within s)"
- assumes "(\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R i) \<in> s)"
+ and "(f has_derivative f'') (at x within s)"
+ and "\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < abs d \<and> abs d < e \<and> (x + d *\<^sub>R i) \<in> s"
shows "f' = f''"
-proof-
+proof -
note as = assms(1,2)[unfolded has_derivative_def]
then interpret f': bounded_linear f' by auto
from as interpret f'': bounded_linear f'' by auto
have "x islimpt s" unfolding islimpt_approachable
- proof(rule,rule)
- fix e::real assume "0<e" guess d
- using assms(3)[rule_format,OF SOME_Basis `e>0`] ..
- thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
- apply(rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI)
- unfolding dist_norm by (auto simp: SOME_Basis nonzero_Basis)
+ proof (rule, rule)
+ fix e :: real
+ assume "e > 0"
+ guess d using assms(3)[rule_format,OF SOME_Basis `e>0`] ..
+ then show "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
+ apply (rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI)
+ unfolding dist_norm
+ apply (auto simp: SOME_Basis nonzero_Basis)
+ done
qed
- hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within)
- unfolding trivial_limit_within by simp
- show ?thesis apply(rule linear_eq_stdbasis)
+ then have *: "netlimit (at x within s) = x"
+ apply -
+ apply (rule netlimit_within)
+ unfolding trivial_limit_within
+ apply simp
+ done
+ show ?thesis
+ apply (rule linear_eq_stdbasis)
unfolding linear_conv_bounded_linear
- apply(rule as(1,2)[THEN conjunct1])+
- proof(rule,rule ccontr)
- fix i :: 'a assume i:"i \<in> Basis" def e \<equiv> "norm (f' i - f'' i)"
+ apply (rule as(1,2)[THEN conjunct1])+
+ proof (rule, rule ccontr)
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ def e \<equiv> "norm (f' i - f'' i)"
assume "f' i \<noteq> f'' i"
- hence "e>0" unfolding e_def by auto
+ then have "e > 0"
+ unfolding e_def by auto
guess d using tendsto_diff [OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
- have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R i)) + f'' (c *\<^sub>R i)))"
+ have *: "norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) =
+ norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R i)) + f'' (c *\<^sub>R i)))"
unfolding scaleR_right_distrib by auto
- also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
+ also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
unfolding f'.scaleR f''.scaleR
- unfolding scaleR_right_distrib scaleR_minus_right by auto
- also have "\<dots> = e" unfolding e_def using c[THEN conjunct1]
+ unfolding scaleR_right_distrib scaleR_minus_right
+ by auto
+ also have "\<dots> = e"
+ unfolding e_def
+ using c[THEN conjunct1]
using norm_minus_cancel[of "f' i - f'' i"]
by (auto simp add: add.commute ab_diff_minus)
- finally show False using c
+ finally show False
+ using c
using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R i"]
unfolding dist_norm
unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
- using i by (auto simp: inverse_eq_divide)
+ using i
+ by (auto simp: inverse_eq_divide)
qed
qed
lemma frechet_derivative_unique_at:
- shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
+ "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
by (rule FDERIV_unique)
lemma frechet_derivative_unique_within_closed_interval:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "x \<in> {a..b}" (is "x\<in>?I")
- assumes "(f has_derivative f' ) (at x within {a..b})"
- assumes "(f has_derivative f'') (at x within {a..b})"
+ assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
+ and "x \<in> {a..b}"
+ and "(f has_derivative f' ) (at x within {a..b})"
+ and "(f has_derivative f'') (at x within {a..b})"
shows "f' = f''"
apply(rule frechet_derivative_unique_within)
apply(rule assms(3,4))+
-proof(rule,rule,rule)
- fix e::real and i :: 'a assume "e>0" and i:"i\<in>Basis"
- thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> {a..b}"
- proof(cases "x\<bullet>i=a\<bullet>i")
- case True thus ?thesis
- apply(rule_tac x="(min (b\<bullet>i - a\<bullet>i) e) / 2" in exI)
+proof (rule, rule, rule)
+ fix e :: real
+ fix i :: 'a
+ assume "e > 0" and i: "i \<in> Basis"
+ then show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> {a..b}"
+ proof (cases "x\<bullet>i = a\<bullet>i")
+ case True
+ then show ?thesis
+ apply (rule_tac x="(min (b\<bullet>i - a\<bullet>i) e) / 2" in exI)
using assms(1)[THEN bspec[where x=i]] and `e>0` and assms(2)
unfolding mem_interval
- using i by (auto simp add: field_simps inner_simps inner_Basis)
- next
+ using i
+ apply (auto simp add: field_simps inner_simps inner_Basis)
+ done
+ next
note * = assms(2)[unfolded mem_interval, THEN bspec, OF i]
- case False moreover have "a \<bullet> i < x \<bullet> i" using False * by auto
+ case False
+ moreover have "a \<bullet> i < x \<bullet> i"
+ using False * by auto
moreover {
have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> a\<bullet>i *2 + x\<bullet>i - a\<bullet>i"
by auto
- also have "\<dots> = a\<bullet>i + x\<bullet>i" by auto
- also have "\<dots> \<le> 2 * (x\<bullet>i)" using * by auto
- finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2" by auto
+ also have "\<dots> = a\<bullet>i + x\<bullet>i"
+ by auto
+ also have "\<dots> \<le> 2 * (x\<bullet>i)"
+ using * by auto
+ finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2"
+ by auto
}
- moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0" using * and `e>0` by auto
- hence "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e" using * by auto
+ moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0"
+ using * and `e>0` by auto
+ then have "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e"
+ using * by auto
ultimately show ?thesis
- apply(rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
+ apply (rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
using assms(1)[THEN bspec, OF i] and `e>0` and assms(2)
unfolding mem_interval
- using i by (auto simp add: field_simps inner_simps inner_Basis)
+ using i
+ apply (auto simp add: field_simps inner_simps inner_Basis)
+ done
qed
qed
lemma frechet_derivative_unique_within_open_interval:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "x \<in> {a<..<b}"
- assumes "(f has_derivative f' ) (at x within {a<..<b})"
- assumes "(f has_derivative f'') (at x within {a<..<b})"
+ and "(f has_derivative f' ) (at x within {a<..<b})"
+ and "(f has_derivative f'') (at x within {a<..<b})"
shows "f' = f''"
proof -
from assms(1) have *: "at x within {a<..<b} = at x"
@@ -422,27 +598,38 @@
qed
lemma frechet_derivative_at:
- shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
- apply(rule frechet_derivative_unique_at[of f],assumption)
- unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
+ "(f has_derivative f') (at x) \<Longrightarrow> f' = frechet_derivative f (at x)"
+ apply (rule frechet_derivative_unique_at[of f])
+ apply assumption
+ unfolding frechet_derivative_works[symmetric]
+ using differentiable_def
+ apply auto
+ done
lemma frechet_derivative_within_closed_interval:
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" and "x \<in> {a..b}"
- assumes "(f has_derivative f') (at x within {a.. b})"
- shows "frechet_derivative f (at x within {a.. b}) = f'"
- apply(rule frechet_derivative_unique_within_closed_interval[where f=f])
- apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
- unfolding differentiable_def using assms(3) by auto
+ fixes f :: "'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
+ and "x \<in> {a..b}"
+ and "(f has_derivative f') (at x within {a..b})"
+ shows "frechet_derivative f (at x within {a..b}) = f'"
+ apply (rule frechet_derivative_unique_within_closed_interval[where f=f])
+ apply (rule assms(1,2))+
+ unfolding frechet_derivative_works[symmetric]
+ unfolding differentiable_def
+ using assms(3)
+ apply auto
+ done
-subsection {* The traditional Rolle theorem in one dimension. *}
+
+subsection {* The traditional Rolle theorem in one dimension *}
lemma linear_componentwise:
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes lf: "linear f"
shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
proof -
- have fA: "finite Basis" by simp
+ have fA: "finite Basis"
+ by simp
have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
by (simp add: inner_setsum_left)
then show ?thesis
@@ -454,16 +641,19 @@
the unfolding of it. *}
lemma jacobian_works:
- "(f::('a::euclidean_space) \<Rightarrow> ('b::euclidean_space)) differentiable net \<longleftrightarrow>
- (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis.
- (\<Sum>j\<in>Basis. frechet_derivative f net (j) \<bullet> i * (h \<bullet> j)) *\<^sub>R i)) net"
- (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net")
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ shows "f differentiable net \<longleftrightarrow>
+ (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis.
+ (\<Sum>j\<in>Basis. frechet_derivative f net j \<bullet> i * (h \<bullet> j)) *\<^sub>R i)) net"
+ (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net")
proof
assume *: ?differentiable
- { fix h i
- have "?SUM h i = frechet_derivative f net h \<bullet> i" using *
- by (auto intro!: setsum_cong
- simp: linear_componentwise[of _ h i] linear_frechet_derivative) }
+ {
+ fix h i
+ have "?SUM h i = frechet_derivative f net h \<bullet> i"
+ using *
+ by (auto intro!: setsum_cong simp: linear_componentwise[of _ h i] linear_frechet_derivative)
+ }
with * show "(f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net"
by (simp add: frechet_derivative_works euclidean_representation)
qed (auto intro!: differentiableI)
@@ -471,54 +661,69 @@
lemma differential_zero_maxmin_component:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes k: "k \<in> Basis"
- and ball: "0 < e" "((\<forall>y \<in> ball x e. (f y)\<bullet>k \<le> (f x)\<bullet>k) \<or> (\<forall>y\<in>ball x e. (f x)\<bullet>k \<le> (f y)\<bullet>k))"
+ and ball: "0 < e" "(\<forall>y \<in> ball x e. (f y)\<bullet>k \<le> (f x)\<bullet>k) \<or> (\<forall>y\<in>ball x e. (f x)\<bullet>k \<le> (f y)\<bullet>k)"
and diff: "f differentiable (at x)"
shows "(\<Sum>j\<in>Basis. (frechet_derivative f (at x) j \<bullet> k) *\<^sub>R j) = (0::'a)" (is "?D k = 0")
proof (rule ccontr)
- assume "?D k \<noteq> 0"
+ assume "\<not> ?thesis"
then obtain j where j: "?D k \<bullet> j \<noteq> 0" "j \<in> Basis"
unfolding euclidean_eq_iff[of _ "0::'a"] by auto
- hence *: "\<bar>?D k \<bullet> j\<bar> / 2 > 0" by auto
+ then have *: "\<bar>?D k \<bullet> j\<bar> / 2 > 0"
+ by auto
note as = diff[unfolded jacobian_works has_derivative_at_alt]
guess e' using as[THEN conjunct2, rule_format, OF *] .. note e' = this
guess d using real_lbound_gt_zero[OF ball(1) e'[THEN conjunct1]] .. note d = this
- { fix c assume "abs c \<le> d"
- hence *:"norm (x + c *\<^sub>R j - x) < e'" using norm_Basis[OF j(2)] d by auto
+ {
+ fix c
+ assume "abs c \<le> d"
+ then have *: "norm (x + c *\<^sub>R j - x) < e'"
+ using norm_Basis[OF j(2)] d by auto
let ?v = "(\<Sum>i\<in>Basis. (\<Sum>l\<in>Basis. ?D i \<bullet> l * ((c *\<^sub>R j :: 'a) \<bullet> l)) *\<^sub>R i)"
- 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
- have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le>
- norm (f (x + c *\<^sub>R j) - f x - ?v)" by (rule Basis_le_norm[OF k])
+ 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
+ have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le> norm (f (x + c *\<^sub>R j) - f x - ?v)"
+ by (rule Basis_le_norm[OF k])
also have "\<dots> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
using e'[THEN conjunct2, rule_format, OF *] and norm_Basis[OF j(2)] j
by simp
- finally have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>" by simp
- hence "\<bar>f (x + c *\<^sub>R j) \<bullet> k - f x \<bullet> k - c * (?D k \<bullet> j)\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
+ finally have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
+ by simp
+ then have "\<bar>f (x + c *\<^sub>R j) \<bullet> k - f x \<bullet> k - c * (?D k \<bullet> j)\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
using j k
- by (simp add: inner_simps field_simps inner_Basis setsum_cases if_dist) }
+ by (simp add: inner_simps field_simps inner_Basis setsum_cases if_dist)
+ }
note * = this
have "x + d *\<^sub>R j \<in> ball x e" "x - d *\<^sub>R j \<in> ball x e"
- unfolding mem_ball dist_norm using norm_Basis[OF j(2)] d by auto
- hence **:"((f (x - d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k) \<or>
- ((f (x - d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k)" using ball by auto
- have ***: "\<And>y y1 y2 d dx::real.
- (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
- show False apply(rule ***[OF **, where dx="d * (?D k \<bullet> j)" and d="\<bar>?D k \<bullet> j\<bar> / 2 * \<bar>d\<bar>"])
+ unfolding mem_ball dist_norm
+ using norm_Basis[OF j(2)] d
+ by auto
+ then have **: "((f (x - d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k) \<or>
+ ((f (x - d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k)"
+ using ball by auto
+ have ***: "\<And>y y1 y2 d dx :: real. 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
+ show False
+ apply (rule ***[OF **, where dx="d * (?D k \<bullet> j)" and d="\<bar>?D k \<bullet> j\<bar> / 2 * \<bar>d\<bar>"])
using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
unfolding mult_minus_left
unfolding abs_mult diff_minus_eq_add scaleR_minus_left
- unfolding algebra_simps by (auto intro: mult_pos_pos)
+ unfolding algebra_simps
+ apply (auto intro: mult_pos_pos)
+ done
qed
text {* In particular if we have a mapping into @{typ "real"}. *}
lemma differential_zero_maxmin:
- fixes f::"'a\<Colon>euclidean_space \<Rightarrow> real"
- assumes "x \<in> s" "open s"
- and deriv: "(f has_derivative f') (at x)"
- and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
+ fixes f::"'a::euclidean_space \<Rightarrow> real"
+ assumes "x \<in> s"
+ and "open s"
+ and deriv: "(f has_derivative f') (at x)"
+ and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
shows "f' = (\<lambda>v. 0)"
proof -
- obtain e where e:"e>0" "ball x e \<subseteq> s"
+ obtain e where e: "e > 0" "ball x e \<subseteq> s"
using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
with differential_zero_maxmin_component[where 'b=real, of 1 e x f] mono deriv
have "(\<Sum>j\<in>Basis. frechet_derivative f (at x) j *\<^sub>R j) = (0::'a)"
@@ -527,529 +732,856 @@
have "\<forall>i\<in>Basis. f' i = 0"
by (simp add: euclidean_eq_iff[of _ "0::'a"] inner_setsum_left_Basis)
with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 1]
- show ?thesis by (simp add: fun_eq_iff)
+ show ?thesis
+ by (simp add: fun_eq_iff)
qed
lemma rolle:
- fixes f::"real\<Rightarrow>real"
- assumes "a < b" and "f a = f b" and "continuous_on {a..b} f"
- assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+ fixes f :: "real \<Rightarrow> real"
+ assumes "a < b"
+ and "f a = f b"
+ and "continuous_on {a..b} f"
+ and "\<forall>x\<in>{a<..<b}. (f has_derivative f' x) (at x)"
shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)"
-proof-
- 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))"
- proof-
- have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto
- hence *:"{a .. b}\<noteq>{}" by auto
+proof -
+ 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)"
+ proof -
+ have "(a + b) / 2 \<in> {a .. b}"
+ using assms(1) by auto
+ then have *: "{a..b} \<noteq> {}"
+ by auto
guess d using continuous_attains_sup[OF compact_interval * assms(3)] .. note d=this
guess c using continuous_attains_inf[OF compact_interval * assms(3)] .. note c=this
show ?thesis
- proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
- case True thus ?thesis
- apply(erule_tac disjE) apply(rule_tac x=d in bexI)
- apply(rule_tac[3] x=c in bexI)
- using d c by auto
+ proof (cases "d \<in> {a<..<b} \<or> c \<in> {a<..<b}")
+ case True
+ then show ?thesis
+ apply (erule_tac disjE)
+ apply (rule_tac x=d in bexI)
+ apply (rule_tac[3] x=c in bexI)
+ using d c
+ apply auto
+ done
next
def e \<equiv> "(a + b) /2"
- case False hence "f d = f c" using d c assms(2) by auto
- hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d"
- using c d apply- apply(erule_tac x=x in ballE)+ by auto
- thus ?thesis
- apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto
+ case False
+ then have "f d = f c"
+ using d c assms(2) by auto
+ then have "\<And>x. x \<in> {a..b} \<Longrightarrow> f x = f d"
+ using c d
+ apply -
+ apply (erule_tac x=x in ballE)+
+ apply auto
+ done
+ then show ?thesis
+ apply (rule_tac x=e in bexI)
+ unfolding e_def
+ using assms(1)
+ apply auto
+ done
qed
qed
then guess x .. note x=this
- hence "f' x = (\<lambda>v. 0)"
- apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
- defer apply(rule open_interval)
- apply(rule assms(4)[unfolded has_derivative_at[THEN sym],THEN bspec[where x=x]],assumption)
- unfolding o_def apply(erule disjE,rule disjI2) by auto
- thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
- apply(drule_tac x=v in fun_cong) using x(1) by auto
+ then have "f' x = (\<lambda>v. 0)"
+ apply (rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
+ defer
+ apply (rule open_interval)
+ apply (rule assms(4)[unfolded has_derivative_at[symmetric],THEN bspec[where x=x]])
+ apply assumption
+ unfolding o_def
+ apply (erule disjE)
+ apply (rule disjI2)
+ apply auto
+ done
+ then show ?thesis
+ apply (rule_tac x=x in bexI)
+ unfolding o_def
+ apply rule
+ apply (drule_tac x=v in fun_cong)
+ using x(1)
+ apply auto
+ done
qed
-subsection {* One-dimensional mean value theorem. *}
+
+subsection {* One-dimensional mean value theorem *}
-lemma mvt: fixes f::"real \<Rightarrow> real"
- assumes "a < b" and "continuous_on {a .. b} f"
+lemma mvt:
+ fixes f :: "real \<Rightarrow> real"
+ assumes "a < b"
+ and "continuous_on {a..b} f"
assumes "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
- shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))"
-proof-
+ shows "\<exists>x\<in>{a<..<b}. f b - f a = (f' x) (b - a)"
+proof -
have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
proof (intro rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"] ballI)
- fix x assume x:"x \<in> {a<..<b}"
- 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)"
+ fix x
+ assume x: "x \<in> {a<..<b}"
+ 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)"
by (intro FDERIV_intros assms(3)[rule_format,OF x] mult_right_has_derivative)
qed (insert assms(1,2), auto intro!: continuous_on_intros simp: field_simps)
then guess x ..
- thus ?thesis apply(rule_tac x=x in bexI)
- apply(drule fun_cong[of _ _ "b - a"]) by auto
+ then show ?thesis
+ apply (rule_tac x=x in bexI)
+ apply (drule fun_cong[of _ _ "b - a"])
+ apply auto
+ done
qed
lemma mvt_simple:
- fixes f::"real \<Rightarrow> real"
- assumes "a<b" and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
+ fixes f :: "real \<Rightarrow> real"
+ assumes "a < b"
+ and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
- apply(rule mvt)
- apply(rule assms(1), rule differentiable_imp_continuous_on)
- unfolding differentiable_on_def differentiable_def defer
+ apply (rule mvt)
+ apply (rule assms(1))
+ apply (rule differentiable_imp_continuous_on)
+ unfolding differentiable_on_def differentiable_def
+ defer
proof
- fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)"
- unfolding has_derivative_within_open[OF x open_interval,THEN sym]
- apply(rule has_derivative_within_subset)
- apply(rule assms(2)[rule_format])
- using x by auto
-qed(insert assms(2), auto)
+ fix x
+ assume x: "x \<in> {a<..<b}"
+ show "(f has_derivative f' x) (at x)"
+ unfolding has_derivative_within_open[OF x open_interval,symmetric]
+ apply (rule has_derivative_within_subset)
+ apply (rule assms(2)[rule_format])
+ using x
+ apply auto
+ done
+qed (insert assms(2), auto)
lemma mvt_very_simple:
- fixes f::"real \<Rightarrow> real"
- assumes "a \<le> b" and "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
+ fixes f :: "real \<Rightarrow> real"
+ assumes "a \<le> b"
+ and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
proof (cases "a = b")
- interpret bounded_linear "f' b" using assms(2) assms(1) by auto
- case True thus ?thesis apply(rule_tac x=a in bexI)
- using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
- unfolding True using zero by auto next
- case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto
+ interpret bounded_linear "f' b"
+ using assms(2) assms(1) by auto
+ case True
+ then show ?thesis
+ apply (rule_tac x=a in bexI)
+ using assms(2)[THEN bspec[where x=a]]
+ unfolding has_derivative_def
+ unfolding True
+ using zero
+ apply auto
+ done
+next
+ case False
+ then show ?thesis
+ using mvt_simple[OF _ assms(2)]
+ using assms(1)
+ by auto
qed
text {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
lemma mvt_general:
- fixes f::"real\<Rightarrow>'a::euclidean_space"
- assumes "a<b" and "continuous_on {a..b} f"
- assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
- shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))"
-proof-
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "a < b"
+ and "continuous_on {a..b} f"
+ and "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+ shows "\<exists>x\<in>{a<..<b}. norm (f b - f a) \<le> norm (f' x (b - a))"
+proof -
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)"
- apply(rule mvt) apply(rule assms(1))
- apply(rule continuous_on_inner continuous_on_intros assms(2) ballI)+
+ apply (rule mvt)
+ apply (rule assms(1))
+ apply (rule continuous_on_inner continuous_on_intros assms(2) ballI)+
unfolding o_def
- apply(rule FDERIV_inner_right)
- using assms(3) by auto
+ apply (rule FDERIV_inner_right)
+ using assms(3)
+ apply auto
+ done
then guess x .. note x=this
- show ?thesis proof(cases "f a = f b")
+ show ?thesis
+ proof (cases "f a = f b")
case False
have "norm (f b - f a) * norm (f b - f a) = (norm (f b - f a))\<^sup>2"
by (simp add: power2_eq_square)
- also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding power2_norm_eq_inner ..
+ also have "\<dots> = (f b - f a) \<bullet> (f b - f a)"
+ unfolding power2_norm_eq_inner ..
also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)"
- using x unfolding inner_simps by (auto simp add: inner_diff_left)
+ using x
+ unfolding inner_simps
+ by (auto simp add: inner_diff_left)
also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))"
by (rule norm_cauchy_schwarz)
- finally show ?thesis using False x(1)
+ finally show ?thesis
+ using False x(1)
by (auto simp add: real_mult_left_cancel)
next
- case True thus ?thesis using assms(1)
- apply (rule_tac x="(a + b) /2" in bexI) by auto
+ case True
+ then show ?thesis
+ using assms(1)
+ apply (rule_tac x="(a + b) /2" in bexI)
+ apply auto
+ done
qed
qed
text {* Still more general bound theorem. *}
lemma differentiable_bound:
- fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)"
- assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
- shows "norm(f x - f y) \<le> B * norm(x - y)"
-proof-
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "convex s"
+ and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
+ and "\<forall>x\<in>s. onorm (f' x) \<le> B"
+ and x: "x \<in> s"
+ and y: "y \<in> s"
+ shows "norm (f x - f y) \<le> B * norm (x - y)"
+proof -
let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
- have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
+ have *: "\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
using assms(1)[unfolded convex_alt,rule_format,OF x y]
unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib
by (auto simp add: algebra_simps)
- hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply-
- apply(rule continuous_on_intros)+
+ then have 1: "continuous_on {0..1} (f \<circ> ?p)"
+ apply -
+ apply (rule continuous_on_intros)+
unfolding continuous_on_eq_continuous_within
- apply(rule,rule differentiable_imp_continuous_within)
- unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
- apply(rule has_derivative_within_subset)
- apply(rule assms(2)[rule_format]) by auto
- 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)"
+ apply rule
+ apply (rule differentiable_imp_continuous_within)
+ unfolding differentiable_def
+ apply (rule_tac x="f' xa" in exI)
+ apply (rule has_derivative_within_subset)
+ apply (rule assms(2)[rule_format])
+ apply auto
+ done
+ 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)"
proof rule
case goal1
let ?u = "x + u *\<^sub>R (y - x)"
- have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
- apply(rule diff_chain_within) apply(rule FDERIV_intros)+
- apply(rule has_derivative_within_subset)
- apply(rule assms(2)[rule_format]) using goal1 * by auto
- thus ?case
- unfolding has_derivative_within_open[OF goal1 open_interval] by auto
+ have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
+ apply (rule diff_chain_within)
+ apply (rule FDERIV_intros)+
+ apply (rule has_derivative_within_subset)
+ apply (rule assms(2)[rule_format])
+ using goal1 *
+ apply auto
+ done
+ then show ?case
+ unfolding has_derivative_within_open[OF goal1 open_interval]
+ by auto
qed
guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
- have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y"
- proof-
+ have **: "\<And>x y. x \<in> s \<Longrightarrow> norm (f' x y) \<le> B * norm y"
+ proof -
case goal1
have "norm (f' x y) \<le> onorm (f' x) * norm y"
- using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
+ by (rule onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]])
also have "\<dots> \<le> B * norm y"
- apply(rule mult_right_mono)
+ apply (rule mult_right_mono)
using assms(3)[rule_format,OF goal1]
- by(auto simp add:field_simps)
- finally show ?case by simp
+ apply (auto simp add: field_simps)
+ done
+ finally show ?case
+ by simp
qed
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)"
- by(auto simp add:norm_minus_commute)
- also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
- also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
- finally show ?thesis by(auto simp add:norm_minus_commute)
+ by (auto simp add: norm_minus_commute)
+ also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))"
+ using u by auto
+ also have "\<dots> \<le> B * norm(y - x)"
+ apply (rule **)
+ using * and u
+ apply auto
+ done
+ finally show ?thesis
+ by (auto simp add: norm_minus_commute)
qed
lemma differentiable_bound_real:
- fixes f::"real \<Rightarrow> real"
- assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
- assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
- shows "norm(f x - f y) \<le> B * norm(x - y)"
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex s"
+ and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
+ and "\<forall>x\<in>s. onorm (f' x) \<le> B"
+ and x: "x \<in> s"
+ and y: "y \<in> s"
+ shows "norm (f x - f y) \<le> B * norm (x - y)"
using differentiable_bound[of s f f' B x y]
- unfolding Ball_def image_iff o_def using assms by auto
+ unfolding Ball_def image_iff o_def
+ using assms
+ by auto
text {* In particular. *}
lemma has_derivative_zero_constant:
- fixes f::"real\<Rightarrow>real"
- assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex s"
+ and "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
shows "\<exists>c. \<forall>x\<in>s. f x = c"
-proof(cases "s={}")
- case False then obtain x where "x\<in>s" by auto
- have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
- thus ?case
- using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
- unfolding onorm_const by auto qed
- thus ?thesis apply(rule_tac x="f x" in exI) by auto
-qed auto
+proof (cases "s={}")
+ case False
+ then obtain x where "x \<in> s"
+ by auto
+ have "\<And>y. y \<in> s \<Longrightarrow> f x = f y"
+ proof -
+ case goal1
+ then show ?case
+ using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x \<in> s`
+ unfolding onorm_const
+ by auto
+ qed
+ then show ?thesis
+ apply (rule_tac x="f x" in exI)
+ apply auto
+ done
+next
+ case True
+ then show ?thesis by auto
+qed
-lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
- assumes "convex s" and "a \<in> s" and "f a = c"
- assumes "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" and "x\<in>s"
+lemma has_derivative_zero_unique:
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex s"
+ and "a \<in> s"
+ and "f a = c"
+ and "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
+ and "x \<in> s"
shows "f x = c"
- using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
+ using has_derivative_zero_constant[OF assms(1,4)]
+ using assms(2-3,5)
+ by auto
-subsection {* Differentiability of inverse function (most basic form). *}
+
+subsection {* Differentiability of inverse function (most basic form) *}
lemma has_derivative_inverse_basic:
- fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
+ fixes f :: "'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
assumes "(f has_derivative f') (at (g y))"
- assumes "bounded_linear g'" and "g' \<circ> f' = id" and "continuous (at y) g"
- assumes "open t" and "y \<in> t" and "\<forall>z\<in>t. f(g z) = z"
+ and "bounded_linear g'"
+ and "g' \<circ> f' = id"
+ and "continuous (at y) g"
+ and "open t"
+ and "y \<in> t"
+ and "\<forall>z\<in>t. f (g z) = z"
shows "(g has_derivative g') (at y)"
-proof-
+proof -
interpret f': bounded_linear f'
using assms unfolding has_derivative_def by auto
- interpret g': bounded_linear g' using assms by auto
+ interpret g': bounded_linear g'
+ using assms by auto
guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
-(* have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
- 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)"
- proof(rule,rule)
+ 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)"
+ proof (rule, rule)
case goal1
- have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
+ have *: "e / C > 0"
+ apply (rule divide_pos_pos)
+ using `e > 0` C
+ apply auto
+ done
guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
- thus ?case apply(rule_tac x=d in exI) apply rule defer
- proof(rule,rule)
- fix z assume as:"norm (z - y) < d" hence "z\<in>t"
+ then show ?case
+ apply (rule_tac x=d in exI)
+ apply rule
+ defer
+ apply rule
+ apply rule
+ proof -
+ fix z
+ assume as: "norm (z - y) < d"
+ then have "z \<in> t"
using d2 d unfolding dist_norm by auto
have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
unfolding g'.diff f'.diff
- unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
+ unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
unfolding assms(7)[rule_format,OF `z\<in>t`]
- apply(subst norm_minus_cancel[THEN sym]) by auto
- also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C"
+ apply (subst norm_minus_cancel[symmetric])
+ apply auto
+ done
+ also have "\<dots> \<le> norm (f (g z) - y - f' (g z - g y)) * C"
by (rule C [THEN conjunct2, rule_format])
also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
- apply(rule mult_right_mono)
- apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]])
- apply(cases "z=y") defer
- apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format])
- using as d C d0 by auto
+ apply (rule mult_right_mono)
+ apply (rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]])
+ apply (cases "z = y")
+ defer
+ apply (rule d1[THEN conjunct2, unfolded dist_norm,rule_format])
+ using as d C d0
+ apply auto
+ done
also have "\<dots> \<le> e * norm (g z - g y)"
using C by (auto simp add: field_simps)
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
by simp
qed auto
qed
- have *:"(0::real) < 1 / 2" by auto
+ have *: "(0::real) < 1 / 2"
+ by auto
guess d using lem1[rule_format,OF *] .. note d=this
- def B\<equiv>"C*2"
- have "B>0" unfolding B_def using C by auto
- have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)"
- proof(rule,rule) case goal1
+ def B\<equiv>"C * 2"
+ have "B > 0"
+ unfolding B_def using C by auto
+ have lem2: "\<forall>z. norm(z - y) < d \<longrightarrow> norm (g z - g y) \<le> B * norm (z - y)"
+ proof (rule, rule)
+ case goal1
have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
- by(rule norm_triangle_sub)
- also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)"
- apply(rule add_left_mono) using d and goal1 by auto
+ by (rule norm_triangle_sub)
+ also have "\<dots> \<le> norm (g' (z - y)) + 1 / 2 * norm (g z - g y)"
+ apply (rule add_left_mono)
+ using d and goal1
+ apply auto
+ done
also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
- apply(rule add_right_mono) using C by auto
- finally show ?case unfolding B_def by(auto simp add:field_simps)
+ apply (rule add_right_mono)
+ using C
+ apply auto
+ done
+ finally show ?case
+ unfolding B_def
+ by (auto simp add: field_simps)
qed
- show ?thesis unfolding has_derivative_at_alt
- proof(rule,rule assms,rule,rule) case goal1
- hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
+ show ?thesis
+ unfolding has_derivative_at_alt
+ apply rule
+ apply (rule assms)
+ apply rule
+ apply rule
+ proof -
+ case goal1
+ then have *: "e / B >0"
+ apply -
+ apply (rule divide_pos_pos)
+ using `B > 0`
+ apply auto
+ done
guess d' using lem1[rule_format,OF *] .. note d'=this
guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
show ?case
- apply(rule_tac x=k in exI,rule) defer
- proof(rule,rule)
- fix z assume as:"norm(z - y) < k"
- hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
+ apply (rule_tac x=k in exI)
+ apply rule
+ defer
+ apply rule
+ apply rule
+ proof -
+ fix z
+ assume as: "norm (z - y) < k"
+ then have "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
using d' k by auto
- also have "\<dots> \<le> e * norm(z - y)"
+ also have "\<dots> \<le> e * norm (z - y)"
unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
- using lem2[THEN spec[where x=z]] using k as using `e>0`
+ using lem2[THEN spec[where x=z]]
+ using k as using `e > 0`
by (auto simp add: field_simps)
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
- by simp qed(insert k, auto)
+ by simp
+ qed(insert k, auto)
qed
qed
text {* Simply rewrite that based on the domain point x. *}
lemma has_derivative_inverse_basic_x:
- fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
- assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
- "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
- shows "(g has_derivative g') (at (f(x)))"
- apply(rule has_derivative_inverse_basic) using assms by auto
+ fixes f :: "'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
+ assumes "(f has_derivative f') (at x)"
+ and "bounded_linear g'"
+ and "g' \<circ> f' = id"
+ and "continuous (at (f x)) g"
+ and "g (f x) = x"
+ and "open t"
+ and "f x \<in> t"
+ and "\<forall>y\<in>t. f (g y) = y"
+ shows "(g has_derivative g') (at (f x))"
+ apply (rule has_derivative_inverse_basic)
+ using assms
+ apply auto
+ done
text {* This is the version in Dieudonne', assuming continuity of f and g. *}
lemma has_derivative_inverse_dieudonne:
- fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
- (**) "x\<in>s" "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "open s"
+ and "open (f ` s)"
+ and "continuous_on s f"
+ and "continuous_on (f ` s) g"
+ and "\<forall>x\<in>s. g (f x) = x"
+ and "x \<in> s"
+ and "(f has_derivative f') (at x)"
+ and "bounded_linear g'"
+ and "g' \<circ> f' = id"
shows "(g has_derivative g') (at (f x))"
- apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
- using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)]
- continuous_on_eq_continuous_at[OF assms(2)] by auto
+ apply (rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
+ using assms(3-6)
+ unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)]
+ apply auto
+ done
text {* Here's the simplest way of not assuming much about g. *}
lemma has_derivative_inverse:
- fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
- "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "compact s"
+ and "x \<in> s"
+ and "f x \<in> interior (f ` s)"
+ and "continuous_on s f"
+ and "\<forall>y\<in>s. g (f y) = y"
+ and "(f has_derivative f') (at x)"
+ and "bounded_linear g'"
+ and "g' \<circ> f' = id"
shows "(g has_derivative g') (at (f x))"
-proof-
- { fix y assume "y\<in>interior (f ` s)"
- then obtain x where "x\<in>s" and *:"y = f x"
- unfolding image_iff using interior_subset by auto
- have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] ..
+proof -
+ {
+ fix y
+ assume "y \<in> interior (f ` s)"
+ then obtain x where "x \<in> s" and *: "y = f x"
+ unfolding image_iff
+ using interior_subset
+ by auto
+ have "f (g y) = y"
+ unfolding * and assms(5)[rule_format,OF `x\<in>s`] ..
} note * = this
show ?thesis
- apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
- apply(rule continuous_on_interior[OF _ assms(3)])
- apply(rule continuous_on_inv[OF assms(4,1)])
- apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
- by(rule, rule *, assumption)
+ apply (rule has_derivative_inverse_basic_x[OF assms(6-8)])
+ apply (rule continuous_on_interior[OF _ assms(3)])
+ apply (rule continuous_on_inv[OF assms(4,1)])
+ apply (rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
+ apply rule
+ apply (rule *)
+ apply assumption
+ done
qed
-subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
+
+subsection {* Proving surjectivity via Brouwer fixpoint theorem *}
lemma brouwer_surjective:
- fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
- assumes "compact t" "convex t" "t \<noteq> {}" "continuous_on t f"
- "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+ assumes "compact t"
+ and "convex t"
+ and "t \<noteq> {}"
+ and "continuous_on t f"
+ and "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t"
+ and "x \<in> s"
shows "\<exists>y\<in>t. f y = x"
-proof-
- have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
- by(auto simp add:algebra_simps)
+proof -
+ have *: "\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
+ by (auto simp add: algebra_simps)
show ?thesis
unfolding *
- apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
- apply(rule continuous_on_intros assms)+ using assms(4-6) by auto
+ apply (rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
+ apply (rule continuous_on_intros assms)+
+ using assms(4-6)
+ apply auto
+ done
qed
lemma brouwer_surjective_cball:
- fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
- assumes "0 < e" "continuous_on (cball a e) f"
- "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+ assumes "e > 0"
+ and "continuous_on (cball a e) f"
+ and "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e"
+ and "x \<in> s"
shows "\<exists>y\<in>cball a e. f y = x"
- apply(rule brouwer_surjective)
- apply(rule compact_cball convex_cball)+
- unfolding cball_eq_empty using assms by auto
+ apply (rule brouwer_surjective)
+ apply (rule compact_cball convex_cball)+
+ unfolding cball_eq_empty
+ using assms
+ apply auto
+ done
text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
lemma sussmann_open_mapping:
- fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
- assumes "open s" "continuous_on s f" "x \<in> s"
- "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
- "t \<subseteq> s" "x \<in> interior t"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
+ assumes "open s"
+ and "continuous_on s f"
+ and "x \<in> s"
+ and "(f has_derivative f') (at x)"
+ and "bounded_linear g'" "f' \<circ> g' = id"
+ and "t \<subseteq> s"
+ and "x \<in> interior t"
shows "f x \<in> interior (f ` t)"
-proof-
- interpret f':bounded_linear f'
- using assms unfolding has_derivative_def by auto
- interpret g':bounded_linear g' using assms by auto
+proof -
+ interpret f': bounded_linear f'
+ using assms
+ unfolding has_derivative_def
+ by auto
+ interpret g': bounded_linear g'
+ using assms
+ by auto
guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this
- hence *:"1/(2*B)>0" by (auto intro!: divide_pos_pos)
+ then have *: "1 / (2 * B) > 0"
+ by (auto intro!: divide_pos_pos)
guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
- have *:"0<e0/B" "0<e1/B"
- apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
+ have *: "0 < e0 / B" "0 < e1 / B"
+ apply (rule_tac[!] divide_pos_pos)
+ using e0 e1 B
+ apply auto
+ done
guess e using real_lbound_gt_zero[OF *] .. note e=this
- have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
- apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
- prefer 3 apply(rule,rule)
+ have "\<forall>z\<in>cball (f x) (e / 2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
+ apply rule
+ apply (rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
+ prefer 3
+ apply rule
+ apply rule
proof-
show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
unfolding g'.diff
- apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
- apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
- apply(rule continuous_on_subset[OF assms(2)])
- apply(rule,unfold image_iff,erule bexE)
+ apply (rule continuous_on_compose[of _ _ f, unfolded o_def])
+ apply (rule continuous_on_intros linear_continuous_on[OF assms(5)])+
+ apply (rule continuous_on_subset[OF assms(2)])
+ apply rule
+ apply (unfold image_iff)
+ apply (erule bexE)
proof-
- fix y z assume as:"y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
+ fix y z
+ assume as: "y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
have "dist x z = norm (g' (f x) - g' y)"
unfolding as(2) and dist_norm by auto
also have "\<dots> \<le> norm (f x - y) * B"
- unfolding g'.diff[THEN sym] using B by auto
+ unfolding g'.diff[symmetric]
+ using B
+ by auto
also have "\<dots> \<le> e * B"
- using as(1)[unfolded mem_cball dist_norm] using B by auto
- also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
- finally have "z\<in>cball x e1" unfolding mem_cball by force
- thus "z \<in> s" using e1 assms(7) by auto
+ using as(1)[unfolded mem_cball dist_norm]
+ using B
+ by auto
+ also have "\<dots> \<le> e1"
+ using e
+ unfolding less_divide_eq
+ using B
+ by auto
+ finally have "z \<in> cball x e1"
+ unfolding mem_cball
+ by force
+ then show "z \<in> s"
+ using e1 assms(7) by auto
qed
next
- fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
- have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
- also have "\<dots> \<le> e * B" apply(rule mult_right_mono)
+ fix y z
+ assume as: "y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
+ have "norm (g' (z - f x)) \<le> norm (z - f x) * B"
+ using B by auto
+ also have "\<dots> \<le> e * B"
+ apply (rule mult_right_mono)
using as(2)[unfolded mem_cball dist_norm] and B
- unfolding norm_minus_commute by auto
- also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
- finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
- have **:"f x + f' (x + g' (z - f x) - x) = z"
- using assms(6)[unfolded o_def id_def,THEN cong] by auto
- 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)"
+ unfolding norm_minus_commute
+ apply auto
+ done
+ also have "\<dots> < e0"
+ using e and B
+ unfolding less_divide_eq
+ by auto
+ finally have *: "norm (x + g' (z - f x) - x) < e0"
+ by auto
+ have **: "f x + f' (x + g' (z - f x) - x) = z"
+ using assms(6)[unfolded o_def id_def,THEN cong]
+ by auto
+ 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)"
using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"]
by (auto simp add: algebra_simps)
also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)"
using e0[THEN conjunct2,rule_format,OF *]
- unfolding algebra_simps ** by auto
+ unfolding algebra_simps **
+ by auto
also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2"
- using as(1)[unfolded mem_cball dist_norm] by auto
+ using as(1)[unfolded mem_cball dist_norm]
+ by auto
also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2"
- using * and B by (auto simp add: field_simps)
- also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
- also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono)
+ using * and B
+ by (auto simp add: field_simps)
+ also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2"
+ by auto
+ also have "\<dots> \<le> e/2 + e/2"
+ apply (rule add_right_mono)
using as(2)[unfolded mem_cball dist_norm]
- unfolding norm_minus_commute by auto
+ unfolding norm_minus_commute
+ apply auto
+ done
finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
- unfolding mem_cball dist_norm by auto
- qed(insert e, auto) note lem = this
- show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
- apply(rule,rule divide_pos_pos) prefer 3
+ unfolding mem_cball dist_norm
+ by auto
+ qed (insert e, auto) note lem = this
+ show ?thesis
+ unfolding mem_interior
+ apply (rule_tac x="e/2" in exI)
+ apply rule
+ apply (rule divide_pos_pos)
+ prefer 3
proof
- fix y assume "y \<in> ball (f x) (e/2)"
- hence *:"y\<in>cball (f x) (e/2)" by auto
+ fix y
+ assume "y \<in> ball (f x) (e / 2)"
+ then have *: "y \<in> cball (f x) (e / 2)"
+ by auto
guess z using lem[rule_format,OF *] .. note z=this
- hence "norm (g' (z - f x)) \<le> norm (z - f x) * B"
- using B by (auto simp add: field_simps)
+ then have "norm (g' (z - f x)) \<le> norm (z - f x) * B"
+ using B
+ by (auto simp add: field_simps)
also have "\<dots> \<le> e * B"
- apply (rule mult_right_mono) using z(1)
- unfolding mem_cball dist_norm norm_minus_commute using B by auto
- also have "\<dots> \<le> e1" using e B unfolding less_divide_eq by auto
- finally have "x + g'(z - f x) \<in> t" apply-
- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
- unfolding mem_cball dist_norm by auto
- thus "y \<in> f ` t" using z by auto
- qed(insert e, auto)
+ apply (rule mult_right_mono)
+ using z(1)
+ unfolding mem_cball dist_norm norm_minus_commute
+ using B
+ apply auto
+ done
+ also have "\<dots> \<le> e1"
+ using e B unfolding less_divide_eq by auto
+ finally have "x + g'(z - f x) \<in> t"
+ apply -
+ apply (rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
+ unfolding mem_cball dist_norm
+ apply auto
+ done
+ then show "y \<in> f ` t"
+ using z by auto
+ qed (insert e, auto)
qed
text {* Hence the following eccentric variant of the inverse function theorem. *)
(* This has no continuity assumptions, but we do need the inverse function. *)
-(* We could put f' o g = I but this happens to fit with the minimal linear *)
+(* We could put f' \<circ> g = I but this happens to fit with the minimal linear *)
(* algebra theory I've set up so far. *}
(* move before left_inverse_linear in Euclidean_Space*)
- lemma right_inverse_linear:
- fixes f::"'a::euclidean_space => 'a"
- assumes lf: "linear f" and gf: "f o g = id"
- shows "linear g"
- proof-
- from gf have fi: "surj f" by (auto simp add: surj_def o_def id_def) metis
- from linear_surjective_isomorphism[OF lf fi]
- obtain h:: "'a => 'a" where
- h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
- have "h = g" apply (rule ext) using gf h(2,3)
- by (simp add: o_def id_def fun_eq_iff) metis
- with h(1) show ?thesis by blast
- qed
-
+lemma right_inverse_linear:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
+ assumes lf: "linear f"
+ and gf: "f \<circ> g = id"
+ shows "linear g"
+proof -
+ from gf have fi: "surj f"
+ by (auto simp add: surj_def o_def id_def) metis
+ from linear_surjective_isomorphism[OF lf fi]
+ obtain h:: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
+ by blast
+ have "h = g"
+ apply (rule ext)
+ using gf h(2,3)
+ apply (simp add: o_def id_def fun_eq_iff)
+ apply metis
+ done
+ with h(1) show ?thesis by blast
+qed
+
lemma has_derivative_inverse_strong:
- fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
- assumes "open s" and "x \<in> s" and "continuous_on s f"
- assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" and "f' o g' = id"
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+ assumes "open s"
+ and "x \<in> s"
+ and "continuous_on s f"
+ and "\<forall>x\<in>s. g (f x) = x"
+ and "(f has_derivative f') (at x)"
+ and "f' \<circ> g' = id"
shows "(g has_derivative g') (at (f x))"
-proof-
- have linf:"bounded_linear f'"
+proof -
+ have linf: "bounded_linear f'"
using assms(5) unfolding has_derivative_def by auto
- hence ling:"bounded_linear g'"
- unfolding linear_conv_bounded_linear[THEN sym]
- apply- apply(rule right_inverse_linear) using assms(6) by auto
- moreover have "g' \<circ> f' = id" using assms(6) linf ling
- unfolding linear_conv_bounded_linear[THEN sym]
- using linear_inverse_left by auto
- moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)"
- apply(rule,rule,rule,rule sussmann_open_mapping )
- apply(rule assms ling)+ by auto
- have "continuous (at (f x)) g" unfolding continuous_at Lim_at
- proof(rule,rule)
- fix e::real assume "e>0"
- hence "f x \<in> interior (f ` (ball x e \<inter> s))"
- using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
- by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
+ then have ling: "bounded_linear g'"
+ unfolding linear_conv_bounded_linear[symmetric]
+ apply -
+ apply (rule right_inverse_linear)
+ using assms(6)
+ apply auto
+ done
+ moreover have "g' \<circ> f' = id"
+ using assms(6) linf ling
+ unfolding linear_conv_bounded_linear[symmetric]
+ using linear_inverse_left
+ by auto
+ moreover have *:"\<forall>t\<subseteq>s. x \<in> interior t \<longrightarrow> f x \<in> interior (f ` t)"
+ apply rule
+ apply rule
+ apply rule
+ apply (rule sussmann_open_mapping)
+ apply (rule assms ling)+
+ apply auto
+ done
+ have "continuous (at (f x)) g"
+ unfolding continuous_at Lim_at
+ proof (rule, rule)
+ fix e :: real
+ assume "e > 0"
+ then have "f x \<in> interior (f ` (ball x e \<inter> s))"
+ using *[rule_format,of "ball x e \<inter> s"] `x \<in> s`
+ by (auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
then guess d unfolding mem_interior .. note d=this
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"
- apply(rule_tac x=d in exI)
- apply(rule,rule d[THEN conjunct1])
- proof(rule,rule) case goal1
- hence "g y \<in> g ` f ` (ball x e \<inter> s)"
+ apply (rule_tac x=d in exI)
+ apply rule
+ apply (rule d[THEN conjunct1])
+ apply rule
+ apply rule
+ proof -
+ case goal1
+ then have "g y \<in> g ` f ` (ball x e \<inter> s)"
using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
- by(auto simp add:dist_commute)
- hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
- thus "dist (g y) (g (f x)) < e"
- using assms(4)[rule_format,OF `x\<in>s`]
+ by (auto simp add: dist_commute)
+ then have "g y \<in> ball x e \<inter> s"
+ using assms(4) by auto
+ then show "dist (g y) (g (f x)) < e"
+ using assms(4)[rule_format,OF `x \<in> s`]
by (auto simp add: dist_commute)
qed
qed
moreover have "f x \<in> interior (f ` s)"
- apply(rule sussmann_open_mapping)
- apply(rule assms ling)+
- using interior_open[OF assms(1)] and `x\<in>s` by auto
+ apply (rule sussmann_open_mapping)
+ apply (rule assms ling)+
+ using interior_open[OF assms(1)] and `x \<in> s`
+ apply auto
+ done
moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y"
- proof- case goal1
- hence "y\<in>f ` s" using interior_subset by auto
+ proof -
+ case goal1
+ then have "y \<in> f ` s"
+ using interior_subset by auto
then guess z unfolding image_iff ..
- thus ?case using assms(4) by auto
+ then show ?case
+ using assms(4) by auto
qed
ultimately show ?thesis
- apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)])
- using assms by auto
+ apply -
+ apply (rule has_derivative_inverse_basic_x[OF assms(5)])
+ using assms
+ apply auto
+ done
qed
text {* A rewrite based on the other domain. *}
lemma has_derivative_inverse_strong_x:
- fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
- assumes "open s" and "g y \<in> s" and "continuous_on s f"
- assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))"
- assumes "f' o g' = id" and "f(g y) = y"
+ fixes f :: "'a::ordered_euclidean_space \<Rightarrow> 'a"
+ assumes "open s"
+ and "g y \<in> s"
+ and "continuous_on s f"
+ and "\<forall>x\<in>s. g (f x) = x"
+ and "(f has_derivative f') (at (g y))"
+ and "f' \<circ> g' = id"
+ and "f (g y) = y"
shows "(g has_derivative g') (at y)"
- using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
+ using has_derivative_inverse_strong[OF assms(1-6)]
+ unfolding assms(7)
+ by simp
text {* On a region. *}
lemma has_derivative_inverse_on:
- fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
- assumes "open s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
- assumes "\<forall>x\<in>s. g(f x) = x" and "f'(x) o g'(x) = id" and "x\<in>s"
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+ assumes "open s"
+ and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
+ and "\<forall>x\<in>s. g (f x) = x"
+ and "f' x \<circ> g' x = id"
+ and "x \<in> s"
shows "(g has_derivative g'(x)) (at (f x))"
- apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f])
- apply(rule assms)+
+ apply (rule has_derivative_inverse_strong[where g'="g' x" and f=f])
+ apply (rule assms)+
unfolding continuous_on_eq_continuous_at[OF assms(1)]
- apply(rule,rule differentiable_imp_continuous_within)
- unfolding differentiable_def using assms by auto
+ apply rule
+ apply (rule differentiable_imp_continuous_within)
+ unfolding differentiable_def
+ using assms
+ apply auto
+ done
text {* Invertible derivative continous at a point implies local
injectivity. It's only for this we need continuity of the derivative,
@@ -1057,269 +1589,381 @@
also continuous. So if we know for some other reason that the inverse
function exists, it's OK. *}
-lemma bounded_linear_sub:
- "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
+lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_linear (\<lambda>x. f x - g x)"
using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g]
by (auto simp add: algebra_simps)
lemma has_derivative_locally_injective:
- fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
- assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
- "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
- "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
- obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)"
-proof-
- interpret bounded_linear g' using assms by auto
+ fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
+ assumes "a \<in> s"
+ and "open s"
+ and "bounded_linear g'"
+ and "g' \<circ> f' a = id"
+ and "\<forall>x\<in>s. (f has_derivative f' x) (at x)"
+ and "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e"
+ obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. f x' = f x \<longrightarrow> x' = x"
+proof -
+ interpret bounded_linear g'
+ using assms by auto
note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
- have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)" defer
- apply(subst euclidean_eq_iff) using f'g' by auto
- hence *:"0 < onorm g'"
- unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastforce
- def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
+ have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)"
+ defer
+ apply (subst euclidean_eq_iff)
+ using f'g'
+ apply auto
+ done
+ then have *: "0 < onorm g'"
+ unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]]
+ by fastforce
+ def k \<equiv> "1 / onorm g' / 2"
+ have *: "k > 0"
+ unfolding k_def using * by auto
guess d1 using assms(6)[rule_format,OF *] .. note d1=this
- from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
- obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
+ from `open s` obtain d2 where "d2 > 0" "ball a d2 \<subseteq> s"
+ using `a\<in>s` ..
+ obtain d2 where "d2 > 0" "ball a d2 \<subseteq> s"
+ using assms(2,1) ..
guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] ..
note d2=this
guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] ..
note d = this
show ?thesis
proof
- show "a\<in>ball a d" using d by auto
+ show "a \<in> ball a d"
+ using d by auto
show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x"
proof (intro strip)
- fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
- def ph \<equiv> "\<lambda>w. w - g'(f w - f x)"
+ fix x y
+ assume as: "x \<in> ball a d" "y \<in> ball a d" "f x = f y"
+ def ph \<equiv> "\<lambda>w. w - g' (f w - f x)"
have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
- unfolding ph_def o_def unfolding diff using f'g'
+ unfolding ph_def o_def
+ unfolding diff
+ using f'g'
by (auto simp add: algebra_simps)
- have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
- apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
- apply(rule_tac[!] ballI)
- proof-
- fix u assume u:"u \<in> ball a d"
- hence "u\<in>s" using d d2 by auto
- have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
- unfolding o_def and diff using f'g' by auto
+ have "norm (ph x - ph y) \<le> (1 / 2) * norm (x - y)"
+ apply (rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
+ apply (rule_tac[!] ballI)
+ proof -
+ fix u
+ assume u: "u \<in> ball a d"
+ then have "u \<in> s"
+ using d d2 by auto
+ have *: "(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
+ unfolding o_def and diff
+ using f'g' by auto
show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
unfolding ph' *
- apply(simp add: comp_def)
- apply(rule bounded_linear.FDERIV[OF assms(3)])
- apply(rule FDERIV_intros) defer
- apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
- apply(rule has_derivative_at_within)
- using assms(5) and `u\<in>s` `a\<in>s`
+ apply (simp add: comp_def)
+ apply (rule bounded_linear.FDERIV[OF assms(3)])
+ apply (rule FDERIV_intros)
+ defer
+ apply (rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
+ apply (rule has_derivative_at_within)
+ using assms(5) and `u \<in> s` `a \<in> s`
apply (auto intro!: FDERIV_intros bounded_linear.FDERIV[of _ "\<lambda>x. x"] derivative_linear)
done
- have **:"bounded_linear (\<lambda>x. f' u x - f' a x)"
- "bounded_linear (\<lambda>x. f' a x - f' u x)"
- apply(rule_tac[!] bounded_linear_sub)
- apply(rule_tac[!] derivative_linear)
- using assms(5) `u\<in>s` `a\<in>s` by auto
+ have **: "bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)"
+ apply (rule_tac[!] bounded_linear_sub)
+ apply (rule_tac[!] derivative_linear)
+ using assms(5) `u \<in> s` `a \<in> s`
+ apply auto
+ done
have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)"
- unfolding * apply(rule onorm_compose)
- unfolding linear_conv_bounded_linear by(rule assms(3) **)+
+ unfolding *
+ apply (rule onorm_compose)
+ unfolding linear_conv_bounded_linear
+ apply (rule assms(3) **)+
+ done
also have "\<dots> \<le> onorm g' * k"
- apply(rule mult_left_mono)
+ apply (rule mult_left_mono)
using d1[THEN conjunct2,rule_format,of u]
using onorm_neg[OF **(1)[unfolded linear_linear]]
using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]]
- by (auto simp add: algebra_simps)
- also have "\<dots> \<le> 1/2" unfolding k_def by auto
- finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption
+ apply (auto simp add: algebra_simps)
+ done
+ also have "\<dots> \<le> 1 / 2"
+ unfolding k_def by auto
+ finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" .
qed
moreover have "norm (ph y - ph x) = norm (y - x)"
- apply(rule arg_cong[where f=norm])
- unfolding ph_def using diff unfolding as by auto
- ultimately show "x = y" unfolding norm_minus_commute by auto
+ apply (rule arg_cong[where f=norm])
+ unfolding ph_def
+ using diff
+ unfolding as
+ apply auto
+ done
+ ultimately show "x = y"
+ unfolding norm_minus_commute by auto
qed
qed auto
qed
-subsection {* Uniformly convergent sequence of derivatives. *}
+
+subsection {* Uniformly convergent sequence of derivatives *}
lemma has_derivative_sequence_lipschitz_lemma:
- fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "convex s"
- assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- assumes "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
- 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)"
-proof (default)+
- fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
- show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
- apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)])
- apply(rule_tac[!] ballI)
- proof-
- fix x assume "x\<in>s"
+ and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ and "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
+ 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)"
+proof rule+
+ fix m n x y
+ assume as: "N \<le> m" "N \<le> n" "x \<in> s" "y \<in> s"
+ show "norm ((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm (x - y)"
+ apply (rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)])
+ apply (rule_tac[!] ballI)
+ proof -
+ fix x
+ assume "x \<in> s"
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)"
- by(rule FDERIV_intros assms(2)[rule_format] `x\<in>s`)+
- { fix h
+ by (rule FDERIV_intros assms(2)[rule_format] `x\<in>s`)+
+ {
+ fix h
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)"
using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"]
- unfolding norm_minus_commute by (auto simp add: algebra_simps)
- also have "\<dots> \<le> e * norm h+ e * norm h"
- using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h]
- using assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
- by(auto simp add:field_simps)
- finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
- thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
- apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
+ unfolding norm_minus_commute
+ by (auto simp add: algebra_simps)
+ also have "\<dots> \<le> e * norm h + e * norm h"
+ using assms(3)[rule_format,OF `N \<le> m` `x \<in> s`, of h]
+ using assms(3)[rule_format,OF `N \<le> n` `x \<in> s`, of h]
+ by (auto simp add: field_simps)
+ finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h"
+ by auto
+ }
+ then show "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
+ apply -
+ apply (rule onorm(2))
+ apply (rule linear_compose_sub)
unfolding linear_conv_bounded_linear
- using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear]
- by auto
+ using assms(2)[rule_format,OF `x \<in> s`, THEN derivative_linear]
+ apply auto
+ done
qed
qed
lemma has_derivative_sequence_lipschitz:
- fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "convex s"
- assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- 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)"
- assumes "0 < e"
- 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)"
-proof(rule,rule)
- case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
+ and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ and "\<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"
+ and "e > 0"
+ 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)"
+proof (rule, rule)
+ case goal1 have *: "2 * (1/2* e) = e" "1/2 * e >0"
+ using `e > 0` by auto
guess N using assms(3)[rule_format,OF *(2)] ..
- thus ?case
- apply(rule_tac x=N in exI)
- apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
- using assms by auto
+ then show ?case
+ apply (rule_tac x=N in exI)
+ apply (rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
+ using assms
+ apply auto
+ done
qed
lemma has_derivative_sequence:
fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "convex s"
- assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- 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)"
- assumes "x0 \<in> s" and "((\<lambda>n. f n x0) ---> l) sequentially"
- shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and>
- (g has_derivative g'(x)) (at x within s)"
-proof-
- 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)"
- apply(rule has_derivative_sequence_lipschitz[where e="42::nat"])
- apply(rule assms)+ by auto
+ and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ and "\<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"
+ and "x0 \<in> s"
+ and "((\<lambda>n. f n x0) ---> l) sequentially"
+ shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)"
+proof -
+ 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)"
+ apply (rule has_derivative_sequence_lipschitz[where e="42::nat"])
+ apply (rule assms)+
+ apply auto
+ done
have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially"
- apply(rule bchoice) unfolding convergent_eq_cauchy
+ apply (rule bchoice)
+ unfolding convergent_eq_cauchy
proof
- fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)"
- proof(cases "x=x0")
- case True thus ?thesis using LIMSEQ_imp_Cauchy[OF assms(5)] by auto
+ fix x
+ assume "x \<in> s"
+ show "Cauchy (\<lambda>n. f n x)"
+ proof (cases "x = x0")
+ case True
+ then show ?thesis
+ using LIMSEQ_imp_Cauchy[OF assms(5)] by auto
next
- case False show ?thesis unfolding Cauchy_def
- proof(rule,rule)
- fix e::real assume "e>0"
- hence *:"e/2>0" "e/2/norm(x-x0)>0"
+ case False
+ show ?thesis
+ unfolding Cauchy_def
+ proof (rule, rule)
+ fix e :: real
+ assume "e > 0"
+ then have *: "e / 2 > 0" "e / 2 / norm (x - x0) > 0"
using False by (auto intro!: divide_pos_pos)
guess M using LIMSEQ_imp_Cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
guess N using lem1[rule_format,OF *(2)] .. note N = this
show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
- apply(rule_tac x="max M N" in exI)
- proof(default+)
- fix m n assume as:"max M N \<le>m" "max M N\<le>n"
- 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))"
- unfolding dist_norm by(rule norm_triangle_sub)
+ apply (rule_tac x="max M N" in exI)
+ proof rule+
+ fix m n
+ assume as: "max M N \<le>m" "max M N\<le>n"
+ 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))"
+ unfolding dist_norm
+ by (rule norm_triangle_sub)
also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False
by auto
also have "\<dots> < e / 2 + e / 2"
- apply(rule add_strict_right_mono)
- using as and M[rule_format] unfolding dist_norm by auto
- finally show "dist (f m x) (f n x) < e" by auto
+ apply (rule add_strict_right_mono)
+ using as and M[rule_format]
+ unfolding dist_norm
+ apply auto
+ done
+ finally show "dist (f m x) (f n x) < e"
+ by auto
qed
qed
qed
qed
then guess g .. note g = this
- 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)"
- proof(rule,rule)
- fix e::real assume *:"e>0"
+ 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)"
+ proof (rule, rule)
+ fix e :: real
+ assume *: "e > 0"
guess N using lem1[rule_format,OF *] .. note N=this
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)"
- apply(rule_tac x=N in exI)
- proof(default+)
- fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
+ apply (rule_tac x=N in exI)
+ proof rule+
+ fix n x y
+ assume as: "N \<le> n" "x \<in> s" "y \<in> s"
have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
unfolding eventually_sequentially
- apply(rule_tac x=N in exI)
- proof(rule,rule)
- fix m assume "N\<le>m"
- thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
+ apply (rule_tac x=N in exI)
+ apply rule
+ apply rule
+ proof -
+ fix m
+ assume "N \<le> m"
+ then show "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
using N[rule_format, of n m x y] and as
by (auto simp add: algebra_simps)
qed
- thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
- apply-
- 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)"])
- apply(rule tendsto_intros g[rule_format] as)+ by assumption
+ then show "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
+ apply -
+ 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)"])
+ apply (rule tendsto_intros g[rule_format] as)+
+ apply assumption
+ done
qed
qed
- show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
- apply(rule,rule,rule g[rule_format],assumption)
- proof fix x assume "x\<in>s"
- have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
+ show ?thesis
+ unfolding has_derivative_within_alt
+ apply (rule_tac x=g in exI)
+ apply rule
+ apply rule
+ apply (rule g[rule_format])
+ apply assumption
+ proof
+ fix x
+ assume "x \<in> s"
+ have lem3: "\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
unfolding LIMSEQ_def
- proof(rule,rule,rule)
- fix u and e::real assume "e>0"
+ proof (rule, rule, rule)
+ fix u
+ fix e :: real
+ assume "e > 0"
show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e"
- proof(cases "u=0")
- case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
- show ?thesis apply(rule_tac x=N in exI) unfolding True
- using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto
+ proof (cases "u = 0")
+ case True
+ guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
+ show ?thesis
+ apply (rule_tac x=N in exI)
+ unfolding True
+ using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0`
+ apply auto
+ done
next
- case False hence *:"e / 2 / norm u > 0"
- using `e>0` by (auto intro!: divide_pos_pos)
+ case False
+ then have *: "e / 2 / norm u > 0"
+ using `e > 0`
+ by (auto intro!: divide_pos_pos)
guess N using assms(3)[rule_format,OF *] .. note N=this
- show ?thesis apply(rule_tac x=N in exI)
- proof(rule,rule) case goal1
- show ?case unfolding dist_norm
+ show ?thesis
+ apply (rule_tac x=N in exI)
+ apply rule
+ apply rule
+ proof -
+ case goal1
+ show ?case
+ unfolding dist_norm
using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
- by (auto simp add:field_simps)
+ by (auto simp add: field_simps)
qed
qed
qed
show "bounded_linear (g' x)"
unfolding linear_linear linear_iff
- apply(rule,rule,rule) defer
- proof(rule,rule)
- fix x' y z::"'m" and c::real
+ apply rule
+ apply rule
+ apply rule
+ defer
+ apply rule
+ apply rule
+ proof -
+ fix x' y z :: 'm
+ fix c :: real
note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
- apply(rule tendsto_unique[OF trivial_limit_sequentially])
- apply(rule lem3[rule_format])
+ apply (rule tendsto_unique[OF trivial_limit_sequentially])
+ apply (rule lem3[rule_format])
unfolding lin[THEN bounded_linear_imp_linear, THEN linear_cmul]
- apply (intro tendsto_intros) by(rule lem3[rule_format])
+ apply (intro tendsto_intros)
+ apply (rule lem3[rule_format])
+ done
show "g' x (y + z) = g' x y + g' x z"
- apply(rule tendsto_unique[OF trivial_limit_sequentially])
- apply(rule lem3[rule_format])
+ apply (rule tendsto_unique[OF trivial_limit_sequentially])
+ apply (rule lem3[rule_format])
unfolding lin[THEN bounded_linear_imp_linear, THEN linear_add]
- apply(rule tendsto_add) by(rule lem3[rule_format])+
+ apply (rule tendsto_add)
+ apply (rule lem3[rule_format])+
+ done
qed
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)"
- proof(rule,rule) case goal1
- have *:"e/3>0" using goal1 by auto
+ proof (rule, rule)
+ case goal1
+ have *: "e / 3 > 0"
+ using goal1 by auto
guess N1 using assms(3)[rule_format,OF *] .. note N1=this
guess N2 using lem2[rule_format,OF *] .. note N2=this
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
- show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1])
- proof(rule,rule)
- fix y assume as:"y \<in> s" "norm (y - x) < d1"
- let ?N ="max N1 N2"
+ show ?case
+ apply (rule_tac x=d1 in exI)
+ apply rule
+ apply (rule d1[THEN conjunct1])
+ apply rule
+ apply rule
+ proof -
+ fix y
+ assume as: "y \<in> s" "norm (y - x) < d1"
+ let ?N = "max N1 N2"
have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)"
- apply(subst norm_minus_cancel[THEN sym])
- using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto
+ apply (subst norm_minus_cancel[symmetric])
+ using N2[rule_format, OF _ `y \<in> s` `x \<in> s`, of ?N]
+ apply auto
+ done
moreover
have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
- using d1 and as by auto
+ using d1 and as
+ by auto
ultimately
- have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
+ have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
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)"]
- by (auto simp add:algebra_simps)
+ by (auto simp add: algebra_simps)
moreover
have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
- using N1 `x\<in>s` by auto
+ using N1 `x \<in> s` by auto
ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
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)"]
- by(auto simp add:algebra_simps)
+ by (auto simp add: algebra_simps)
qed
qed
qed
@@ -1328,122 +1972,172 @@
text {* Can choose to line up antiderivatives if we want. *}
lemma has_antiderivative_sequence:
- fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "convex s"
- assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- 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"
- shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
-proof(cases "s={}")
- case False then obtain a where "a\<in>s" by auto
- 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
+ and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ and "\<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"
+ shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g' x) (at x within s)"
+proof (cases "s = {}")
+ case False
+ then obtain a where "a \<in> s"
+ by auto
+ 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
show ?thesis
- apply(rule *)
- apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
- apply(rule,rule)
- apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
- apply(rule `a\<in>s`) by auto
+ apply (rule *)
+ apply (rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
+ apply rule
+ apply rule
+ apply (rule has_derivative_add_const, rule assms(2)[rule_format])
+ apply assumption
+ apply (rule `a \<in> s`)
+ apply auto
+ done
qed auto
lemma has_antiderivative_limit:
- fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
+ fixes g' :: "'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
assumes "convex s"
- 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))"
- shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
-proof-
- 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))"
- apply(rule) using assms(2)
- apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
+ and "\<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)"
+ shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g' x) (at x within s)"
+proof -
+ 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)"
+ apply rule
+ using assms(2)
+ apply (erule_tac x="inverse (real (Suc n))" in allE)
+ apply auto
+ done
guess f using *[THEN choice] .. note * = this
- guess f' using *[THEN choice] .. note f=this
- show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer
- proof(rule,rule)
- fix e::real assume "0<e" guess N using reals_Archimedean[OF `e>0`] .. note N=this
+ guess f' using *[THEN choice] .. note f = this
+ show ?thesis
+ apply (rule has_antiderivative_sequence[OF assms(1), of f f'])
+ defer
+ apply rule
+ apply rule
+ proof -
+ fix e :: real
+ assume "e > 0"
+ guess N using reals_Archimedean[OF `e>0`] .. note N=this
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"
- apply(rule_tac x=N in exI)
- proof(default+)
+ apply (rule_tac x=N in exI)
+ proof rule+
case goal1
- have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
- using goal1(1) by(auto simp add:field_simps)
+ have *: "inverse (real (Suc n)) \<le> e"
+ apply (rule order_trans[OF _ N[THEN less_imp_le]])
+ using goal1(1)
+ apply (auto simp add: field_simps)
+ done
show ?case
- using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
- apply(rule order_trans) using N * apply(cases "h=0") by auto
+ using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
+ apply (rule order_trans)
+ using N *
+ apply (cases "h = 0")
+ apply auto
+ done
qed
- qed(insert f,auto)
+ qed (insert f, auto)
qed
-subsection {* Differentiation of a series. *}
+
+subsection {* Differentiation of a series *}
-definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
-(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
+definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> nat set \<Rightarrow> bool"
+ (infixl "sums'_seq" 12)
+ where "(f sums_seq l) s \<longleftrightarrow> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
lemma has_derivative_series:
- fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes "convex s"
- assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- 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)"
- assumes "x\<in>s" and "((\<lambda>n. f n x) sums_seq l) k"
- 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)"
+ and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ and "\<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"
+ and "x \<in> s"
+ and "((\<lambda>n. f n x) sums_seq l) k"
+ 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)"
unfolding sums_seq_def
- apply(rule has_derivative_sequence[OF assms(1) _ assms(3)])
- apply(rule, rule)
- apply(rule has_derivative_setsum)
- apply(rule assms(2)[rule_format])
+ apply (rule has_derivative_sequence[OF assms(1) _ assms(3)])
+ apply rule
+ apply rule
+ apply (rule has_derivative_setsum)
+ apply (rule assms(2)[rule_format])
apply assumption
- using assms(4-5) unfolding sums_seq_def by auto
+ using assms(4-5)
+ unfolding sums_seq_def
+ apply auto
+ done
-subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
+
+text {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
-definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real filter \<Rightarrow> bool)"
-(infixl "has'_vector'_derivative" 12) where
- "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
+definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
+ (infixl "has'_vector'_derivative" 12)
+ where "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
-definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
+definition "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"
lemma vector_derivative_works:
- fixes f::"real \<Rightarrow> 'a::real_normed_vector"
- shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
+ fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
+ shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net"
+ (is "?l = ?r")
proof
- assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
- then interpret bounded_linear f' by auto
- show ?r unfolding vector_derivative_def has_vector_derivative_def
- apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
- using f' unfolding scaleR[THEN sym] by auto
+ assume ?l
+ guess f' using `?l`[unfolded differentiable_def] .. note f' = this
+ then interpret bounded_linear f'
+ by auto
+ show ?r
+ unfolding vector_derivative_def has_vector_derivative_def
+ apply -
+ apply (rule someI_ex,rule_tac x="f' 1" in exI)
+ using f'
+ unfolding scaleR[symmetric]
+ apply auto
+ done
next
- assume ?r thus ?l
+ assume ?r
+ then show ?l
unfolding vector_derivative_def has_vector_derivative_def differentiable_def
by auto
qed
lemma has_vector_derivative_withinI_DERIV:
- assumes f: "DERIV f x :> y" shows "(f has_vector_derivative y) (at x within s)"
+ assumes f: "DERIV f x :> y"
+ shows "(f has_vector_derivative y) (at x within s)"
unfolding has_vector_derivative_def real_scaleR_def
apply (rule has_derivative_at_within)
using DERIV_conv_has_derivative[THEN iffD1, OF f]
- apply (subst mult_commute) .
+ apply (subst mult_commute)
+ apply assumption
+ done
lemma vector_derivative_unique_at:
assumes "(f has_vector_derivative f') (at x)"
- assumes "(f has_vector_derivative f'') (at x)"
+ and "(f has_vector_derivative f'') (at x)"
shows "f' = f''"
-proof-
+proof -
have "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
using assms [unfolded has_vector_derivative_def]
by (rule frechet_derivative_unique_at)
- thus ?thesis unfolding fun_eq_iff by auto
+ then show ?thesis
+ unfolding fun_eq_iff by auto
qed
lemma vector_derivative_unique_within_closed_interval:
- assumes "a < b" and "x \<in> {a..b}"
+ assumes "a < b"
+ and "x \<in> {a..b}"
assumes "(f has_vector_derivative f') (at x within {a..b})"
assumes "(f has_vector_derivative f'') (at x within {a..b})"
shows "f' = f''"
-proof-
- have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
- apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
- using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2)
- by auto
+proof -
+ have *: "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
+ apply (rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
+ using assms(3-)[unfolded has_vector_derivative_def]
+ using assms(1-2)
+ apply auto
+ done
show ?thesis
- proof(rule ccontr)
+ proof (rule ccontr)
assume **: "f' \<noteq> f''"
with * have "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1"
by (auto simp: fun_eq_iff)
@@ -1453,76 +2147,106 @@
qed
lemma vector_derivative_at:
- shows "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
- apply(rule vector_derivative_unique_at) defer apply assumption
- unfolding vector_derivative_works[THEN sym] differentiable_def
- unfolding has_vector_derivative_def by auto
+ "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
+ apply (rule vector_derivative_unique_at)
+ defer
+ apply assumption
+ unfolding vector_derivative_works[symmetric] differentiable_def
+ unfolding has_vector_derivative_def
+ apply auto
+ done
lemma vector_derivative_within_closed_interval:
- assumes "a < b" and "x \<in> {a..b}"
+ assumes "a < b"
+ and "x \<in> {a..b}"
assumes "(f has_vector_derivative f') (at x within {a..b})"
shows "vector_derivative f (at x within {a..b}) = f'"
- apply(rule vector_derivative_unique_within_closed_interval)
+ apply (rule vector_derivative_unique_within_closed_interval)
using vector_derivative_works[unfolded differentiable_def]
- using assms by(auto simp add:has_vector_derivative_def)
+ using assms
+ apply (auto simp add:has_vector_derivative_def)
+ done
-lemma has_vector_derivative_within_subset:
- "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
- unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
+lemma has_vector_derivative_within_subset:
+ "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
+ (f has_vector_derivative f') (at x within t)"
+ unfolding has_vector_derivative_def
+ apply (rule has_derivative_within_subset)
+ apply auto
+ done
-lemma has_vector_derivative_const:
- "((\<lambda>x. c) has_vector_derivative 0) net"
- unfolding has_vector_derivative_def using has_derivative_const by auto
+lemma has_vector_derivative_const: "((\<lambda>x. c) has_vector_derivative 0) net"
+ unfolding has_vector_derivative_def
+ using has_derivative_const
+ by auto
lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
- unfolding has_vector_derivative_def using has_derivative_id by auto
+ unfolding has_vector_derivative_def
+ using has_derivative_id
+ by auto
lemma has_vector_derivative_cmul:
- "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
+ "(f has_vector_derivative f') net \<Longrightarrow>
+ ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
unfolding has_vector_derivative_def
apply (drule scaleR_right_has_derivative)
- by (auto simp add: algebra_simps)
+ apply (auto simp add: algebra_simps)
+ done
lemma has_vector_derivative_cmul_eq:
assumes "c \<noteq> 0"
shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
- apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
- apply(rule has_vector_derivative_cmul) using assms by auto
+ apply rule
+ apply (drule has_vector_derivative_cmul[where c="1/c"])
+ defer
+ apply (rule has_vector_derivative_cmul)
+ using assms
+ apply auto
+ done
lemma has_vector_derivative_neg:
- "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
- unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
+ "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. - f x) has_vector_derivative (- f')) net"
+ unfolding has_vector_derivative_def
+ apply (drule has_derivative_neg)
+ apply auto
+ done
lemma has_vector_derivative_add:
assumes "(f has_vector_derivative f') net"
- assumes "(g has_vector_derivative g') net"
- shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
+ and "(g has_vector_derivative g') net"
+ shows "((\<lambda>x. f x + g x) has_vector_derivative (f' + g')) net"
using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
- unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
+ unfolding has_vector_derivative_def
+ unfolding scaleR_right_distrib
+ by auto
lemma has_vector_derivative_sub:
assumes "(f has_vector_derivative f') net"
- assumes "(g has_vector_derivative g') net"
- shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
+ and "(g has_vector_derivative g') net"
+ shows "((\<lambda>x. f x - g x) has_vector_derivative (f' - g')) net"
using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
- unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
+ unfolding has_vector_derivative_def scaleR_right_diff_distrib
+ by auto
lemma has_vector_derivative_bilinear_within:
assumes "(f has_vector_derivative f') (at x within s)"
- assumes "(g has_vector_derivative g') (at x within s)"
+ and "(g has_vector_derivative g') (at x within s)"
assumes "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)"
-proof-
- interpret bounded_bilinear h using assms by auto
- show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
+proof -
+ interpret bounded_bilinear h
+ using assms by auto
+ show ?thesis
+ using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
unfolding o_def has_vector_derivative_def
- using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib
+ using assms(3)
+ unfolding scaleR_right scaleR_left scaleR_right_distrib
by auto
qed
lemma has_vector_derivative_bilinear_at:
assumes "(f has_vector_derivative f') (at x)"
- assumes "(g has_vector_derivative g') (at x)"
+ and "(g has_vector_derivative g') (at x)"
assumes "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
using has_vector_derivative_bilinear_within[OF assms] .
@@ -1533,40 +2257,54 @@
by (rule has_derivative_at_within)
lemma has_vector_derivative_transform_within:
- assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+ assumes "0 < d"
+ and "x \<in> s"
+ and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
assumes "(f has_vector_derivative f') (at x within s)"
shows "(g has_vector_derivative f') (at x within s)"
- using assms unfolding has_vector_derivative_def
+ using assms
+ unfolding has_vector_derivative_def
by (rule has_derivative_transform_within)
lemma has_vector_derivative_transform_at:
- assumes "0 < d" and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
- assumes "(f has_vector_derivative f') (at x)"
+ assumes "0 < d"
+ and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
+ and "(f has_vector_derivative f') (at x)"
shows "(g has_vector_derivative f') (at x)"
- using assms unfolding has_vector_derivative_def
+ using assms
+ unfolding has_vector_derivative_def
by (rule has_derivative_transform_at)
lemma has_vector_derivative_transform_within_open:
- assumes "open s" and "x \<in> s" and "\<forall>y\<in>s. f y = g y"
- assumes "(f has_vector_derivative f') (at x)"
+ assumes "open s"
+ and "x \<in> s"
+ and "\<forall>y\<in>s. f y = g y"
+ and "(f has_vector_derivative f') (at x)"
shows "(g has_vector_derivative f') (at x)"
- using assms unfolding has_vector_derivative_def
+ using assms
+ unfolding has_vector_derivative_def
by (rule has_derivative_transform_within_open)
lemma vector_diff_chain_at:
assumes "(f has_vector_derivative f') (at x)"
- assumes "(g has_vector_derivative g') (at (f x))"
+ and "(g has_vector_derivative g') (at (f x))"
shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
- using assms(2) unfolding has_vector_derivative_def apply-
- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
- unfolding o_def real_scaleR_def scaleR_scaleR .
+ using assms(2)
+ unfolding has_vector_derivative_def
+ apply -
+ apply (drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
+ apply (simp only: o_def real_scaleR_def scaleR_scaleR)
+ done
lemma vector_diff_chain_within:
assumes "(f has_vector_derivative f') (at x within s)"
- assumes "(g has_vector_derivative g') (at (f x) within f ` s)"
- shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
- using assms(2) unfolding has_vector_derivative_def apply-
- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
- unfolding o_def real_scaleR_def scaleR_scaleR .
+ and "(g has_vector_derivative g') (at (f x) within f ` s)"
+ shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
+ using assms(2)
+ unfolding has_vector_derivative_def
+ apply -
+ apply (drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
+ apply (simp only: o_def real_scaleR_def scaleR_scaleR)
+ done
end