(* Title: HOL/Multivariate_Analysis/Derivative.thy Author: John Harrison Author: Robert Himmelmann, TU Muenchen (translation from HOL Light) *) section ‹Multivariate calculus in Euclidean space› theory Derivative imports Brouwer_Fixpoint Operator_Norm Uniform_Limit Bounded_Linear_Function begin lemma onorm_inner_left: assumes "bounded_linear r" shows "onorm (λx. r x ∙ f) ≤ onorm r * norm f" proof (rule onorm_bound) fix x have "norm (r x ∙ f) ≤ norm (r x) * norm f" by (simp add: Cauchy_Schwarz_ineq2) also have "… ≤ onorm r * norm x * norm f" by (intro mult_right_mono onorm assms norm_ge_zero) finally show "norm (r x ∙ f) ≤ onorm r * norm f * norm x" by (simp add: ac_simps) qed (intro mult_nonneg_nonneg norm_ge_zero onorm_pos_le assms) lemma onorm_inner_right: assumes "bounded_linear r" shows "onorm (λx. f ∙ r x) ≤ norm f * onorm r" apply (subst inner_commute) apply (rule onorm_inner_left[OF assms, THEN order_trans]) apply simp done declare has_derivative_bounded_linear[dest] subsection ‹Derivatives› subsubsection ‹Combining theorems.› lemmas has_derivative_id = has_derivative_ident lemmas has_derivative_neg = has_derivative_minus lemmas has_derivative_sub = has_derivative_diff lemmas scaleR_right_has_derivative = has_derivative_scaleR_right lemmas scaleR_left_has_derivative = has_derivative_scaleR_left lemmas inner_right_has_derivative = has_derivative_inner_right lemmas inner_left_has_derivative = has_derivative_inner_left lemmas mult_right_has_derivative = has_derivative_mult_right lemmas mult_left_has_derivative = has_derivative_mult_left lemma has_derivative_add_const: "(f has_derivative f') net ⟹ ((λx. f x + c) has_derivative f') net" by (intro derivative_eq_intros) auto subsection ‹Derivative with composed bilinear function.› lemma has_derivative_bilinear_within: assumes "(f has_derivative f') (at x within s)" and "(g has_derivative g') (at x within s)" and "bounded_bilinear h" shows "((λx. h (f x) (g x)) has_derivative (λd. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" using bounded_bilinear.FDERIV[OF assms(3,1,2)] . lemma has_derivative_bilinear_at: assumes "(f has_derivative f') (at x)" and "(g has_derivative g') (at x)" and "bounded_bilinear h" shows "((λx. h (f x) (g x)) has_derivative (λd. h (f x) (g' d) + h (f' d) (g x))) (at x)" using has_derivative_bilinear_within[of f f' x UNIV g g' h] assms by simp text ‹These are the only cases we'll care about, probably.› lemma has_derivative_within: "(f has_derivative f') (at x within s) ⟷ bounded_linear f' ∧ ((λy. (1 / norm(y - x)) *⇩_{R}(f y - (f x + f' (y - x)))) ⤏ 0) (at x within s)" unfolding has_derivative_def Lim by (auto simp add: netlimit_within field_simps) lemma has_derivative_at: "(f has_derivative f') (at x) ⟷ bounded_linear f' ∧ ((λy. (1 / (norm(y - x))) *⇩_{R}(f y - (f x + f' (y - x)))) ⤏ 0) (at x)" using has_derivative_within [of f f' x UNIV] by simp text ‹More explicit epsilon-delta forms.› lemma has_derivative_within': "(f has_derivative f')(at x within s) ⟷ 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 dist_norm unfolding diff_0_right by (simp add: diff_diff_eq) 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)" using has_derivative_within' [of f f' x UNIV] by simp 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 blast lemma has_derivative_within_open: "a ∈ s ⟹ open s ⟹ (f has_derivative f') (at a within s) ⟷ (f has_derivative f') (at a)" by (simp only: at_within_interior interior_open) lemma has_derivative_right: fixes f :: "real ⇒ real" and y :: "real" shows "(f has_derivative (op * y)) (at x within ({x <..} ∩ I)) ⟷ ((λt. (f x - f t) / (x - t)) ⤏ y) (at x within ({x <..} ∩ I))" proof - have "((λt. (f t - (f x + y * (t - x))) / ¦t - x¦) ⤏ 0) (at x within ({x<..} ∩ I)) ⟷ ((λt. (f t - f x) / (t - x) - y) ⤏ 0) (at x within ({x<..} ∩ I))" by (intro Lim_cong_within) (auto simp add: diff_divide_distrib add_divide_distrib) also have "… ⟷ ((λt. (f t - f x) / (t - x)) ⤏ y) (at x within ({x<..} ∩ I))" by (simp add: Lim_null[symmetric]) also have "… ⟷ ((λt. (f x - f t) / (x - t)) ⤏ y) (at x within ({x<..} ∩ I))" by (intro Lim_cong_within) (simp_all add: field_simps) finally show ?thesis by (simp add: bounded_linear_mult_right has_derivative_within) qed subsubsection ‹Caratheodory characterization› lemma DERIV_within_iff: "(f has_field_derivative D) (at a within s) ⟷ ((λz. (f z - f a) / (z - a)) ⤏ D) (at a within s)" proof - have 1: "⋀w y. ~(w = a) ==> y / (w - a) - D = (y - (w - a)*D)/(w - a)" by (metis divide_diff_eq_iff eq_iff_diff_eq_0 mult.commute) show ?thesis apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right) apply (simp add: LIM_zero_iff [where l = D, symmetric]) apply (simp add: Lim_within dist_norm) apply (simp add: nonzero_norm_divide [symmetric]) apply (simp add: 1 diff_diff_eq ac_simps) done qed lemma DERIV_caratheodory_within: "(f has_field_derivative l) (at x within s) ⟷ (∃g. (∀z. f z - f x = g z * (z - x)) ∧ continuous (at x within s) g ∧ g x = l)" (is "?lhs = ?rhs") proof assume ?lhs show ?rhs proof (intro exI conjI) let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))" show "∀z. f z - f x = ?g z * (z-x)" by simp show "continuous (at x within s) ?g" using ‹?lhs› by (auto simp add: continuous_within DERIV_within_iff cong: Lim_cong_within) show "?g x = l" by simp qed next assume ?rhs then obtain g where "(∀z. f z - f x = g z * (z-x))" and "continuous (at x within s) g" and "g x = l" by blast thus ?lhs by (auto simp add: continuous_within DERIV_within_iff cong: Lim_cong_within) qed subsubsection ‹Limit transformation for derivatives› lemma has_derivative_transform_within: assumes "(f has_derivative f') (at x within s)" and "0 < d" and "x ∈ s" and "⋀x'. ⟦x' ∈ s; dist x' x < d⟧ ⟹ f x' = g x'" shows "(g has_derivative f') (at x within s)" using assms unfolding has_derivative_within by (force simp add: intro: Lim_transform_within) lemma has_derivative_transform_within_open: assumes "(f has_derivative f') (at x)" and "open s" and "x ∈ s" and "⋀x. x∈s ⟹ f x = g x" shows "(g has_derivative f') (at x)" using assms unfolding has_derivative_at by (force simp add: intro: Lim_transform_within_open) subsection ‹Differentiability› definition differentiable_on :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒ 'a set ⇒ bool" (infix "differentiable'_on" 50) 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_at_imp_differentiable_on: "(⋀x. x ∈ s ⟹ f differentiable at x) ⟹ f differentiable_on s" by (metis differentiable_at_withinI differentiable_on_def) corollary differentiable_iff_scaleR: fixes f :: "real ⇒ 'a::real_normed_vector" shows "f differentiable F ⟷ (∃d. (f has_derivative (λx. x *⇩_{R}d)) F)" by (auto simp: differentiable_def dest: has_derivative_linear linear_imp_scaleR) 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 (metis at_within_interior interior_open) lemma differentiable_transform_within: assumes "f differentiable (at x within s)" and "0 < d" and "x ∈ s" and "⋀x'. ⟦x'∈s; dist x' x < d⟧ ⟹ f x' = g x'" shows "g differentiable (at x within s)" using assms has_derivative_transform_within unfolding differentiable_def by blast 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 unfolding some_eq_ex[of "λ f' . (f has_derivative f') net"] .. lemma linear_frechet_derivative: "f differentiable net ⟹ linear (frechet_derivative f net)" unfolding frechet_derivative_works has_derivative_def by (auto intro: bounded_linear.linear) subsection ‹Differentiability implies continuity› lemma differentiable_imp_continuous_within: "f differentiable (at x within s) ⟹ continuous (at x within s) f" by (auto simp: differentiable_def intro: has_derivative_continuous) 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 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 text ‹Results about neighborhoods filter.› lemma eventually_nhds_metric_le: "eventually P (nhds a) = (∃d>0. ∀x. dist x a ≤ d ⟶ P x)" unfolding eventually_nhds_metric by (safe, rule_tac x="d / 2" in exI, auto) lemma le_nhds: "F ≤ nhds a ⟷ (∀S. open S ∧ a ∈ S ⟶ eventually (λx. x ∈ S) F)" unfolding le_filter_def eventually_nhds by (fast elim: eventually_mono) lemma le_nhds_metric: "F ≤ nhds a ⟷ (∀e>0. eventually (λx. dist x a < e) F)" unfolding le_filter_def eventually_nhds_metric by (fast elim: eventually_mono) lemma le_nhds_metric_le: "F ≤ nhds a ⟷ (∀e>0. eventually (λx. dist x a ≤ e) F)" unfolding le_filter_def eventually_nhds_metric_le by (fast elim: eventually_mono) text ‹Several results are easier using a "multiplied-out" variant. (I got this idea from Dieudonne's proof of the chain rule).› lemma has_derivative_within_alt: "(f has_derivative f') (at x within s) ⟷ 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))" unfolding has_derivative_within filterlim_def le_nhds_metric_le eventually_filtermap eventually_at dist_norm diff_diff_eq by (force simp add: linear_0 bounded_linear.linear pos_divide_le_eq) lemma has_derivative_within_alt2: "(f has_derivative f') (at x within s) ⟷ bounded_linear f' ∧ (∀e>0. eventually (λy. norm (f y - f x - f' (y - x)) ≤ e * norm (y - x)) (at x within s))" unfolding has_derivative_within filterlim_def le_nhds_metric_le eventually_filtermap eventually_at dist_norm diff_diff_eq by (force simp add: linear_0 bounded_linear.linear pos_divide_le_eq) lemma has_derivative_at_alt: "(f has_derivative f') (at x) ⟷ 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] by simp subsection ‹The chain rule› lemma diff_chain_within[derivative_intros]: assumes "(f has_derivative f') (at x within s)" and "(g has_derivative g') (at (f x) within (f ` s))" shows "((g ∘ f) has_derivative (g' ∘ f'))(at x within s)" using has_derivative_in_compose[OF assms] by (simp add: comp_def) lemma diff_chain_at[derivative_intros]: "(f has_derivative f') (at x) ⟹ (g has_derivative g') (at (f x)) ⟹ ((g ∘ f) has_derivative (g' ∘ f')) (at x)" using has_derivative_compose[of f f' x UNIV g g'] by (simp add: comp_def) subsection ‹Composition rules stated just for differentiability› lemma differentiable_chain_at: "f differentiable (at x) ⟹ g differentiable (at (f x)) ⟹ (g ∘ 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 ∘ f) differentiable (at x within s)" unfolding differentiable_def by (meson diff_chain_within) subsection ‹Uniqueness of derivative› text ‹ 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 :: "'a::euclidean_space ⇒ 'b::real_normed_vector" assumes "(f has_derivative f') (at x within s)" and "(f has_derivative f'') (at x within s)" and "∀i∈Basis. ∀e>0. ∃d. 0 < ¦d¦ ∧ ¦d¦ < e ∧ (x + d *⇩_{R}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 "e > 0" obtain d where "0 < ¦d¦" and "¦d¦ < e" and "x + d *⇩_{R}(SOME i. i ∈ Basis) ∈ s" using assms(3) SOME_Basis ‹e>0› by blast then show "∃x'∈s. x' ≠ x ∧ dist x' x < e" apply (rule_tac x="x + d *⇩_{R}(SOME i. i ∈ Basis)" in bexI) unfolding dist_norm apply (auto simp: SOME_Basis nonzero_Basis) done qed then have *: "netlimit (at x within s) = x" apply (auto intro!: netlimit_within) by (metis trivial_limit_within) show ?thesis apply (rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear apply (rule as(1,2)[THEN conjunct1])+ proof (rule, rule ccontr) fix i :: 'a assume i: "i ∈ Basis" def e ≡ "norm (f' i - f'' i)" assume "f' i ≠ f'' i" then have "e > 0" unfolding e_def by auto obtain d where d: "0 < d" "(⋀xa. xa∈s ⟶ 0 < dist xa x ∧ dist xa x < d ⟶ dist ((f xa - f x - f' (xa - x)) /⇩_{R}norm (xa - x) - (f xa - f x - f'' (xa - x)) /⇩_{R}norm (xa - x)) (0 - 0) < e)" using tendsto_diff [OF as(1,2)[THEN conjunct2]] unfolding * Lim_within using ‹e>0› by blast obtain c where c: "0 < ¦c¦" "¦c¦ < d ∧ x + c *⇩_{R}i ∈ s" using assms(3) i d(1) by blast have *: "norm (- ((1 / ¦c¦) *⇩_{R}f' (c *⇩_{R}i)) + (1 / ¦c¦) *⇩_{R}f'' (c *⇩_{R}i)) = norm ((1 / ¦c¦) *⇩_{R}(- (f' (c *⇩_{R}i)) + f'' (c *⇩_{R}i)))" unfolding scaleR_right_distrib by auto also have "… = norm ((1 / ¦c¦) *⇩_{R}(c *⇩_{R}(- (f' i) + f'' i)))" unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto also have "… = e" unfolding e_def using c(1) using norm_minus_cancel[of "f' i - f'' i"] by auto finally show False using c using d(2)[of "x + c *⇩_{R}i"] unfolding dist_norm unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib using i by (auto simp: inverse_eq_divide) qed qed lemma frechet_derivative_unique_at: "(f has_derivative f') (at x) ⟹ (f has_derivative f'') (at x) ⟹ f' = f''" by (rule has_derivative_unique) lemma frechet_derivative_unique_within_closed_interval: fixes f::"'a::euclidean_space ⇒ 'b::real_normed_vector" assumes "∀i∈Basis. a∙i < b∙i" and "x ∈ cbox a b" and "(f has_derivative f' ) (at x within cbox a b)" and "(f has_derivative f'') (at x within cbox a b)" shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof (rule, rule, rule) fix e :: real fix i :: 'a assume "e > 0" and i: "i ∈ Basis" then show "∃d. 0 < ¦d¦ ∧ ¦d¦ < e ∧ x + d *⇩_{R}i ∈ cbox a b" proof (cases "x∙i = a∙i") case True then show ?thesis apply (rule_tac x="(min (b∙i - a∙i) e) / 2" in exI) using assms(1)[THEN bspec[where x=i]] and ‹e>0› and assms(2) unfolding mem_box using i apply (auto simp add: field_simps inner_simps inner_Basis) done next note * = assms(2)[unfolded mem_box, THEN bspec, OF 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 then have "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 bspec, OF i] and ‹e>0› and assms(2) unfolding mem_box using i apply (auto simp add: field_simps inner_simps inner_Basis) done qed qed lemma frechet_derivative_unique_within_open_interval: fixes f::"'a::euclidean_space ⇒ 'b::real_normed_vector" assumes "x ∈ box a b" and "(f has_derivative f' ) (at x within box a b)" and "(f has_derivative f'') (at x within box a b)" shows "f' = f''" proof - from assms(1) have *: "at x within box a b = at x" by (metis at_within_interior interior_open open_box) from assms(2,3) [unfolded *] show "f' = f''" by (rule frechet_derivative_unique_at) qed lemma frechet_derivative_at: "(f has_derivative f') (at x) ⟹ f' = frechet_derivative f (at x)" apply (rule frechet_derivative_unique_at[of f]) apply assumption unfolding frechet_derivative_works[symmetric] using differentiable_def apply auto done lemma frechet_derivative_within_cbox: fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector" assumes "∀i∈Basis. a∙i < b∙i" and "x ∈ cbox a b" and "(f has_derivative f') (at x within cbox a b)" shows "frechet_derivative f (at x within cbox a b) = f'" using assms by (metis Derivative.differentiableI frechet_derivative_unique_within_closed_interval frechet_derivative_works) subsection ‹The traditional Rolle theorem in one dimension› text ‹Derivatives of local minima and maxima are zero.› lemma has_derivative_local_min: fixes f :: "'a::real_normed_vector ⇒ real" assumes deriv: "(f has_derivative f') (at x)" assumes min: "eventually (λy. f x ≤ f y) (at x)" shows "f' = (λh. 0)" proof fix h :: 'a interpret f': bounded_linear f' using deriv by (rule has_derivative_bounded_linear) show "f' h = 0" proof (cases "h = 0") assume "h ≠ 0" from min obtain d where d1: "0 < d" and d2: "∀y∈ball x d. f x ≤ f y" unfolding eventually_at by (force simp: dist_commute) have "FDERIV (λr. x + r *⇩_{R}h) 0 :> (λr. r *⇩_{R}h)" by (intro derivative_eq_intros) auto then have "FDERIV (λr. f (x + r *⇩_{R}h)) 0 :> (λk. f' (k *⇩_{R}h))" by (rule has_derivative_compose, simp add: deriv) then have "DERIV (λr. f (x + r *⇩_{R}h)) 0 :> f' h" unfolding has_field_derivative_def by (simp add: f'.scaleR mult_commute_abs) moreover have "0 < d / norm h" using d1 and ‹h ≠ 0› by simp moreover have "∀y. ¦0 - y¦ < d / norm h ⟶ f (x + 0 *⇩_{R}h) ≤ f (x + y *⇩_{R}h)" using ‹h ≠ 0› by (auto simp add: d2 dist_norm pos_less_divide_eq) ultimately show "f' h = 0" by (rule DERIV_local_min) qed (simp add: f'.zero) qed lemma has_derivative_local_max: fixes f :: "'a::real_normed_vector ⇒ real" assumes "(f has_derivative f') (at x)" assumes "eventually (λy. f y ≤ f x) (at x)" shows "f' = (λh. 0)" using has_derivative_local_min [of "λx. - f x" "λh. - f' h" "x"] using assms unfolding fun_eq_iff by simp lemma differential_zero_maxmin: fixes f::"'a::real_normed_vector ⇒ real" assumes "x ∈ s" and "open s" and deriv: "(f has_derivative f') (at x)" and mono: "(∀y∈s. f y ≤ f x) ∨ (∀y∈s. f x ≤ f y)" shows "f' = (λv. 0)" using mono proof assume "∀y∈s. f y ≤ f x" with ‹x ∈ s› and ‹open s› have "eventually (λy. f y ≤ f x) (at x)" unfolding eventually_at_topological by auto with deriv show ?thesis by (rule has_derivative_local_max) next assume "∀y∈s. f x ≤ f y" with ‹x ∈ s› and ‹open s› have "eventually (λy. f x ≤ f y) (at x)" unfolding eventually_at_topological by auto with deriv show ?thesis by (rule has_derivative_local_min) qed lemma differential_zero_maxmin_component: (* TODO: delete? *) fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space" assumes k: "k ∈ Basis" and ball: "0 < e" "(∀y ∈ ball x e. (f y)∙k ≤ (f x)∙k) ∨ (∀y∈ball x e. (f x)∙k ≤ (f y)∙k)" and diff: "f differentiable (at x)" shows "(∑j∈Basis. (frechet_derivative f (at x) j ∙ k) *⇩_{R}j) = (0::'a)" (is "?D k = 0") proof - let ?f' = "frechet_derivative f (at x)" have "x ∈ ball x e" using ‹0 < e› by simp moreover have "open (ball x e)" by simp moreover have "((λx. f x ∙ k) has_derivative (λh. ?f' h ∙ k)) (at x)" using bounded_linear_inner_left diff[unfolded frechet_derivative_works] by (rule bounded_linear.has_derivative) ultimately have "(λh. frechet_derivative f (at x) h ∙ k) = (λv. 0)" using ball(2) by (rule differential_zero_maxmin) then show ?thesis unfolding fun_eq_iff by simp qed lemma rolle: fixes f :: "real ⇒ real" assumes "a < b" and "f a = f b" and "continuous_on {a .. b} f" and "∀x∈{a <..< b}. (f has_derivative f' x) (at x)" shows "∃x∈{a <..< b}. f' x = (λv. 0)" proof - have "∃x∈box a b. (∀y∈box a b. f x ≤ f y) ∨ (∀y∈box a b. f y ≤ f x)" proof - have "(a + b) / 2 ∈ {a .. b}" using assms(1) by auto then have *: "{a .. b} ≠ {}" by auto obtain d where d: "d ∈cbox a b" "∀y∈cbox a b. f y ≤ f d" using continuous_attains_sup[OF compact_Icc * assms(3)] by auto obtain c where c: "c ∈ cbox a b" "∀y∈cbox a b. f c ≤ f y" using continuous_attains_inf[OF compact_Icc * assms(3)] by auto show ?thesis proof (cases "d ∈ box a b ∨ c ∈ box a b") case True then show ?thesis by (metis c(2) d(2) box_subset_cbox subset_iff) next def e ≡ "(a + b) /2" case False then have "f d = f c" using d c assms(2) by auto then have "⋀x. x ∈ {a..b} ⟹ f x = f d" using c d by force then show ?thesis apply (rule_tac x=e in bexI) unfolding e_def using assms(1) apply auto done qed qed then obtain x where x: "x ∈ {a <..< b}" "(∀y∈{a <..< b}. f x ≤ f y) ∨ (∀y∈{a <..< b}. f y ≤ f x)" by auto then have "f' x = (λv. 0)" apply (rule_tac differential_zero_maxmin[of x "box a b" f "f' x"]) using assms apply auto done then show ?thesis by (metis x(1)) qed subsection ‹One-dimensional mean value theorem› lemma mvt: fixes f :: "real ⇒ real" assumes "a < b" and "continuous_on {a..b} f" assumes "∀x∈{a<..<b}. (f has_derivative (f' x)) (at x)" shows "∃x∈{a<..<b}. f b - f a = (f' x) (b - a)" proof - have "∃x∈{a <..< b}. (λxa. f' x xa - (f b - f a) / (b - a) * xa) = (λv. 0)" proof (intro rolle[OF assms(1), of "λx. f x - (f b - f a) / (b - a) * x"] ballI) fix x assume x: "x ∈ {a <..< b}" show "((λx. f x - (f b - f a) / (b - a) * x) has_derivative (λxa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)" by (intro derivative_intros assms(3)[rule_format,OF x]) qed (insert assms(1,2), auto intro!: continuous_intros simp: field_simps) then obtain x where "x ∈ {a <..< b}" "(λxa. f' x xa - (f b - f a) / (b - a) * xa) = (λv. 0)" .. then show ?thesis by (metis (hide_lams) assms(1) diff_gt_0_iff_gt eq_iff_diff_eq_0 zero_less_mult_iff nonzero_mult_divide_cancel_right not_real_square_gt_zero times_divide_eq_left) qed lemma mvt_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 (rule mvt) have "f differentiable_on {a..b}" using assms(2) unfolding differentiable_on_def differentiable_def by fast then show "continuous_on {a..b} f" by (rule differentiable_imp_continuous_on) show "∀x∈{a<..<b}. (f has_derivative f' x) (at x)" proof fix x assume x: "x ∈ {a <..< b}" show "(f has_derivative f' x) (at x)" unfolding at_within_open[OF x open_greaterThanLessThan,symmetric] apply (rule has_derivative_within_subset) apply (rule assms(2)[rule_format]) using x apply auto done qed qed (rule assms(1)) 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 then show ?thesis apply (rule_tac x=a in bexI) using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def unfolding True using zero apply auto done next case False then show ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed text ‹A nice generalization (see Havin's proof of 5.19 from Rudin's book).› lemma mvt_general: fixes f :: "real ⇒ 'a::real_inner" assumes "a < b" and "continuous_on {a .. b} f" and "∀x∈{a<..<b}. (f has_derivative f'(x)) (at x)" shows "∃x∈{a<..<b}. norm (f b - f a) ≤ norm (f' x (b - a))" proof - have "∃x∈{a<..<b}. (f b - f a) ∙ f b - (f b - f a) ∙ f a = (f b - f a) ∙ f' x (b - a)" apply (rule mvt) apply (rule assms(1)) apply (intro continuous_intros assms(2)) using assms(3) apply (fast intro: has_derivative_inner_right) done then obtain x where x: "x ∈ {a<..<b}" "(f b - f a) ∙ f b - (f b - f a) ∙ f a = (f b - f a) ∙ f' x (b - a)" .. 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) 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(2) by (simp only: inner_diff_right) 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: mult_left_cancel) next case True then show ?thesis using assms(1) apply (rule_tac x="(a + b) /2" in bexI) apply auto done qed qed subsection ‹More general bound theorems› lemma differentiable_bound_general: fixes f :: "real ⇒ 'a::real_normed_vector" assumes "a < b" and f_cont: "continuous_on {a .. b} f" and phi_cont: "continuous_on {a .. b} φ" and f': "⋀x. a < x ⟹ x < b ⟹ (f has_vector_derivative f' x) (at x)" and phi': "⋀x. a < x ⟹ x < b ⟹ (φ has_vector_derivative φ' x) (at x)" and bnd: "⋀x. a < x ⟹ x < b ⟹ norm (f' x) ≤ φ' x" shows "norm (f b - f a) ≤ φ b - φ a" proof - { fix x assume x: "a < x" "x < b" have "0 ≤ norm (f' x)" by simp also have "… ≤ φ' x" using x by (auto intro!: bnd) finally have "0 ≤ φ' x" . } note phi'_nonneg = this note f_tendsto = assms(2)[simplified continuous_on_def, rule_format] note phi_tendsto = assms(3)[simplified continuous_on_def, rule_format] { fix e::real assume "e > 0" def e2 ≡ "e / 2" with ‹e > 0› have "e2 > 0" by simp let ?le = "λx1. norm (f x1 - f a) ≤ φ x1 - φ a + e * (x1 - a) + e" def A ≡ "{x2. a ≤ x2 ∧ x2 ≤ b ∧ (∀x1∈{a ..< x2}. ?le x1)}" have A_subset: "A ⊆ {a .. b}" by (auto simp: A_def) { fix x2 assume a: "a ≤ x2" "x2 ≤ b" and le: "∀x1∈{a..<x2}. ?le x1" have "?le x2" using ‹e > 0› proof cases assume "x2 ≠ a" with a have "a < x2" by simp have "at x2 within {a <..<x2}≠ bot" using ‹a < x2› by (auto simp: trivial_limit_within islimpt_in_closure) moreover have "((λx1. (φ x1 - φ a) + e * (x1 - a) + e) ⤏ (φ x2 - φ a) + e * (x2 - a) + e) (at x2 within {a <..<x2})" "((λx1. norm (f x1 - f a)) ⤏ norm (f x2 - f a)) (at x2 within {a <..<x2})" using a by (auto intro!: tendsto_eq_intros f_tendsto phi_tendsto intro: tendsto_within_subset[where S="{a .. b}"]) moreover have "eventually (λx. x > a) (at x2 within {a <..<x2})" by (auto simp: eventually_at_filter) hence "eventually ?le (at x2 within {a <..<x2})" unfolding eventually_at_filter by eventually_elim (insert le, auto) ultimately show ?thesis by (rule tendsto_le) qed simp } note le_cont = this have "a ∈ A" using assms by (auto simp: A_def) hence [simp]: "A ≠ {}" by auto have A_ivl: "⋀x1 x2. x2 ∈ A ⟹ x1 ∈ {a ..x2} ⟹ x1 ∈ A" by (simp add: A_def) have [simp]: "bdd_above A" by (auto simp: A_def) def y ≡ "Sup A" have "y ≤ b" unfolding y_def by (simp add: cSup_le_iff) (simp add: A_def) have leI: "⋀x x1. a ≤ x1 ⟹ x ∈ A ⟹ x1 < x ⟹ ?le x1" by (auto simp: A_def intro!: le_cont) have y_all_le: "∀x1∈{a..<y}. ?le x1" by (auto simp: y_def less_cSup_iff leI) have "a ≤ y" by (metis ‹a ∈ A› ‹bdd_above A› cSup_upper y_def) have "y ∈ A" using y_all_le ‹a ≤ y› ‹y ≤ b› by (auto simp: A_def) hence "A = {a .. y}" using A_subset by (auto simp: subset_iff y_def cSup_upper intro: A_ivl) from le_cont[OF ‹a ≤ y› ‹y ≤ b› y_all_le] have le_y: "?le y" . { assume "a ≠ y" with ‹a ≤ y› have "a < y" by simp have "y = b" proof (rule ccontr) assume "y ≠ b" hence "y < b" using ‹y ≤ b› by simp let ?F = "at y within {y..<b}" from f' phi' have "(f has_vector_derivative f' y) ?F" and "(φ has_vector_derivative φ' y) ?F" using ‹a < y› ‹y < b› by (auto simp add: at_within_open[of _ "{a<..<b}"] has_vector_derivative_def intro!: has_derivative_subset[where s="{a<..<b}" and t="{y..<b}"]) hence "∀⇩_{F}x1 in ?F. norm (f x1 - f y - (x1 - y) *⇩_{R}f' y) ≤ e2 * ¦x1 - y¦" "∀⇩_{F}x1 in ?F. norm (φ x1 - φ y - (x1 - y) *⇩_{R}φ' y) ≤ e2 * ¦x1 - y¦" using ‹e2 > 0› by (auto simp: has_derivative_within_alt2 has_vector_derivative_def) moreover have "∀⇩_{F}x1 in ?F. y ≤ x1" "∀⇩_{F}x1 in ?F. x1 < b" by (auto simp: eventually_at_filter) ultimately have "∀⇩_{F}x1 in ?F. norm (f x1 - f y) ≤ (φ x1 - φ y) + e * ¦x1 - y¦" (is "∀⇩_{F}x1 in ?F. ?le' x1") proof eventually_elim case (elim x1) from norm_triangle_ineq2[THEN order_trans, OF elim(1)] have "norm (f x1 - f y) ≤ norm (f' y) * ¦x1 - y¦ + e2 * ¦x1 - y¦" by (simp add: ac_simps) also have "norm (f' y) ≤ φ' y" using bnd ‹a < y› ‹y < b› by simp also from elim have "φ' y * ¦x1 - y¦ ≤ φ x1 - φ y + e2 * ¦x1 - y¦" by (simp add: ac_simps) finally have "norm (f x1 - f y) ≤ φ x1 - φ y + e2 * ¦x1 - y¦ + e2 * ¦x1 - y¦" by (auto simp: mult_right_mono) thus ?case by (simp add: e2_def) qed moreover have "?le' y" by simp ultimately obtain S where S: "open S" "y ∈ S" "⋀x. x∈S ⟹ x ∈ {y..<b} ⟹ ?le' x" unfolding eventually_at_topological by metis from ‹open S› obtain d where d: "⋀x. dist x y < d ⟹ x ∈ S" "d > 0" by (force simp: dist_commute open_dist ball_def dest!: bspec[OF _ ‹y ∈ S›]) def d' ≡ "min ((y + b)/2) (y + (d/2))" have "d' ∈ A" unfolding A_def proof safe show "a ≤ d'" using ‹a < y› ‹0 < d› ‹y < b› by (simp add: d'_def) show "d' ≤ b" using ‹y < b› by (simp add: d'_def min_def) fix x1 assume x1: "x1 ∈ {a..<d'}" { assume "x1 < y" hence "?le x1" using ‹x1 ∈ {a..<d'}› y_all_le by auto } moreover { assume "x1 ≥ y" hence x1': "x1 ∈ S" "x1 ∈ {y..<b}" using x1 by (auto simp: d'_def dist_real_def intro!: d) have "norm (f x1 - f a) ≤ norm (f x1 - f y) + norm (f y - f a)" by (rule order_trans[OF _ norm_triangle_ineq]) simp also note S(3)[OF x1'] also note le_y finally have "?le x1" using ‹x1 ≥ y› by (auto simp: algebra_simps) } ultimately show "?le x1" by arith qed hence "d' ≤ y" unfolding y_def by (rule cSup_upper) simp thus False using ‹d > 0› ‹y < b› by (simp add: d'_def min_def split: split_if_asm) qed } moreover { assume "a = y" with ‹a < b› have "y < b" by simp with ‹a = y› f_cont phi_cont ‹e2 > 0› have 1: "∀⇩_{F}x in at y within {y..b}. dist (f x) (f y) < e2" and 2: "∀⇩_{F}x in at y within {y..b}. dist (φ x) (φ y) < e2" by (auto simp: continuous_on_def tendsto_iff) have 3: "eventually (λx. y < x) (at y within {y..b})" by (auto simp: eventually_at_filter) have 4: "eventually (λx::real. x < b) (at y within {y..b})" using _ ‹y < b› by (rule order_tendstoD) (auto intro!: tendsto_eq_intros) from 1 2 3 4 have eventually_le: "eventually (λx. ?le x) (at y within {y .. b})" proof eventually_elim case (elim x1) have "norm (f x1 - f a) = norm (f x1 - f y)" by (simp add: ‹a = y›) also have "norm (f x1 - f y) ≤ e2" using elim ‹a = y› by (auto simp : dist_norm intro!: less_imp_le) also have "… ≤ e2 + (φ x1 - φ a + e2 + e * (x1 - a))" using ‹0 < e› elim by (intro add_increasing2[OF add_nonneg_nonneg order.refl]) (auto simp: ‹a = y› dist_norm intro!: mult_nonneg_nonneg) also have "… = φ x1 - φ a + e * (x1 - a) + e" by (simp add: e2_def) finally show "?le x1" . qed from this[unfolded eventually_at_topological] ‹?le y› obtain S where S: "open S" "y ∈ S" "⋀x. x∈S ⟹ x ∈ {y..b} ⟹ ?le x" by metis from ‹open S› obtain d where d: "⋀x. dist x y < d ⟹ x ∈ S" "d > 0" by (force simp: dist_commute open_dist ball_def dest!: bspec[OF _ ‹y ∈ S›]) def d' ≡ "min b (y + (d/2))" have "d' ∈ A" unfolding A_def proof safe show "a ≤ d'" using ‹a = y› ‹0 < d› ‹y < b› by (simp add: d'_def) show "d' ≤ b" by (simp add: d'_def) fix x1 assume "x1 ∈ {a..<d'}" hence "x1 ∈ S" "x1 ∈ {y..b}" by (auto simp: ‹a = y› d'_def dist_real_def intro!: d ) thus "?le x1" by (rule S) qed hence "d' ≤ y" unfolding y_def by (rule cSup_upper) simp hence "y = b" using ‹d > 0› ‹y < b› by (simp add: d'_def) } ultimately have "y = b" by auto with le_y have "norm (f b - f a) ≤ φ b - φ a + e * (b - a + 1)" by (simp add: algebra_simps) } note * = this { fix e::real assume "e > 0" hence "norm (f b - f a) ≤ φ b - φ a + e" using *[of "e / (b - a + 1)"] ‹a < b› by simp } thus ?thesis by (rule field_le_epsilon) qed lemma differentiable_bound: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "convex s" and "∀x∈s. (f has_derivative f' x) (at x within s)" and "∀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 = "λu. x + u *⇩_{R}(y - x)" let ?φ = "λh. h * B * norm (x - y)" have *: "⋀u. u∈{0..1} ⟹ x + u *⇩_{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) have 0: "continuous_on (?p ` {0..1}) f" using * unfolding continuous_on_eq_continuous_within apply - apply rule apply (rule differentiable_imp_continuous_within) unfolding differentiable_def apply (rule_tac x="f' xa" in exI) apply (rule has_derivative_within_subset) apply (rule assms(2)[rule_format]) apply auto done from * have 1: "continuous_on {0 .. 1} (f ∘ ?p)" by (intro continuous_intros 0)+ { fix u::real assume u: "u ∈{0 <..< 1}" let ?u = "?p u" interpret linear "(f' ?u)" using u by (auto intro!: has_derivative_linear assms(2)[rule_format] *) have "(f ∘ ?p has_derivative (f' ?u) ∘ (λu. 0 + u *⇩_{R}(y - x))) (at u within box 0 1)" apply (rule diff_chain_within) apply (rule derivative_intros)+ apply (rule has_derivative_within_subset) apply (rule assms(2)[rule_format]) using u * apply auto done hence "((f ∘ ?p) has_vector_derivative f' ?u (y - x)) (at u)" by (simp add: has_derivative_within_open[OF u open_greaterThanLessThan] scaleR has_vector_derivative_def o_def) } note 2 = this { have "continuous_on {0..1} ?φ" by (rule continuous_intros)+ } note 3 = this { fix u::real assume u: "u ∈{0 <..< 1}" have "(?φ has_vector_derivative B * norm (x - y)) (at u)" by (auto simp: has_vector_derivative_def intro!: derivative_eq_intros) } note 4 = this { fix u::real assume u: "u ∈{0 <..< 1}" let ?u = "?p u" interpret bounded_linear "(f' ?u)" using u by (auto intro!: has_derivative_bounded_linear assms(2)[rule_format] *) have "norm (f' ?u (y - x)) ≤ onorm (f' ?u) * norm (y - x)" by (rule onorm) fact also have "onorm (f' ?u) ≤ B" using u by (auto intro!: assms(3)[rule_format] *) finally have "norm ((f' ?u) (y - x)) ≤ B * norm (x - y)" by (simp add: mult_right_mono norm_minus_commute) } note 5 = this have "norm (f x - f y) = norm ((f ∘ (λu. x + u *⇩_{R}(y - x))) 1 - (f ∘ (λu. x + u *⇩_{R}(y - x))) 0)" by (auto simp add: norm_minus_commute) also from differentiable_bound_general[OF zero_less_one 1, OF 3 2 4 5] have "norm ((f ∘ ?p) 1 - (f ∘ ?p) 0) ≤ B * norm (x - y)" by simp finally show ?thesis . qed lemma differentiable_bound_segment: fixes f::"'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "⋀t. t ∈ {0..1} ⟹ x0 + t *⇩_{R}a ∈ G" assumes f': "⋀x. x ∈ G ⟹ (f has_derivative f' x) (at x within G)" assumes B: "∀x∈{0..1}. onorm (f' (x0 + x *⇩_{R}a)) ≤ B" shows "norm (f (x0 + a) - f x0) ≤ norm a * B" proof - let ?G = "(λx. x0 + x *⇩_{R}a) ` {0..1}" have "?G = op + x0 ` (λx. x *⇩_{R}a) ` {0..1}" by auto also have "convex …" by (intro convex_translation convex_scaled convex_real_interval) finally have "convex ?G" . moreover have "?G ⊆ G" "x0 ∈ ?G" "x0 + a ∈ ?G" using assms by (auto intro: image_eqI[where x=1]) ultimately show ?thesis using has_derivative_subset[OF f' ‹?G ⊆ G›] B differentiable_bound[of "(λx. x0 + x *⇩_{R}a) ` {0..1}" f f' B "x0 + a" x0] by (auto simp: ac_simps) qed lemma differentiable_bound_linearization: fixes f::"'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "⋀t. t ∈ {0..1} ⟹ a + t *⇩_{R}(b - a) ∈ S" assumes f'[derivative_intros]: "⋀x. x ∈ S ⟹ (f has_derivative f' x) (at x within S)" assumes B: "∀x∈S. onorm (f' x - f' x0) ≤ B" assumes "x0 ∈ S" shows "norm (f b - f a - f' x0 (b - a)) ≤ norm (b - a) * B" proof - def g ≡ "λx. f x - f' x0 x" have g: "⋀x. x ∈ S ⟹ (g has_derivative (λi. f' x i - f' x0 i)) (at x within S)" unfolding g_def using assms by (auto intro!: derivative_eq_intros bounded_linear.has_derivative[OF has_derivative_bounded_linear, OF f']) from B have B: "∀x∈{0..1}. onorm (λi. f' (a + x *⇩_{R}(b - a)) i - f' x0 i) ≤ B" using assms by (auto simp: fun_diff_def) from differentiable_bound_segment[OF assms(1) g B] ‹x0 ∈ S› show ?thesis by (simp add: g_def field_simps linear_sub[OF has_derivative_linear[OF f']]) qed text ‹In particular.› lemma has_derivative_zero_constant: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "convex s" and "⋀x. x ∈ s ⟹ (f has_derivative (λh. 0)) (at x within s)" shows "∃c. ∀x∈s. f x = c" proof - { fix x y assume "x ∈ s" "y ∈ s" then have "norm (f x - f y) ≤ 0 * norm (x - y)" using assms by (intro differentiable_bound[of s]) (auto simp: onorm_zero) then have "f x = f y" by simp } then show ?thesis by metis qed lemma has_field_derivative_zero_constant: assumes "convex s" "⋀x. x ∈ s ⟹ (f has_field_derivative 0) (at x within s)" shows "∃c. ∀x∈s. f (x) = (c :: 'a :: real_normed_field)" proof (rule has_derivative_zero_constant) have A: "op * 0 = (λ_. 0 :: 'a)" by (intro ext) simp fix x assume "x ∈ s" thus "(f has_derivative (λh. 0)) (at x within s)" using assms(2)[of x] by (simp add: has_field_derivative_def A) qed fact lemma has_derivative_zero_unique: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "convex s" and "⋀x. x ∈ s ⟹ (f has_derivative (λh. 0)) (at x within s)" and "x ∈ s" "y ∈ s" shows "f x = f y" using has_derivative_zero_constant[OF assms(1,2)] assms(3-) by force lemma has_derivative_zero_unique_connected: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "open s" "connected s" assumes f: "⋀x. x ∈ s ⟹ (f has_derivative (λx. 0)) (at x)" assumes "x ∈ s" "y ∈ s" shows "f x = f y" proof (rule connected_local_const[where f=f, OF ‹connected s› ‹x∈s› ‹y∈s›]) show "∀a∈s. eventually (λb. f a = f b) (at a within s)" proof fix a assume "a ∈ s" with ‹open s› obtain e where "0 < e" "ball a e ⊆ s" by (rule openE) then have "∃c. ∀x∈ball a e. f x = c" by (intro has_derivative_zero_constant) (auto simp: at_within_open[OF _ open_ball] f convex_ball) with ‹0<e› have "∀x∈ball a e. f a = f x" by auto then show "eventually (λb. f a = f b) (at a within s)" using ‹0<e› unfolding eventually_at_topological by (intro exI[of _ "ball a e"]) auto qed qed subsection ‹Differentiability of inverse function (most basic form)› lemma has_derivative_inverse_basic: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "(f has_derivative f') (at (g y))" and "bounded_linear g'" and "g' ∘ f' = id" and "continuous (at y) g" and "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 obtain C where C: "0 < C" "⋀x. norm (g' x) ≤ norm x * C" using bounded_linear.pos_bounded[OF assms(2)] by blast 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) fix e :: real assume "e > 0" with C(1) have *: "e / C > 0" by auto obtain d0 where d0: "0 < d0" "∀ya. norm (ya - g y) < d0 ⟶ norm (f ya - f (g y) - f' (ya - g y)) ≤ e / C * norm (ya - g y)" using assms(1) unfolding has_derivative_at_alt using * by blast obtain d1 where d1: "0 < d1" "∀x. 0 < dist x y ∧ dist x y < d1 ⟶ dist (g x) (g y) < d0" using assms(4) unfolding continuous_at Lim_at using d0(1) by blast obtain d2 where d2: "0 < d2" "∀ya. dist ya y < d2 ⟶ ya ∈ t" using assms(5) unfolding open_dist using assms(6) by blast obtain d where d: "0 < d" "d < d1" "d < d2" using real_lbound_gt_zero[OF d1(1) d2(1)] by blast then show "∃d>0. ∀z. norm (z - y) < d ⟶ norm (g z - g y - g' (z - y)) ≤ e * norm (g z - g y)" apply (rule_tac x=d in exI) apply rule defer apply rule apply rule proof - fix z assume as: "norm (z - y) < d" then have "z ∈ 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[symmetric]) apply auto done also have "… ≤ norm (f (g z) - y - f' (g z - g y)) * C" by (rule C(2)) also have "… ≤ (e / C) * norm (g z - g y) * C" apply (rule mult_right_mono) apply (rule d0(2)[rule_format,unfolded assms(7)[rule_format,OF ‹y∈t›]]) apply (cases "z = y") defer apply (rule d1(2)[unfolded dist_norm,rule_format]) using as d C d0 apply auto done 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 obtain d where d: "0 < d" "∀z. norm (z - y) < d ⟶ norm (g z - g y - g' (z - y)) ≤ 1 / 2 * norm (g z - g y)" using lem1 * by blast def B ≡ "C * 2" have "B > 0" unfolding B_def using C by auto have lem2: "norm (g z - g y) ≤ B * norm (z - y)" if z: "norm(z - y) < d" for z proof - 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 z apply auto done also have "… ≤ norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply (rule add_right_mono) using C apply auto done finally show "norm (g z - g y) ≤ B * norm (z - y)" unfolding B_def by (auto simp add: field_simps) qed show ?thesis unfolding has_derivative_at_alt apply rule apply (rule assms) apply rule apply rule proof - fix e :: real assume "e > 0" then have *: "e / B > 0" by (metis ‹B > 0› divide_pos_pos) obtain d' where d': "0 < d'" "∀z. norm (z - y) < d' ⟶ norm (g z - g y - g' (z - y)) ≤ e / B * norm (g z - g y)" using lem1 * by blast obtain k where k: "0 < k" "k < d" "k < d'" using real_lbound_gt_zero[OF d(1) d'(1)] by blast show "∃d>0. ∀ya. norm (ya - y) < d ⟶ norm (g ya - g y - g' (ya - y)) ≤ e * norm (ya - y)" apply (rule_tac x=k in exI) apply auto proof - fix z assume as: "norm (z - y) < k" then have "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[of 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 text ‹Simply rewrite that based on the domain point x.› lemma has_derivative_inverse_basic_x: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "(f has_derivative f') (at x)" and "bounded_linear g'" and "g' ∘ f' = id" and "continuous (at (f x)) g" and "g (f x) = x" and "open t" and "f x ∈ t" and "∀y∈t. f (g y) = y" shows "(g has_derivative g') (at (f x))" apply (rule has_derivative_inverse_basic) using assms apply auto done text ‹This is the version in Dieudonne', assuming continuity of f and g.› lemma has_derivative_inverse_dieudonne: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "open s" and "open (f ` s)" and "continuous_on s f" and "continuous_on (f ` s) g" and "∀x∈s. g (f x) = x" and "x ∈ s" and "(f has_derivative f') (at x)" and "bounded_linear g'" and "g' ∘ 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)] apply auto done text ‹Here's the simplest way of not assuming much about g.› lemma has_derivative_inverse: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "compact s" and "x ∈ s" and "f x ∈ interior (f ` s)" and "continuous_on s f" and "∀y∈s. g (f y) = y" and "(f has_derivative f') (at x)" and "bounded_linear g'" and "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_inv[OF assms(4,1)]) apply (rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ apply (metis *) done qed subsection ‹Proving surjectivity via Brouwer fixpoint theorem› lemma brouwer_surjective: fixes f :: "'n::euclidean_space ⇒ 'n" assumes "compact t" and "convex t" and "t ≠ {}" and "continuous_on t f" and "∀x∈s. ∀y∈t. x + (y - f y) ∈ t" and "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 "λy. x + (y - f y)"]) apply (rule continuous_intros assms)+ using assms(4-6) apply auto done qed lemma brouwer_surjective_cball: fixes f :: "'n::euclidean_space ⇒ 'n" assumes "e > 0" and "continuous_on (cball a e) f" and "∀x∈s. ∀y∈cball a e. x + (y - f y) ∈ cball a e" and "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 apply auto done text ‹See Sussmann: "Multidifferential calculus", Theorem 2.1.1› lemma sussmann_open_mapping: fixes f :: "'a::real_normed_vector ⇒ 'b::euclidean_space" assumes "open s" and "continuous_on s f" and "x ∈ s" and "(f has_derivative f') (at x)" and "bounded_linear g'" "f' ∘ g' = id" and "t ⊆ s" and "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 obtain B where B: "0 < B" "∀x. norm (g' x) ≤ norm x * B" using bounded_linear.pos_bounded[OF assms(5)] by blast hence *: "1 / (2 * B) > 0" by auto obtain e0 where e0: "0 < e0" "∀y. norm (y - x) < e0 ⟶ norm (f y - f x - f' (y - x)) ≤ 1 / (2 * B) * norm (y - x)" using assms(4) unfolding has_derivative_at_alt using * by blast obtain e1 where e1: "0 < e1" "cball x e1 ⊆ t" using assms(8) unfolding mem_interior_cball by blast have *: "0 < e0 / B" "0 < e1 / B" using e0 e1 B by auto obtain e where e: "0 < e" "e < e0 / B" "e < e1 / B" using real_lbound_gt_zero[OF *] by blast have "∀z∈cball (f x) (e / 2). ∃y∈cball (f x) e. f (x + g' (y - f x)) = z" apply rule apply (rule brouwer_surjective_cball[where s="cball (f x) (e/2)"]) prefer 3 apply rule apply rule proof- show "continuous_on (cball (f x) e) (λy. f (x + g' (y - f x)))" unfolding g'.diff apply (rule continuous_on_compose[of _ _ f, unfolded o_def]) apply (rule continuous_intros linear_continuous_on[OF assms(5)])+ apply (rule continuous_on_subset[OF assms(2)]) apply rule apply (unfold image_iff) apply (erule bexE) proof- fix y z assume as: "y ∈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[symmetric] 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 then show "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 apply auto done 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: algebra_simps) also have "… ≤ 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0(2)[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 apply auto done 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 apply (rule divide_pos_pos) prefer 3 proof fix y assume "y ∈ ball (f x) (e / 2)" then have *: "y ∈ cball (f x) (e / 2)" by auto obtain z where z: "z ∈ cball (f x) e" "f (x + g' (z - f x)) = y" using lem * by blast then have "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 apply auto done also have "… ≤ e1" using e B unfolding less_divide_eq by auto finally have "x + g'(z - f x) ∈ t" apply - apply (rule e1(2)[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm apply auto done then show "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' ∘ g = I› but this happens to fit with the minimal linear algebra theory I've set up so far.› (* move before left_inverse_linear in Euclidean_Space*) lemma right_inverse_linear: fixes f :: "'a::euclidean_space ⇒ 'a" assumes lf: "linear f" and gf: "f ∘ g = id" shows "linear g" proof - from gf have fi: "surj f" by (auto simp add: surj_def o_def id_def) metis from linear_surjective_isomorphism[OF lf fi] obtain h:: "'a ⇒ 'a" where h: "linear h" "∀x. h (f x) = x" "∀x. f (h x) = x" by blast have "h = g" apply (rule ext) using gf h(2,3) apply (simp add: o_def id_def fun_eq_iff) apply metis done with h(1) show ?thesis by blast qed lemma has_derivative_inverse_strong: fixes f :: "'n::euclidean_space ⇒ 'n" assumes "open s" and "x ∈ s" and "continuous_on s f" and "∀x∈s. g (f x) = x" and "(f has_derivative f') (at x)" and "f' ∘ 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 then have ling: "bounded_linear g'" unfolding linear_conv_bounded_linear[symmetric] apply - apply (rule right_inverse_linear) using assms(6) apply auto done moreover have "g' ∘ f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[symmetric] using linear_inverse_left by auto moreover have *:"∀t⊆s. x ∈ interior t ⟶ f x ∈ interior (f ` t)" apply clarify apply (rule sussmann_open_mapping) apply (rule assms ling)+ apply auto done have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof (rule, rule) fix e :: real assume "e > 0" then have "f x ∈ 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 obtain d where d: "0 < d" "ball (f x) d ⊆ f ` (ball x e ∩ s)" unfolding mem_interior by blast 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 apply (rule d(1)) apply rule apply rule proof - fix y assume "0 < dist y (f x) ∧ dist y (f x) < d" then have "g y ∈ g ` f ` (ball x e ∩ s)" using d(2)[unfolded subset_eq,THEN bspec[where x=y]] by (auto simp add: dist_commute) then have "g y ∈ ball x e ∩ s" using assms(4) by auto then show "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› apply auto done moreover have "f (g y) = y" if "y ∈ interior (f ` s)" for y proof - from that have "y ∈ f ` s" using interior_subset by auto then obtain z where "z ∈ s" "y = f z" unfolding image_iff .. then show ?thesis using assms(4) by auto qed ultimately show ?thesis using assms by (metis has_derivative_inverse_basic_x open_interior) qed text ‹A rewrite based on the other domain.› lemma has_derivative_inverse_strong_x: fixes f :: "'a::euclidean_space ⇒ 'a" assumes "open s" and "g y ∈ s" and "continuous_on s f" and "∀x∈s. g (f x) = x" and "(f has_derivative f') (at (g y))" and "f' ∘ 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 text ‹On a region.› lemma has_derivative_inverse_on: fixes f :: "'n::euclidean_space ⇒ 'n" assumes "open s" and "∀x∈s. (f has_derivative f'(x)) (at x)" and "∀x∈s. g (f x) = x" and "f' x ∘ 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)+ unfolding continuous_on_eq_continuous_at[OF assms(1)] apply rule apply (rule differentiable_imp_continuous_within) unfolding differentiable_def using assms apply auto done text ‹Invertible derivative continous at a point implies local injectivity. It's only for this we need continuity of the derivative, 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 has_derivative_locally_injective: fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space" assumes "a ∈ s" and "open s" and "bounded_linear g'" and "g' ∘ f' a = id" and "∀x∈s. (f has_derivative f' x) (at x)" and "∀e>0. ∃d>0. ∀x. dist a x < d ⟶ onorm (λv. 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 (∑Basis)) = (∑Basis)" "(∑Basis) ≠ (0::'n)" defer apply (subst euclidean_eq_iff) using f'g' apply auto done then have *: "0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)] by fastforce def k ≡ "1 / onorm g' / 2" have *: "k > 0" unfolding k_def using * by auto obtain d1 where d1: "0 < d1" "⋀x. dist a x < d1 ⟹ onorm (λv. f' x v - f' a v) < k" using assms(6) * by blast 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) .. obtain d2 where d2: "0 < d2" "ball a d2 ⊆ s" using assms(2) unfolding open_contains_ball using ‹a∈s› by blast obtain d where d: "0 < d" "d < d1" "d < d2" using real_lbound_gt_zero[OF d1(1) d2(1)] by blast 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 ≡ "λw. w - g' (f w - f x)" have ph':"ph = g' ∘ (λw. 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'="λx v. v - g'(f' x v)"]) apply (rule_tac[!] ballI) proof - fix u assume u: "u ∈ ball a d" then have "u ∈ s" using d d2 by auto have *: "(λv. v - g' (f' u v)) = g' ∘ (λw. f' a w - f' u w)" unfolding o_def and diff using f'g' by auto show "(ph has_derivative (λv. v - g' (f' u v))) (at u within ball a d)" unfolding ph' * apply (simp add: comp_def) apply (rule bounded_linear.has_derivative[OF assms(3)]) apply (rule derivative_intros) defer apply (rule has_derivative_sub[where g'="λx.0",unfolded diff_0_right]) apply (rule has_derivative_at_within) using assms(5) and ‹u ∈ s› ‹a ∈ s› apply (auto intro!: derivative_intros bounded_linear.has_derivative[of _ "λx. x"] has_derivative_bounded_linear) done have **: "bounded_linear (λx. f' u x - f' a x)" "bounded_linear (λx. f' a x - f' u x)" apply (rule_tac[!] bounded_linear_sub) apply (rule_tac[!] has_derivative_bounded_linear) using assms(5) ‹u ∈ s› ‹a ∈ s› apply auto done have "onorm (λv. v - g' (f' u v)) ≤ onorm g' * onorm (λw. f' a w - f' u w)" unfolding * apply (rule onorm_compose) apply (rule assms(3) **)+ done also have "… ≤ onorm g' * k" apply (rule mult_left_mono) using d1(2)[of u] using onorm_neg[where f="λx. f' u x - f' a x"] using d and u and onorm_pos_le[OF assms(3)] apply (auto simp add: algebra_simps) done also have "… ≤ 1 / 2" unfolding k_def by auto finally show "onorm (λv. v - g' (f' u v)) ≤ 1 / 2" . qed moreover have "norm (ph y - ph x) = norm (y - x)" apply (rule arg_cong[where f=norm]) unfolding ph_def using diff unfolding as apply auto done 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 ⇒ 'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "convex s" and "∀n. ∀x∈s. ((f n) has_derivative (f' n x)) (at x within s)" and "∀n≥N. ∀x∈s. ∀h. norm (f' n x h - g' x h) ≤ e * norm h" and "0 ≤ e" 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 rule+ 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'="λx 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 "((λa. f m a - f n a) has_derivative (λh. f' m x h - f' n x h)) (at x within s)" by (rule derivative_intros assms(2)[rule_format] ‹x∈s›)+ show "onorm (λh. f' m x h - f' n x h) ≤ 2 * e" proof (rule onorm_bound) 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 show "norm (f' m x h - f' n x h) ≤ 2 * e * norm h" by auto qed (simp add: ‹0 ≤ e›) qed qed lemma has_derivative_sequence_lipschitz: fixes f :: "nat ⇒ 'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "convex s" and "∀n. ∀x∈s. ((f n) has_derivative (f' n x)) (at x within s)" and "∀e>0. ∃N. ∀n≥N. ∀x∈s. ∀h. norm (f' n x h - g' x h) ≤ e * norm h" 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) fix e :: real assume "e > 0" then have *: "2 * (1/2* e) = e" "1/2 * e >0" by auto obtain N where "∀n≥N. ∀x∈s. ∀h. norm (f' n x h - g' x h) ≤ 1 / 2 * e * norm h" using assms(3) *(2) by blast then show "∃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_tac x=N in exI) apply (rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *]) using assms ‹e > 0› apply auto done qed lemma has_derivative_sequence: fixes f :: "nat ⇒ 'a::real_normed_vector ⇒ 'b::banach" assumes "convex s" and "∀n. ∀x∈s. ((f n) has_derivative (f' n x)) (at x within s)" and "∀e>0. ∃N. ∀n≥N. ∀x∈s. ∀h. norm (f' n x h - g' x h) ≤ e * norm h" and "x0 ∈ s" and "((λn. f n x0) ⤏ l) sequentially" shows "∃g. ∀x∈s. ((λn. 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)" using assms(1,2,3) by (rule has_derivative_sequence_lipschitz) have "∃g. ∀x∈s. ((λn. f n x) ⤏ g x) sequentially" apply (rule bchoice) unfolding convergent_eq_cauchy proof fix x assume "x ∈ s" show "Cauchy (λn. f n x)" proof (cases "x = x0") case True then show ?thesis using LIMSEQ_imp_Cauchy[OF assms(5)] by auto next case False show ?thesis unfolding Cauchy_def proof (rule, rule) fix e :: real assume "e > 0" hence *: "e / 2 > 0" "e / 2 / norm (x - x0) > 0" using False by auto obtain M where M: "∀m≥M. ∀n≥M. dist (f m x0) (f n x0) < e / 2" using LIMSEQ_imp_Cauchy[OF assms(5)] unfolding Cauchy_def using *(1) by blast obtain N where N: "∀m≥N. ∀n≥N. ∀xa∈s. ∀y∈s. norm (f m xa - f n xa - (f m y - f n y)) ≤ e / 2 / norm (x - x0) * norm (xa - y)" using lem1 *(2) by blast show "∃M. ∀m≥M. ∀n≥M. dist (f m x) (f n x) < e" apply (rule_tac x="max M N" in exI) proof rule+ 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 apply auto done finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed then obtain g where g: "∀x∈s. (λn. f n x) ⇢ g x" .. 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" obtain N where 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)" using lem1 * by blast 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 rule+ fix n x y assume as: "N ≤ n" "x ∈ s" "y ∈ s" have "((λm. norm (f n x - f n y - (f m x - f m y))) ⤏ norm (f n x - f n y - (g x - g y))) sequentially" by (intro tendsto_intros g[rule_format] as) moreover have "eventually (λm. norm (f n x - f n y - (f m x - f m y)) ≤ e * norm (x - y)) sequentially" unfolding eventually_sequentially apply (rule_tac x=N in exI) apply rule apply rule proof - fix m assume "N ≤ m" then show "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 ultimately show "norm (f n x - f n y - (g x - g y)) ≤ e * norm (x - y)" by (rule tendsto_ge_const[OF trivial_limit_sequentially]) qed qed have "∀x∈s. ((λn. f n x) ⤏ g x) sequentially ∧ (g has_derivative g' x) (at x within s)" unfolding has_derivative_within_alt2 proof (intro ballI conjI) fix x assume "x ∈ s" then show "((λn. f n x) ⤏ g x) sequentially" by (simp add: g) have lem3: "∀u. ((λn. f' n x u) ⤏ g' x u) sequentially" unfolding filterlim_def le_nhds_metric_le eventually_filtermap dist_norm proof (intro allI impI) fix u fix e :: real assume "e > 0" show "eventually (λn. norm (f' n x u - g' x u) ≤ e) sequentially" proof (cases "u = 0") case True have "eventually (λn. norm (f' n x u - g' x u) ≤ e * norm u) sequentially" using assms(3)[folded eventually_sequentially] and ‹0 < e› and ‹x ∈ s› by (fast elim: eventually_mono) then show ?thesis using ‹u = 0› and ‹0 < e› by (auto elim: eventually_mono) next case False with ‹0 < e› have "0 < e / norm u" by simp then have "eventually (λn. norm (f' n x u - g' x u) ≤ e / norm u * norm u) sequentially" using assms(3)[folded eventually_sequentially] and ‹x ∈ s› by (fast elim: eventually_mono) then show ?thesis using ‹u ≠ 0› by simp qed qed show "bounded_linear (g' x)" proof fix x' y z :: 'a fix c :: real note lin = assms(2)[rule_format,OF ‹x∈s›,THEN has_derivative_bounded_linear] show "g' x (c *⇩_{R}x') = c *⇩_{R}g' x x'" apply (rule tendsto_unique[OF trivial_limit_sequentially]) apply (rule lem3[rule_format]) unfolding lin[THEN bounded_linear.linear, THEN linear_cmul] apply (intro tendsto_intros) apply (rule lem3[rule_format]) done show "g' x (y + z) = g' x y + g' x z" apply (rule tendsto_unique[OF trivial_limit_sequentially]) apply (rule lem3[rule_format]) unfolding lin[THEN bounded_linear.linear, THEN linear_add] apply (rule tendsto_add) apply (rule lem3[rule_format])+ done obtain N where N: "∀h. norm (f' N x h - g' x h) ≤ 1 * norm h" using assms(3) ‹x ∈ s› by (fast intro: zero_less_one) have "bounded_linear (f' N x)" using assms(2) ‹x ∈ s› by fast from bounded_linear.bounded [OF this] obtain K where K: "∀h. norm (f' N x h) ≤ norm h * K" .. { fix h have "norm (g' x h) = norm (f' N x h - (f' N x h - g' x h))" by simp also have "… ≤ norm (f' N x h) + norm (f' N x h - g' x h)" by (rule norm_triangle_ineq4) also have "… ≤ norm h * K + 1 * norm h" using N K by (fast intro: add_mono) finally have "norm (g' x h) ≤ norm h * (K + 1)" by (simp add: ring_distribs) } then show "∃K. ∀h. norm (g' x h) ≤ norm h * K" by fast qed show "∀e>0. eventually (λy. norm (g y - g x - g' x (y - x)) ≤ e * norm (y - x)) (at x within s)" proof (rule, rule) fix e :: real assume "e > 0" then have *: "e / 3 > 0" by auto obtain N1 where N1: "∀n≥N1. ∀x∈s. ∀h. norm (f' n x h - g' x h) ≤ e / 3 * norm h" using assms(3) * by blast obtain N2 where N2: "∀n≥N2. ∀x∈s. ∀y∈s. norm (f n x - f n y - (g x - g y)) ≤ e / 3 * norm (x - y)" using lem2 * by blast let ?N = "max N1 N2" have "eventually (λy. norm (f ?N y - f ?N x - f' ?N x (y - x)) ≤ e / 3 * norm (y - x)) (at x within s)" using assms(2)[unfolded has_derivative_within_alt2] and ‹x ∈ s› and * by fast moreover have "eventually (λy. y ∈ s) (at x within s)" unfolding eventually_at by (fast intro: zero_less_one) ultimately show "∀⇩_{F}y in at x within s. norm (g y - g x - g' x (y - x)) ≤ e * norm (y - x)" proof (rule eventually_elim2) fix y assume "y ∈ s" assume "norm (f ?N y - f ?N x - f' ?N x (y - x)) ≤ e / 3 * norm (y - x)" moreover have "norm (g y - g x - (f ?N y - f ?N x)) ≤ e / 3 * norm (y - x)" using N2[rule_format, OF _ ‹y ∈ s› ‹x ∈ s›] by (simp add: norm_minus_commute) 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 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 then show ?thesis by fast qed text ‹Can choose to line up antiderivatives if we want.› lemma has_antiderivative_sequence: fixes f :: "nat ⇒ 'a::real_normed_vector ⇒ 'b::banach" assumes "convex s" and "∀n. ∀x∈s. ((f n) has_derivative (f' n x)) (at x within s)" and "∀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 "λn x. f n x + (f 0 a - f n a)"]) apply (metis assms(2) has_derivative_add_const) apply (rule ‹a ∈ s›) apply auto done qed auto lemma has_antiderivative_limit: fixes g' :: "'a::real_normed_vector ⇒ 'a ⇒ 'b::banach" assumes "convex s" and "∀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)" by (simp add: assms(2)) obtain f where *: "∀x. ∃f'. ∀xa∈s. (f x has_derivative f' xa) (at xa within s) ∧ (∀h. norm (f' xa h - g' xa h) ≤ inverse (real (Suc x)) * norm h)" using *[THEN choice] .. obtain f' where f: "∀x. ∀xa∈s. (f x has_derivative f' x xa) (at xa within s) ∧ (∀h. norm (f' x xa h - g' xa h) ≤ inverse (real (Suc x)) * norm h)" using *[THEN choice] .. show ?thesis apply (rule has_antiderivative_sequence[OF assms(1), of f f']) defer apply rule apply rule proof - fix e :: real assume "e > 0" obtain N where N: "inverse (real (Suc N)) < e" using reals_Archimedean[OF ‹e>0›] .. 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) apply rule apply rule apply rule apply rule proof - fix n x h assume n: "N ≤ n" and x: "x ∈ s" have *: "inverse (real (Suc n)) ≤ e" apply (rule order_trans[OF _ N[THEN less_imp_le]]) using n apply (auto simp add: field_simps) done show "norm (f' n x h - g' x h) ≤ e * norm h" using f[rule_format,THEN conjunct2, OF x, of n, THEN spec[where x=h]] apply (rule order_trans) using N * apply (cases "h = 0") apply auto done qed qed (insert f, auto) qed subsection ‹Differentiation of a series› lemma has_derivative_series: fixes f :: "nat ⇒ 'a::real_normed_vector ⇒ 'b::banach" assumes "convex s" and "⋀n x. x ∈ s ⟹ ((f n) has_derivative (f' n x)) (at x within s)" and "∀e>0. ∃N. ∀n≥N. ∀x∈s. ∀h. norm (setsum (λi. f' i x h) {..<n} - g' x h) ≤ e * norm h" and "x ∈ s" and "(λn. f n x) sums l" shows "∃g. ∀x∈s. (λn. f n x) sums (g x) ∧ (g has_derivative g' x) (at x within s)" unfolding sums_def apply (rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply (metis assms(2) has_derivative_setsum) using assms(4-5) unfolding sums_def apply auto done lemma has_field_derivative_series: fixes f :: "nat ⇒ ('a :: {real_normed_field,banach}) ⇒ 'a" assumes "convex s" assumes "⋀n x. x ∈ s ⟹ (f n has_field_derivative f' n x) (at x within s)" assumes "uniform_limit s (λn x. ∑i<n. f' i x) g' sequentially" assumes "x0 ∈ s" "summable (λn. f n x0)" shows "∃g. ∀x∈s. (λn. f n x) sums g x ∧ (g has_field_derivative g' x) (at x within s)" unfolding has_field_derivative_def proof (rule has_derivative_series) show "∀e>0. ∃N. ∀n≥N. ∀x∈s. ∀h. norm ((∑i<n. f' i x * h) - g' x * h) ≤ e * norm h" proof (intro allI impI) fix e :: real assume "e > 0" with assms(3) obtain N where N: "⋀n x. n ≥ N ⟹ x ∈ s ⟹ norm ((∑i<n. f' i x) - g' x) < e" unfolding uniform_limit_iff eventually_at_top_linorder dist_norm by blast { fix n :: nat and x h :: 'a assume nx: "n ≥ N" "x ∈ s" have "norm ((∑i<n. f' i x * h) - g' x * h) = norm ((∑i<n. f' i x) - g' x) * norm h" by (simp add: norm_mult [symmetric] ring_distribs setsum_left_distrib) also from N[OF nx] have "norm ((∑i<n. f' i x) - g' x) ≤ e" by simp hence "norm ((∑i<n. f' i x) - g' x) * norm h ≤ e * norm h" by (intro mult_right_mono) simp_all finally have "norm ((∑i<n. f' i x * h) - g' x * h) ≤ e * norm h" . } thus "∃N. ∀n≥N. ∀x∈s. ∀h. norm ((∑i<n. f' i x * h) - g' x * h) ≤ e * norm h" by blast qed qed (insert assms, auto simp: has_field_derivative_def) lemma has_field_derivative_series': fixes f :: "nat ⇒ ('a :: {real_normed_field,banach}) ⇒ 'a" assumes "convex s" assumes "⋀n x. x ∈ s ⟹ (f n has_field_derivative f' n x) (at x within s)" assumes "uniformly_convergent_on s (λn x. ∑i<n. f' i x)" assumes "x0 ∈ s" "summable (λn. f n x0)" "x ∈ interior s" shows "summable (λn. f n x)" "((λx. ∑n. f n x) has_field_derivative (∑n. f' n x)) (at x)" proof - from ‹x ∈ interior s› have "x ∈ s" using interior_subset by blast def g' ≡ "λx. ∑i. f' i x" from assms(3) have "uniform_limit s (λn x. ∑i<n. f' i x) g' sequentially" by (simp add: uniformly_convergent_uniform_limit_iff suminf_eq_lim g'_def) from has_field_derivative_series[OF assms(1,2) this assms(4,5)] obtain g where g: "⋀x. x ∈ s ⟹ (λn. f n x) sums g x" "⋀x. x ∈ s ⟹ (g has_field_derivative g' x) (at x within s)" by blast from g(1)[OF ‹x ∈ s›] show "summable (λn. f n x)" by (simp add: sums_iff) from g(2)[OF ‹x ∈ s›] ‹x ∈ interior s› have "(g has_field_derivative g' x) (at x)" by (simp add: at_within_interior[of x s]) also have "(g has_field_derivative g' x) (at x) ⟷ ((λx. ∑n. f n x) has_field_derivative g' x) (at x)" using eventually_nhds_in_nhd[OF ‹x ∈ interior s›] interior_subset[of s] g(1) by (intro DERIV_cong_ev) (auto elim!: eventually_mono simp: sums_iff) finally show "((λx. ∑n. f n x) has_field_derivative g' x) (at x)" . qed lemma differentiable_series: fixes f :: "nat ⇒ ('a :: {real_normed_field,banach}) ⇒ 'a" assumes "convex s" "open s" assumes "⋀n x. x ∈ s ⟹ (f n has_field_derivative f' n x) (at x)" assumes "uniformly_convergent_on s (λn x. ∑i<n. f' i x)" assumes "x0 ∈ s" "summable (λn. f n x0)" and x: "x ∈ s" shows "summable (λn. f n x)" and "(λx. ∑n. f n x) differentiable (at x)" proof - from assms(4) obtain g' where A: "uniform_limit s (λn x. ∑i<n. f' i x) g' sequentially" unfolding uniformly_convergent_on_def by blast from x and ‹open s› have s: "at x within s = at x" by (rule at_within_open) have "∃g. ∀x∈s. (λn. f n x) sums g x ∧ (g has_field_derivative g' x) (at x within s)" by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within) then obtain g where g: "⋀x. x ∈ s ⟹ (λn. f n x) sums g x" "⋀x. x ∈ s ⟹ (g has_field_derivative g' x) (at x within s)" by blast from g[OF x] show "summable (λn. f n x)" by (auto simp: summable_def) from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)" by (simp add: has_field_derivative_def s) have "((λx. ∑n. f n x) has_derivative op * (g' x)) (at x)" by (rule has_derivative_transform_within_open[OF g' ‹open s› x]) (insert g, auto simp: sums_iff) thus "(λx. ∑n. f n x) differentiable (at x)" unfolding differentiable_def by (auto simp: summable_def differentiable_def has_field_derivative_def) qed lemma differentiable_series': fixes f :: "nat ⇒ ('a :: {real_normed_field,banach}) ⇒ 'a" assumes "convex s" "open s" assumes "⋀n x. x ∈ s ⟹ (f n has_field_derivative f' n x) (at x)" assumes "uniformly_convergent_on s (λn x. ∑i<n. f' i x)" assumes "x0 ∈ s" "summable (λn. f n x0)" shows "(λx. ∑n. f n x) differentiable (at x0)" using differentiable_series[OF assms, of x0] ‹x0 ∈ s› by blast+ text ‹Considering derivative @{typ "real ⇒ 'b::real_normed_vector"} as a vector.› definition "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)" lemma vector_derivative_unique_within: assumes not_bot: "at x within s ≠ bot" and f': "(f has_vector_derivative f') (at x within s)" and f'': "(f has_vector_derivative f'') (at x within s)" shows "f' = f''" proof - have "(λx. x *⇩_{R}f') = (λx. x *⇩_{R}f'')" proof (rule frechet_derivative_unique_within) show "∀i∈Basis. ∀e>0. ∃d. 0 < ¦d¦ ∧ ¦d¦ < e ∧ x + d *⇩_{R}i ∈ s" proof clarsimp fix e :: real assume "0 < e" with islimpt_approachable_real[of x s] not_bot obtain x' where "x' ∈ s" "x' ≠ x" "¦x' - x¦ < e" by (auto simp add: trivial_limit_within) then show "∃d. d ≠ 0 ∧ ¦d¦ < e ∧ x + d ∈ s" by (intro exI[of _ "x' - x"]) auto qed qed (insert f' f'', auto simp: has_vector_derivative_def) then show ?thesis unfolding fun_eq_iff by (metis scaleR_one) qed lemma vector_derivative_unique_at: "(f has_vector_derivative f') (at x) ⟹ (f has_vector_derivative f'') (at x) ⟹ f' = f''" by (rule vector_derivative_unique_within) auto lemma differentiableI_vector: "(f has_vector_derivative y) F ⟹ f differentiable F" by (auto simp: differentiable_def has_vector_derivative_def) lemma vector_derivative_works: "f differentiable net ⟷ (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r") proof assume ?l obtain f' where f': "(f has_derivative f') net" using ‹?l› unfolding differentiable_def .. then interpret bounded_linear f' by auto show ?r unfolding vector_derivative_def has_vector_derivative_def by (rule someI[of _ "f' 1"]) (simp add: scaleR[symmetric] f') qed (auto simp: vector_derivative_def has_vector_derivative_def differentiable_def) lemma vector_derivative_within: assumes not_bot: "at x within s ≠ bot" and y: "(f has_vector_derivative y) (at x within s)" shows "vector_derivative f (at x within s) = y" using y by (intro vector_derivative_unique_within[OF not_bot vector_derivative_works[THEN iffD1] y]) (auto simp: differentiable_def has_vector_derivative_def) lemma frechet_derivative_eq_vector_derivative: assumes "f differentiable (at x)" shows "(frechet_derivative f (at x)) = (λr. r *⇩_{R}vector_derivative f (at x))" using assms by (auto simp: differentiable_iff_scaleR vector_derivative_def has_vector_derivative_def intro: someI frechet_derivative_at [symmetric]) lemma has_real_derivative: fixes f :: "real ⇒ real" assumes "(f has_derivative f') F" obtains c where "(f has_real_derivative c) F" proof - obtain c where "f' = (λx. x * c)" by (metis assms has_derivative_bounded_linear real_bounded_linear) then show ?thesis by (metis assms that has_field_derivative_def mult_commute_abs) qed lemma has_real_derivative_iff: fixes f :: "real ⇒ real" shows "(∃c. (f has_real_derivative c) F) = (∃D. (f has_derivative D) F)" by (metis has_field_derivative_def has_real_derivative) definition deriv :: "('a ⇒ 'a::real_normed_field) ⇒ 'a ⇒ 'a" where "deriv f x ≡ SOME D. DERIV f x :> D" lemma DERIV_imp_deriv: "DERIV f x :> f' ⟹ deriv f x = f'" unfolding deriv_def by (metis some_equality DERIV_unique) lemma DERIV_deriv_iff_has_field_derivative: "DERIV f x :> deriv f x ⟷ (∃f'. (f has_field_derivative f') (at x))" by (auto simp: has_field_derivative_def DERIV_imp_deriv) lemma DERIV_deriv_iff_real_differentiable: fixes x :: real shows "DERIV f x :> deriv f x ⟷ f differentiable at x" unfolding differentiable_def by (metis DERIV_imp_deriv has_real_derivative_iff) lemma real_derivative_chain: fixes x :: real shows "f differentiable at x ⟹ g differentiable at (f x) ⟹ deriv (g o f) x = deriv g (f x) * deriv f x" by (metis DERIV_deriv_iff_real_differentiable DERIV_chain DERIV_imp_deriv) lemma field_derivative_eq_vector_derivative: "(deriv f x) = vector_derivative f (at x)" by (simp add: mult.commute deriv_def vector_derivative_def has_vector_derivative_def has_field_derivative_def) lemma islimpt_closure_open: fixes s :: "'a::perfect_space set" assumes "open s" and t: "t = closure s" "x ∈ t" shows "x islimpt t" proof cases assume "x ∈ s" { fix T assume "x ∈ T" "open T" then have "open (s ∩ T)" using ‹open s› by auto then have "s ∩ T ≠ {x}" using not_open_singleton[of x] by auto with ‹x ∈ T› ‹x ∈ s› have "∃y∈t. y ∈ T ∧ y ≠ x" using closure_subset[of s] by (auto simp: t) } then show ?thesis by (auto intro!: islimptI) next assume "x ∉ s" with t show ?thesis unfolding t closure_def by (auto intro: islimpt_subset) qed lemma vector_derivative_unique_within_closed_interval: assumes ab: "a < b" "x ∈ cbox a b" assumes D: "(f has_vector_derivative f') (at x within cbox a b)" "(f has_vector_derivative f'') (at x within cbox a b)" shows "f' = f''" using ab by (intro vector_derivative_unique_within[OF _ D]) (auto simp: trivial_limit_within intro!: islimpt_closure_open[where s="{a <..< b}"]) lemma vector_derivative_at: "(f has_vector_derivative f') (at x) ⟹ vector_derivative f (at x) = f'" by (intro vector_derivative_within at_neq_bot) lemma has_vector_derivative_id_at [simp]: "vector_derivative (λx. x) (at a) = 1" by (simp add: vector_derivative_at) lemma vector_derivative_minus_at [simp]: "f differentiable at a ⟹ vector_derivative (λx. - f x) (at a) = - vector_derivative f (at a)" by (simp add: vector_derivative_at has_vector_derivative_minus vector_derivative_works [symmetric]) lemma vector_derivative_add_at [simp]: "⟦f differentiable at a; g differentiable at a⟧ ⟹ vector_derivative (λx. f x + g x) (at a) = vector_derivative f (at a) + vector_derivative g (at a)" by (simp add: vector_derivative_at has_vector_derivative_add vector_derivative_works [symmetric]) lemma vector_derivative_diff_at [simp]: "⟦f differentiable at a; g differentiable at a⟧ ⟹ vector_derivative (λx. f x - g x) (at a) = vector_derivative f (at a) - vector_derivative g (at a)" by (simp add: vector_derivative_at has_vector_derivative_diff vector_derivative_works [symmetric]) lemma vector_derivative_mult_at [simp]: fixes f g :: "real ⇒ 'a :: real_normed_algebra" shows "⟦f differentiable at a; g differentiable at a⟧ ⟹ vector_derivative (λx. f x * g x) (at a) = f a * vector_derivative g (at a) + vector_derivative f (at a) * g a" by (simp add: vector_derivative_at has_vector_derivative_mult vector_derivative_works [symmetric]) lemma vector_derivative_scaleR_at [simp]: "⟦f differentiable at a; g differentiable at a⟧ ⟹ vector_derivative (λx. f x *⇩_{R}g x) (at a) = f a *⇩_{R}vector_derivative g (at a) + vector_derivative f (at a) *⇩_{R}g a" apply (rule vector_derivative_at) apply (rule has_vector_derivative_scaleR) apply (auto simp: vector_derivative_works has_vector_derivative_def has_field_derivative_def mult_commute_abs) done lemma vector_derivative_within_closed_interval: assumes ab: "a < b" "x ∈ cbox a b" assumes f: "(f has_vector_derivative f') (at x within cbox a b)" shows "vector_derivative f (at x within cbox a b) = f'" by (intro vector_derivative_unique_within_closed_interval[OF ab _ f] vector_derivative_works[THEN iffD1] differentiableI_vector) fact 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)" by (auto simp: has_vector_derivative_def intro: has_derivative_within_subset) 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) lemma has_vector_derivative_weaken: fixes x D and f g s t assumes f: "(f has_vector_derivative D) (at x within t)" and "x ∈ s" "s ⊆ t" and "⋀x. x ∈ s ⟹ f x = g x" shows "(g has_vector_derivative D) (at x within s)" proof - have "(f has_vector_derivative D) (at x within s) ⟷ (g has_vector_derivative D) (at x within s)" unfolding has_vector_derivative_def has_derivative_iff_norm using assms by (intro conj_cong Lim_cong_within refl) auto then show ?thesis using has_vector_derivative_within_subset[OF f ‹s ⊆ t›] by simp qed lemma has_vector_derivative_transform_within: assumes "(f has_vector_derivative f') (at x within s)" and "0 < d" and "x ∈ s" and "⋀x'. ⟦x'∈s; dist x' x < d⟧ ⟹ f x' = g x'" 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_within_open: assumes "(f has_vector_derivative f') (at x)" and "open s" and "x ∈ s" and "⋀y. y∈s ⟹ f y = g y" 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)" and "(g has_vector_derivative g') (at (f x))" shows "((g ∘ f) has_vector_derivative (f' *⇩_{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]]) apply (simp only: o_def real_scaleR_def scaleR_scaleR) done lemma vector_diff_chain_within: assumes "(f has_vector_derivative f') (at x within s)" and "(g has_vector_derivative g') (at (f x) within f ` s)" shows "((g ∘ f) has_vector_derivative (f' *⇩_{R}g')) (at x within s)" using assms(2) unfolding has_vector_derivative_def apply - apply (drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]]) apply (simp only: o_def real_scaleR_def scaleR_scaleR) done lemma vector_derivative_const_at [simp]: "vector_derivative (λx. c) (at a) = 0" by (simp add: vector_derivative_at) lemma vector_derivative_at_within_ivl: "(f has_vector_derivative f') (at x) ⟹ a ≤ x ⟹ x ≤ b ⟹ a<b ⟹ vector_derivative f (at x within {a..b}) = f'" using has_vector_derivative_at_within vector_derivative_within_closed_interval by fastforce lemma vector_derivative_chain_at: assumes "f differentiable at x" "(g differentiable at (f x))" shows "vector_derivative (g ∘ f) (at x) = vector_derivative f (at x) *⇩_{R}vector_derivative g (at (f x))" by (metis vector_diff_chain_at vector_derivative_at vector_derivative_works assms) subsection ‹Relation between convexity and derivative› (* TODO: Generalise to real vector spaces? *) lemma convex_on_imp_above_tangent: assumes convex: "convex_on A f" and connected: "connected A" assumes c: "c ∈ interior A" and x : "x ∈ A" assumes deriv: "(f has_field_derivative f') (at c within A)" shows "f x - f c ≥ f' * (x - c)" proof (cases x c rule: linorder_cases) assume xc: "x > c" let ?A' = "interior A ∩ {c<..}" from c have "c ∈ interior A ∩ closure {c<..}" by auto also have "… ⊆ closure (interior A ∩ {c<..})" by (intro open_inter_closure_subset) auto finally have "at c within ?A' ≠ bot" by (subst at_within_eq_bot_iff) auto moreover from deriv have "((λy. (f y - f c) / (y - c)) ⤏ f') (at c within ?A')" unfolding DERIV_within_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le) moreover from eventually_at_right_real[OF xc] have "eventually (λy. (f y - f c) / (y - c) ≤ (f x - f c) / (x - c)) (at_right c)" proof eventually_elim fix y assume y: "y ∈ {c<..<x}" with convex connected x c have "f y ≤ (f x - f c) / (x - c) * (y - c) + f c" using interior_subset[of A] by (intro convex_onD_Icc' convex_on_subset[OF convex] connected_contains_Icc) auto hence "f y - f c ≤ (f x - f c) / (x - c) * (y - c)" by simp thus "(f y - f c) / (y - c) ≤ (f x - f c) / (x - c)" using y xc by (simp add: divide_simps) qed hence "eventually (λy. (f y - f c) / (y - c) ≤ (f x - f c) / (x - c)) (at c within ?A')" by (blast intro: filter_leD at_le) ultimately have "f' ≤ (f x - f c) / (x - c)" by (rule tendsto_ge_const) thus ?thesis using xc by (simp add: field_simps) next assume xc: "x < c" let ?A' = "interior A ∩ {..<c}" from c have "c ∈ interior A ∩ closure {..<c}" by auto also have "… ⊆ closure (interior A ∩ {..<c})" by (intro open_inter_closure_subset) auto finally have "at c within ?A' ≠ bot" by (subst at_within_eq_bot_iff) auto moreover from deriv have "((λy. (f y - f c) / (y - c)) ⤏ f') (at c within ?A')" unfolding DERIV_within_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le) moreover from eventually_at_left_real[OF xc] have "eventually (λy. (f y - f c) / (y - c) ≥ (f x - f c) / (x - c)) (at_left c)" proof eventually_elim fix y assume y: "y ∈ {x<..<c}" with convex connected x c have "f y ≤ (f x - f c) / (c - x) * (c - y) + f c" using interior_subset[of A] by (intro convex_onD_Icc'' convex_on_subset[OF convex] connected_contains_Icc) auto hence "f y - f c ≤ (f x - f c) * ((c - y) / (c - x))" by simp also have "(c - y) / (c - x) = (y - c) / (x - c)" using y xc by (simp add: field_simps) finally show "(f y - f c) / (y - c) ≥ (f x - f c) / (x - c)" using y xc by (simp add: divide_simps) qed hence "eventually (λy. (f y - f c) / (y - c) ≥ (f x - f c) / (x - c)) (at c within ?A')" by (blast intro: filter_leD at_le) ultimately have "f' ≥ (f x - f c) / (x - c)" by (rule tendsto_le_const) thus ?thesis using xc by (simp add: field_simps) qed simp_all subsection ‹Partial derivatives› lemma eventually_at_Pair_within_TimesI1: fixes x::"'a::metric_space" assumes "∀⇩_{F}x' in at x within X. P x'" assumes "P x" shows "∀⇩_{F}(x', y') in at (x, y) within X × Y. P x'" proof - from assms[unfolded eventually_at_topological] obtain S where S: "open S" "x ∈ S" "⋀x'. x' ∈ X ⟹ x' ∈ S ⟹ P x'" by metis show "∀⇩_{F}(x', y') in at (x, y) within X × Y. P x'" unfolding eventually_at_topological by (auto intro!: exI[where x="S × UNIV"] S open_Times) qed lemma eventually_at_Pair_within_TimesI2: fixes x::"'a::metric_space" assumes "∀⇩_{F}y' in at y within Y. P y'" assumes "P y" shows "∀⇩_{F}(x', y') in at (x, y) within X × Y. P y'" proof - from assms[unfolded eventually_at_topological] obtain S where S: "open S" "y ∈ S" "⋀y'. y' ∈ Y ⟹ y' ∈ S ⟹ P y'" by metis show "∀⇩_{F}(x', y') in at (x, y) within X × Y. P y'" unfolding eventually_at_topological by (auto intro!: exI[where x="UNIV × S"] S open_Times) qed lemma has_derivative_partialsI: assumes fx: "⋀x y. x ∈ X ⟹ y ∈ Y ⟹ ((λx. f x y) has_derivative blinfun_apply (fx x y)) (at x within X)" assumes fy: "⋀x y. x ∈ X ⟹ y ∈ Y ⟹ ((λy. f x y) has_derivative blinfun_apply (fy x y)) (at y within Y)" assumes fx_cont: "continuous_on (X × Y) (λ(x, y). fx x y)" assumes fy_cont: "continuous_on (X × Y) (λ(x, y). fy x y)" assumes "x ∈ X" "y ∈ Y" assumes "convex X" "convex Y" shows "((λ(x, y). f x y) has_derivative (λ(tx, ty). fx x y tx + fy x y ty)) (at (x, y) within X × Y)" proof (safe intro!: has_derivativeI tendstoI, goal_cases) case (2 e') def e≡"e' / 9" have "e > 0" using ‹e' > 0› by (simp add: e_def) have "(x, y) ∈ X × Y" using assms by auto from fy_cont[unfolded continuous_on_eq_continuous_within, rule_format, OF this, unfolded continuous_within, THEN tendstoD, OF ‹e > 0›] have "∀⇩_{F}(x', y') in at (x, y) within X × Y. dist (fy x' y') (fy x y) < e" by (auto simp: split_beta') from this[unfolded eventually_at] obtain d' where "d' > 0" "⋀x' y'. x' ∈ X ⟹ y' ∈ Y ⟹ (x', y') ≠ (x, y) ⟹ dist (x', y') (x, y) < d' ⟹ dist (fy x' y') (fy x y) < e" by auto then have d': "x' ∈ X ⟹ y' ∈ Y ⟹ dist (x', y') (x, y) < d' ⟹ dist (fy x' y') (fy x y) < e" for x' y' using ‹0 < e› by (cases "(x', y') = (x, y)") auto def d ≡ "d' / sqrt 2" have "d > 0" using ‹0 < d'› by (simp add: d_def) have d: "x' ∈ X ⟹ y' ∈ Y ⟹ dist x' x < d ⟹ dist y' y < d ⟹ dist (fy x' y') (fy x y) < e" for x' y' by (auto simp: dist_prod_def d_def intro!: d' real_sqrt_sum_squares_less) let ?S = "ball y d ∩ Y" have "convex ?S" by (auto intro!: convex_Int ‹convex Y›) { fix x'::'a and y'::'b assume x': "x' ∈ X" and y': "y' ∈ Y" assume dx': "dist x' x < d" and dy': "dist y' y < d" have "norm (fy x' y' - fy x' y) ≤ dist (fy x' y') (fy x y) + dist (fy x' y) (fy x y)" by norm also have "dist (fy x' y') (fy x y) < e" by (rule d; fact) also have "dist (fy x' y) (fy x y) < e" by (auto intro!: d simp: dist_prod_def x' ‹d > 0› ‹y ∈ Y› dx') finally have "norm (fy x' y' - fy x' y) < e + e" by arith then have "onorm (blinfun_apply (fy x' y') - blinfun_apply (fy x' y)) < e + e" by (auto simp: norm_blinfun.rep_eq blinfun.diff_left[abs_def] fun_diff_def) } note onorm = this have ev_mem: "∀⇩_{F}(x', y') in at (x, y) within X × Y. (x', y') ∈ X × Y" using ‹x ∈ X› ‹y ∈ Y› by (auto simp: eventually_at intro!: zero_less_one) moreover have ev_dist: "∀⇩_{F}xy in at (x, y) within X × Y. dist xy (x, y) < d" if "d > 0" for d using eventually_at_ball[OF that] by (rule eventually_elim2) (auto simp: dist_commute intro!: eventually_True) note ev_dist[OF ‹0 < d›] ultimately have "∀⇩_{F}(x', y') in at (x, y) within X × Y. norm (f x' y' - f x' y - (fy x' y) (y' - y)) ≤ norm (y' - y) * (e + e)" proof (eventually_elim, safe) fix x' y' assume "x' ∈ X" and y': "y' ∈ Y" assume dist: "dist (x', y') (x, y) < d" then have dx: "dist x' x < d" and dy: "dist y' y < d" unfolding dist_prod_def fst_conv snd_conv atomize_conj by (metis le_less_trans real_sqrt_sum_squares_ge1 real_sqrt_sum_squares_ge2) { fix t::real assume "t ∈ {0 .. 1}" then have "y + t *⇩_{R}(y' - y) ∈ closed_segment y y'" by (auto simp: closed_segment_def algebra_simps intro!: exI[where x=t]) also have "… ⊆ ball y d ∩ Y" using ‹y ∈ Y› ‹0 < d› dy y' by (intro ‹convex ?S›[unfolded convex_contains_segment, rule_format, of y y']) (auto simp: dist_commute) finally have "y + t *⇩_{R}(y' - y) ∈ ?S" . } note seg = this have "∀x∈ball y d ∩ Y. onorm (blinfun_apply (fy x' x) - blinfun_apply (fy x' y)) ≤ e + e" by (safe intro!: onorm less_imp_le ‹x' ∈ X› dx) (auto simp: dist_commute ‹0 < d› ‹y ∈ Y›) with seg has_derivative_within_subset[OF assms(2)[OF ‹x' ∈ X›]] show "norm (f x' y' - f x' y - (fy x' y) (y' - y)) ≤ norm (y' - y) * (e + e)" by (rule differentiable_bound_linearization[where S="?S"]) (auto intro!: ‹0 < d› ‹y ∈ Y›) qed moreover let ?le = "λx'. norm (f x' y - f x y - (fx x y) (x' - x)) ≤ norm (x' - x) * e" from fx[OF ‹x ∈ X› ‹y ∈ Y›, unfolded has_derivative_within, THEN conjunct2, THEN tendstoD, OF ‹0 < e›] have "∀⇩_{F}x' in at x within X. ?le x'" by eventually_elim (auto simp: dist_norm divide_simps blinfun.bilinear_simps field_simps split: split_if_asm) then have "∀⇩_{F}(x', y') in at (x, y) within X × Y. ?le x'" by (rule eventually_at_Pair_within_TimesI1) (simp add: blinfun.bilinear_simps) moreover have "∀⇩_{F}(x', y') in at (x, y) within X × Y. norm ((x', y') - (x, y)) ≠ 0" unfolding norm_eq_zero right_minus_eq by (auto simp: eventually_at intro!: zero_less_one) moreover from fy_cont[unfolded continuous_on_eq_continuous_within, rule_format, OF SigmaI[OF ‹x ∈ X› ‹y ∈ Y›], unfolded continuous_within, THEN tendstoD, OF ‹0 < e›] have "∀⇩_{F}x' in at x within X. norm (fy x' y - fy x y) < e" unfolding eventually_at using ‹y ∈ Y› by (auto simp: dist_prod_def dist_norm) then have "∀⇩_{F}(x', y') in at (x, y) within X × Y. norm (fy x' y - fy x y) < e" by (rule eventually_at_Pair_within_TimesI1) (simp add: blinfun.bilinear_simps ‹0 < e›) ultimately have "∀⇩_{F}(x', y') in at (x, y) within X × Y. norm ((f x' y' - f x y - (fx x y (x' - x) + fy x y (y' - y))) /⇩_{R}norm ((x', y') - (x, y))) < e'" apply eventually_elim proof safe fix x' y' have "norm (f x' y' - f x y - (fx x y (x' - x) + fy x y (y' - y))) ≤ norm (f x' y' - f x' y - fy x' y (y' - y)) + norm (fy x y (y' - y) - fy x' y (y' - y)) + norm (f x' y - f x y - fx x y (x' - x))" by norm also assume nz: "norm ((x', y') - (x, y)) ≠ 0" and nfy: "norm (fy x' y - fy x y) < e" assume "norm (f x' y' - f x' y - blinfun_apply (fy x' y) (y' - y)) ≤ norm (y' - y) * (e + e)" also assume "norm (f x' y - f x y - blinfun_apply (fx x y) (x' - x)) ≤ norm (x' - x) * e" also have "norm ((fy x y) (y' - y) - (fy x' y) (y' - y)) ≤ norm ((fy x y) - (fy x' y)) * norm (y' - y)" by (auto simp: blinfun.bilinear_simps[symmetric] intro!: norm_blinfun) also have "… ≤ (e + e) * norm (y' - y)" using ‹e > 0› nfy by (auto simp: norm_minus_commute intro!: mult_right_mono) also have "norm (x' - x) * e ≤ norm (x' - x) * (e + e)" using ‹0 < e› by simp also have "norm (y' - y) * (e + e) + (e + e) * norm (y' - y) + norm (x' - x) * (e + e) ≤ (norm (y' - y) + norm (x' - x)) * (4 * e)" using ‹e > 0› by (simp add: algebra_simps) also have "… ≤ 2 * norm ((x', y') - (x, y)) * (4 * e)" using ‹0 < e› real_sqrt_sum_squares_ge1[of "norm (x' - x)" "norm (y' - y)"] real_sqrt_sum_squares_ge2[of "norm (y' - y)" "norm (x' - x)"] by (auto intro!: mult_right_mono simp: norm_prod_def simp del: real_sqrt_sum_squares_ge1 real_sqrt_sum_squares_ge2) also have "… ≤ norm ((x', y') - (x, y)) * (8 * e)" by simp also have "… < norm ((x', y') - (x, y)) * e'" using ‹0 < e'› nz by (auto simp: e_def) finally show "norm ((f x' y' - f x y - (fx x y (x' - x) + fy x y (y' - y))) /⇩_{R}norm ((x', y') - (x, y))) < e'" by (auto simp: divide_simps dist_norm mult.commute) qed then show ?case by eventually_elim (auto simp: dist_norm field_simps) qed (auto intro!: bounded_linear_intros simp: split_beta') end