diff --git a/src/HOL/Multivariate_Analysis/Derivative.thy b/src/HOL/Multivariate_Analysis/Derivative.thy --- a/src/HOL/Multivariate_Analysis/Derivative.thy +++ b/src/HOL/Multivariate_Analysis/Derivative.thy @@ -1,1335 +1,1334 @@ (* Title: HOL/Library/Convex_Euclidean_Space.thy Author: John Harrison Translation from HOL light: Robert Himmelmann, TU Muenchen *) header {* Multivariate calculus in Euclidean space. *} theory Derivative imports Brouwer_Fixpoint RealVector begin (* Because I do not want to type this all the time *) lemmas linear_linear = linear_conv_bounded_linear[THEN sym] subsection {* Derivatives *} text {* The definition is slightly tricky since we make it work over nets of a particular form. This lets us prove theorems generally and use "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *} definition has_derivative :: "('a::real_normed_vector \ 'b::real_normed_vector) \ ('a \ 'b) \ ('a net \ bool)" (infixl "(has'_derivative)" 12) where "(f has_derivative f') net \ bounded_linear f' \ ((\y. (1 / (norm (y - netlimit net))) *\<^sub>R (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net" lemma derivative_linear[dest]:"(f has_derivative f') net \ bounded_linear f'" unfolding has_derivative_def by auto lemma FDERIV_conv_has_derivative:"FDERIV f (x::'a::{real_normed_vector,perfect_space}) :> f' = (f has_derivative f') (at x)" (is "?l = ?r") proof assume ?l note as = this[unfolded fderiv_def] show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule) fix e::real assume "e>0" guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] .. thus "\d>0. \xa. 0 < dist xa x \ dist xa x < d \ dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e" apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE) unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed next assume ?r note as = this[unfolded has_derivative_def] show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule) fix e::real assume "e>0" guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] .. thus "\s>0. \xa. xa \ 0 \ dist xa 0 < s \ dist (norm (f (x + xa) - f x - f' xa) / norm xa) 0 < e" apply- apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE) unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed qed subsection {* These are the only cases we'll care about, probably. *} lemma has_derivative_within: "(f has_derivative f') (at x within s) \ bounded_linear f' \ ((\y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)" unfolding has_derivative_def and Lim by(auto simp add:netlimit_within) lemma has_derivative_at: "(f has_derivative f') (at x) \ bounded_linear f' \ ((\y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)" apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto subsection {* More explicit epsilon-delta forms. *} lemma has_derivative_within': "(f has_derivative f')(at x within s) \ bounded_linear f' \ (\e>0. \d>0. \x'\s. 0 < norm(x' - x) \ norm(x' - x) < d \ norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)" unfolding has_derivative_within Lim_within vector_dist_norm unfolding diff_0_right norm_mul by(simp add: group_simps) lemma has_derivative_at': "(f has_derivative f') (at x) \ bounded_linear f' \ (\e>0. \d>0. \x'. 0 < norm(x' - x) \ norm(x' - x) < d \ norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)" apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within' by auto lemma has_derivative_at_within: "(f has_derivative f') (at x) \ (f has_derivative f') (at x within s)" unfolding has_derivative_within' has_derivative_at' by meson lemma has_derivative_within_open: "a \ s \ open s \ ((f has_derivative f') (at a within s) \ (f has_derivative f') (at a))" unfolding has_derivative_within has_derivative_at using Lim_within_open by auto subsection {* Derivatives on real = Derivatives on real^1 *} lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by(auto simp add:vec1_dest_vec1_simps) lemma bounded_linear_vec1_dest_vec1: fixes f::"real \ real" shows "linear (vec1 \ f \ dest_vec1) = bounded_linear f" (is "?l = ?r") proof- { assume ?l guess K using linear_bounded[OF `?l`] .. hence "\K. \x. \f x\ \ \x\ * K" apply(rule_tac x=K in exI) unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) } thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def unfolding vec1_dest_vec1_simps by auto qed lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\real" shows "((vec1 \ f \ dest_vec1) has_derivative (vec1 \ f' \ dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)" unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear] unfolding o_def Lim_within Ball_def unfolding forall_vec1 unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\real" shows "((vec1 \ f \ dest_vec1) has_derivative (vec1 \ f' \ dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)" using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto lemma bounded_linear_vec1: fixes f::"'a::real_normed_vector\real" shows "bounded_linear f = bounded_linear (vec1 \ f)" unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def unfolding vec1_dest_vec1_simps by auto lemma bounded_linear_dest_vec1: fixes f::"real\'a::real_normed_vector" shows "bounded_linear f = bounded_linear (f \ dest_vec1)" unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def unfolding vec1_dest_vec1_simps by auto lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\real" shows "(f has_derivative f') (at x) = ((vec1 \ f) has_derivative (vec1 \ f')) (at x)" unfolding has_derivative_at unfolding bounded_linear_vec1[unfolded linear_conv_bounded_linear] unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto lemma has_derivative_within_dest_vec1:fixes f::"real\'a::real_normed_vector" shows "((f \ dest_vec1) has_derivative (f' \ dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)" unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto lemma has_derivative_at_dest_vec1:fixes f::"real\'a::real_normed_vector" shows "((f \ dest_vec1) has_derivative (f' \ dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)" using has_derivative_within_dest_vec1[where s=UNIV] by(auto simp add:within_UNIV) lemma derivative_is_linear: fixes f::"real^'a \ real^'b" shows "(f has_derivative f') net \ linear f'" unfolding has_derivative_def and linear_conv_bounded_linear by auto subsection {* Combining theorems. *} lemma (in bounded_linear) has_derivative: "(f has_derivative f) net" unfolding has_derivative_def apply(rule,rule bounded_linear_axioms) unfolding diff by(simp add: Lim_const) lemma has_derivative_id: "((\x. x) has_derivative (\h. h)) net" apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp lemma has_derivative_const: "((\x. c) has_derivative (\h. 0)) net" unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const) lemma (in bounded_linear) cmul: shows "bounded_linear (\x. (c::real) *\<^sub>R f x)" proof guess K using pos_bounded .. thus "\K. \x. norm ((c::real) *\<^sub>R f x) \ norm x * K" apply(rule_tac x="abs c * K" in exI) proof fix x case goal1 hence "abs c * norm (f x) \ abs c * (norm x * K)" apply-apply(erule conjE,erule_tac x=x in allE) apply(rule mult_left_mono) by auto thus ?case by(auto simp add:field_simps) qed qed(auto simp add: scaleR.add_right add scaleR) lemma has_derivative_cmul: assumes "(f has_derivative f') net" shows "((\x. c *\<^sub>R f(x)) has_derivative (\h. c *\<^sub>R f'(h))) net" unfolding has_derivative_def apply(rule,rule bounded_linear.cmul) using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]] unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto lemma has_derivative_cmul_eq: assumes "c \ 0" shows "(((\x. c *\<^sub>R f(x)) has_derivative (\h. c *\<^sub>R f'(h))) net \ (f has_derivative f') net)" apply(rule) defer apply(rule has_derivative_cmul,assumption) apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto lemma has_derivative_neg: "(f has_derivative f') net \ ((\x. -(f x)) has_derivative (\h. -(f' h))) net" apply(drule has_derivative_cmul[where c="-1"]) by auto 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- 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:group_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:group_simps) lemma has_derivative_setsum: assumes "finite s" "\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) 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 subsection {* somewhat different results for derivative of scalar multiplier. *} lemma has_derivative_vmul_component: fixes c::"real^'a \ real^'b" and v::"real^'c" 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)) = (c y $ k - (c (netlimit net) $ k + c' (y - netlimit net) $ k)) *\<^sub>R v" unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto show ?thesis unfolding has_derivative_def and * and linear_conv_bounded_linear[symmetric] apply(rule,rule linear_vmul_component[of c' k v, unfolded smult_conv_scaleR]) defer apply(subst vector_smult_lzero[THEN sym, of v]) unfolding scaleR_scaleR smult_conv_scaleR apply(rule Lim_vmul) using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net") apply(rule,assumption,rule disjI2,rule,rule) proof- have *:"\x. x - vec 0 = (x::real^'n)" by auto have **:"\d x. d * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k)) = (d *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net) ))) $k" by(auto simp add:field_simps) fix e assume "\ trivial_limit net" "0 < (e::real)" then have "eventually (\x. dist ((1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e) net" using assms[unfolded has_derivative_def Lim] by auto thus "eventually (\x. dist (1 / norm (x - netlimit net) * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k))) 0 < e) net" proof (rule eventually_elim1) case goal1 thus ?case apply - unfolding vector_dist_norm apply(rule le_less_trans) prefer 2 apply assumption unfolding * ** and norm_vec1 using component_le_norm[of "(1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net))) - 0" k] by auto qed qed(insert assms[unfolded has_derivative_def], auto simp add:linear_conv_bounded_linear) qed lemma has_derivative_vmul_within: fixes c::"real \ real" and v::"real^'a" assumes "(c has_derivative c') (at x within s)" shows "((\x. (c x) *\<^sub>R v) has_derivative (\x. (c' x) *\<^sub>R v)) (at x within s)" proof- have *:"\c. (\x. (vec1 \ c \ dest_vec1) x $ 1 *\<^sub>R v) = (\x. (c x) *\<^sub>R v) \ dest_vec1" unfolding o_def by auto show ?thesis using has_derivative_vmul_component[of "vec1 \ c \ dest_vec1" "vec1 \ c' \ dest_vec1" "at (vec1 x) within vec1 ` s" 1 v] unfolding * and has_derivative_within_vec1_dest_vec1 unfolding has_derivative_within_dest_vec1 using assms by auto qed lemma has_derivative_vmul_at: fixes c::"real \ real" and v::"real^'a" assumes "(c has_derivative c') (at x)" shows "((\x. (c x) *\<^sub>R v) has_derivative (\x. (c' x) *\<^sub>R v)) (at x)" using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV) lemma has_derivative_lift_dot: assumes "(f has_derivative f') net" shows "((\x. inner v (f x)) has_derivative (\t. inner v (f' t))) net" proof- show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule) apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption) 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 subsection {* limit transformation for derivatives. *} lemma has_derivative_transform_within: assumes "0 < d" "x \ s" "\x'\s. dist x' x < d \ f x' = g x'" "(f has_derivative f') (at x within s)" shows "(g has_derivative f') (at x within s)" using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption) apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto lemma has_derivative_transform_at: assumes "0 < d" "\x'. dist x' x < d \ f x' = g x'" "(f has_derivative f') (at x)" shows "(g has_derivative f') (at x)" apply(subst within_UNIV[THEN sym]) apply(rule has_derivative_transform_within[OF assms(1)]) using assms(2-3) unfolding within_UNIV by auto lemma has_derivative_transform_within_open: assumes "open s" "x \ s" "\y\s. f y = g y" "(f has_derivative f') (at x)" shows "(g has_derivative f') (at x)" using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption) apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto subsection {* differentiability. *} definition differentiable :: "('a::real_normed_vector \ 'b::real_normed_vector) \ 'a net \ bool" (infixr "differentiable" 30) where "f differentiable net \ (\f'. (f has_derivative f') net)" definition differentiable_on :: "('a::real_normed_vector \ 'b::real_normed_vector) \ 'a set \ bool" (infixr "differentiable'_on" 30) where "f differentiable_on s \ (\x\s. f differentiable (at x within s))" lemma differentiableI: "(f has_derivative f') net \ f differentiable net" unfolding differentiable_def by auto 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))" unfolding differentiable_def has_derivative_within_open[OF assms] by auto lemma differentiable_at_imp_differentiable_on: "(\x\(s::(real^'n) set). f differentiable at x) \ f differentiable_on s" unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI) 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_transform_within: assumes "0 < d" "x \ s" "\x'\s. dist x' x < d \ f x' = g x'" "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)]) 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 subsection {* Frechet derivative and Jacobian matrix. *} definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)" lemma frechet_derivative_works: "f differentiable net \ (f has_derivative (frechet_derivative f net)) net" unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\ f' . (f has_derivative f') net"] .. lemma linear_frechet_derivative: fixes f::"real^'a \ real^'b" shows "f differentiable net \ linear(frechet_derivative f net)" unfolding frechet_derivative_works has_derivative_def unfolding linear_conv_bounded_linear by auto definition "jacobian f net = matrix(frechet_derivative f net)" lemma jacobian_works: "(f::(real^'a) \ (real^'b)) differentiable net \ (f has_derivative (\h. (jacobian f net) *v h)) net" apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption subsection {* Differentiability implies continuity. *} 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 vector_dist_norm diff_0_right norm_mul 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 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 vector_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" 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" unfolding differentiable_on_def continuous_on_eq_continuous_within using differentiable_imp_continuous_within by blast lemma has_derivative_within_subset: "(f has_derivative f') (at x within s) \ t \ s \ (f has_derivative f') (at x within t)" unfolding has_derivative_within using Lim_within_subset by blast lemma differentiable_within_subset: "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" unfolding differentiable_on_def using differentiable_within_subset by blast lemma differentiable_on_empty: "f differentiable_on {}" unfolding differentiable_on_def by auto subsection {* Several results are easier using a "multiplied-out" variant. *) (* (I got this idea from Dieudonne's proof of the chain rule). *} lemma has_derivative_within_alt: "(f has_derivative f') (at x within s) \ 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- 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 False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\s`] unfolding vector_dist_norm diff_0_right norm_mul using as(3) using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded vector_dist_norm] by(auto simp add:linear_0 linear_sub group_simps) thus ?thesis by(auto simp add:group_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 vector_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:group_simps) qed auto qed lemma has_derivative_at_alt: "(f has_derivative f') (at (x::real^'n)) \ bounded_linear f' \ (\e>0. \d>0. \y. norm(y - x) < d \ norm(f y - f x - f'(y - x)) \ e * norm(y - x))" using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto 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))" 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]]) 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) note assms = assms[unfolded has_derivative_within_alt] fix e::real assume "0 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this def d0 \ "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto def d \ "(min d0 (de / (B1 + 1))) / 2" have "de * 2 / (B1 + 1) > de / (B1 + 1)" apply(rule divide_strict_right_mono) using B1 de by auto hence d:"d>0" "d < d1" "d < d2" "d < (de / (B1 + 1))" unfolding d_def using d0 d1 d2 de B1 by(auto intro!: divide_pos_pos simp add:min_less_iff_disj not_less) show "\d>0. \y\s. norm (y - x) < d \ norm ((g \ f) y - (g \ f) x - (g' \ f') (y - x)) \ e * norm (y - x)" apply(rule_tac x=d in exI) proof(rule,rule `d>0`,rule,rule) fix y assume as:"y \ s" "norm (y - x) < d" hence 1:"norm (f y - f x - f' (y - x)) \ 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:group_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:group_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 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 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) 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:group_simps f'.diff g'.diff g'.add) also have "\ \ B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:group_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 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 subsection {* Composition rules stated just for differentiability. *} lemma differentiable_const[intro]: "(\z. c) differentiable (net::'a::real_normed_vector net)" unfolding differentiable_def using has_derivative_const by auto lemma differentiable_id[intro]: "(\z. z) differentiable (net::'a::real_normed_vector net)" 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 net)" 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 net)" 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 net)" unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\z. f' z + f'a z" in exI) apply(rule has_derivative_add) by auto lemma differentiable_sub: "f differentiable net \ g differentiable net \ (\z. f z - g z) differentiable (net::'a::real_normed_vector net)" 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: fixes f::"'a \ (real^'n \real^'n)" assumes "finite s" "\a\s. (f a) differentiable net" 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 lemma differentiable_setsum_numseg: fixes f::"_ \ (real^'n \real^'n)" shows "\i. m \ i \ i \ n \ (f i) differentiable net \ (\x. setsum (\a. f a x) {m::nat..n}) differentiable net" apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto lemma differentiable_chain_at: "f differentiable (at x) \ g differentiable (at(f x)) \ (g o f) differentiable (at x)" unfolding differentiable_def by(meson diff_chain_at) lemma differentiable_chain_within: "f differentiable (at x within s) \ g differentiable (at(f x) within (f ` s)) \ (g o f) differentiable (at x within s)" unfolding differentiable_def by(meson diff_chain_within) subsection {* Uniqueness of derivative. *) (* *) (* The general result is a bit messy because we need approachability of the *) (* limit point from any direction. But OK for nontrivial intervals etc. *} lemma frechet_derivative_unique_within: fixes f::"real^'a \ real^'b" assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)" "(\i::'a::finite. \e>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) guess a using UNIV_witness[where 'a='a] .. fix e::real assume "00`,of a] .. thus "\x'\s. x' \ x \ dist x' x < e" apply(rule_tac x="x + d*\<^sub>R basis a" in bexI) using basis_nonzero[of a] norm_basis[of a] unfolding vector_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 ccontr) fix i::'a def e \ "norm (f' (basis i) - f'' (basis i))" 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 d[THEN conjunct1],of i] .. 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 norm_mul using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by(auto simp add:group_simps) finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"] using norm_basis[of i] unfolding vector_dist_norm unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib by auto qed qed lemma frechet_derivative_unique_at: fixes f::"real^'a \ real^'b" shows "(f has_derivative f') (at x) \ (f has_derivative f'') (at x) \ f' = f''" apply(rule frechet_derivative_unique_within[of f f' x UNIV f'']) unfolding within_UNIV apply(assumption)+ apply(rule,rule,rule) apply(rule_tac x="e/2" in exI) by auto lemma "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::"real^'a \ real^'b" assumes "\i. a$i < b$i" "x \ {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) fix e::real and i::'a assume "e>0" 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 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 "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) using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2) unfolding mem_interval by(auto simp add:field_simps) qed qed lemma frechet_derivative_unique_within_open_interval: fixes f::"real^'a \ real^'b" assumes "x \ {a<..0" note * = assms(1)[unfolded mem_interval,THEN spec[where x=i]] have "a $ i < x $ i" using * 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 "\d. 0 < \d\ \ \d\ < e \ x + d *\<^sub>R basis i \ {a<..0` and assms(1) unfolding mem_interval by(auto simp add:field_simps) qed lemma frechet_derivative_at: fixes f::"real^'a \ real^'b" shows "(f has_derivative f') (at x) \ (f' = frechet_derivative f (at x))" apply(rule frechet_derivative_unique_at[of f],assumption) unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto lemma frechet_derivative_within_closed_interval: fixes f::"real^'a \ real^'b" assumes "\i. a$i < b$i" "x \ {a..b}" "(f has_derivative f') (at x within {a.. b})" shows "frechet_derivative f (at x within {a.. b}) = f'" apply(rule frechet_derivative_unique_within_closed_interval[where f=f]) apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym] unfolding differentiable_def using assms(3) by auto subsection {* Component of the differential must be zero if it exists at a local *) (* maximum or minimum for that corresponding component. *} lemma differential_zero_maxmin_component: fixes f::"real^'a \ real^'b" assumes "0 < e" "((\y \ ball x e. (f y)$k \ (f x)$k) \ (\y\ball x e. (f x)$k \ (f y)$k))" "f differentiable (at x)" shows "jacobian f (at x) $ k = 0" proof(rule ccontr) def D \ "jacobian f (at x)" assume "jacobian f (at x) $ k \ 0" then obtain j where j:"D$k$j \ 0" unfolding Cart_eq D_def by auto hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto note as = assms(3)[unfolded jacobian_works has_derivative_at_alt] guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this { fix c assume "abs c \ d" hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto have "\(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\ \ norm (f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j))" by(rule component_le_norm) also have "\ \ \D $ k $ j\ / 2 * \c\" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto finally have "\(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\ \ \D $ k $ j\ / 2 * \c\" by simp hence "\f (x + c *\<^sub>R basis j) $ k - f x $ k - c * D $ k $ j\ \ \D $ k $ j\ / 2 * \c\" unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric] unfolding dot_rmult dot_basis unfolding smult_conv_scaleR by simp } note * = this have "x + d *\<^sub>R basis j \ ball x e" "x - d *\<^sub>R basis j \ ball x e" unfolding mem_ball vector_dist_norm using norm_basis[of j] d by auto hence **:"((f (x - d *\<^sub>R basis j))$k \ (f x)$k \ (f (x + d *\<^sub>R basis j))$k \ (f x)$k) \ ((f (x - d *\<^sub>R basis j))$k \ (f x)$k \ (f (x + d *\<^sub>R basis j))$k \ (f x)$k)" using assms(2) by auto 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 group_simps by (auto intro: mult_pos_pos) qed subsection {* In particular if we have a mapping into R^1. *} lemma differential_zero_maxmin: fixes f::"real^'a \ real" assumes "x \ s" "open s" "(f has_derivative f') (at x)" "(\y\s. f y \ f x) \ (\y\s. f x \ f y)" shows "f' = (\v. 0)" proof- note deriv = assms(3)[unfolded has_derivative_at_vec1] obtain e where e:"e>0" "ball x e \ s" using assms(2)[unfolded open_contains_ball] and assms(1) by auto hence **:"(jacobian (vec1 \ f) (at x)) $ 1 = 0" using differential_zero_maxmin_component[of e x "\x. vec1 (f x)" 1] using assms(4) and assms(3)[unfolded has_derivative_at_vec1 o_def] unfolding differentiable_def o_def by auto have *:"jacobian (vec1 \ f) (at x) = matrix (vec1 \ f')" unfolding jacobian_def and frechet_derivative_at[OF deriv] .. have "vec1 \ f' = (\x. 0)" apply(rule) unfolding matrix_works[OF derivative_is_linear[OF deriv],THEN sym] unfolding Cart_eq matrix_vector_mul_component using **[unfolded *] by auto thus ?thesis apply-apply(rule,subst vec1_eq[THEN sym]) unfolding o_def apply(drule fun_cong) by auto qed subsection {* The traditional Rolle theorem in one dimension. *} lemma vec1_le[simp]:fixes a::real shows "vec1 a \ vec1 b \ a \ b" unfolding vector_le_def by auto lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \ a < b" unfolding vector_less_def by auto 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 guess d using continuous_attains_sup[OF compact_real_interval * assms(3)] .. note d=this guess c using continuous_attains_inf[OF compact_real_interval * assms(3)] .. note c=this 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 then guess x .. note x=this hence "f' x \ dest_vec1 = (\v. 0)" apply(rule_tac differential_zero_maxmin[of "vec1 x" "vec1 ` {a<.. dest_vec1" "(f' x) \ dest_vec1"]) unfolding vec1_interval defer apply(rule open_interval) apply(rule assms(4)[unfolded has_derivative_at_dest_vec1[THEN sym],THEN bspec[where x=x]],assumption) unfolding o_def apply(erule disjE,rule disjI2) by(auto simp add: vector_less_def) thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule apply(drule_tac x="vec1 v" in fun_cong) unfolding vec1_dest_vec1 using x(1) by auto qed subsection {* One-dimensional mean value theorem. *} lemma mvt: fixes f::"real \ real" assumes "a < b" "continuous_on {a .. b} f" "\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 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])+ 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 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") 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 subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *} lemma inner_eq_dot: fixes a::"real^'n" shows "a \ b = inner a b" unfolding inner_vector_def dot_def by auto lemma mvt_general: fixes f::"real\real^'n" 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))unfolding inner_eq_dot 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:class_semiring.semiring_rules) also have "\ = (f b - f a) \ (f b - f a)" unfolding norm_pow_2 .. also have "\ = (f b - f a) \ f' x (b - a)" using x by auto 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::"real^'a \ real^'b" 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- 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:group_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 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_real] 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 "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 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 (** move this **) declare norm_vec1[simp] lemma onorm_vec1: fixes f::"real \ real" shows "onorm (\x. vec1 (f (dest_vec1 x))) = onorm f" proof- have "\x::real^1. norm x = 1 \ x\{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:Cart_eq) hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by(auto simp add:norm_vec1) have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto have "\x::real. norm x = 1 \ x\{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto have 4:"{norm (f x) |x. norm x = 1} = (\x. norm (f x)) ` {x. norm x=1}" by auto show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max norm_vec1) 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" shows "norm(f x - f y) \ B * norm(x - y)" using differentiable_bound[of "vec1 ` s" "vec1 \ f \ dest_vec1" "\x. vec1 \ (f' (dest_vec1 x)) \ dest_vec1" B "vec1 x" "vec1 y"] unfolding Ball_def forall_vec1 unfolding has_derivative_within_vec1_dest_vec1 image_iff unfolding convex_vec1 unfolding o_def vec1_dest_vec1_simps onorm_vec1 using assms by auto subsection {* In particular. *} 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={}") 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_vec1[of "\x. 0", THEN sym] onorm_const norm_vec1 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 subsection {* Differentiability of inverse function (most basic form). *} lemma has_derivative_inverse_basic: fixes f::"real^'b \ real^'c" 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 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 *:"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 vector_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 vector_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 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 mult_frac_num 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::"real^'b \ real^'c" 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)))" apply(rule has_derivative_inverse_basic) using assms by auto subsection {* This is the version in Dieudonne', assuming continuity of f and g. *} lemma has_derivative_inverse_dieudonne: fixes f::"real^'a \ real^'b" 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 subsection {* Here's the simplest way of not assuming much about g. *} lemma has_derivative_inverse: fixes f::"real^'a \ real^'b" 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- { 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 subsection {* Proving surjectivity via Brouwer fixpoint theorem. *} lemma brouwer_surjective: fixes f::"real^'n \ real^'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:group_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::"real^'n \ real^'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 text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *} lemma sussmann_open_mapping: fixes f::"real^'a \ real^'b" 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 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 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 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- 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 vector_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 vector_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 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 vector_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 "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:group_simps) also have "\ \ 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding group_simps ** by auto also have "\ \ 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball vector_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 vector_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 vector_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 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 vector_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]) unfolding mem_cball vector_dist_norm by auto 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. *) (* We could put f' o g = I but this happens to fit with the minimal linear *) (* algebra theory I've set up so far. *} lemma has_derivative_inverse_strong: fixes f::"real^'n \ real^'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] 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 ) apply(rule assms ling)+ by auto have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule) fix e::real assume "e>0" hence "f x \ 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]] 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 subsection {* A rewrite based on the other domain. *} lemma has_derivative_inverse_strong_x: fixes f::"real^'n \ real^'n" 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" 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::"real^'n \ real^'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" 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)+ unfolding continuous_on_eq_continuous_at[OF assms(1)] 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. *} 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:group_simps) lemma has_derivative_locally_injective: fixes f::"real^'n \ real^'m" 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- interpret bounded_linear g' using assms by auto note f'g' = assms(4)[unfolded id_def o_def,THEN cong] have "g' (f' a 1) = 1" using f'g' by auto 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) 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:group_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 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` 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:group_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]) unfolding ph_def using diff unfolding as by auto 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 \ real^'m \ real^'n" 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)+ 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)" 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:group_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] 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 lemma has_derivative_sequence_lipschitz: fixes f::"nat \ real^'m \ real^'n" 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) 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 lemma has_derivative_sequence: fixes f::"nat\real^'m\real^'n" 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- 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) 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+) 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 vector_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 vector_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+) 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:group_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 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") 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) 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 vector_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::"real^'m" and c::real note lin = assms(2)[rule_format,OF `x\s`,THEN derivative_linear] show "g' x (c *s x') = c *s g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially]) apply(rule lem3[rule_format]) unfolding smult_conv_scaleR 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 Lim_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 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:group_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:group_simps) qed qed qed qed subsection {* Can choose to line up antiderivatives if we want. *} lemma has_antiderivative_sequence: fixes f::"nat\ real^'m \ real^'n" 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_limit: fixes g'::"real^'m \ real^'m \ real^'n" 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- 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) 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 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 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 \ real^'m \ real^'n" 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" 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) using assms(4-5) unfolding sums_seq_def by auto subsection {* Derivative with composed bilinear function. *} lemma has_derivative_bilinear_within: fixes h::"real^'m \ real^'n \ real^'p" and f::"real^'q \ real^'m" 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] 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 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 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 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) 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 pordered_semiring_class.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 vector_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 vector_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))" unfolding linear_linear linear_def unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR unfolding h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right by auto thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within unfolding smult_conv_scaleR 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 lemma has_derivative_bilinear_at: fixes h::"real^'m \ real^'n \ real^'p" and f::"real^'p \ real^'m" assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "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 subsection {* Considering derivative R(^1)->R^n as a vector. *} definition has_vector_derivative :: "(real \ 'b::real_normed_vector) \ ('b) \ (real net \ bool)" (infixl "has'_vector'_derivative" 12) where "(f has_vector_derivative f') net \ (f has_derivative (\x. x *\<^sub>R f')) net" definition "vector_derivative f net \ (SOME f'. (f has_vector_derivative f') net)" 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 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 lemma vector_derivative_unique_at: fixes f::"real\real^'n" assumes "(f has_vector_derivative f') (at x)" "(f has_vector_derivative f'') (at x)" shows "f' = f''" proof- have *:"(\x. x *\<^sub>R f') \ dest_vec1 = (\x. x *\<^sub>R f'') \ dest_vec1" apply(rule frechet_derivative_unique_at) using assms[unfolded has_vector_derivative_def] unfolding has_derivative_at_dest_vec1[THEN sym] by auto show ?thesis proof(rule ccontr) assume "f' \ f''" moreover hence "((\x. x *\<^sub>R f') \ dest_vec1) (vec1 1) = ((\x. x *\<^sub>R f'') \ dest_vec1) (vec1 1)" using * by auto ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed lemma vector_derivative_unique_within_closed_interval: fixes f::"real \ real^'n" 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- have *:"(\x. x *\<^sub>R f') \ dest_vec1 = (\x. x *\<^sub>R f'') \ dest_vec1" apply(rule frechet_derivative_unique_within_closed_interval[of "vec1 a" "vec1 b"]) using assms(3-)[unfolded has_vector_derivative_def] unfolding has_derivative_within_dest_vec1[THEN sym] vec1_interval using assms(1-2) by auto show ?thesis proof(rule ccontr) assume "f' \ f''" moreover hence "((\x. x *\<^sub>R f') \ dest_vec1) (vec1 1) = ((\x. x *\<^sub>R f'') \ dest_vec1) (vec1 1)" using * by auto ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed lemma vector_derivative_at: fixes f::"real \ real^'a" shows "(f has_vector_derivative f') (at x) \ vector_derivative f (at x) = f'" apply(rule vector_derivative_unique_at) defer apply assumption unfolding vector_derivative_works[THEN sym] differentiable_def unfolding has_vector_derivative_def by auto lemma vector_derivative_within_closed_interval: fixes f::"real \ real^'a" assumes "a < b" "x \ {a..b}" "(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] using assms by(auto simp add:has_vector_derivative_def) -lemma has_vector_derivative_within_subset: fixes f::"real \ real^'a" shows +lemma has_vector_derivative_within_subset: "(f has_vector_derivative f') (at x within s) \ t \ s \ (f has_vector_derivative f') (at x within t)" unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto -lemma has_vector_derivative_const: fixes c::"real^'n" shows +lemma has_vector_derivative_const: "((\x. c) has_vector_derivative 0) net" unfolding has_vector_derivative_def using has_derivative_const by auto 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: fixes f::"real \ real^'a" - shows "(f has_vector_derivative f') net \ ((\x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net" +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:group_simps) -lemma has_vector_derivative_cmul_eq: fixes f::"real \ real^'a" 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" 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" 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" 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: fixes h::"real^'m \ real^'n \ real^'p" 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- interpret bounded_bilinear h using assms by auto show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def has_derivative_within_dest_vec1[THEN sym]], where h=h] unfolding o_def vec1_dest_vec1 has_vector_derivative_def unfolding has_derivative_within_dest_vec1[unfolded o_def, where f="\x. h (f x) (g x)" and f'="\d. h (f x) (d *\<^sub>R g') + h (d *\<^sub>R f') (g x)"] using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed lemma has_vector_derivative_bilinear_at: fixes h::"real^'m \ real^'n \ real^'p" assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "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_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)" shows "(g has_vector_derivative f') (at x within s)" 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)" shows "(g has_vector_derivative f') (at x)" 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)" shows "(g has_vector_derivative f') (at x)" 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))" 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]]) 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)" shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)" using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]]) unfolding o_def scaleR.scaleR_left by auto end