# HG changeset patch # User huffman # Date 1312900638 25200 # Node ID 2362a970e3485ef18d8da42da32148639cb2da96 # Parent 5469da57ab773c235c013201ec42df8de64493a8 Derivative.thy: clean up formatting diff -r 5469da57ab77 -r 2362a970e348 src/HOL/Multivariate_Analysis/Derivative.thy --- a/src/HOL/Multivariate_Analysis/Derivative.thy Mon Aug 08 21:17:52 2011 -0700 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy Tue Aug 09 07:37:18 2011 -0700 @@ -171,32 +171,33 @@ lemma has_derivative_neg_eq: "((\x. -(f x)) has_derivative (\h. -(f' h))) net \ (f has_derivative f') net" apply(rule, drule_tac[!] has_derivative_neg) by auto -lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net" - shows "((\x. f(x) + g(x)) has_derivative (\h. f'(h) + g'(h))) net" proof- +lemma has_derivative_add: + assumes "(f has_derivative f') net" and "(g has_derivative g') net" + shows "((\x. f(x) + g(x)) has_derivative (\h. f'(h) + g'(h))) net" +proof- note as = assms[unfolded has_derivative_def] show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add) using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as - by (auto simp add:algebra_simps scaleR_right_diff_distrib scaleR_right_distrib) qed + by (auto simp add:algebra_simps scaleR_right_diff_distrib scaleR_right_distrib) +qed lemma has_derivative_add_const:"(f has_derivative f') net \ ((\x. f x + c) has_derivative f') net" apply(drule has_derivative_add) apply(rule has_derivative_const) by auto lemma has_derivative_sub: - "(f has_derivative f') net \ (g has_derivative g') net \ ((\x. f(x) - g(x)) has_derivative (\h. f'(h) - g'(h))) net" - apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:algebra_simps) + assumes "(f has_derivative f') net" and "(g has_derivative g') net" + shows "((\x. f(x) - g(x)) has_derivative (\h. f'(h) - g'(h))) net" + unfolding diff_minus by (intro has_derivative_add has_derivative_neg assms) -lemma has_derivative_setsum: assumes "finite s" "\a\s. ((f a) has_derivative (f' a)) net" +lemma has_derivative_setsum: + assumes "finite s" and "\a\s. ((f a) has_derivative (f' a)) net" shows "((\x. setsum (\a. f a x) s) has_derivative (\h. setsum (\a. f' a h) s)) net" - apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1)) -proof- fix x F assume as:"finite F" "x \ F" "x\s" "((\x. \a\F. f a x) has_derivative (\h. \a\F. f' a h)) net" - thus "((\xa. \a\insert x F. f a xa) has_derivative (\h. \a\insert x F. f' a h)) net" - unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto -qed(auto intro!: has_derivative_const) + using assms by (induct, simp_all add: has_derivative_const has_derivative_add) lemma has_derivative_setsum_numseg: "\i. m \ i \ i \ n \ ((f i) has_derivative (f' i)) net \ ((\x. setsum (\i. f i x) {m..n::nat}) has_derivative (\h. setsum (\i. f' i h) {m..n})) net" - apply(rule has_derivative_setsum) by auto + by (rule has_derivative_setsum) simp_all subsection {* somewhat different results for derivative of scalar multiplier. *} @@ -211,7 +212,8 @@ unfolding euclidean_component_def by (rule inner.bounded_linear_right) -lemma has_derivative_vmul_component: fixes c::"'a::real_normed_vector \ 'b::euclidean_space" and v::"'c::real_normed_vector" +lemma has_derivative_vmul_component: + fixes c::"'a::real_normed_vector \ 'b::euclidean_space" and v::"'c::real_normed_vector" assumes "(c has_derivative c') net" shows "((\x. c(x)$$k *\<^sub>R v) has_derivative (\x. (c' x)$$k *\<^sub>R v)) net" proof- have *:"\y. (c y $$ k *\<^sub>R v - (c (netlimit net) $$ k *\<^sub>R v + c' (y - netlimit net) $$ k *\<^sub>R v)) = @@ -261,9 +263,11 @@ apply(drule Lim_inner[where a=v]) unfolding o_def by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed -lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const - has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul - bounded_linear.has_derivative has_derivative_lift_dot +lemmas has_derivative_intros = + has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id + has_derivative_const has_derivative_neg has_derivative_vmul_component + has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul + bounded_linear.has_derivative has_derivative_lift_dot subsection {* limit transformation for derivatives. *} @@ -303,22 +307,28 @@ lemma differentiable_at_withinI: "f differentiable (at x) \ f differentiable (at x within s)" unfolding differentiable_def using has_derivative_at_within by blast -lemma differentiable_within_open: assumes "a \ s" "open s" shows - "f differentiable (at a within s) \ (f differentiable (at a))" +lemma differentiable_within_open: (* TODO: delete *) + assumes "a \ s" and "open s" + shows "f differentiable (at a within s) \ (f differentiable (at a))" using assms by (simp only: at_within_interior interior_open) -lemma differentiable_on_eq_differentiable_at: "open s \ (f differentiable_on s \ (\x\s. f differentiable at x))" - unfolding differentiable_on_def by(auto simp add: differentiable_within_open) +lemma differentiable_on_eq_differentiable_at: + "open s \ (f differentiable_on s \ (\x\s. f differentiable at x))" + unfolding differentiable_on_def + by (auto simp add: at_within_interior interior_open) lemma differentiable_transform_within: - assumes "0 < d" "x \ s" "\x'\s. dist x' x < d \ f x' = g x'" "f differentiable (at x within s)" + assumes "0 < d" and "x \ s" and "\x'\s. dist x' x < d \ f x' = g x'" + assumes "f differentiable (at x within s)" shows "g differentiable (at x within s)" - using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)]) + using assms(4) unfolding differentiable_def + by (auto intro!: has_derivative_transform_within[OF assms(1-3)]) lemma differentiable_transform_at: assumes "0 < d" "\x'. dist x' x < d \ f x' = g x'" "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. *} @@ -330,34 +340,50 @@ lemma linear_frechet_derivative: shows "f differentiable net \ linear(frechet_derivative f net)" - unfolding frechet_derivative_works has_derivative_def by (auto intro: bounded_linear_imp_linear) + 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 \ 'b::real_normed_vector" +lemma Lim_mul_norm_within: + fixes f::"'a::real_normed_vector \ 'b::real_normed_vector" shows "(f ---> 0) (at a within s) \ ((\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 + 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]) -lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)" - shows "continuous (at x within s) f" proof- - from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this +lemma differentiable_imp_continuous_within: + assumes "f differentiable (at x within s)" + shows "continuous (at x within s) f" +proof- + from assms guess f' unfolding differentiable_def has_derivative_within .. + note f'=this then interpret bounded_linear f' by auto have *:"\xa. x\xa \ (f' \ (\y. y - x)) xa + norm (xa - x) *\<^sub>R ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) - ((f' \ (\y. y - x)) x + 0) = f xa - f x" using zero by auto have **:"continuous (at x within s) (f' \ (\y. y - x))" apply(rule continuous_within_compose) apply(rule continuous_intros)+ by(rule linear_continuous_within[OF f'[THEN conjunct1]]) - show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within] - apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and dist_norm - apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI) - by(auto simp add:zero * elim!:allE) qed + show ?thesis unfolding continuous_within + using f'[THEN conjunct2, THEN Lim_mul_norm_within] + apply- apply(drule Lim_add) + apply(rule **[unfolded continuous_within]) + unfolding Lim_within and dist_norm + apply (rule, rule) + apply (erule_tac x=e in allE) + apply (erule impE | assumption)+ + apply (erule exE, rule_tac x=d in exI) + by (auto simp add: zero * elim!: allE) +qed -lemma differentiable_imp_continuous_at: "f differentiable at x \ continuous (at x) f" +lemma differentiable_imp_continuous_at: + "f differentiable at x \ continuous (at x) f" by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV]) -lemma differentiable_imp_continuous_on: "f differentiable_on s \ continuous_on s f" +lemma differentiable_imp_continuous_on: + "f differentiable_on s \ continuous_on s f" unfolding differentiable_on_def continuous_on_eq_continuous_within using differentiable_imp_continuous_within by blast @@ -369,7 +395,8 @@ "f differentiable (at x within t) \ s \ t \ f differentiable (at x within s)" unfolding differentiable_def using has_derivative_within_subset by blast -lemma differentiable_on_subset: "f differentiable_on t \ s \ t \ f differentiable_on s" +lemma differentiable_on_subset: + "f differentiable_on t \ s \ t \ f differentiable_on s" unfolding differentiable_on_def using differentiable_within_subset by blast lemma differentiable_on_empty: "f differentiable_on {}" @@ -381,27 +408,43 @@ lemma has_derivative_within_alt: "(f has_derivative f') (at x within s) \ bounded_linear f' \ (\e>0. \d>0. \y\s. norm(y - x) < d \ norm(f(y) - f(x) - f'(y - x)) \ e * norm(y - x))" (is "?lhs \ ?rhs") -proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,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) proof- +proof + assume ?lhs thus ?rhs + unfolding has_derivative_within apply-apply(erule conjE,rule,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) + proof- fix x y e d assume as:"0 < e" "0 < d" "norm (y - x) < d" "\xa\s. 0 < dist xa x \ dist xa x < d \ dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \ s" "bounded_linear f'" then interpret bounded_linear f' by auto show "norm (f y - f x - f' (y - x)) \ e * norm (y - x)" proof(cases "y=x") - case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next + case True thus ?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\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) 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) unfolding dist_norm diff_0_right norm_scaleR - apply(erule_tac x=xa in ballE,erule impE) proof- + thus ?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) + 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 \ s" "0 < norm (y - x) \ norm (y - x) < d" "norm (f y - f x - f' (y - x)) \ e / 2 * norm (y - x)" thus "\1 / norm (y - x)\ * 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) qed auto qed + apply(rule_tac le_less_trans[of _ "e/2"]) + by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps) + qed auto +qed lemma has_derivative_at_alt: "(f has_derivative f') (at x) \ bounded_linear f' \ @@ -411,11 +454,14 @@ subsection {* The chain rule. *} lemma diff_chain_within: - assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))" + 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)" - unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]]) + unfolding has_derivative_within_alt + apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]]) apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption) - apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule) + apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) +proof(rule,rule) note assms = assms[unfolded has_derivative_within_alt] fix e::real assume "0 min (norm (y - x)) (e / 2 / B2 * norm (y - x))" using d1 d2 d by auto have "norm (f y - f x) \ norm (f y - f x - f' (y - x)) + norm (f' (y - x))" - using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:algebra_simps) - also have "\ \ norm (f y - f x - f' (y - x)) + B1 * norm (y - x)" apply(rule add_left_mono) using B1 by(auto simp add:algebra_simps) - also have "\ \ min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)" apply(rule add_right_mono) using d1 d2 d as by auto + using norm_triangle_sub[of "f y - f x" "f' (y - x)"] + by(auto simp add:algebra_simps) + also have "\ \ norm (f y - f x - f' (y - x)) + B1 * norm (y - x)" + apply(rule add_left_mono) using B1 by(auto simp add:algebra_simps) + also have "\ \ min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)" + apply(rule add_right_mono) using d1 d2 d as by auto also have "\ \ norm (y - x) + B1 * norm (y - x)" by auto also have "\ = norm (y - x) * (1 + B1)" by(auto simp add:field_simps) finally have 3:"norm (f y - f x) \ norm (y - x) * (1 + B1)" by auto - hence "norm (f y - f x) \ d * (1 + B1)" apply- apply(rule order_trans,assumption,rule mult_right_mono) using as B1 by auto + hence "norm (f y - f x) \ d * (1 + B1)" apply- + apply(rule order_trans,assumption,rule mult_right_mono) + using as B1 by auto also have "\ < de" using d B1 by(auto simp add:field_simps) finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \ e / 2 / (1 + B1) * norm (f y - f x)" - apply-apply(rule de[THEN conjunct2,rule_format]) using `y\s` using d as by auto + apply-apply(rule de[THEN conjunct2,rule_format]) + using `y\s` using d as by auto also have "\ = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto - also have "\ \ e / 2 * norm (y - x)" apply(rule mult_left_mono) using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq) + also have "\ \ e / 2 * norm (y - x)" apply(rule mult_left_mono) + using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq) finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \ e / 2 * norm (y - x)" by auto interpret g': bounded_linear g' using assms(2) by auto interpret f': bounded_linear f' using assms(1) by auto have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))" by(auto simp add:algebra_simps f'.diff g'.diff g'.add) - also have "\ \ B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:algebra_simps) - also have "\ \ B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto + also have "\ \ B2 * norm (f y - f x - f' (y - x))" using B2 + by (auto simp add: algebra_simps) + also have "\ \ B2 * (e / 2 / B2 * norm (y - x))" + apply (rule mult_left_mono) using as d1 d2 d B2 by auto also have "\ \ e / 2 * norm (y - x)" using B2 by auto finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \ e / 2 * norm (y - x)" by auto - have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \ e * norm (y - x)" using 5 4 by auto - thus "norm ((g \ f) y - (g \ f) x - (g' \ f') (y - x)) \ e * norm (y - x)" unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub) by assumption qed qed + have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \ e * norm (y - x)" + using 5 4 by auto + thus "norm ((g \ f) y - (g \ f) x - (g' \ f') (y - x)) \ e * norm (y - x)" + unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub) + by assumption + qed +qed lemma diff_chain_at: "(f has_derivative f') (at x) \ (g has_derivative g') (at (f x)) \ ((g o f) has_derivative (g' o f')) (at x)" - using diff_chain_within[of f f' x UNIV g g'] using has_derivative_within_subset[of g g' "f x" UNIV "range f"] unfolding within_UNIV by auto + using diff_chain_within[of f f' x UNIV g g'] + using has_derivative_within_subset[of g g' "f x" UNIV "range f"] + unfolding within_UNIV by auto subsection {* Composition rules stated just for differentiability. *} -lemma differentiable_const[intro]: "(\z. c) differentiable (net::'a::real_normed_vector filter)" +lemma differentiable_const [intro]: + "(\z. c) differentiable (net::'a::real_normed_vector filter)" unfolding differentiable_def using has_derivative_const by auto -lemma differentiable_id[intro]: "(\z. z) differentiable (net::'a::real_normed_vector filter)" +lemma differentiable_id [intro]: + "(\z. z) differentiable (net::'a::real_normed_vector filter)" unfolding differentiable_def using has_derivative_id by auto -lemma differentiable_cmul[intro]: "f differentiable net \ (\x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)" - unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto +lemma differentiable_cmul [intro]: + "f differentiable net \ + (\x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)" + unfolding differentiable_def + apply(erule exE, drule has_derivative_cmul) by auto -lemma differentiable_neg[intro]: "f differentiable net \ (\z. -(f z)) differentiable (net::'a::real_normed_vector filter)" - unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto +lemma differentiable_neg [intro]: + "f differentiable net \ + (\z. -(f z)) differentiable (net::'a::real_normed_vector filter)" + unfolding differentiable_def + apply(erule exE, drule has_derivative_neg) by auto lemma differentiable_add: "f differentiable net \ g differentiable net \ (\z. f z + g z) differentiable (net::'a::real_normed_vector filter)" @@ -488,14 +558,18 @@ lemma differentiable_sub: "f differentiable net \ g differentiable net \ (\z. f z - g z) differentiable (net::'a::real_normed_vector filter)" - unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\z. f' z - f'a z" in exI) - apply(rule has_derivative_sub) by auto + unfolding differentiable_def apply(erule exE)+ + apply(rule_tac x="\z. f' z - f'a z" in exI) + apply(rule has_derivative_sub) by auto lemma differentiable_setsum: assumes "finite s" "\a\s. (f a) differentiable net" - shows "(\x. setsum (\a. f a x) s) differentiable net" proof- + shows "(\x. setsum (\a. f a x) s) differentiable net" +proof- guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] .. - thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed + thus ?thesis unfolding differentiable_def apply- + apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto +qed lemma differentiable_setsum_numseg: shows "\i. m \ i \ i \ n \ (f i) differentiable net \ (\x. setsum (\a. f a x) {m::nat..n}) differentiable net" @@ -517,63 +591,102 @@ limit point from any direction. But OK for nontrivial intervals etc. *} -lemma frechet_derivative_unique_within: fixes f::"'a::euclidean_space \ 'b::real_normed_vector" - assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)" - "(\ie>0. \d. 0 < abs(d) \ abs(d) < e \ (x + d *\<^sub>R basis i) \ s)" shows "f' = f''" proof- +lemma frechet_derivative_unique_within: + fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" + assumes "(f has_derivative f') (at x within s)" + assumes "(f has_derivative f'') (at x within s)" + assumes "(\ie>0. \d. 0 < abs(d) \ abs(d) < e \ (x + d *\<^sub>R basis i) \ s)" + shows "f' = f''" +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 "00`] .. - thus "\x'\s. x' \ x \ dist x' x < e" apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI) - unfolding dist_norm by auto 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) unfolding linear_conv_bounded_linear - apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule,rule ccontr) + 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 "00`] .. + thus "\x'\s. x' \ x \ dist x' x < e" + apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI) + unfolding dist_norm by auto + 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) + unfolding linear_conv_bounded_linear + apply(rule as(1,2)[THEN conjunct1])+ + proof(rule,rule,rule ccontr) fix i assume i:"i "norm (f' (basis i) - f'' (basis i))" - assume "f' (basis i) \ f'' (basis i)" hence "e>0" unfolding e_def by auto + assume "f' (basis i) \ f'' (basis i)" + hence "e>0" unfolding e_def by auto guess d using Lim_sub[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 / \c\) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \c\) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))" unfolding scaleR_right_distrib by auto also have "\ = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))" - unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto - also have "\ = e" unfolding e_def using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by (auto simp add: add.commute ab_diff_minus) - finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis 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 qed qed + unfolding f'.scaleR f''.scaleR + unfolding scaleR_right_distrib scaleR_minus_right by auto + also have "\ = e" unfolding e_def using c[THEN conjunct1] + using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] + by (auto simp add: add.commute ab_diff_minus) + finally show False using c + using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis 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 + qed +qed lemma frechet_derivative_unique_at: shows "(f has_derivative f') (at x) \ (f has_derivative f'') (at x) \ f' = f''" unfolding FDERIV_conv_has_derivative [symmetric] by (rule FDERIV_unique) -lemma continuous_isCont: "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def +lemma continuous_isCont: "isCont f x = continuous (at x) f" + unfolding isCont_def LIM_def unfolding continuous_at Lim_at unfolding dist_nz by auto -lemma frechet_derivative_unique_within_closed_interval: fixes f::"'a::ordered_euclidean_space \ 'b::real_normed_vector" - assumes "\i {a..b}" (is "x\?I") 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,rule) +lemma frechet_derivative_unique_within_closed_interval: + fixes f::"'a::ordered_euclidean_space \ 'b::real_normed_vector" + assumes "\i {a..b}" (is "x\?I") + assumes "(f has_derivative f' ) (at x within {a..b})" + assumes "(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,rule) fix e::real and i assume "e>0" and i:"id. 0 < \d\ \ \d\ < e \ x + d *\<^sub>R basis i \ {a..b}" proof(cases "x$$i=a$$i") - case True thus ?thesis apply(rule_tac x="(min (b$$i - a$$i) e) / 2" in exI) + thus "\d. 0 < \d\ \ \d\ < e \ x + d *\<^sub>R basis i \ {a..b}" + proof(cases "x$$i=a$$i") + case True thus ?thesis + apply(rule_tac x="(min (b$$i - a$$i) e) / 2" in exI) using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2) - unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps) + unfolding mem_interval euclidean_simps basis_component + using i by (auto simp add: field_simps) next note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]] case False moreover have "a $$ i < x $$ i" using False * by auto - moreover { have "a $$ i * 2 + min (x $$ i - a $$ i) e \ a$$i *2 + x$$i - a$$i" by auto - also have "\ = a$$i + x$$i" by auto also have "\ \ 2 * x$$i" using * by auto - finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \ x $$ i * 2" by auto } + moreover { + have "a $$ i * 2 + min (x $$ i - a $$ i) e \ a$$i *2 + x$$i - a$$i" + by auto + also have "\ = a$$i + x$$i" by auto + also have "\ \ 2 * x$$i" using * by auto + finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \ x $$ i * 2" by auto + } moreover have "min (x $$ i - a $$ i) e \ 0" using * and `e>0` by auto hence "x $$ i * 2 \ b $$ i * 2 + min (x $$ i - a $$ i) e" using * by auto - ultimately show ?thesis apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI) + ultimately show ?thesis + apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI) using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2) - unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps) qed qed + unfolding mem_interval euclidean_simps basis_component + using i by (auto simp add: field_simps) + qed +qed -lemma frechet_derivative_unique_within_open_interval: fixes f::"'a::ordered_euclidean_space \ 'b::real_normed_vector" - assumes "x \ {a<.. 'b::real_normed_vector" + assumes "x \ {a<.. 'b::real_normed_vector" - assumes "\i {a..b}" "(f has_derivative f') (at x within {a.. b})" +lemma frechet_derivative_within_closed_interval: + fixes f::"'a::ordered_euclidean_space \ 'b::real_normed_vector" + assumes "\i {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] @@ -660,8 +775,10 @@ have ***: "\y y1 y2 d dx::real. (y1\y\y2\y) \ (y\y1\y\y2) \ d < abs dx \ abs(y1 - y - - dx) \ d \ (abs (y2 - y - dx) \ d) \ False" by arith show False apply(rule ***[OF **, where dx="d * ?D k $$ j" and d="\?D k $$ j\ / 2 * \d\"]) - 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) + 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) qed subsection {* In particular if we have a mapping into @{typ "real"}. *} @@ -673,7 +790,8 @@ and mono: "(\y\s. f y \ f x) \ (\y\s. f x \ f y)" shows "f' = (\v. 0)" proof - - obtain e where e:"e>0" "ball x e \ s" using `open s`[unfolded open_contains_ball] and `x \ s` by auto + obtain e where e:"e>0" "ball x e \ s" + using `open s`[unfolded open_contains_ball] and `x \ s` by auto with differential_zero_maxmin_component[where 'b=real, of 0 e x f, simplified] have "(\\ j. frechet_derivative f (at x) (basis j)) = (0::'a)" unfolding differentiable_def using mono deriv by auto @@ -685,177 +803,283 @@ qed lemma rolle: fixes f::"real\real" - assumes "a < b" "f a = f b" "continuous_on {a..b} f" - "\x\{a<..x\{a<..v. 0)" proof- - have "\x\{a<..y\{a<.. f y) \ (\y\{a<.. f x))" proof- - have "(a + b) / 2 \ {a .. b}" using assms(1) by auto hence *:"{a .. b}\{}" by auto + assumes "a < b" and "f a = f b" and "continuous_on {a..b} f" + assumes "\x\{a<..x\{a<..v. 0)" +proof- + have "\x\{a<..y\{a<.. f y) \ (\y\{a<.. f x))" + proof- + have "(a + b) / 2 \ {a .. b}" using assms(1) by auto + hence *:"{a .. b}\{}" 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\{a<.. c\{a<.. "(a + b) /2" + show ?thesis + proof(cases "d\{a<.. c\{a<.. "(a + b) /2" case False hence "f d = f c" using d c assms(2) by auto - hence "\x. x\{a..b} \ 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 qed qed + hence "\x. x\{a..b} \ 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 + qed + qed then guess x .. note x=this - hence "f' x = (\v. 0)" apply(rule_tac differential_zero_maxmin[of x "{a<..v. 0)" + apply(rule_tac differential_zero_maxmin[of x "{a<.. real" - assumes "a < b" "continuous_on {a .. b} f" "\x\{a<..x\{a<..x\{a<..x\{a<..x\{a<..xa. f' x xa - (f b - f a) / (b - a) * xa) = (\v. 0)" - apply(rule rolle[OF assms(1), of "\x. f x - (f b - f a) / (b - a) * x"]) defer - apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof + apply(rule rolle[OF assms(1), of "\x. f x - (f b - f a) / (b - a) * x"]) + defer + apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ + proof fix x assume x:"x \ {a<..x. f x - (f b - f a) / (b - a) * x) has_derivative (\xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)" by(rule has_derivative_intros assms(3)[rule_format,OF x] - has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+ + has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+ qed(insert assms(1), auto simp add:field_simps) - then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed + then guess x .. + thus ?thesis apply(rule_tac x=x in bexI) + apply(drule fun_cong[of _ _ "b - a"]) by auto +qed -lemma mvt_simple: fixes f::"real \ real" - assumes "ax\{a..b}. (f has_derivative f' x) (at x within {a..b})" +lemma mvt_simple: + fixes f::"real \ real" + assumes "ax\{a..b}. (f has_derivative f' x) (at x within {a..b})" shows "\x\{a<.. {a<.. real" - assumes "a \ b" "\x\{a..b}. (f has_derivative f'(x)) (at x within {a..b})" - shows "\x\{a..b}. f b - f a = f' x (b - a)" proof(cases "a = b") +lemma mvt_very_simple: + fixes f::"real \ real" + assumes "a \ b" and "\x\{a..b}. (f has_derivative f'(x)) (at x within {a..b})" + shows "\x\{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 qed + case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto +qed subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *} -lemma mvt_general: fixes f::"real\'a::euclidean_space" - assumes "ax\{a<..x\{a<.. norm(f'(x) (b - a))" proof- +lemma mvt_general: + fixes f::"real\'a::euclidean_space" + assumes "ax\{a<..x\{a<.. norm(f'(x) (b - a))" +proof- have "\x\{a<.. (f b - f a) \ f) b - (op \ (f b - f a) \ f) a = (f b - f a) \ f' x (b - a)" - apply(rule mvt) apply(rule assms(1)) apply(rule continuous_on_inner continuous_on_intros assms(2))+ - unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto + apply(rule mvt) apply(rule assms(1)) + apply(rule continuous_on_inner continuous_on_intros assms(2))+ + unfolding o_def apply(rule,rule has_derivative_lift_dot) + using assms(3) by auto then guess x .. note x=this 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)^2" by(simp add: power2_eq_square) + have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" + by (simp add: power2_eq_square) also have "\ = (f b - f a) \ (f b - f a)" unfolding power2_norm_eq_inner .. - also have "\ = (f b - f a) \ f' x (b - a)" using x unfolding inner_simps by (auto simp add: inner_diff_left) - also have "\ \ norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz) - 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 qed qed + also have "\ = (f b - f a) \ f' x (b - a)" + using x unfolding inner_simps by (auto simp add: inner_diff_left) + also have "\ \ norm (f b - f a) * norm (f' x (b - a))" + by (rule norm_cauchy_schwarz) + 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 + qed +qed subsection {* Still more general bound theorem. *} -lemma differentiable_bound: fixes f::"'a::euclidean_space \ 'b::euclidean_space" - assumes "convex s" "\x\s. (f has_derivative f'(x)) (at x within s)" "\x\s. onorm(f' x) \ B" and x:"x\s" and y:"y\s" - shows "norm(f x - f y) \ B * norm(x - y)" proof- +lemma differentiable_bound: + fixes f::"'a::euclidean_space \ 'b::euclidean_space" + assumes "convex s" and "\x\s. (f has_derivative f'(x)) (at x within s)" + assumes "\x\s. onorm(f' x) \ B" and x:"x\s" and y:"y\s" + shows "norm(f x - f y) \ B * norm(x - y)" +proof- let ?p = "\u. x + u *\<^sub>R (y - x)" have *:"\u. u\{0..1} \ x + u *\<^sub>R (y - x) \ 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 \ ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+ - unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within) + 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 \ ?p)" apply- + apply(rule continuous_on_intros continuous_on_vmul)+ + 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:"\u\{0<..<1}. ((f \ ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \ (\u. 0 + u *\<^sub>R (y - x))) (at u)" proof rule case goal1 + apply(rule has_derivative_within_subset) + apply(rule assms(2)[rule_format]) by auto + have 2:"\u\{0<..<1}. ((f \ ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \ (\u. 0 + u *\<^sub>R (y - x))) (at u)" + proof rule + case goal1 let ?u = "x + u *\<^sub>R (y - x)" have "(f \ ?p has_derivative (f' ?u) \ (\u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})" apply(rule diff_chain_within) apply(rule has_derivative_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 qed + 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 + qed guess u using mvt_general[OF zero_less_one 1 2] .. note u = this - have **:"\x y. x\s \ norm (f' x y) \ B * norm y" proof- case goal1 + have **:"\x y. x\s \ norm (f' x y) \ B * norm y" + proof- + case goal1 have "norm (f' x y) \ onorm (f' x) * norm y" using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption - also have "\ \ B * norm y" apply(rule mult_right_mono) - using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps) - finally show ?case by simp qed + also have "\ \ B * norm y" + apply(rule mult_right_mono) + using assms(3)[rule_format,OF goal1] + by(auto simp add:field_simps) + finally show ?case by simp + qed have "norm (f x - f y) = norm ((f \ (\u. x + u *\<^sub>R (y - x))) 1 - (f \ (\u. x + u *\<^sub>R (y - x))) 0)" by(auto simp add:norm_minus_commute) also have "\ \ norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto also have "\ \ B * norm(y - x)" apply(rule **) using * and u by auto - finally show ?thesis by(auto simp add:norm_minus_commute) qed + finally show ?thesis by(auto simp add:norm_minus_commute) +qed -lemma differentiable_bound_real: fixes f::"real \ real" - assumes "convex s" "\x\s. (f has_derivative f' x) (at x within s)" "\x\s. onorm(f' x) \ B" and x:"x\s" and y:"y\s" +lemma differentiable_bound_real: + fixes f::"real \ real" + assumes "convex s" and "\x\s. (f has_derivative f' x) (at x within s)" + assumes "\x\s. onorm(f' x) \ B" and x:"x\s" and y:"y\s" shows "norm(f x - f y) \ 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 subsection {* In particular. *} -lemma has_derivative_zero_constant: fixes f::"real\real" +lemma has_derivative_zero_constant: + fixes f::"real\real" assumes "convex s" "\x\s. (f has_derivative (\h. 0)) (at x within s)" - shows "\c. \x\s. f x = c" proof(cases "s={}") + shows "\c. \x\s. f x = c" +proof(cases "s={}") case False then obtain x where "x\s" by auto have "\y. y\s \ f x = f y" proof- case goal1 - thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\s` - unfolding onorm_const by auto qed - thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto + thus ?case + using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\s` + unfolding onorm_const by auto qed + thus ?thesis apply(rule_tac x="f x" in exI) by auto +qed auto lemma has_derivative_zero_unique: fixes f::"real\real" - assumes "convex s" "a \ s" "f a = c" "\x\s. (f has_derivative (\h. 0)) (at x within s)" "x\s" - shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto + assumes "convex s" and "a \ s" and "f a = c" + assumes "\x\s. (f has_derivative (\h. 0)) (at x within s)" and "x\s" + shows "f x = c" + using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto subsection {* Differentiability of inverse function (most basic form). *} -lemma has_derivative_inverse_basic: fixes f::"'b::euclidean_space \ 'c::euclidean_space" - assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \ f' = id" "continuous (at y) g" - "open t" "y \ t" "\z\t. f(g z) = z" - shows "(g has_derivative g') (at y)" proof- - interpret f': bounded_linear f' using assms unfolding has_derivative_def by auto +lemma has_derivative_inverse_basic: + fixes f::"'b::euclidean_space \ 'c::euclidean_space" + assumes "(f has_derivative f') (at (g y))" + assumes "bounded_linear g'" and "g' \ f' = id" and "continuous (at y) g" + assumes "open t" and "y \ t" and "\z\t. f(g z) = z" + shows "(g has_derivative g') (at y)" +proof- + interpret f': bounded_linear f' + using assms unfolding has_derivative_def 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:"\x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*) - have lem1:"\e>0. \d>0. \z. norm(z - y) < d \ norm(g z - g y - g'(z - y)) \ e * norm(g z - g y)" proof(rule,rule) case goal1 + have lem1:"\e>0. \d>0. \z. norm(z - y) < d \ norm(g z - g y - g'(z - y)) \ 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 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\t" using d2 d unfolding dist_norm by auto + 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\t" + using d2 d unfolding dist_norm by auto have "norm (g z - g y - g' (z - y)) \ 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(7)[rule_format,OF `z\t`] apply(subst norm_minus_cancel[THEN sym]) by auto - also have "\ \ norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format]) - also have "\ \ (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\t`]]) apply(cases "z=y") defer - apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format]) using as d C d0 by auto - also have "\ \ e * norm (g z - g y)" using C by(auto simp add:field_simps) - finally show "norm (g z - g y - g' (z - y)) \ e * norm (g z - g y)" by simp qed auto qed - have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\"C*2" + unfolding g'.diff f'.diff + unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] + unfolding assms(7)[rule_format,OF `z\t`] + apply(subst norm_minus_cancel[THEN sym]) by auto + also have "\ \ norm(f (g z) - y - f' (g z - g y)) * C" + by (rule C [THEN conjunct2, rule_format]) + also have "\ \ (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\t`]]) + apply(cases "z=y") defer + apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format]) + using as d C d0 by auto + also have "\ \ e * norm (g z - g y)" + using C by (auto simp add: field_simps) + finally show "norm (g z - g y - g' (z - y)) \ e * norm (g z - g y)" + by simp + qed auto + qed + have *:"(0::real) < 1 / 2" by auto + guess d using lem1[rule_format,OF *] .. note d=this + def B\"C*2" have "B>0" unfolding B_def using C by auto - have lem2:"\z. norm(z - y) < d \ norm(g z - g y) \ B * norm(z - y)" proof(rule,rule) case goal1 - have "norm (g z - g y) \ norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" by(rule norm_triangle_sub) - also have "\ \ norm(g' (z - y)) + 1 / 2 * norm (g z - g y)" apply(rule add_left_mono) using d and goal1 by auto - also have "\ \ 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) qed - show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1 + have lem2:"\z. norm(z - y) < d \ norm(g z - g y) \ B * norm(z - y)" + proof(rule,rule) case goal1 + have "norm (g z - g y) \ norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" + by(rule norm_triangle_sub) + also have "\ \ norm(g' (z - y)) + 1 / 2 * norm (g z - g y)" + apply(rule add_left_mono) using d and goal1 by auto + also have "\ \ 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) + 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 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)) \ e / B * norm(g z - g y)" using d' k by auto - also have "\ \ 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` by(auto simp add:field_simps) - finally show "norm (g z - g y - g' (z - y)) \ e * norm (z - y)" by simp qed(insert k, auto) qed qed + 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)) \ e / B * norm(g z - g y)" + using d' k by auto + also have "\ \ 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` + by (auto simp add: field_simps) + finally show "norm (g z - g y - g' (z - y)) \ e * norm (z - y)" + by simp qed(insert k, auto) + qed +qed subsection {* Simply rewrite that based on the domain point x. *} -lemma has_derivative_inverse_basic_x: fixes f::"'b::euclidean_space \ 'c::euclidean_space" +lemma has_derivative_inverse_basic_x: + fixes f::"'b::euclidean_space \ '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 \ t" "\y\t. f(g y) = y" shows "(g has_derivative g') (at (f(x)))" @@ -863,95 +1087,147 @@ subsection {* This is the version in Dieudonne', assuming continuity of f and g. *} -lemma has_derivative_inverse_dieudonne: fixes f::"'a::euclidean_space \ 'b::euclidean_space" +lemma has_derivative_inverse_dieudonne: + fixes f::"'a::euclidean_space \ 'b::euclidean_space" assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\x\s. g(f x) = x" (**) "x\s" "(f has_derivative f') (at x)" "bounded_linear g'" "g' o 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 + using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)] + continuous_on_eq_continuous_at[OF assms(2)] by auto subsection {* Here's the simplest way of not assuming much about g. *} -lemma has_derivative_inverse: fixes f::"'a::euclidean_space \ 'b::euclidean_space" +lemma has_derivative_inverse: + fixes f::"'a::euclidean_space \ 'b::euclidean_space" assumes "compact s" "x \ s" "f x \ interior(f ` s)" "continuous_on s f" "\y\s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \ f' = id" - shows "(g has_derivative g') (at (f x))" proof- + shows "(g has_derivative g') (at (f x))" +proof- { fix y assume "y\interior (f ` s)" - then obtain x where "x\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\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_inverse[OF assms(4,1)]) - apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption) qed + then obtain x where "x\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\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_inverse[OF assms(4,1)]) + apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ + by(rule, rule *, assumption) +qed subsection {* Proving surjectivity via Brouwer fixpoint theorem. *} -lemma brouwer_surjective: fixes f::"'n::ordered_euclidean_space \ 'n" +lemma brouwer_surjective: + fixes f::"'n::ordered_euclidean_space \ 'n" assumes "compact t" "convex t" "t \ {}" "continuous_on t f" "\x\s. \y\t. x + (y - f y) \ t" "x\s" - shows "\y\t. f y = x" proof- - have *:"\x y. f y = x \ x + (y - f y) = y" by(auto simp add:algebra_simps) - show ?thesis unfolding * apply(rule brouwer[OF assms(1-3), of "\y. x + (y - f y)"]) - apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed + shows "\y\t. f y = x" +proof- + have *:"\x y. f y = x \ x + (y - f y) = y" + by(auto simp add:algebra_simps) + show ?thesis + unfolding * + apply(rule brouwer[OF assms(1-3), of "\y. x + (y - f y)"]) + apply(rule continuous_on_intros assms)+ using assms(4-6) by auto +qed -lemma brouwer_surjective_cball: fixes f::"'n::ordered_euclidean_space \ 'n" +lemma brouwer_surjective_cball: + fixes f::"'n::ordered_euclidean_space \ 'n" assumes "0 < e" "continuous_on (cball a e) f" "\x\s. \y\cball a e. x + (y - f y) \ cball a e" "x\s" - shows "\y\cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+ - unfolding cball_eq_empty using assms by auto + shows "\y\cball a e. f y = x" + apply(rule brouwer_surjective) + apply(rule compact_cball convex_cball)+ + unfolding cball_eq_empty using assms by auto text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *} -lemma sussmann_open_mapping: fixes f::"'a::euclidean_space \ 'b::ordered_euclidean_space" +lemma sussmann_open_mapping: + fixes f::"'a::euclidean_space \ 'b::ordered_euclidean_space" assumes "open s" "continuous_on s f" "x \ s" "(f has_derivative f') (at x)" "bounded_linear g'" "f' \ g' = id" "t \ s" "x \ interior t" - shows "f x \ interior (f ` t)" proof- - interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto + shows "f x \ 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 - guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos) + guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this + hence *:"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 *:"0z\cball (f x) (e/2). \y\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) proof- - show "continuous_on (cball (f x) e) (\y. f (x + g' (y - f x)))" unfolding g'.diff + prefer 3 apply(rule,rule) + proof- + show "continuous_on (cball (f x) e) (\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) proof- + apply(rule continuous_on_subset[OF assms(2)]) + apply(rule,unfold image_iff,erule bexE) + proof- fix y z assume as:"y \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 "\ \ norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto - also have "\ \ e * B" using as(1)[unfolded mem_cball dist_norm] using B by auto + have "dist x z = norm (g' (f x) - g' y)" + unfolding as(2) and dist_norm by auto + also have "\ \ norm (f x - y) * B" + unfolding g'.diff[THEN sym] using B by auto + also have "\ \ e * B" + using as(1)[unfolded mem_cball dist_norm] using B by auto also have "\ \ e1" using e unfolding less_divide_eq using B by auto finally have "z\cball x e1" unfolding mem_cball by force - thus "z \ s" using e1 assms(7) by auto qed next + thus "z \ s" using e1 assms(7) by auto + qed + next fix y z assume as:"y \ cball (f x) (e / 2)" "z \ cball (f x) e" have "norm (g' (z - f x)) \ norm (z - f x) * B" using B by auto - also have "\ \ 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 "\ \ 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 "\ < 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 **:"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))))) \ 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 "\ \ 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding algebra_simps ** by auto - also have "\ \ 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball dist_norm] by auto - also have "\ \ 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps) + using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] + by (auto simp add: algebra_simps) + also have "\ \ 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" + using e0[THEN conjunct2,rule_format,OF *] + unfolding algebra_simps ** by auto + also have "\ \ 1 / (B * 2) * norm (g' (z - f x)) + e/2" + using as(1)[unfolded mem_cball dist_norm] by auto + also have "\ \ 1 / (B * 2) * B * norm (z - f x) + e/2" + using * and B by (auto simp add: field_simps) also have "\ \ 1 / 2 * norm (z - f x) + e/2" by auto - also have "\ \ e/2 + e/2" apply(rule add_right_mono) using as(2)[unfolded mem_cball dist_norm] unfolding norm_minus_commute by auto - finally show "y + (z - f (x + g' (z - f x))) \ cball (f x) e" unfolding mem_cball dist_norm by auto + also have "\ \ e/2 + e/2" apply(rule add_right_mono) + using as(2)[unfolded mem_cball dist_norm] + unfolding norm_minus_commute by auto + finally show "y + (z - f (x + g' (z - f x))) \ 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 proof - fix y assume "y \ ball (f x) (e/2)" hence *:"y\cball (f x) (e/2)" by auto + apply(rule,rule divide_pos_pos) prefer 3 + proof + fix y assume "y \ ball (f x) (e/2)" + hence *:"y\cball (f x) (e/2)" by auto guess z using lem[rule_format,OF *] .. note z=this - hence "norm (g' (z - f x)) \ norm (z - f x) * B" using B by(auto simp add:field_simps) - also have "\ \ e * B" apply(rule mult_right_mono) using z(1) unfolding mem_cball dist_norm norm_minus_commute using B by auto + hence "norm (g' (z - f x)) \ norm (z - f x) * B" + using B by (auto simp add: field_simps) + also have "\ \ e * B" + apply (rule mult_right_mono) using z(1) + unfolding mem_cball dist_norm norm_minus_commute using B by auto also have "\ \ e1" using e B unfolding less_divide_eq by auto - finally have "x + g'(z - f x) \ t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format]) + finally have "x + g'(z - f x) \ t" apply- + apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm by auto - thus "y \ f ` t" using z by auto qed(insert e, auto) qed + thus "y \ 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. *) @@ -960,7 +1236,8 @@ (* move before left_inverse_linear in Euclidean_Space*) - lemma right_inverse_linear: fixes f::"'a::euclidean_space => 'a" + lemma right_inverse_linear: + fixes f::"'a::euclidean_space => 'a" assumes lf: "linear f" and gf: "f o g = id" shows "linear g" proof- @@ -973,275 +1250,476 @@ with h(1) show ?thesis by blast qed -lemma has_derivative_inverse_strong: fixes f::"'n::ordered_euclidean_space \ 'n" - assumes "open s" "x \ s" "continuous_on s f" - "\x\s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id" - shows "(g has_derivative g') (at (f x))" 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' \ f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym] +lemma has_derivative_inverse_strong: + fixes f::"'n::ordered_euclidean_space \ 'n" + assumes "open s" and "x \ s" and "continuous_on s f" + assumes "\x\s. g(f x) = x" "(f has_derivative f') (at x)" and "f' o g' = id" + shows "(g has_derivative g') (at (f x))" +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' \ f' = id" using assms(6) linf ling + unfolding linear_conv_bounded_linear[THEN sym] using linear_inverse_left by auto - moreover have *:"\t\s. x\interior t \ f x \ interior (f ` t)" apply(rule,rule,rule,rule sussmann_open_mapping ) + moreover have *:"\t\s. x\interior t \ f x \ 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) + have "continuous (at (f x)) g" unfolding continuous_at Lim_at + proof(rule,rule) fix e::real assume "e>0" - hence "f x \ interior (f ` (ball x e \ s))" using *[rule_format,of "ball x e \ s"] `x\s` + hence "f x \ interior (f ` (ball x e \ s))" + using *[rule_format,of "ball x e \ s"] `x\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 "\d>0. \y. 0 < dist y (f x) \ dist y (f x) < d \ 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 \ g ` f ` (ball x e \ s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]] + apply(rule_tac x=d in exI) + apply(rule,rule d[THEN conjunct1]) + proof(rule,rule) case goal1 + hence "g y \ g ` f ` (ball x e \ s)" + using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]] by(auto simp add:dist_commute) hence "g y \ ball x e \ s" using assms(4) by auto - thus "dist (g y) (g (f x)) < e" using assms(4)[rule_format,OF `x\s`] by(auto simp add:dist_commute) qed qed - moreover have "f x \ interior (f ` s)" apply(rule sussmann_open_mapping) - apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\s` by auto - moreover have "\y. y \ interior (f ` s) \ f (g y) = y" proof- case goal1 - hence "y\f ` s" using interior_subset by auto then guess z unfolding image_iff .. - thus ?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 qed + thus "dist (g y) (g (f x)) < e" + using assms(4)[rule_format,OF `x\s`] + by (auto simp add: dist_commute) + qed + qed + moreover have "f x \ interior (f ` s)" + apply(rule sussmann_open_mapping) + apply(rule assms ling)+ + using interior_open[OF assms(1)] and `x\s` by auto + moreover have "\y. y \ interior (f ` s) \ f (g y) = y" + proof- case goal1 + hence "y\f ` s" using interior_subset by auto + then guess z unfolding image_iff .. + thus ?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 +qed subsection {* A rewrite based on the other domain. *} -lemma has_derivative_inverse_strong_x: fixes f::"'a::ordered_euclidean_space \ 'a" - assumes "open s" "g y \ s" "continuous_on s f" - "\x\s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y" +lemma has_derivative_inverse_strong_x: + fixes f::"'a::ordered_euclidean_space \ 'a" + assumes "open s" and "g y \ s" and "continuous_on s f" + assumes "\x\s. g(f x) = x" "(f has_derivative f') (at (g y))" + assumes "f' o 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 subsection {* On a region. *} -lemma has_derivative_inverse_on: fixes f::"'n::ordered_euclidean_space \ 'n" - assumes "open s" "\x\s. (f has_derivative f'(x)) (at x)" "\x\s. g(f x) = x" "f'(x) o g'(x) = id" "x\s" +lemma has_derivative_inverse_on: + fixes f::"'n::ordered_euclidean_space \ 'n" + assumes "open s" and "\x\s. (f has_derivative f'(x)) (at x)" + assumes "\x\s. g(f x) = x" and "f'(x) o g'(x) = id" and "x\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_at) unfolding differentiable_def using assms by auto + apply(rule,rule differentiable_imp_continuous_at) + unfolding differentiable_def using assms by auto -subsection {* Invertible derivative continous at a point implies local injectivity. *) -(* It's only for this we need continuity of the derivative, except of course *) -(* if we want the fact that the inverse derivative is also continuous. So if *) -(* we know for some other reason that the inverse function exists, it's OK. *} +text {* Invertible derivative continous at a point implies local +injectivity. It's only for this we need continuity of the derivative, +except of course if we want the fact that the inverse derivative is +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 \ bounded_linear g ==> bounded_linear (\x. f x - g x)" - using bounded_linear_add[of f "\x. - g x"] bounded_linear_minus[of g] by(auto simp add:algebra_simps) +lemma bounded_linear_sub: + "bounded_linear f \ bounded_linear g ==> bounded_linear (\x. f x - g x)" + using bounded_linear_add[of f "\x. - g x"] bounded_linear_minus[of g] + by (auto simp add: algebra_simps) -lemma has_derivative_locally_injective: fixes f::"'n::euclidean_space \ 'm::euclidean_space" +lemma has_derivative_locally_injective: + fixes f::"'n::euclidean_space \ 'm::euclidean_space" assumes "a \ s" "open s" "bounded_linear g'" "g' o f'(a) = id" "\x\s. (f has_derivative f'(x)) (at x)" "\e>0. \d>0. \x. dist a x < d \ onorm(\v. f' x v - f' a v) < e" - obtains t where "a \ t" "open t" "\x\t. \x'\t. (f x' = f x) \ (x' = x)" proof- + obtains t where "a \ t" "open t" "\x\t. \x'\t. (f x' = f x) \ (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 (\\ i.1)) = (\\ i.1)" "(\\ i.1) \ (0::'n)" defer apply(subst euclidean_eq) using f'g' by auto - hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp + hence *:"0 < onorm g'" + unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp def k \ "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 \ s" using `a\s` .. obtain d2 where "d2>0" "ball a d2 \ s" using assms(2,1) .. - guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\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\ball a d" using d by auto - show "\x\ball a d. \x'\ball a d. f x' = f x \ x' = x" proof(intro strip) + guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\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\ball a d" using d by auto + show "\x\ball a d. \x'\ball a d. f x' = f x \ x' = x" + proof (intro strip) fix x y assume as:"x\ball a d" "y\ball a d" "f x = f y" - def ph \ "\w. w - g'(f w - f x)" have ph':"ph = g' \ (\w. f' a w - (f w - f x))" - unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:algebra_simps) + def ph \ "\w. w - g'(f w - f x)" + have ph':"ph = g' \ (\w. f' a w - (f w - f x))" + unfolding ph_def o_def unfolding diff using f'g' + by (auto simp add: algebra_simps) have "norm (ph x - ph y) \ (1/2) * norm (x - y)" apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\x v. v - g'(f' x v)"]) - apply(rule_tac[!] ballI) proof- fix u assume u:"u \ ball a d" hence "u\s" using d d2 by auto - have *:"(\v. v - g' (f' u v)) = g' \ (\w. f' a w - f' u w)" unfolding o_def and diff using f'g' by auto + apply(rule_tac[!] ballI) + proof- + fix u assume u:"u \ ball a d" + hence "u\s" using d d2 by auto + have *:"(\v. v - g' (f' u v)) = g' \ (\w. f' a w - f' u w)" + unfolding o_def and diff using f'g' by auto show "(ph has_derivative (\v. v - g' (f' u v))) (at u within ball a d)" - unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)]) - apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\x.0",unfolded diff_0_right]) - apply(rule has_derivative_at_within) using assms(5) and `u\s` `a\s` + unfolding ph' * apply(rule diff_chain_within) defer + apply(rule bounded_linear.has_derivative[OF assms(3)]) + apply(rule has_derivative_intros) defer + apply(rule has_derivative_sub[where g'="\x.0",unfolded diff_0_right]) + apply(rule has_derivative_at_within) + using assms(5) and `u\s` `a\s` by(auto intro!: has_derivative_intros derivative_linear) - have **:"bounded_linear (\x. f' u x - f' a x)" "bounded_linear (\x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub) - apply(rule_tac[!] derivative_linear) using assms(5) `u\s` `a\s` by auto - have "onorm (\v. v - g' (f' u v)) \ onorm g' * onorm (\w. f' a w - f' u w)" unfolding * apply(rule onorm_compose) - unfolding linear_conv_bounded_linear by(rule assms(3) **)+ - also have "\ \ onorm g' * k" 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) + have **:"bounded_linear (\x. f' u x - f' a x)" + "bounded_linear (\x. f' a x - f' u x)" + apply(rule_tac[!] bounded_linear_sub) + apply(rule_tac[!] derivative_linear) + using assms(5) `u\s` `a\s` by auto + have "onorm (\v. v - g' (f' u v)) \ onorm g' * onorm (\w. f' a w - f' u w)" + unfolding * apply(rule onorm_compose) + unfolding linear_conv_bounded_linear by(rule assms(3) **)+ + also have "\ \ onorm g' * k" + 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 "\ \ 1/2" unfolding k_def by auto - finally show "onorm (\v. v - g' (f' u v)) \ 1 / 2" by assumption qed - moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm]) + finally show "onorm (\v. v - g' (f' u v)) \ 1 / 2" by assumption + 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 qed qed auto qed + ultimately show "x = y" unfolding norm_minus_commute by auto + qed + qed auto +qed subsection {* Uniformly convergent sequence of derivatives. *} -lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \ 'm::euclidean_space \ 'n::euclidean_space" - assumes "convex s" "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" - "\n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm(h)" - shows "\m\N. \n\N. \x\s. \y\s. norm((f m x - f n x) - (f m y - f n y)) \ 2 * e * norm(x - y)" proof(default)+ +lemma has_derivative_sequence_lipschitz_lemma: + fixes f::"nat \ 'm::euclidean_space \ 'n::euclidean_space" + assumes "convex s" + assumes "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" + assumes "\n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm(h)" + shows "\m\N. \n\N. \x\s. \y\s. norm((f m x - f n x) - (f m y - f n y)) \ 2 * e * norm(x - y)" +proof (default)+ fix m n x y assume as:"N\m" "N\n" "x\s" "y\s" show "norm((f m x - f n x) - (f m y - f n y)) \ 2 * e * norm(x - y)" - apply(rule differentiable_bound[where f'="\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\s" show "((\a. f m a - f n a) has_derivative (\h. f' m x h - f' n x h)) (at x within s)" + apply(rule differentiable_bound[where f'="\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\s" + show "((\a. f m a - f n a) has_derivative (\h. f' m x h - f' n x h)) (at x within s)" by(rule has_derivative_intros assms(2)[rule_format] `x\s`)+ - { fix h have "norm (f' m x h - f' n x h) \ 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 "\ \ e * norm h+ e * norm h" using assms(3)[rule_format,OF `N\m` `x\s`, of h] assms(3)[rule_format,OF `N\n` `x\s`, of h] + { fix h + have "norm (f' m x h - f' n x h) \ 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 "\ \ e * norm h+ e * norm h" + using assms(3)[rule_format,OF `N\m` `x\s`, of h] + using assms(3)[rule_format,OF `N\n` `x\s`, of h] by(auto simp add:field_simps) finally have "norm (f' m x h - f' n x h) \ 2 * e * norm h" by auto } - thus "onorm (\h. f' m x h - f' n x h) \ 2 * e" apply-apply(rule onorm(2)) apply(rule linear_compose_sub) - unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\s`, THEN derivative_linear] by auto qed qed + thus "onorm (\h. f' m x h - f' n x h) \ 2 * e" + apply-apply(rule onorm(2)) apply(rule linear_compose_sub) + unfolding linear_conv_bounded_linear + using assms(2)[rule_format,OF `x\s`, THEN derivative_linear] + by auto + qed +qed -lemma has_derivative_sequence_lipschitz: fixes f::"nat \ 'm::euclidean_space \ 'n::euclidean_space" - assumes "convex s" "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" - "\e>0. \N. \n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm(h)" "0 < e" - shows "\e>0. \N. \m\N. \n\N. \x\s. \y\s. norm((f m x - f n x) - (f m y - f n y)) \ e * norm(x - y)" proof(rule,rule) +lemma has_derivative_sequence_lipschitz: + fixes f::"nat \ 'm::euclidean_space \ 'n::euclidean_space" + assumes "convex s" + assumes "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" + assumes "\e>0. \N. \n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm(h)" + assumes "0 < e" + shows "\e>0. \N. \m\N. \n\N. \x\s. \y\s. norm((f m x - f n x) - (f m y - f n y)) \ 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 qed + 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 +qed -lemma has_derivative_sequence: fixes f::"nat\ 'm::euclidean_space \ 'n::euclidean_space" - assumes "convex s" "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" - "\e>0. \N. \n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm(h)" - "x0 \ s" "((\n. f n x0) ---> l) sequentially" - shows "\g. \x\s. ((\n. f n x) ---> g x) sequentially \ (g has_derivative g'(x)) (at x within s)" proof- +lemma has_derivative_sequence: + fixes f::"nat\ 'm::euclidean_space \ 'n::euclidean_space" + assumes "convex s" + assumes "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" + assumes "\e>0. \N. \n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm(h)" + assumes "x0 \ s" and "((\n. f n x0) ---> l) sequentially" + shows "\g. \x\s. ((\n. f n x) ---> g x) sequentially \ + (g has_derivative g'(x)) (at x within s)" +proof- have lem1:"\e>0. \N. \m\N. \n\N. \x\s. \y\s. norm((f m x - f n x) - (f m y - f n y)) \ e * norm(x - y)" - apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto - have "\g. \x\s. ((\n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof - fix x assume "x\s" show "Cauchy (\n. f n x)" proof(cases "x=x0") - case True thus ?thesis using convergent_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" using False by(auto intro!:divide_pos_pos) + apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) + apply(rule assms)+ by auto + have "\g. \x\s. ((\n. f n x) ---> g x) sequentially" + apply(rule bchoice) unfolding convergent_eq_cauchy + proof + fix x assume "x\s" show "Cauchy (\n. f n x)" + proof(cases "x=x0") + case True thus ?thesis using convergent_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" + using False by (auto intro!: divide_pos_pos) guess M using convergent_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 " \M. \m\M. \n\M. dist (f m x) (f n x) < e" apply(rule_tac x="max M N" in exI) proof(default+) + show "\M. \m\M. \n\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 \m" "max M N\n" have "dist (f m x) (f n x) \ 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 "\ \ norm (f m x0 - f n x0) + e / 2" using N[rule_format,OF _ _ `x\s` `x0\s`, of m n] and as and False by auto - also have "\ < 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 qed qed qed qed + also have "\ \ norm (f m x0 - f n x0) + e / 2" + using N[rule_format,OF _ _ `x\s` `x0\s`, of m n] and as and False + by auto + also have "\ < 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 + qed + qed + qed + qed then guess g .. note g = this - have lem2:"\e>0. \N. \n\N. \x\s. \y\s. norm((f n x - f n y) - (g x - g y)) \ e * norm(x - y)" proof(rule,rule) - fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this - show "\N. \n\N. \x\s. \y\s. norm (f n x - f n y - (g x - g y)) \ e * norm (x - y)" apply(rule_tac x=N in exI) proof(default+) + have lem2:"\e>0. \N. \n\N. \x\s. \y\s. norm((f n x - f n y) - (g x - g y)) \ e * norm(x - y)" + proof(rule,rule) + fix e::real assume *:"e>0" + guess N using lem1[rule_format,OF *] .. note N=this + show "\N. \n\N. \x\s. \y\s. norm (f n x - f n y - (g x - g y)) \ e * norm (x - y)" + apply(rule_tac x=N in exI) + proof(default+) fix n x y assume as:"N \ n" "x \ s" "y \ s" - have "eventually (\xa. norm (f n x - f n y - (f xa x - f xa y)) \ e * norm (x - y)) sequentially" - unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule) - fix m assume "N\m" thus "norm (f n x - f n y - (f m x - f m y)) \ 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)) \ e * norm (x - y)" apply- + have "eventually (\xa. norm (f n x - f n y - (f xa x - f xa y)) \ e * norm (x - y)) sequentially" + unfolding eventually_sequentially + apply(rule_tac x=N in exI) + proof(rule,rule) + fix m assume "N\m" + thus "norm (f n x - f n y - (f m x - f m y)) \ 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)) \ e * norm (x - y)" + apply- apply(rule Lim_norm_ubound[OF trivial_limit_sequentially, where f="\m. (f n x - f n y) - (f m x - f m y)"]) - apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed + apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption + 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\s" - have lem3:"\u. ((\n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule) - fix u and e::real assume "e>0" show "\N. \n\N. dist (f' n x u) (g' x u) < e" proof(cases "u=0") + apply(rule,rule,rule g[rule_format],assumption) + proof fix x assume "x\s" + have lem3:"\u. ((\n. f' n x u) ---> g' x u) sequentially" + unfolding Lim_sequentially + proof(rule,rule,rule) + fix u and e::real assume "e>0" + show "\N. \n\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\s`,of _ 0] and `e>0` by auto next - case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos) + using N[rule_format,OF _ `x\s`,of _ 0] and `e>0` by auto + next + case False hence *:"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 using N[rule_format,OF goal1 `x\s`, of u] False `e>0` - by (auto simp add:field_simps) qed qed qed - show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule) + show ?thesis apply(rule_tac x=N in exI) + proof(rule,rule) case goal1 + show ?case unfolding dist_norm + using N[rule_format,OF goal1 `x\s`, of u] False `e>0` + by (auto simp add:field_simps) + qed + qed + qed + show "bounded_linear (g' x)" + unfolding linear_linear linear_def + apply(rule,rule,rule) defer + proof(rule,rule) fix x' y z::"'m" and c::real note lin = assms(2)[rule_format,OF `x\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]) + 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]) unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format] apply(rule Lim_cmul) by(rule lem3[rule_format]) - show "g' x (y + z) = g' x y + g' x z" apply(rule tendsto_unique[OF trivial_limit_sequentially]) - apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format] - apply(rule Lim_add) by(rule lem3[rule_format])+ qed - show "\e>0. \d>0. \y\s. norm (y - x) < d \ norm (g y - g x - g' x (y - x)) \ e * norm (y - x)" proof(rule,rule) case goal1 - have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this + show "g' x (y + z) = g' x y + g' x z" + apply(rule tendsto_unique[OF trivial_limit_sequentially]) + apply(rule lem3[rule_format]) + unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format] + apply(rule Lim_add) by(rule lem3[rule_format])+ + qed + show "\e>0. \d>0. \y\s. norm (y - x) < d \ norm (g y - g x - g' x (y - x)) \ e * norm (y - x)" + 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\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 \ s" "norm (y - x) < d1" let ?N ="max N1 N2" - have "norm (g y - g x - (f ?N y - f ?N x)) \ e /3 * norm (y - x)" apply(subst norm_minus_cancel[THEN sym]) - using N2[rule_format, OF _ `y\s` `x\s`, of ?N] by auto moreover - have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \ e / 3 * norm (y - x)" using d1 and as by auto ultimately + show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) + proof(rule,rule) + fix y assume as:"y \ s" "norm (y - x) < d1" + let ?N ="max N1 N2" + have "norm (g y - g x - (f ?N y - f ?N x)) \ e /3 * norm (y - x)" + apply(subst norm_minus_cancel[THEN sym]) + using N2[rule_format, OF _ `y\s` `x\s`, of ?N] by auto + moreover + have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \ e / 3 * norm (y - x)" + using d1 and as by auto + ultimately have "norm (g y - g x - f' ?N x (y - x)) \ 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) moreover - have " norm (f' ?N x (y - x) - g' x (y - x)) \ e / 3 * norm (y - x)" using N1 `x\s` by auto + 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) + moreover + have " norm (f' ?N x (y - x) - g' x (y - x)) \ e / 3 * norm (y - x)" + using N1 `x\s` by auto ultimately show "norm (g y - g x - g' x (y - x)) \ 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) - qed qed qed qed + 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) + qed + qed + qed +qed subsection {* Can choose to line up antiderivatives if we want. *} -lemma has_antiderivative_sequence: fixes f::"nat\ 'm::euclidean_space \ 'n::euclidean_space" - assumes "convex s" "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" - "\e>0. \N. \n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm h" - shows "\g. \x\s. (g has_derivative g'(x)) (at x within s)" proof(cases "s={}") - case False then obtain a where "a\s" by auto have *:"\P Q. \g. \x\s. P g x \ Q g x \ \g. \x\s. Q g x" by auto - show ?thesis apply(rule *) apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\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\s`) by(auto intro!: Lim_const) qed auto +lemma has_antiderivative_sequence: + fixes f::"nat\ 'm::euclidean_space \ 'n::euclidean_space" + assumes "convex s" + assumes "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" + assumes "\e>0. \N. \n\N. \x\s. \h. norm(f' n x h - g' x h) \ e * norm h" + shows "\g. \x\s. (g has_derivative g'(x)) (at x within s)" +proof(cases "s={}") + case False then obtain a where "a\s" by auto + have *:"\P Q. \g. \x\s. P g x \ Q g x \ \g. \x\s. Q g x" by auto + show ?thesis + apply(rule *) + apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\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\s`) by(auto intro!: Lim_const) +qed auto -lemma has_antiderivative_limit: fixes g'::"'m::euclidean_space \ 'm \ 'n::euclidean_space" - assumes "convex s" "\e>0. \f f'. \x\s. (f has_derivative (f' x)) (at x within s) \ (\h. norm(f' x h - g' x h) \ e * norm(h))" - shows "\g. \x\s. (g has_derivative g'(x)) (at x within s)" proof- +lemma has_antiderivative_limit: + fixes g'::"'m::euclidean_space \ 'm \ 'n::euclidean_space" + assumes "convex s" + assumes "\e>0. \f f'. \x\s. (f has_derivative (f' x)) (at x within s) \ (\h. norm(f' x h - g' x h) \ e * norm(h))" + shows "\g. \x\s. (g has_derivative g'(x)) (at x within s)" +proof- have *:"\n. \f f'. \x\s. (f has_derivative (f' x)) (at x within s) \ (\h. norm(f' x h - g' x h) \ inverse (real (Suc n)) * norm(h))" - apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto - 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) + apply(rule) using assms(2) + apply(erule_tac x="inverse (real (Suc n))" in allE) by auto + 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 "00`] .. note N=this - show "\N. \n\N. \x\s. \h. norm (f' n x h - g' x h) \ e * norm h" apply(rule_tac x=N in exI) proof(default+) case goal1 + show "\N. \n\N. \x\s. \h. norm (f' n x h - g' x h) \ e * norm h" + apply(rule_tac x=N in exI) + proof(default+) + case goal1 have *:"inverse (real (Suc n)) \ e" apply(rule order_trans[OF _ N[THEN less_imp_le]]) using goal1(1) by(auto simp add:field_simps) - 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 qed qed(insert f,auto) qed + 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 + qed + qed(insert f,auto) +qed subsection {* Differentiation of a series. *} definition sums_seq :: "(nat \ 'a::real_normed_vector) \ 'a \ (nat set) \ bool" (infixl "sums'_seq" 12) where "(f sums_seq l) s \ ((\n. setsum f (s \ {0..n})) ---> l) sequentially" -lemma has_derivative_series: fixes f::"nat \ 'm::euclidean_space \ 'n::euclidean_space" - assumes "convex s" "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" - "\e>0. \N. \n\N. \x\s. \h. norm(setsum (\i. f' i x h) (k \ {0..n}) - g' x h) \ e * norm(h)" - "x\s" "((\n. f n x) sums_seq l) k" +lemma has_derivative_series: + fixes f::"nat \ 'm::euclidean_space \ 'n::euclidean_space" + assumes "convex s" + assumes "\n. \x\s. ((f n) has_derivative (f' n x)) (at x within s)" + assumes "\e>0. \N. \n\N. \x\s. \h. norm(setsum (\i. f' i x h) (k \ {0..n}) - g' x h) \ e * norm(h)" + assumes "x\s" and "((\n. f n x) sums_seq l) k" shows "\g. \x\s. ((\n. f n x) sums_seq (g x)) k \ (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) defer apply(rule,rule assms(2)[rule_format],assumption) + unfolding sums_seq_def + apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) + apply(rule,rule) + apply(rule has_derivative_setsum) defer + apply(rule,rule assms(2)[rule_format],assumption) using assms(4-5) unfolding sums_seq_def by auto subsection {* Derivative with composed bilinear function. *} lemma has_derivative_bilinear_within: - assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h" - shows "((\x. h (f x) (g x)) has_derivative (\d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" proof- - have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within]) - using assms(2) unfolding differentiable_def by auto moreover - interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto - interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto - interpret h:bounded_bilinear h using assms by auto - have "((\y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym] - apply(rule Lim_linear[of "\y. y - x" 0 "at x within s" f']) using Lim_sub[OF Lim_within_id Lim_const, of x x s] + assumes "(f has_derivative f') (at x within s)" + assumes "(g has_derivative g') (at x within s)" + assumes "bounded_bilinear h" + shows "((\x. h (f x) (g x)) has_derivative (\d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" +proof- + have "(g ---> g x) (at x within s)" + apply(rule differentiable_imp_continuous_within[unfolded continuous_within]) + using assms(2) unfolding differentiable_def by auto + moreover + interpret f':bounded_linear f' + using assms unfolding has_derivative_def by auto + interpret g':bounded_linear g' + using assms unfolding has_derivative_def by auto + interpret h:bounded_bilinear h + using assms by auto + have "((\y. f' (y - x)) ---> 0) (at x within s)" + unfolding f'.zero[THEN sym] + apply(rule Lim_linear[of "\y. y - x" 0 "at x within s" f']) + using Lim_sub[OF Lim_within_id Lim_const, of x x s] unfolding id_def using assms(1) unfolding has_derivative_def by auto hence "((\y. f x + f' (y - x)) ---> f x) (at x within s)" - using Lim_add[OF Lim_const, of "\y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately + using Lim_add[OF Lim_const, of "\y. f' (y - x)" 0 "at x within s" "f x"] + by auto + ultimately have *:"((\x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x)))) + h ((1/ (norm (x' - x))) *\<^sub>R (f x' - (f x + f' (x' - x)))) (g x')) ---> h (f x) 0 + h 0 (g x)) (at x within s)" - apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2) unfolding has_derivative_within by auto + apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) + using assms(1-2) unfolding has_derivative_within by auto guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this guess C using f'.pos_bounded .. note C=this guess D using g'.pos_bounded .. note D=this have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos) - have **:"((\y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)" unfolding Lim_within proof(rule,rule) case goal1 + have **:"((\y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)" + unfolding Lim_within + proof(rule,rule) case goal1 hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos) - thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE) + thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) + proof(rule,rule,erule conjE) fix y assume as:"y \ s" "0 < dist y x" "dist y x < e / (B * C * D)" have "norm (h (f' (y - x)) (g' (y - x))) \ norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto - also have "\ \ (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono) - apply(rule mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg) - also have "\ = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps) - also have "\ < e * norm (y - x)" apply(rule mult_strict_right_mono) - using as(3)[unfolded dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps) + also have "\ \ (norm (y - x) * C) * (D * norm (y - x)) * B" + apply(rule mult_right_mono) + apply(rule mult_mono) using B C D + by (auto simp add: field_simps intro!:mult_nonneg_nonneg) + also have "\ = (B * C * D * norm (y - x)) * norm (y - x)" + by (auto simp add: field_simps) + also have "\ < e * norm (y - x)" + apply(rule mult_strict_right_mono) + using as(3)[unfolded dist_norm] and as(2) + unfolding pos_less_divide_eq[OF bcd] by (auto simp add: field_simps) finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e" - unfolding dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed + unfolding dist_norm apply-apply(cases "y = x") + by(auto simp add: field_simps) + qed + qed have "bounded_linear (\d. h (f x) (g' d) + h (f' d) (g x))" apply (rule bounded_linear_add) apply (rule bounded_linear_compose [OF h.bounded_linear_right `bounded_linear g'`]) @@ -1250,12 +1728,17 @@ thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left - scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed + scaleR_right_diff_distrib h.zero_right h.zero_left + by(auto simp add:field_simps) +qed lemma has_derivative_bilinear_at: - assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h" + assumes "(f has_derivative f') (at x)" + assumes "(g has_derivative g') (at x)" + assumes "bounded_bilinear h" shows "((\x. h (f x) (g x)) has_derivative (\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] unfolding within_UNIV using assms by auto + using has_derivative_bilinear_within[of f f' x UNIV g g' h] + unfolding within_UNIV using assms by auto subsection {* Considering derivative @{typ "real \ 'b\real_normed_vector"} as a vector. *} @@ -1265,14 +1748,20 @@ definition "vector_derivative f net \ (SOME f'. (f has_vector_derivative f') net)" -lemma vector_derivative_works: fixes f::"real \ 'a::real_normed_vector" +lemma vector_derivative_works: + fixes f::"real \ 'a::real_normed_vector" shows "f differentiable net \ (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 +proof + assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this then interpret bounded_linear f' by auto thus ?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 -next assume ?r thus ?l unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed +next + assume ?r thus ?l + unfolding vector_derivative_def has_vector_derivative_def differentiable_def + by auto +qed lemma vector_derivative_unique_at: assumes "(f has_vector_derivative f') (at x)" @@ -1285,16 +1774,26 @@ thus ?thesis unfolding fun_eq_iff by auto qed -lemma vector_derivative_unique_within_closed_interval: fixes f::"real \ 'n::ordered_euclidean_space" - assumes "a < b" "x \ {a..b}" - "(f has_vector_derivative f') (at x within {a..b})" - "(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof- +lemma vector_derivative_unique_within_closed_interval: + fixes f::"real \ 'n::ordered_euclidean_space" + assumes "a < b" and "x \ {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 *:"(\x. x *\<^sub>R f') = (\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 - show ?thesis proof(rule ccontr) assume "f' \ f''" moreover - hence "(\x. x *\<^sub>R f') 1 = (\x. x *\<^sub>R f'') 1" using * by (auto simp: fun_eq_iff) - ultimately show False unfolding o_def by auto qed qed + using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2) + by auto + show ?thesis + proof(rule ccontr) + assume "f' \ f''" + moreover + hence "(\x. x *\<^sub>R f') 1 = (\x. x *\<^sub>R f'') 1" + using * by (auto simp: fun_eq_iff) + ultimately show False unfolding o_def by auto + qed +qed lemma vector_derivative_at: shows "(f has_vector_derivative f') (at x) \ vector_derivative f (at x) = f'" @@ -1302,8 +1801,10 @@ unfolding vector_derivative_works[THEN sym] differentiable_def unfolding has_vector_derivative_def by auto -lemma vector_derivative_within_closed_interval: fixes f::"real \ 'a::ordered_euclidean_space" - assumes "a < b" "x \ {a..b}" "(f has_vector_derivative f') (at x within {a..b})" +lemma vector_derivative_within_closed_interval: + fixes f::"real \ 'a::ordered_euclidean_space" + assumes "a < b" and "x \ {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) using vector_derivative_works[unfolded differentiable_def] @@ -1320,71 +1821,95 @@ lemma has_vector_derivative_id: "((\x::real. x) has_vector_derivative 1) net" unfolding has_vector_derivative_def using has_derivative_id by auto -lemma has_vector_derivative_cmul: "(f has_vector_derivative f') net \ ((\x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net" - unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:algebra_simps) +lemma has_vector_derivative_cmul: + "(f has_vector_derivative f') net \ ((\x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net" + unfolding has_vector_derivative_def apply(drule has_derivative_cmul) + by (auto simp add: algebra_simps) -lemma has_vector_derivative_cmul_eq: assumes "c \ 0" +lemma has_vector_derivative_cmul_eq: + assumes "c \ 0" shows "(((\x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \ (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 lemma has_vector_derivative_neg: - "(f has_vector_derivative f') net \ ((\x. -(f x)) has_vector_derivative (- f')) net" + "(f has_vector_derivative f') net \ ((\x. -(f x)) has_vector_derivative (- f')) net" unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto lemma has_vector_derivative_add: - assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net" + assumes "(f has_vector_derivative f') net" + assumes "(g has_vector_derivative g') net" shows "((\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 lemma has_vector_derivative_sub: - assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net" + assumes "(f has_vector_derivative f') net" + assumes "(g has_vector_derivative g') net" shows "((\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 lemma has_vector_derivative_bilinear_within: - assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h" - shows "((\x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)" proof- + assumes "(f has_vector_derivative f') (at x within s)" + assumes "(g has_vector_derivative g') (at x within s)" + assumes "bounded_bilinear h" + shows "((\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] unfolding o_def has_vector_derivative_def - using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed + 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)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h" + assumes "(f has_vector_derivative f') (at x)" + assumes "(g has_vector_derivative g') (at x)" + assumes "bounded_bilinear h" shows "((\x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)" apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto -lemma has_vector_derivative_at_within: "(f has_vector_derivative f') (at x) \ (f has_vector_derivative f') (at x within s)" - unfolding has_vector_derivative_def apply(rule has_derivative_at_within) by auto +lemma has_vector_derivative_at_within: + "(f has_vector_derivative f') (at x) \ (f has_vector_derivative f') (at x within s)" + unfolding has_vector_derivative_def + by (rule has_derivative_at_within) auto lemma has_vector_derivative_transform_within: - assumes "0 < d" "x \ s" "\x'\s. dist x' x < d \ f x' = g x'" "(f has_vector_derivative f') (at x within s)" + assumes "0 < d" and "x \ s" and "\x'\s. dist x' x < d \ 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 by(rule has_derivative_transform_within) + using assms unfolding has_vector_derivative_def + by (rule has_derivative_transform_within) lemma has_vector_derivative_transform_at: - assumes "0 < d" "\x'. dist x' x < d \ f x' = g x'" "(f has_vector_derivative f') (at x)" + assumes "0 < d" and "\x'. dist x' x < d \ f x' = g x'" + assumes "(f has_vector_derivative f') (at x)" shows "(g has_vector_derivative f') (at x)" - using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at) + using assms unfolding has_vector_derivative_def + by (rule has_derivative_transform_at) lemma has_vector_derivative_transform_within_open: - assumes "open s" "x \ s" "\y\s. f y = g y" "(f has_vector_derivative f') (at x)" + assumes "open s" and "x \ s" and "\y\s. f y = g y" + assumes "(f has_vector_derivative f') (at x)" shows "(g has_vector_derivative f') (at x)" - using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within_open) + 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)" "(g has_vector_derivative g') (at (f x))" + assumes "(f has_vector_derivative f') (at x)" + assumes "(g has_vector_derivative g') (at (f x))" shows "((g \ 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]]) + 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 scaleR.scaleR_left by auto lemma vector_diff_chain_within: - assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)" + 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]]) + 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 scaleR.scaleR_left by auto end