(* Title: HOL/Analysis/Derivative.thy Author: John Harrison Author: Robert Himmelmann, TU Muenchen (translation from HOL Light); tidied by LCP *) section ‹Derivative› theory Derivative imports Bounded_Linear_Function Line_Segment Convex_Euclidean_Space begin declare bounded_linear_inner_left [intro] declare has_derivative_bounded_linear[dest] subsection ‹Derivatives› 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✐‹tag unimportant› ‹Derivative with composed bilinear function› text ‹More explicit epsilon-delta forms.› proposition 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 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_componentwise_within: "(f has_derivative f') (at a within S) ⟷ (∀i ∈ Basis. ((λx. f x ∙ i) has_derivative (λx. f' x ∙ i)) (at a within S))" apply (simp add: has_derivative_within) apply (subst tendsto_componentwise_iff) apply (simp add: ball_conj_distrib inner_diff_left inner_left_distrib flip: bounded_linear_componentwise_iff) done lemma has_derivative_at_withinI: "(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_right: fixes f :: "real ⇒ real" and y :: "real" shows "(f has_derivative ((*) 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_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 has_field_derivative_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 has_field_derivative_iff cong: Lim_cong_within) qed subsection ‹Differentiability› definition✐‹tag important› 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_onD: "⟦f differentiable_on S; x ∈ S⟧ ⟹ f differentiable (at x within S)" using differentiable_on_def by blast lemma differentiable_at_withinI: "f differentiable (at x) ⟹ f differentiable (at x within s)" unfolding differentiable_def using has_derivative_at_withinI 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✐‹tag unimportant› 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_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 lemma differentiable_on_ident [simp, derivative_intros]: "(λx. x) differentiable_on S" by (simp add: differentiable_at_imp_differentiable_on) lemma differentiable_on_id [simp, derivative_intros]: "id differentiable_on S" by (simp add: id_def) lemma differentiable_on_const [simp, derivative_intros]: "(λz. c) differentiable_on S" by (simp add: differentiable_on_def) lemma differentiable_on_mult [simp, derivative_intros]: fixes f :: "'M::real_normed_vector ⇒ 'a::real_normed_algebra" shows "⟦f differentiable_on S; g differentiable_on S⟧ ⟹ (λz. f z * g z) differentiable_on S" unfolding differentiable_on_def differentiable_def using differentiable_def differentiable_mult by blast lemma differentiable_on_compose: "⟦g differentiable_on S; f differentiable_on (g ` S)⟧ ⟹ (λx. f (g x)) differentiable_on S" by (simp add: differentiable_in_compose differentiable_on_def) lemma bounded_linear_imp_differentiable_on: "bounded_linear f ⟹ f differentiable_on S" by (simp add: differentiable_on_def bounded_linear_imp_differentiable) lemma linear_imp_differentiable_on: fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector" shows "linear f ⟹ f differentiable_on S" by (simp add: differentiable_on_def linear_imp_differentiable) lemma differentiable_on_minus [simp, derivative_intros]: "f differentiable_on S ⟹ (λz. -(f z)) differentiable_on S" by (simp add: differentiable_on_def) lemma differentiable_on_add [simp, derivative_intros]: "⟦f differentiable_on S; g differentiable_on S⟧ ⟹ (λz. f z + g z) differentiable_on S" by (simp add: differentiable_on_def) lemma differentiable_on_diff [simp, derivative_intros]: "⟦f differentiable_on S; g differentiable_on S⟧ ⟹ (λz. f z - g z) differentiable_on S" by (simp add: differentiable_on_def) lemma differentiable_on_inverse [simp, derivative_intros]: fixes f :: "'a :: real_normed_vector ⇒ 'b :: real_normed_field" shows "f differentiable_on S ⟹ (⋀x. x ∈ S ⟹ f x ≠ 0) ⟹ (λx. inverse (f x)) differentiable_on S" by (simp add: differentiable_on_def) lemma differentiable_on_scaleR [derivative_intros, simp]: "⟦f differentiable_on S; g differentiable_on S⟧ ⟹ (λx. f x *⇩_{R}g x) differentiable_on S" unfolding differentiable_on_def by (blast intro: differentiable_scaleR) lemma has_derivative_sqnorm_at [derivative_intros, simp]: "((λx. (norm x)⇧^{2}) has_derivative (λx. 2 *⇩_{R}(a ∙ x))) (at a)" using bounded_bilinear.FDERIV [of "(∙)" id id a _ id id] by (auto simp: inner_commute dot_square_norm bounded_bilinear_inner) lemma differentiable_sqnorm_at [derivative_intros, simp]: fixes a :: "'a :: {real_normed_vector,real_inner}" shows "(λx. (norm x)⇧^{2}) differentiable (at a)" by (force simp add: differentiable_def intro: has_derivative_sqnorm_at) lemma differentiable_on_sqnorm [derivative_intros, simp]: fixes S :: "'a :: {real_normed_vector,real_inner} set" shows "(λx. (norm x)⇧^{2}) differentiable_on S" by (simp add: differentiable_at_imp_differentiable_on) lemma differentiable_norm_at [derivative_intros, simp]: fixes a :: "'a :: {real_normed_vector,real_inner}" shows "a ≠ 0 ⟹ norm differentiable (at a)" using differentiableI has_derivative_norm by blast lemma differentiable_on_norm [derivative_intros, simp]: fixes S :: "'a :: {real_normed_vector,real_inner} set" shows "0 ∉ S ⟹ norm differentiable_on S" by (metis differentiable_at_imp_differentiable_on differentiable_norm_at) subsection ‹Frechet derivative and Jacobian matrix› definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)" proposition 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) lemma frechet_derivative_const [simp]: "frechet_derivative (λx. c) (at a) = (λx. 0)" using differentiable_const frechet_derivative_works has_derivative_const has_derivative_unique by blast lemma frechet_derivative_id [simp]: "frechet_derivative id (at a) = id" using differentiable_def frechet_derivative_works has_derivative_id has_derivative_unique by blast lemma frechet_derivative_ident [simp]: "frechet_derivative (λx. x) (at a) = (λx. x)" by (metis eq_id_iff frechet_derivative_id) subsection ‹Differentiability implies continuity› proposition 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 lemma has_derivative_continuous_on: "(⋀x. x ∈ s ⟹ (f has_derivative f' x) (at x within s)) ⟹ continuous_on s f" by (auto intro!: differentiable_imp_continuous_on differentiableI simp: differentiable_on_def) 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› proposition 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)" by (meson diff_chain_within has_derivative_at_withinI) lemma has_vector_derivative_shift: "(f has_vector_derivative D x) (at x) ⟹ ((+) d ∘ f has_vector_derivative D x) (at x)" using diff_chain_at [OF _ shift_has_derivative_id] by (simp add: has_derivative_iff_Ex has_vector_derivative_def) lemma has_vector_derivative_within_open: "a ∈ S ⟹ open S ⟹ (f has_vector_derivative f') (at a within S) ⟷ (f has_vector_derivative f') (at a)" by (simp only: at_within_interior interior_open) lemma field_vector_diff_chain_within: assumes Df: "(f has_vector_derivative f') (at x within S)" and Dg: "(g has_field_derivative g') (at (f x) within f ` S)" shows "((g ∘ f) has_vector_derivative (f' * g')) (at x within S)" using diff_chain_within[OF Df[unfolded has_vector_derivative_def] Dg [unfolded has_field_derivative_def]] by (auto simp: o_def mult.commute has_vector_derivative_def) lemma vector_derivative_diff_chain_within: assumes Df: "(f has_vector_derivative f') (at x within S)" and Dg: "(g has_derivative g') (at (f x) within f`S)" shows "((g ∘ f) has_vector_derivative (g' f')) (at x within S)" using diff_chain_within[OF Df[unfolded has_vector_derivative_def] Dg] linear.scaleR[OF has_derivative_linear[OF Dg]] unfolding has_vector_derivative_def o_def by (auto simp: o_def mult.commute has_vector_derivative_def) subsection✐‹tag unimportant› ‹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✐‹tag important› ‹ The general result is a bit messy because we need approachability of the limit point from any direction. But OK for nontrivial intervals etc. › proposition frechet_derivative_unique_within: fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector" assumes 1: "(f has_derivative f') (at x within S)" and 2: "(f has_derivative f'') (at x within S)" and S: "⋀i e. ⟦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 (intro allI impI) 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" by (rule_tac x="x + d *⇩_{R}(SOME i. i ∈ Basis)" in bexI) (auto simp: dist_norm SOME_Basis nonzero_Basis) qed then have *: "netlimit (at x within S) = x" by (simp add: Lim_ident_at trivial_limit_within) show ?thesis proof (rule linear_eq_stdbasis) show "linear f'" "linear f''" unfolding linear_conv_bounded_linear using as by auto next fix i :: 'a assume i: "i ∈ Basis" define e where "e = norm (f' i - f'' i)" show "f' i = f'' i" proof (rule ccontr) assume "f' i ≠ f'' i" then have "e > 0" unfolding e_def by auto obtain d where d: "0 < d" "(⋀y. y∈S ⟶ 0 < dist y x ∧ dist y x < d ⟶ dist ((f y - f x - f' (y - x)) /⇩_{R}norm (y - x) - (f y - f x - f'' (y - x)) /⇩_{R}norm (y - 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 qed proposition frechet_derivative_unique_within_closed_interval: fixes f::"'a::euclidean_space ⇒ 'b::real_normed_vector" assumes ab: "⋀i. i∈Basis ⟹ a∙i < b∙i" and x: "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''" proof (rule frechet_derivative_unique_within) 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 with ab[of i] ‹e>0› x i show ?thesis by (rule_tac x="(min (b∙i - a∙i) e) / 2" in exI) (auto simp add: mem_box field_simps inner_simps inner_Basis) next case False moreover have "a ∙ i < x ∙ i" using False i mem_box(2) x by force 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 ‹a ∙ i < x ∙ i› 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" by (simp add: ‹0 < e› ‹a ∙ i < x ∙ i› less_eq_real_def) then have "x ∙ i * 2 ≤ b ∙ i * 2 + min (x ∙ i - a ∙ i) e" using i mem_box(2) x by force ultimately show ?thesis using ab[of i] ‹e>0› x i by (rule_tac x="- (min (x∙i - a∙i) e) / 2" in exI) (auto simp add: mem_box field_simps inner_simps inner_Basis) qed qed (use assms in auto) lemma frechet_derivative_unique_within_open_interval: fixes f::"'a::euclidean_space ⇒ 'b::real_normed_vector" assumes x: "x ∈ box a b" and f: "(f has_derivative f' ) (at x within box a b)" "(f has_derivative f'') (at x within box a b)" shows "f' = f''" by (metis at_within_open assms has_derivative_unique open_box) lemma frechet_derivative_at: "(f has_derivative f') (at x) ⟹ f' = frechet_derivative f (at x)" using differentiable_def frechet_derivative_works has_derivative_unique by blast lemma frechet_derivative_compose: "frechet_derivative (f o g) (at x) = frechet_derivative (f) (at (g x)) o frechet_derivative g (at x)" if "g differentiable at x" "f differentiable at (g x)" by (metis diff_chain_at frechet_derivative_at frechet_derivative_works that) lemma frechet_derivative_within_cbox: fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector" assumes "⋀i. 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) lemma frechet_derivative_transform_within_open: "frechet_derivative f (at x) = frechet_derivative g (at x)" if "f differentiable at x" "open X" "x ∈ X" "⋀x. x ∈ X ⟹ f x = g x" by (meson frechet_derivative_at frechet_derivative_works has_derivative_transform_within_open that) subsection ‹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") case False 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 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: 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 subsection ‹One-dimensional mean value theorem› lemma mvt_simple: fixes f :: "real ⇒ real" assumes "a < b" and derf: "⋀x. ⟦a ≤ x; x ≤ 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 derf unfolding differentiable_on_def differentiable_def by force then show "continuous_on {a..b} f" by (rule differentiable_imp_continuous_on) show "(f has_derivative f' x) (at x)" if "a < x" "x < b" for x by (metis at_within_Icc_at derf leI order.asym that) qed (use assms in auto) lemma mvt_very_simple: fixes f :: "real ⇒ real" assumes "a ≤ b" and derf: "⋀x. ⟦a ≤ x; x ≤ 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 by auto case True then show ?thesis by force next case False then show ?thesis using mvt_simple[OF _ derf] by (metis ‹a ≤ b› atLeastAtMost_iff dual_order.order_iff_strict greaterThanLessThan_iff) 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 contf: "continuous_on {a..b} f" and derf: "⋀x. ⟦a < x; x < 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 [OF ‹a < b›, where f = "λx. (f b - f a) ∙ f x"]) apply (intro continuous_intros contf) using derf apply (auto 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 ‹a < b› by (rule_tac x="(a + b) /2" in bexI) auto qed qed subsection ‹More general bound theorems› proposition 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" define e2 where "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" define A where "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) define y where "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" . have "y = b" proof (cases "a = y") case True 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›]) define d' where "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 then show "y = b" using ‹d > 0› ‹y < b› by (simp add: d'_def) next case False with ‹a ≤ y› have "a < y" by simp show "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 have "φ' y * ¦x1 - y¦ ≤ φ x1 - φ y + e2 * ¦x1 - y¦" using elim 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›]) define d' where "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'}" show "?le x1" proof (cases "x1 < y") case True then show ?thesis using ‹y ∈ A› local.leI x1 by auto next case False 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 show "?le x1" using False by (auto simp: algebra_simps) qed 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: if_split_asm) qed qed with le_y have "norm (f b - f a) ≤ φ b - φ a + e * (b - a + 1)" by (simp add: algebra_simps) } note * = this show ?thesis proof (rule field_le_epsilon) fix e::real assume "e > 0" then show "norm (f b - f a) ≤ φ b - φ a + e" using *[of "e / (b - a + 1)"] ‹a < b› by simp qed qed lemma differentiable_bound: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "convex S" and derf: "⋀x. x∈S ⟹ (f has_derivative f' x) (at x within S)" and B: "⋀x. 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 *: "x + u *⇩_{R}(y - x) ∈ S" if "u ∈ {0..1}" for u proof - have "u *⇩_{R}y = u *⇩_{R}(y - x) + u *⇩_{R}x" by (simp add: scale_right_diff_distrib) then show "x + u *⇩_{R}(y - x) ∈ S" using that ‹convex S› x y by (simp add: convex_alt) (metis pth_b(2) pth_c(1) scaleR_collapse) qed have "⋀z. z ∈ (λu. x + u *⇩_{R}(y - x)) ` {0..1} ⟹ (f has_derivative f' z) (at z within (λu. x + u *⇩_{R}(y - x)) ` {0..1})" by (auto intro: * has_derivative_subset [OF derf]) then have "continuous_on (?p ` {0..1}) f" unfolding continuous_on_eq_continuous_within by (meson has_derivative_continuous) with * have 1: "continuous_on {0 .. 1} (f ∘ ?p)" by (intro continuous_intros)+ { fix u::real assume u: "u ∈{0 <..< 1}" let ?u = "?p u" interpret linear "(f' ?u)" using u by (auto intro!: has_derivative_linear derf *) have "(f ∘ ?p has_derivative (f' ?u) ∘ (λu. 0 + u *⇩_{R}(y - x))) (at u within box 0 1)" by (intro derivative_intros has_derivative_subset [OF derf]) (use u * in auto) hence "((f ∘ ?p) has_vector_derivative f' ?u (y - x)) (at u)" by (simp add: at_within_open[OF u open_greaterThanLessThan] scaleR has_vector_derivative_def o_def) } note 2 = this have 3: "continuous_on {0..1} ?φ" by (rule continuous_intros)+ have 4: "(?φ has_vector_derivative B * norm (x - y)) (at u)" for u by (auto simp: has_vector_derivative_def intro!: derivative_eq_intros) { 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 derf *) have "norm (f' ?u (y - x)) ≤ onorm (f' ?u) * norm (y - x)" by (rule onorm) (rule bounded_linear) 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 field_differentiable_bound: fixes S :: "'a::real_normed_field set" assumes cvs: "convex S" and df: "⋀z. z ∈ S ⟹ (f has_field_derivative f' z) (at z within S)" and dn: "⋀z. z ∈ S ⟹ norm (f' z) ≤ B" and "x ∈ S" "y ∈ S" shows "norm(f x - f y) ≤ B * norm(x - y)" proof (rule differentiable_bound [OF cvs]) show "⋀x. x ∈ S ⟹ (f has_derivative (*) (f' x)) (at x within S)" by (simp add: df has_field_derivative_imp_has_derivative) show "⋀x. x ∈ S ⟹ onorm ((*) (f' x)) ≤ B" by (metis (no_types, opaque_lifting) dn norm_mult onorm_le order.refl order_trans) qed (use assms in auto) 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. 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 = (+) 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 (force simp: ac_simps) qed lemma differentiable_bound_linearization: fixes f::"'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes S: "⋀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. 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 - define g where [abs_def]: "g x = f x - f' x0 x" for 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 "∀x∈{0..1}. onorm (λi. f' (a + x *⇩_{R}(b - a)) i - f' x0 i) ≤ B" using assms by (auto simp: fun_diff_def) with differentiable_bound_segment[OF S g] ‹x0 ∈ S› show ?thesis by (simp add: g_def field_simps linear_diff[OF has_derivative_linear[OF f']]) qed lemma vector_differentiable_bound_linearization: fixes f::"real ⇒ 'b::real_normed_vector" assumes f': "⋀x. x ∈ S ⟹ (f has_vector_derivative f' x) (at x within S)" assumes "closed_segment a b ⊆ S" assumes B: "⋀x. x ∈ S ⟹ norm (f' x - f' x0) ≤ B" assumes "x0 ∈ S" shows "norm (f b - f a - (b - a) *⇩_{R}f' x0) ≤ norm (b - a) * B" using assms by (intro differentiable_bound_linearization[of a b S f "λx h. h *⇩_{R}f' x" x0 B]) (force simp: closed_segment_real_eq has_vector_derivative_def scaleR_diff_right[symmetric] mult.commute[of B] intro!: onorm_le mult_left_mono)+ 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: "(*) 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_vector_derivative_zero_constant: assumes "convex s" assumes "⋀x. x ∈ s ⟹ (f has_vector_derivative 0) (at x within s)" obtains c where "⋀x. x ∈ s ⟹ f x = c" using has_derivative_zero_constant[of s f] assms by (auto simp: has_vector_derivative_def) 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) 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 derf: "(f has_derivative f') (at (g y))" and ling': "bounded_linear g'" and "g' ∘ f' = id" and contg: "continuous (at y) g" and "open T" and "y ∈ T" and fg: "⋀z. 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 (intro allI impI) fix e :: real assume "e > 0" with C(1) have *: "e / C > 0" by auto obtain d0 where "0 < d0" and d0: "⋀u. norm (u - g y) < d0 ⟹ norm (f u - f (g y) - f' (u - g y)) ≤ e / C * norm (u - g y)" using derf * unfolding has_derivative_at_alt by blast obtain d1 where "0 < d1" and d1: "⋀x. ⟦0 < dist x y; dist x y < d1⟧ ⟹ dist (g x) (g y) < d0" using contg ‹0 < d0› unfolding continuous_at Lim_at by blast obtain d2 where "0 < d2" and d2: "⋀u. dist u y < d2 ⟹ u ∈ T" using ‹open T› ‹y ∈ T› unfolding open_dist by blast obtain d where d: "0 < d" "d < d1" "d < d2" using field_lbound_gt_zero[OF ‹0 < d1› ‹0 < d2›] by blast show "∃d>0. ∀z. norm (z - y) < d ⟶ norm (g z - g y - g' (z - y)) ≤ e * norm (g z - g y)" proof (intro exI allI impI conjI) 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] fg[OF ‹z∈T›] by (simp add: norm_minus_commute) 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" proof - have "norm (g z - g y) < d0" by (metis as cancel_comm_monoid_add_class.diff_cancel d(2) ‹0 < d0› d1 diff_gt_0_iff_gt diff_strict_mono dist_norm dist_self zero_less_dist_iff) then show ?thesis by (metis C(1) ‹y ∈ T› d0 fg mult_le_cancel_iff1) qed 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 (use d in auto) qed have *: "(0::real) < 1 / 2" by auto obtain d where "0 < d" and 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 define B where "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)" by (rule add_left_mono) (use d z in auto) also have "… ≤ norm (z - y) * C + 1 / 2 * norm (g z - g y)" by (rule add_right_mono) (use C in auto) 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 proof (intro conjI assms allI impI) fix e :: real assume "e > 0" then have *: "e / B > 0" by (metis ‹B > 0› divide_pos_pos) obtain d' where "0 < d'" and 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 field_lbound_gt_zero[OF ‹0 < d› ‹0 < d'›] by blast show "∃d>0. ∀ya. norm (ya - y) < d ⟶ norm (g ya - g y - g' (ya - y)) ≤ e * norm (ya - y)" proof (intro exI allI impI conjI) 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] k as ‹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 (use k in auto) qed qed text✐‹tag unimportant›‹Inverse function theorem for complex derivatives› lemma has_field_derivative_inverse_basic: shows "DERIV f (g y) :> f' ⟹ f' ≠ 0 ⟹ continuous (at y) g ⟹ open t ⟹ y ∈ t ⟹ (⋀z. z ∈ t ⟹ f (g z) = z) ⟹ DERIV g y :> inverse (f')" unfolding has_field_derivative_def by (rule has_derivative_inverse_basic) (auto simp: bounded_linear_mult_right) 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. y ∈ T ⟹ f (g y) = y" shows "(g has_derivative g') (at (f x))" by (rule has_derivative_inverse_basic) (use assms in auto) 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 fS: "open (f ` S)" and A: "continuous_on S f" "continuous_on (f ` S) g" "⋀x. x ∈ S ⟹ g (f x) = x" "x ∈ S" and B: "(f has_derivative f') (at x)" "bounded_linear g'" "g' ∘ f' = id" shows "(g has_derivative g') (at (f x))" using A fS continuous_on_eq_continuous_at by (intro has_derivative_inverse_basic_x[OF B _ _ fS]) force+ text ‹Here's the simplest way of not assuming much about g.› proposition has_derivative_inverse: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector" assumes "compact S" and "x ∈ S" and fx: "f x ∈ interior (f ` S)" and "continuous_on S f" and gf: "⋀y. y ∈ S ⟹ g (f y) = y" and B: "(f has_derivative f') (at x)" "bounded_linear g'" "g' ∘ f' = id" shows "(g has_derivative g') (at (f x))" proof - have *: "⋀y. y ∈ interior (f ` S) ⟹ f (g y) = y" by (metis gf image_iff interior_subset subsetCE) show ?thesis using assms * continuous_on_interior continuous_on_inv fx by (intro has_derivative_inverse_basic_x[OF B, where T = "interior (f`S)"]) blast+ qed text ‹Invertible derivative continuous 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.› proposition has_derivative_locally_injective: fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space" assumes "a ∈ S" and "open S" and bling: "bounded_linear g'" and "g' ∘ f' a = id" and derf: "⋀x. x ∈ S ⟹ (f has_derivative f' x) (at x)" and "⋀e. e > 0 ⟹ ∃d>0. ∀x. dist a x < d ⟶ onorm (λv. f' x v - f' a v) < e" obtains r where "r > 0" "ball a r ⊆ S" "inj_on f (ball a r)" 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)" using f'g' by auto then have *: "0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)] by fastforce define k where "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 < d2" "ball a d2 ⊆ S" using ‹0 < d2› ‹ball a d2 ⊆ S› by blast obtain d where d: "0 < d" "d < d1" "d < d2" using field_lbound_gt_zero[OF d1(1) d2(1)] by blast show ?thesis proof show "0 < d" by (fact d) show "ball a d ⊆ S" using ‹d < d2› ‹ball a d2 ⊆ S› by auto show "inj_on f (ball a d)" unfolding inj_on_def proof (intro strip) fix x y assume as: "x ∈ ball a d" "y ∈ ball a d" "f x = f y" define ph where [abs_def]: "ph w = w - g' (f w - f x)" for w have ph':"ph = g' ∘ (λw. f' a w - (f w - f x))" unfolding ph_def o_def by (simp add: diff f'g') have "norm (ph x - ph y) ≤ (1 / 2) * norm (x - y)" proof (rule differentiable_bound[OF convex_ball _ _ as(1-2)]) 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 have blin: "bounded_linear (f' a)" using ‹a ∈ S› derf by blast show "(ph has_derivative (λv. v - g' (f' u v))) (at u within ball a d)" unfolding ph' * comp_def by