(* Title: HOL/Deriv.thy Author: Jacques D. Fleuriot, University of Cambridge, 1998 Author: Brian Huffman Author: Lawrence C Paulson, 2004 Author: Benjamin Porter, 2005 *) section ‹Differentiation› theory Deriv imports Limits begin subsection ‹Frechet derivative› definition has_derivative :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒ ('a ⇒ 'b) ⇒ 'a filter ⇒ bool" (infix "(has'_derivative)" 50) where "(f has_derivative f') F ⟷ bounded_linear f' ∧ ((λy. ((f y - f (Lim F (λx. x))) - f' (y - Lim F (λx. x))) /⇩_{R}norm (y - Lim F (λx. x))) ⤏ 0) F" text ‹ Usually the filter \<^term>‹F› is \<^term>‹at x within s›. \<^term>‹(f has_derivative D) (at x within s)› means: \<^term>‹D› is the derivative of function \<^term>‹f› at point \<^term>‹x› within the set \<^term>‹s›. Where \<^term>‹s› is used to express left or right sided derivatives. In most cases \<^term>‹s› is either a variable or \<^term>‹UNIV›. › 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 tendsto_iff by (subst eventually_Lim_ident_at) (auto simp add: field_simps) lemma has_derivative_eq_rhs: "(f has_derivative f') F ⟹ f' = g' ⟹ (f has_derivative g') F" by simp definition has_field_derivative :: "('a::real_normed_field ⇒ 'a) ⇒ 'a ⇒ 'a filter ⇒ bool" (infix "(has'_field'_derivative)" 50) where "(f has_field_derivative D) F ⟷ (f has_derivative (*) D) F" lemma DERIV_cong: "(f has_field_derivative X) F ⟹ X = Y ⟹ (f has_field_derivative Y) F" by simp definition has_vector_derivative :: "(real ⇒ 'b::real_normed_vector) ⇒ 'b ⇒ real filter ⇒ bool" (infix "has'_vector'_derivative" 50) where "(f has_vector_derivative f') net ⟷ (f has_derivative (λx. x *⇩_{R}f')) net" lemma has_vector_derivative_eq_rhs: "(f has_vector_derivative X) F ⟹ X = Y ⟹ (f has_vector_derivative Y) F" by simp named_theorems derivative_intros "structural introduction rules for derivatives" setup ‹ let val eq_thms = @{thms has_derivative_eq_rhs DERIV_cong has_vector_derivative_eq_rhs} fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms in Global_Theory.add_thms_dynamic (\<^binding>‹derivative_eq_intros›, fn context => Named_Theorems.get (Context.proof_of context) \<^named_theorems>‹derivative_intros› |> map_filter eq_rule) end › text ‹ The following syntax is only used as a legacy syntax. › abbreviation (input) FDERIV :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒ 'a ⇒ ('a ⇒ 'b) ⇒ bool" ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where "FDERIV f x :> f' ≡ (f has_derivative f') (at x)" lemma has_derivative_bounded_linear: "(f has_derivative f') F ⟹ bounded_linear f'" by (simp add: has_derivative_def) lemma has_derivative_linear: "(f has_derivative f') F ⟹ linear f'" using bounded_linear.linear[OF has_derivative_bounded_linear] . lemma has_derivative_ident[derivative_intros, simp]: "((λx. x) has_derivative (λx. x)) F" by (simp add: has_derivative_def) lemma has_derivative_id [derivative_intros, simp]: "(id has_derivative id) (at a)" by (metis eq_id_iff has_derivative_ident) lemma has_derivative_const[derivative_intros, simp]: "((λx. c) has_derivative (λx. 0)) F" by (simp add: has_derivative_def) lemma (in bounded_linear) bounded_linear: "bounded_linear f" .. lemma (in bounded_linear) has_derivative: "(g has_derivative g') F ⟹ ((λx. f (g x)) has_derivative (λx. f (g' x))) F" unfolding has_derivative_def by (auto simp add: bounded_linear_compose [OF bounded_linear] scaleR diff dest: tendsto) lemmas has_derivative_scaleR_right [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_scaleR_right] lemmas has_derivative_scaleR_left [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_scaleR_left] lemmas has_derivative_mult_right [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_mult_right] lemmas has_derivative_mult_left [derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_mult_left] lemmas has_derivative_of_real[derivative_intros, simp] = bounded_linear.has_derivative[OF bounded_linear_of_real] lemma has_derivative_add[simp, derivative_intros]: assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F" shows "((λx. f x + g x) has_derivative (λx. f' x + g' x)) F" unfolding has_derivative_def proof safe let ?x = "Lim F (λx. x)" let ?D = "λf f' y. ((f y - f ?x) - f' (y - ?x)) /⇩_{R}norm (y - ?x)" have "((λx. ?D f f' x + ?D g g' x) ⤏ (0 + 0)) F" using f g by (intro tendsto_add) (auto simp: has_derivative_def) then show "(?D (λx. f x + g x) (λx. f' x + g' x) ⤏ 0) F" by (simp add: field_simps scaleR_add_right scaleR_diff_right) qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear) lemma has_derivative_sum[simp, derivative_intros]: "(⋀i. i ∈ I ⟹ (f i has_derivative f' i) F) ⟹ ((λx. ∑i∈I. f i x) has_derivative (λx. ∑i∈I. f' i x)) F" by (induct I rule: infinite_finite_induct) simp_all lemma has_derivative_minus[simp, derivative_intros]: "(f has_derivative f') F ⟹ ((λx. - f x) has_derivative (λx. - f' x)) F" using has_derivative_scaleR_right[of f f' F "-1"] by simp lemma has_derivative_diff[simp, derivative_intros]: "(f has_derivative f') F ⟹ (g has_derivative g') F ⟹ ((λx. f x - g x) has_derivative (λx. f' x - g' x)) F" by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus) lemma has_derivative_at_within: "(f has_derivative f') (at x within s) ⟷ (bounded_linear f' ∧ ((λy. ((f y - f x) - f' (y - x)) /⇩_{R}norm (y - x)) ⤏ 0) (at x within s))" proof (cases "at x within s = bot") case True then show ?thesis by (metis (no_types, lifting) has_derivative_within tendsto_bot) next case False then show ?thesis by (simp add: Lim_ident_at has_derivative_def) qed lemma has_derivative_iff_norm: "(f has_derivative f') (at x within s) ⟷ bounded_linear f' ∧ ((λy. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ⤏ 0) (at x within s)" using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric] by (simp add: has_derivative_at_within divide_inverse ac_simps) lemma has_derivative_at: "(f has_derivative D) (at x) ⟷ (bounded_linear D ∧ (λh. norm (f (x + h) - f x - D h) / norm h) ─0→ 0)" by (simp add: has_derivative_iff_norm LIM_offset_zero_iff) lemma field_has_derivative_at: fixes x :: "'a::real_normed_field" shows "(f has_derivative (*) D) (at x) ⟷ (λh. (f (x + h) - f x) / h) ─0→ D" (is "?lhs = ?rhs") proof - have "?lhs = (λh. norm (f (x + h) - f x - D * h) / norm h) ─0 → 0" by (simp add: bounded_linear_mult_right has_derivative_at) also have "... = (λy. norm ((f (x + y) - f x - D * y) / y)) ─0→ 0" by (simp cong: LIM_cong flip: nonzero_norm_divide) also have "... = (λy. norm ((f (x + y) - f x) / y - D / y * y)) ─0→ 0" by (simp only: diff_divide_distrib times_divide_eq_left [symmetric]) also have "... = ?rhs" by (simp add: tendsto_norm_zero_iff LIM_zero_iff cong: LIM_cong) finally show ?thesis . qed lemma has_derivative_iff_Ex: "(f has_derivative f') (at x) ⟷ bounded_linear f' ∧ (∃e. (∀h. f (x+h) = f x + f' h + e h) ∧ ((λh. norm (e h) / norm h) ⤏ 0) (at 0))" unfolding has_derivative_at by force lemma has_derivative_at_within_iff_Ex: assumes "x ∈ S" "open S" shows "(f has_derivative f') (at x within S) ⟷ bounded_linear f' ∧ (∃e. (∀h. x+h ∈ S ⟶ f (x+h) = f x + f' h + e h) ∧ ((λh. norm (e h) / norm h) ⤏ 0) (at 0))" (is "?lhs = ?rhs") proof safe show "bounded_linear f'" if "(f has_derivative f') (at x within S)" using has_derivative_bounded_linear that by blast show "∃e. (∀h. x + h ∈ S ⟶ f (x + h) = f x + f' h + e h) ∧ (λh. norm (e h) / norm h) ─0→ 0" if "(f has_derivative f') (at x within S)" by (metis (full_types) assms that has_derivative_iff_Ex at_within_open) show "(f has_derivative f') (at x within S)" if "bounded_linear f'" and eq [rule_format]: "∀h. x + h ∈ S ⟶ f (x + h) = f x + f' h + e h" and 0: "(λh. norm (e (h::'a)::'b) / norm h) ─0→ 0" for e proof - have 1: "f y - f x = f' (y-x) + e (y-x)" if "y ∈ S" for y using eq [of "y-x"] that by simp have 2: "((λy. norm (e (y-x)) / norm (y - x)) ⤏ 0) (at x within S)" by (simp add: "0" assms tendsto_offset_zero_iff) have "((λy. norm (f y - f x - f' (y - x)) / norm (y - x)) ⤏ 0) (at x within S)" by (simp add: Lim_cong_within 1 2) then show ?thesis by (simp add: has_derivative_iff_norm ‹bounded_linear f'›) qed qed lemma has_derivativeI: "bounded_linear f' ⟹ ((λy. ((f y - f x) - f' (y - x)) /⇩_{R}norm (y - x)) ⤏ 0) (at x within s) ⟹ (f has_derivative f') (at x within s)" by (simp add: has_derivative_at_within) lemma has_derivativeI_sandwich: assumes e: "0 < e" and bounded: "bounded_linear f'" and sandwich: "(⋀y. y ∈ s ⟹ y ≠ x ⟹ dist y x < e ⟹ norm ((f y - f x) - f' (y - x)) / norm (y - x) ≤ H y)" and "(H ⤏ 0) (at x within s)" shows "(f has_derivative f') (at x within s)" unfolding has_derivative_iff_norm proof safe show "((λy. norm (f y - f x - f' (y - x)) / norm (y - x)) ⤏ 0) (at x within s)" proof (rule tendsto_sandwich[where f="λx. 0"]) show "(H ⤏ 0) (at x within s)" by fact show "eventually (λn. norm (f n - f x - f' (n - x)) / norm (n - x) ≤ H n) (at x within s)" unfolding eventually_at using e sandwich by auto qed (auto simp: le_divide_eq) qed fact lemma has_derivative_subset: "(f has_derivative f') (at x within s) ⟹ t ⊆ s ⟹ (f has_derivative f') (at x within t)" by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset) lemma has_derivative_within_singleton_iff: "(f has_derivative g) (at x within {x}) ⟷ bounded_linear g" by (auto intro!: has_derivativeI_sandwich[where e=1] has_derivative_bounded_linear) 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 within t)" and "open s" and "x ∈ s" and "⋀x. x∈s ⟹ f x = g x" shows "(g has_derivative f') (at x within t)" using assms unfolding has_derivative_within by (force simp add: intro: Lim_transform_within_open) lemma has_derivative_transform: assumes "x ∈ s" "⋀x. x ∈ s ⟹ g x = f x" assumes "(f has_derivative f') (at x within s)" shows "(g has_derivative f') (at x within s)" using assms by (intro has_derivative_transform_within[OF _ zero_less_one, where g=g]) auto lemma has_derivative_transform_eventually: assumes "(f has_derivative f') (at x within s)" "(∀⇩_{F}x' in at x within s. f x' = g x')" assumes "f x = g x" "x ∈ s" shows "(g has_derivative f') (at x within s)" using assms proof - from assms(2,3) obtain d where "d > 0" "⋀x'. x' ∈ s ⟹ dist x' x < d ⟹ f x' = g x'" by (force simp: eventually_at) from has_derivative_transform_within[OF assms(1) this(1) assms(4) this(2)] show ?thesis . qed lemma has_field_derivative_transform_within: assumes "(f has_field_derivative f') (at a within S)" and "0 < d" and "a ∈ S" and "⋀x. ⟦x ∈ S; dist x a < d⟧ ⟹ f x = g x" shows "(g has_field_derivative f') (at a within S)" using assms unfolding has_field_derivative_def by (metis has_derivative_transform_within) lemma has_field_derivative_transform_within_open: assumes "(f has_field_derivative f') (at a)" and "open S" "a ∈ S" and "⋀x. x ∈ S ⟹ f x = g x" shows "(g has_field_derivative f') (at a)" using assms unfolding has_field_derivative_def by (metis has_derivative_transform_within_open) subsection ‹Continuity› lemma has_derivative_continuous: assumes f: "(f has_derivative f') (at x within s)" shows "continuous (at x within s) f" proof - from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear) note F.tendsto[tendsto_intros] let ?L = "λf. (f ⤏ 0) (at x within s)" have "?L (λy. norm ((f y - f x) - f' (y - x)) / norm (y - x))" using f unfolding has_derivative_iff_norm by blast then have "?L (λy. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m) by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros) also have "?m ⟷ ?L (λy. norm ((f y - f x) - f' (y - x)))" by (intro filterlim_cong) (simp_all add: eventually_at_filter) finally have "?L (λy. (f y - f x) - f' (y - x))" by (rule tendsto_norm_zero_cancel) then have "?L (λy. ((f y - f x) - f' (y - x)) + f' (y - x))" by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero) then have "?L (λy. f y - f x)" by simp from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis by (simp add: continuous_within) qed subsection ‹Composition› lemma tendsto_at_iff_tendsto_nhds_within: "f x = y ⟹ (f ⤏ y) (at x within s) ⟷ (f ⤏ y) (inf (nhds x) (principal s))" unfolding tendsto_def eventually_inf_principal eventually_at_filter by (intro ext all_cong imp_cong) (auto elim!: eventually_mono) lemma has_derivative_in_compose: assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at (f x) within (f`s))" shows "((λx. g (f x)) has_derivative (λx. g' (f' x))) (at x within s)" proof - from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear) from g interpret G: bounded_linear g' by (rule has_derivative_bounded_linear) from F.bounded obtain kF where kF: "⋀x. norm (f' x) ≤ norm x * kF" by fast from G.bounded obtain kG where kG: "⋀x. norm (g' x) ≤ norm x * kG" by fast note G.tendsto[tendsto_intros] let ?L = "λf. (f ⤏ 0) (at x within s)" let ?D = "λf f' x y. (f y - f x) - f' (y - x)" let ?N = "λf f' x y. norm (?D f f' x y) / norm (y - x)" let ?gf = "λx. g (f x)" and ?gf' = "λx. g' (f' x)" define Nf where "Nf = ?N f f' x" define Ng where [abs_def]: "Ng y = ?N g g' (f x) (f y)" for y show ?thesis proof (rule has_derivativeI_sandwich[of 1]) show "bounded_linear (λx. g' (f' x))" using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear) next fix y :: 'a assume neq: "y ≠ x" have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)" by (simp add: G.diff G.add field_simps) also have "… ≤ norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))" by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def) also have "… ≤ Nf y * kG + Ng y * (Nf y + kF)" proof (intro add_mono mult_left_mono) have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))" by simp also have "… ≤ norm (?D f f' x y) + norm (f' (y - x))" by (rule norm_triangle_ineq) also have "… ≤ norm (?D f f' x y) + norm (y - x) * kF" using kF by (intro add_mono) simp finally show "norm (f y - f x) / norm (y - x) ≤ Nf y + kF" by (simp add: neq Nf_def field_simps) qed (use kG in ‹simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps›) finally show "?N ?gf ?gf' x y ≤ Nf y * kG + Ng y * (Nf y + kF)" . next have [tendsto_intros]: "?L Nf" using f unfolding has_derivative_iff_norm Nf_def .. from f have "(f ⤏ f x) (at x within s)" by (blast intro: has_derivative_continuous continuous_within[THEN iffD1]) then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))" unfolding filterlim_def by (simp add: eventually_filtermap eventually_at_filter le_principal) have "((?N g g' (f x)) ⤏ 0) (at (f x) within f`s)" using g unfolding has_derivative_iff_norm .. then have g': "((?N g g' (f x)) ⤏ 0) (inf (nhds (f x)) (principal (f`s)))" by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp have [tendsto_intros]: "?L Ng" unfolding Ng_def by (rule filterlim_compose[OF g' f']) show "((λy. Nf y * kG + Ng y * (Nf y + kF)) ⤏ 0) (at x within s)" by (intro tendsto_eq_intros) auto qed simp qed lemma has_derivative_compose: "(f has_derivative f') (at x within s) ⟹ (g has_derivative g') (at (f x)) ⟹ ((λx. g (f x)) has_derivative (λx. g' (f' x))) (at x within s)" by (blast intro: has_derivative_in_compose has_derivative_subset) lemma has_derivative_in_compose2: assumes "⋀x. x ∈ t ⟹ (g has_derivative g' x) (at x within t)" assumes "f ` s ⊆ t" "x ∈ s" assumes "(f has_derivative f') (at x within s)" shows "((λx. g (f x)) has_derivative (λy. g' (f x) (f' y))) (at x within s)" using assms by (auto intro: has_derivative_subset intro!: has_derivative_in_compose[of f f' x s g]) lemma (in bounded_bilinear) FDERIV: assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" shows "((λx. f x ** g x) has_derivative (λh. f x ** g' h + f' h ** g x)) (at x within s)" proof - from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]] obtain KF where norm_F: "⋀x. norm (f' x) ≤ norm x * KF" by fast from pos_bounded obtain K where K: "0 < K" and norm_prod: "⋀a b. norm (a ** b) ≤ norm a * norm b * K" by fast let ?D = "λf f' y. f y - f x - f' (y - x)" let ?N = "λf f' y. norm (?D f f' y) / norm (y - x)" define Ng where "Ng = ?N g g'" define Nf where "Nf = ?N f f'" let ?fun1 = "λy. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)" let ?fun2 = "λy. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K" let ?F = "at x within s" show ?thesis proof (rule has_derivativeI_sandwich[of 1]) show "bounded_linear (λh. f x ** g' h + f' h ** g x)" by (intro bounded_linear_add bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left] has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f]) next from g have "(g ⤏ g x) ?F" by (intro continuous_within[THEN iffD1] has_derivative_continuous) moreover from f g have "(Nf ⤏ 0) ?F" "(Ng ⤏ 0) ?F" by (simp_all add: has_derivative_iff_norm Ng_def Nf_def) ultimately have "(?fun2 ⤏ norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F" by (intro tendsto_intros) (simp_all add: LIM_zero_iff) then show "(?fun2 ⤏ 0) ?F" by simp next fix y :: 'd assume "y ≠ x" have "?fun1 y = norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)" by (simp add: diff_left diff_right add_left add_right field_simps) also have "… ≤ (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K + norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)" by (intro divide_right_mono mult_mono' order_trans [OF norm_triangle_ineq add_mono] order_trans [OF norm_prod mult_right_mono] mult_nonneg_nonneg order_refl norm_ge_zero norm_F K [THEN order_less_imp_le]) also have "… = ?fun2 y" by (simp add: add_divide_distrib Ng_def Nf_def) finally show "?fun1 y ≤ ?fun2 y" . qed simp qed lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult] lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR] lemma has_derivative_prod[simp, derivative_intros]: fixes f :: "'i ⇒ 'a::real_normed_vector ⇒ 'b::real_normed_field" shows "(⋀i. i ∈ I ⟹ (f i has_derivative f' i) (at x within S)) ⟹ ((λx. ∏i∈I. f i x) has_derivative (λy. ∑i∈I. f' i y * (∏j∈I - {i}. f j x))) (at x within S)" proof (induct I rule: infinite_finite_induct) case infinite then show ?case by simp next case empty then show ?case by simp next case (insert i I) let ?P = "λy. f i x * (∑i∈I. f' i y * (∏j∈I - {i}. f j x)) + (f' i y) * (∏i∈I. f i x)" have "((λx. f i x * (∏i∈I. f i x)) has_derivative ?P) (at x within S)" using insert by (intro has_derivative_mult) auto also have "?P = (λy. ∑i'∈insert i I. f' i' y * (∏j∈insert i I - {i'}. f j x))" using insert(1,2) by (auto simp add: sum_distrib_left insert_Diff_if intro!: ext sum.cong) finally show ?case using insert by simp qed lemma has_derivative_power[simp, derivative_intros]: fixes f :: "'a :: real_normed_vector ⇒ 'b :: real_normed_field" assumes f: "(f has_derivative f') (at x within S)" shows "((λx. f x^n) has_derivative (λy. of_nat n * f' y * f x^(n - 1))) (at x within S)" using has_derivative_prod[OF f, of "{..< n}"] by (simp add: prod_constant ac_simps) lemma has_derivative_inverse': fixes x :: "'a::real_normed_div_algebra" assumes x: "x ≠ 0" shows "(inverse has_derivative (λh. - (inverse x * h * inverse x))) (at x within S)" (is "(_ has_derivative ?f) _") proof (rule has_derivativeI_sandwich) show "bounded_linear (λh. - (inverse x * h * inverse x))" by (simp add: bounded_linear_minus bounded_linear_mult_const bounded_linear_mult_right) show "0 < norm x" using x by simp have "(inverse ⤏ inverse x) (at x within S)" using tendsto_inverse tendsto_ident_at x by auto then show "((λy. norm (inverse y - inverse x) * norm (inverse x)) ⤏ 0) (at x within S)" by (simp add: LIM_zero_iff tendsto_mult_left_zero tendsto_norm_zero) next fix y :: 'a assume h: "y ≠ x" "dist y x < norm x" then have "y ≠ 0" by auto have "norm (inverse y - inverse x - ?f (y -x)) / norm (y - x) = norm (- (inverse y * (y - x) * inverse x - inverse x * (y - x) * inverse x)) / norm (y - x)" by (simp add: ‹y ≠ 0› inverse_diff_inverse x) also have "... = norm ((inverse y - inverse x) * (y - x) * inverse x) / norm (y - x)" by (simp add: left_diff_distrib norm_minus_commute) also have "… ≤ norm (inverse y - inverse x) * norm (y - x) * norm (inverse x) / norm (y - x)" by (simp add: norm_mult) also have "… = norm (inverse y - inverse x) * norm (inverse x)" by simp finally show "norm (inverse y - inverse x - ?f (y -x)) / norm (y - x) ≤ norm (inverse y - inverse x) * norm (inverse x)" . qed lemma has_derivative_inverse[simp, derivative_intros]: fixes f :: "_ ⇒ 'a::real_normed_div_algebra" assumes x: "f x ≠ 0" and f: "(f has_derivative f') (at x within S)" shows "((λx. inverse (f x)) has_derivative (λh. - (inverse (f x) * f' h * inverse (f x)))) (at x within S)" using has_derivative_compose[OF f has_derivative_inverse', OF x] . lemma has_derivative_divide[simp, derivative_intros]: fixes f :: "_ ⇒ 'a::real_normed_div_algebra" assumes f: "(f has_derivative f') (at x within S)" and g: "(g has_derivative g') (at x within S)" assumes x: "g x ≠ 0" shows "((λx. f x / g x) has_derivative (λh. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within S)" using has_derivative_mult[OF f has_derivative_inverse[OF x g]] by (simp add: field_simps) lemma has_derivative_power_int': fixes x :: "'a::real_normed_field" assumes x: "x ≠ 0" shows "((λx. power_int x n) has_derivative (λy. y * (of_int n * power_int x (n - 1)))) (at x within S)" proof (cases n rule: int_cases4) case (nonneg n) thus ?thesis using x by (cases "n = 0") (auto intro!: derivative_eq_intros simp: field_simps power_int_diff fun_eq_iff simp flip: power_Suc) next case (neg n) thus ?thesis using x by (auto intro!: derivative_eq_intros simp: field_simps power_int_diff power_int_minus simp flip: power_Suc power_Suc2 power_add) qed lemma has_derivative_power_int[simp, derivative_intros]: fixes f :: "_ ⇒ 'a::real_normed_field" assumes x: "f x ≠ 0" and f: "(f has_derivative f') (at x within S)" shows "((λx. power_int (f x) n) has_derivative (λh. f' h * (of_int n * power_int (f x) (n - 1)))) (at x within S)" using has_derivative_compose[OF f has_derivative_power_int', OF x] . text ‹Conventional form requires mult-AC laws. Types real and complex only.› lemma has_derivative_divide'[derivative_intros]: fixes f :: "_ ⇒ 'a::real_normed_field" assumes f: "(f has_derivative f') (at x within S)" and g: "(g has_derivative g') (at x within S)" and x: "g x ≠ 0" shows "((λx. f x / g x) has_derivative (λh. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)" proof - have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) = (f' h * g x - f x * g' h) / (g x * g x)" for h by (simp add: field_simps x) then show ?thesis using has_derivative_divide [OF f g] x by simp qed subsection ‹Uniqueness› text ‹ This can not generally shown for \<^const>‹has_derivative›, as we need to approach the point from all directions. There is a proof in ‹Analysis› for ‹euclidean_space›. › lemma has_derivative_at2: "(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 lemma has_derivative_zero_unique: assumes "((λx. 0) has_derivative F) (at x)" shows "F = (λh. 0)" proof - interpret F: bounded_linear F using assms by (rule has_derivative_bounded_linear) let ?r = "λh. norm (F h) / norm h" have *: "?r ─0→ 0" using assms unfolding has_derivative_at by simp show "F = (λh. 0)" proof show "F h = 0" for h proof (rule ccontr) assume **: "¬ ?thesis" then have h: "h ≠ 0" by (auto simp add: F.zero) with ** have "0 < ?r h" by simp from LIM_D [OF * this] obtain S where S: "0 < S" and r: "⋀x. x ≠ 0 ⟹ norm x < S ⟹ ?r x < ?r h" by auto from dense [OF S] obtain t where t: "0 < t ∧ t < S" .. let ?x = "scaleR (t / norm h) h" have "?x ≠ 0" and "norm ?x < S" using t h by simp_all then have "?r ?x < ?r h" by (rule r) then show False using t h by (simp add: F.scaleR) qed qed qed lemma has_derivative_unique: assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'" proof - have "((λx. 0) has_derivative (λh. F h - F' h)) (at x)" using has_derivative_diff [OF assms] by simp then have "(λh. F h - F' h) = (λh. 0)" by (rule has_derivative_zero_unique) then show "F = F'" unfolding fun_eq_iff right_minus_eq . qed lemma has_derivative_Uniq: "∃⇩_{≤}⇩_{1}F. (f has_derivative F) (at x)" by (simp add: Uniq_def has_derivative_unique) subsection ‹Differentiability predicate› definition differentiable :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒ 'a filter ⇒ bool" (infix "differentiable" 50) where "f differentiable F ⟷ (∃D. (f has_derivative D) F)" lemma differentiable_subset: "f differentiable (at x within s) ⟹ t ⊆ s ⟹ f differentiable (at x within t)" unfolding differentiable_def by (blast intro: has_derivative_subset) lemmas differentiable_within_subset = differentiable_subset lemma differentiable_ident [simp, derivative_intros]: "(λx. x) differentiable F" unfolding differentiable_def by (blast intro: has_derivative_ident) lemma differentiable_const [simp, derivative_intros]: "(λz. a) differentiable F" unfolding differentiable_def by (blast intro: has_derivative_const) lemma differentiable_in_compose: "f differentiable (at (g x) within (g`s)) ⟹ g differentiable (at x within s) ⟹ (λx. f (g x)) differentiable (at x within s)" unfolding differentiable_def by (blast intro: has_derivative_in_compose) lemma differentiable_compose: "f differentiable (at (g x)) ⟹ g differentiable (at x within s) ⟹ (λx. f (g x)) differentiable (at x within s)" by (blast intro: differentiable_in_compose differentiable_subset) lemma differentiable_add [simp, derivative_intros]: "f differentiable F ⟹ g differentiable F ⟹ (λx. f x + g x) differentiable F" unfolding differentiable_def by (blast intro: has_derivative_add) lemma differentiable_sum[simp, derivative_intros]: assumes "finite s" "∀a∈s. (f a) differentiable net" shows "(λx. sum (λa. f a x) s) differentiable net" proof - from bchoice[OF assms(2)[unfolded differentiable_def]] show ?thesis by (auto intro!: has_derivative_sum simp: differentiable_def) qed lemma differentiable_minus [simp, derivative_intros]: "f differentiable F ⟹ (λx. - f x) differentiable F" unfolding differentiable_def by (blast intro: has_derivative_minus) lemma differentiable_diff [simp, derivative_intros]: "f differentiable F ⟹ g differentiable F ⟹ (λx. f x - g x) differentiable F" unfolding differentiable_def by (blast intro: has_derivative_diff) lemma differentiable_mult [simp, derivative_intros]: fixes f g :: "'a::real_normed_vector ⇒ 'b::real_normed_algebra" shows "f differentiable (at x within s) ⟹ g differentiable (at x within s) ⟹ (λx. f x * g x) differentiable (at x within s)" unfolding differentiable_def by (blast intro: has_derivative_mult) lemma differentiable_cmult_left_iff [simp]: fixes c::"'a::real_normed_field" shows "(λt. c * q t) differentiable at t ⟷ c = 0 ∨ (λt. q t) differentiable at t" (is "?lhs = ?rhs") proof assume L: ?lhs {assume "c ≠ 0" then have "q differentiable at t" using differentiable_mult [OF differentiable_const L, of concl: "1/c"] by auto } then show ?rhs by auto qed auto lemma differentiable_cmult_right_iff [simp]: fixes c::"'a::real_normed_field" shows "(λt. q t * c) differentiable at t ⟷ c = 0 ∨ (λt. q t) differentiable at t" (is "?lhs = ?rhs") by (simp add: mult.commute flip: differentiable_cmult_left_iff) lemma differentiable_inverse [simp, derivative_intros]: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_field" shows "f differentiable (at x within s) ⟹ f x ≠ 0 ⟹ (λx. inverse (f x)) differentiable (at x within s)" unfolding differentiable_def by (blast intro: has_derivative_inverse) lemma differentiable_divide [simp, derivative_intros]: fixes f g :: "'a::real_normed_vector ⇒ 'b::real_normed_field" shows "f differentiable (at x within s) ⟹ g differentiable (at x within s) ⟹ g x ≠ 0 ⟹ (λx. f x / g x) differentiable (at x within s)" unfolding divide_inverse by simp lemma differentiable_power [simp, derivative_intros]: fixes f g :: "'a::real_normed_vector ⇒ 'b::real_normed_field" shows "f differentiable (at x within s) ⟹ (λx. f x ^ n) differentiable (at x within s)" unfolding differentiable_def by (blast intro: has_derivative_power) lemma differentiable_power_int [simp, derivative_intros]: fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_field" shows "f differentiable (at x within s) ⟹ f x ≠ 0 ⟹ (λx. power_int (f x) n) differentiable (at x within s)" unfolding differentiable_def by (blast intro: has_derivative_power_int) lemma differentiable_scaleR [simp, derivative_intros]: "f differentiable (at x within s) ⟹ g differentiable (at x within s) ⟹ (λx. f x *⇩_{R}g x) differentiable (at x within s)" unfolding differentiable_def by (blast intro: has_derivative_scaleR) lemma has_derivative_imp_has_field_derivative: "(f has_derivative D) F ⟹ (⋀x. x * D' = D x) ⟹ (f has_field_derivative D') F" unfolding has_field_derivative_def by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute) lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F ⟹ (f has_derivative (*) D) F" by (simp add: has_field_derivative_def) lemma DERIV_subset: "(f has_field_derivative f') (at x within s) ⟹ t ⊆ s ⟹ (f has_field_derivative f') (at x within t)" by (simp add: has_field_derivative_def has_derivative_subset) lemma has_field_derivative_at_within: "(f has_field_derivative f') (at x) ⟹ (f has_field_derivative f') (at x within s)" using DERIV_subset by blast abbreviation (input) DERIV :: "('a::real_normed_field ⇒ 'a) ⇒ 'a ⇒ 'a ⇒ bool" ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where "DERIV f x :> D ≡ (f has_field_derivative D) (at x)" abbreviation has_real_derivative :: "(real ⇒ real) ⇒ real ⇒ real filter ⇒ bool" (infix "(has'_real'_derivative)" 50) where "(f has_real_derivative D) F ≡ (f has_field_derivative D) F" lemma real_differentiable_def: "f differentiable at x within s ⟷ (∃D. (f has_real_derivative D) (at x within s))" proof safe assume "f differentiable at x within s" then obtain f' where *: "(f has_derivative f') (at x within s)" unfolding differentiable_def by auto then obtain c where "f' = ((*) c)" by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff) with * show "∃D. (f has_real_derivative D) (at x within s)" unfolding has_field_derivative_def by auto qed (auto simp: differentiable_def has_field_derivative_def) lemma real_differentiableE [elim?]: assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)" using assms by (auto simp: real_differentiable_def) lemma has_field_derivative_iff: "(f has_field_derivative D) (at x within S) ⟷ ((λy. (f y - f x) / (y - x)) ⤏ D) (at x within S)" proof - have "((λy. norm (f y - f x - D * (y - x)) / norm (y - x)) ⤏ 0) (at x within S) = ((λy. (f y - f x) / (y - x) - D) ⤏ 0) (at x within S)" by (smt (verit, best) Lim_cong_within divide_diff_eq_iff norm_divide right_minus_eq tendsto_norm_zero_iff) then show ?thesis by (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right LIM_zero_iff) qed lemma DERIV_def: "DERIV f x :> D ⟷ (λh. (f (x + h) - f x) / h) ─0→ D" unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff .. text ‹due to Christian Pardillo Laursen, replacing a proper epsilon-delta horror› lemma field_derivative_lim_unique: assumes f: "(f has_field_derivative df) (at z)" and s: "s ⇢ 0" "⋀n. s n ≠ 0" and a: "(λn. (f (z + s n) - f z) / s n) ⇢ a" shows "df = a" proof - have "((λk. (f (z + k) - f z) / k) ⤏ df) (at 0)" using f by (simp add: DERIV_def) with s have "((λn. (f (z + s n) - f z) / s n) ⇢ df)" by (simp flip: LIMSEQ_SEQ_conv) then show ?thesis using a by (rule LIMSEQ_unique) qed lemma mult_commute_abs: "(λx. x * c) = (*) c" for c :: "'a::ab_semigroup_mult" by (simp add: fun_eq_iff mult.commute) lemma DERIV_compose_FDERIV: fixes f::"real⇒real" assumes "DERIV f (g x) :> f'" assumes "(g has_derivative g') (at x within s)" shows "((λx. f (g x)) has_derivative (λx. g' x * f')) (at x within s)" using assms has_derivative_compose[of g g' x s f "(*) f'"] by (auto simp: has_field_derivative_def ac_simps) subsection ‹Vector derivative› text ‹It's for real derivatives only, and not obviously generalisable to field derivatives› lemma has_real_derivative_iff_has_vector_derivative: "(f has_real_derivative y) F ⟷ (f has_vector_derivative y) F" unfolding has_vector_derivative_def has_field_derivative_def real_scaleR_def mult_commute_abs .. lemma has_field_derivative_subset: "(f has_field_derivative y) (at x within s) ⟹ t ⊆ s ⟹ (f has_field_derivative y) (at x within t)" by (fact DERIV_subset) lemma has_vector_derivative_const[simp, derivative_intros]: "((λx. c) has_vector_derivative 0) net" by (auto simp: has_vector_derivative_def) lemma has_vector_derivative_id[simp, derivative_intros]: "((λx. x) has_vector_derivative 1) net" by (auto simp: has_vector_derivative_def) lemma has_vector_derivative_minus[derivative_intros]: "(f has_vector_derivative f') net ⟹ ((λx. - f x) has_vector_derivative (- f')) net" by (auto simp: has_vector_derivative_def) lemma has_vector_derivative_add[derivative_intros]: "(f has_vector_derivative f') net ⟹ (g has_vector_derivative g') net ⟹ ((λx. f x + g x) has_vector_derivative (f' + g')) net" by (auto simp: has_vector_derivative_def scaleR_right_distrib) lemma has_vector_derivative_sum[derivative_intros]: "(⋀i. i ∈ I ⟹ (f i has_vector_derivative f' i) net) ⟹ ((λx. ∑i∈I. f i x) has_vector_derivative (∑i∈I. f' i)) net" by (auto simp: has_vector_derivative_def fun_eq_iff scaleR_sum_right intro!: derivative_eq_intros) lemma has_vector_derivative_diff[derivative_intros]: "(f has_vector_derivative f') net ⟹ (g has_vector_derivative g') net ⟹ ((λx. f x - g x) has_vector_derivative (f' - g')) net" by (auto simp: has_vector_derivative_def scaleR_diff_right) lemma has_vector_derivative_add_const: "((λt. g t + z) has_vector_derivative f') net = ((λt. g t) has_vector_derivative f') net" apply (intro iffI) apply (force dest: has_vector_derivative_diff [where g = "λt. z", OF _ has_vector_derivative_const]) apply (force dest: has_vector_derivative_add [OF _ has_vector_derivative_const]) done lemma has_vector_derivative_diff_const: "((λt. g t - z) has_vector_derivative f') net = ((λt. g t) has_vector_derivative f') net" using has_vector_derivative_add_const [where z = "-z"] by simp lemma (in bounded_linear) has_vector_derivative: assumes "(g has_vector_derivative g') F" shows "((λx. f (g x)) has_vector_derivative f g') F" using has_derivative[OF assms[unfolded has_vector_derivative_def]] by (simp add: has_vector_derivative_def scaleR) lemma (in bounded_bilinear) has_vector_derivative: assumes "(f has_vector_derivative f') (at x within s)" and "(g has_vector_derivative g') (at x within s)" shows "((λx. f x ** g x) has_vector_derivative (f x ** g' + f' ** g x)) (at x within s)" using FDERIV[OF assms(1-2)[unfolded has_vector_derivative_def]] by (simp add: has_vector_derivative_def scaleR_right scaleR_left scaleR_right_distrib) lemma has_vector_derivative_scaleR[derivative_intros]: "(f has_field_derivative f') (at x within s) ⟹ (g has_vector_derivative g') (at x within s) ⟹ ((λx. f x *⇩_{R}g x) has_vector_derivative (f x *⇩_{R}g' + f' *⇩_{R}g x)) (at x within s)" unfolding has_real_derivative_iff_has_vector_derivative by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR]) lemma has_vector_derivative_mult[derivative_intros]: "(f has_vector_derivative f') (at x within s) ⟹ (g has_vector_derivative g') (at x within s) ⟹ ((λx. f x * g x) has_vector_derivative (f x * g' + f' * g x)) (at x within s)" for f g :: "real ⇒ 'a::real_normed_algebra" by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_mult]) lemma has_vector_derivative_of_real[derivative_intros]: "(f has_field_derivative D) F ⟹ ((λx. of_real (f x)) has_vector_derivative (of_real D)) F" by (rule bounded_linear.has_vector_derivative[OF bounded_linear_of_real]) (simp add: has_real_derivative_iff_has_vector_derivative) lemma has_vector_derivative_real_field: "(f has_field_derivative f') (at (of_real a)) ⟹ ((λx. f (of_real x)) has_vector_derivative f') (at a within s)" using has_derivative_compose[of of_real of_real a _ f "(*) f'"] by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def) lemma has_vector_derivative_continuous: "(f has_vector_derivative D) (at x within s) ⟹ continuous (at x within s) f" by (auto intro: has_derivative_continuous simp: has_vector_derivative_def) lemma continuous_on_vector_derivative: "(⋀x. x ∈ S ⟹ (f has_vector_derivative f' x) (at x within S)) ⟹ continuous_on S f" by (auto simp: continuous_on_eq_continuous_within intro!: has_vector_derivative_continuous) lemma has_vector_derivative_mult_right[derivative_intros]: fixes a :: "'a::real_normed_algebra" shows "(f has_vector_derivative x) F ⟹ ((λx. a * f x) has_vector_derivative (a * x)) F" by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_right]) lemma has_vector_derivative_mult_left[derivative_intros]: fixes a :: "'a::real_normed_algebra" shows "(f has_vector_derivative x) F ⟹ ((λx. f x * a) has_vector_derivative (x * a)) F" by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_left]) lemma has_vector_derivative_divide[derivative_intros]: fixes a :: "'a::real_normed_field" shows "(f has_vector_derivative x) F ⟹ ((λx. f x / a) has_vector_derivative (x / a)) F" using has_vector_derivative_mult_left [of f x F "inverse a"] by (simp add: field_class.field_divide_inverse) subsection ‹Derivatives› lemma DERIV_D: "DERIV f x :> D ⟹ (λh. (f (x + h) - f x) / h) ─0→ D" by (simp add: DERIV_def) lemma has_field_derivativeD: "(f has_field_derivative D) (at x within S) ⟹ ((λy. (f y - f x) / (y - x)) ⤏ D) (at x within S)" by (simp add: has_field_derivative_iff) lemma DERIV_const [simp, derivative_intros]: "((λx. k) has_field_derivative 0) F" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto lemma DERIV_ident [simp, derivative_intros]: "((λx. x) has_field_derivative 1) F" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto lemma field_differentiable_add[derivative_intros]: "(f has_field_derivative f') F ⟹ (g has_field_derivative g') F ⟹ ((λz. f z + g z) has_field_derivative f' + g') F" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add]) (auto simp: has_field_derivative_def field_simps mult_commute_abs) corollary DERIV_add: "(f has_field_derivative D) (at x within s) ⟹ (g has_field_derivative E) (at x within s) ⟹ ((λx. f x + g x) has_field_derivative D + E) (at x within s)" by (rule field_differentiable_add) lemma field_differentiable_minus[derivative_intros]: "(f has_field_derivative f') F ⟹ ((λz. - (f z)) has_field_derivative -f') F" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus]) (auto simp: has_field_derivative_def field_simps mult_commute_abs) corollary DERIV_minus: "(f has_field_derivative D) (at x within s) ⟹ ((λx. - f x) has_field_derivative -D) (at x within s)" by (rule field_differentiable_minus) lemma field_differentiable_diff[derivative_intros]: "(f has_field_derivative f') F ⟹ (g has_field_derivative g') F ⟹ ((λz. f z - g z) has_field_derivative f' - g') F" by (simp only: diff_conv_add_uminus field_differentiable_add field_differentiable_minus) corollary DERIV_diff: "(f has_field_derivative D) (at x within s) ⟹ (g has_field_derivative E) (at x within s) ⟹ ((λx. f x - g x) has_field_derivative D - E) (at x within s)" by (rule field_differentiable_diff) lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) ⟹ continuous (at x within s) f" by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp corollary DERIV_isCont: "DERIV f x :> D ⟹ isCont f x" by (rule DERIV_continuous) lemma DERIV_atLeastAtMost_imp_continuous_on: assumes "⋀x. ⟦a ≤ x; x ≤ b⟧ ⟹ ∃y. DERIV f x :> y" shows "continuous_on {a..b} f" by (meson DERIV_isCont assms atLeastAtMost_iff continuous_at_imp_continuous_at_within continuous_on_eq_continuous_within) lemma DERIV_continuous_on: "(⋀x. x ∈ s ⟹ (f has_field_derivative (D x)) (at x within s)) ⟹ continuous_on s f" unfolding continuous_on_eq_continuous_within by (intro continuous_at_imp_continuous_on ballI DERIV_continuous) lemma DERIV_mult': "(f has_field_derivative D) (at x within s) ⟹ (g has_field_derivative E) (at x within s) ⟹ ((λx. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult]) (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative) lemma DERIV_mult[derivative_intros]: "(f has_field_derivative Da) (at x within s) ⟹ (g has_field_derivative Db) (at x within s) ⟹ ((λx. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult]) (auto simp: field_simps dest: has_field_derivative_imp_has_derivative) text ‹Derivative of linear multiplication› lemma DERIV_cmult: "(f has_field_derivative D) (at x within s) ⟹ ((λx. c * f x) has_field_derivative c * D) (at x within s)" by (drule DERIV_mult' [OF DERIV_const]) simp lemma DERIV_cmult_right: "(f has_field_derivative D) (at x within s) ⟹ ((λx. f x * c) has_field_derivative D * c) (at x within s)" using DERIV_cmult by (auto simp add: ac_simps) lemma DERIV_cmult_Id [simp]: "((*) c has_field_derivative c) (at x within s)" using DERIV_ident [THEN DERIV_cmult, where c = c and x = x] by simp lemma DERIV_cdivide: "(f has_field_derivative D) (at x within s) ⟹ ((λx. f x / c) has_field_derivative D / c) (at x within s)" using DERIV_cmult_right[of f D x s "1 / c"] by simp lemma DERIV_unique: "DERIV f x :> D ⟹ DERIV f x :> E ⟹ D = E" unfolding DERIV_def by (rule LIM_unique) lemma DERIV_Uniq: "∃⇩_{≤}⇩_{1}D. DERIV f x :> D" by (simp add: DERIV_unique Uniq_def) lemma DERIV_sum[derivative_intros]: "(⋀ n. n ∈ S ⟹ ((λx. f x n) has_field_derivative (f' x n)) F) ⟹ ((λx. sum (f x) S) has_field_derivative sum (f' x) S) F" by (rule has_derivative_imp_has_field_derivative [OF has_derivative_sum]) (auto simp: sum_distrib_left mult_commute_abs dest: has_field_derivative_imp_has_derivative) lemma DERIV_inverse'[derivative_intros]: assumes "(f has_field_derivative D) (at x within s)" and "f x ≠ 0" shows "((λx. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)" proof - have "(f has_derivative (λx. x * D)) = (f has_derivative (*) D)" by (rule arg_cong [of "λx. x * D"]) (simp add: fun_eq_iff) with assms have "(f has_derivative (λx. x * D)) (at x within s)" by (auto dest!: has_field_derivative_imp_has_derivative) then show ?thesis using ‹f x ≠ 0› by (auto intro: has_derivative_imp_has_field_derivative has_derivative_inverse) qed text ‹Power of ‹-1›› lemma DERIV_inverse: "x ≠ 0 ⟹ ((λx. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)" by (drule DERIV_inverse' [OF DERIV_ident]) simp text ‹Derivative of inverse› lemma DERIV_inverse_fun: "(f has_field_derivative d) (at x within s) ⟹ f x ≠ 0 ⟹ ((λx. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)" by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib) text ‹Derivative of quotient› lemma DERIV_divide[derivative_intros]: "(f has_field_derivative D) (at x within s) ⟹ (g has_field_derivative E) (at x within s) ⟹ g x ≠ 0 ⟹ ((λx. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide]) (auto dest: has_field_derivative_imp_has_derivative simp: field_simps) lemma DERIV_quotient: "(f has_field_derivative d) (at x within s) ⟹ (g has_field_derivative e) (at x within s)⟹ g x ≠ 0 ⟹ ((λy. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)" by (drule (2) DERIV_divide) (simp add: mult.commute) lemma DERIV_power_Suc: "(f has_field_derivative D) (at x within s) ⟹ ((λx. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power]) (auto simp: has_field_derivative_def) lemma DERIV_power[derivative_intros]: "(f has_field_derivative D) (at x within s) ⟹ ((λx. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)" by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power]) (auto simp: has_field_derivative_def) lemma DERIV_pow: "((λx. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)" using DERIV_power [OF DERIV_ident] by simp lemma DERIV_power_int [derivative_intros]: assumes [derivative_intros]: "(f has_field_derivative d) (at x within s)" and [simp]: "f x ≠ 0" shows "((λx. power_int (f x) n) has_field_derivative (of_int n * power_int (f x) (n - 1) * d)) (at x within s)" proof (cases n rule: int_cases4) case (nonneg n) thus ?thesis by (cases "n = 0") (auto intro!: derivative_eq_intros simp: field_simps power_int_diff simp flip: power_Suc power_Suc2 power_add) next case (neg n) thus ?thesis by (auto intro!: derivative_eq_intros simp: field_simps power_int_diff power_int_minus simp flip: power_Suc power_Suc2 power_add) qed lemma DERIV_chain': "(f has_field_derivative D) (at x within s) ⟹ DERIV g (f x) :> E ⟹ ((λx. g (f x)) has_field_derivative E * D) (at x within s)" using has_derivative_compose[of f "(*) D" x s g "(*) E"] by (simp only: has_field_derivative_def mult_commute_abs ac_simps) corollary DERIV_chain2: "DERIV f (g x) :> Da ⟹ (g has_field_derivative Db) (at x within s) ⟹ ((λx. f (g x)) has_field_derivative Da * Db) (at x within s)" by (rule DERIV_chain') text ‹Standard version› lemma DERIV_chain: "DERIV f (g x) :> Da ⟹ (g has_field_derivative Db) (at x within s) ⟹ (f ∘ g has_field_derivative Da * Db) (at x within s)" by (drule (1) DERIV_chain', simp add: o_def mult.commute) lemma DERIV_image_chain: "(f has_field_derivative Da) (at (g x) within (g ` s)) ⟹ (g has_field_derivative Db) (at x within s) ⟹ (f ∘ g has_field_derivative Da * Db) (at x within s)" using has_derivative_in_compose [of g "(*) Db" x s f "(*) Da "] by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps) (*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*) lemma DERIV_chain_s: assumes "(⋀x. x ∈ s ⟹ DERIV g x :> g'(x))" and "DERIV f x :> f'" and "f x ∈ s" shows "DERIV (λx. g(f x)) x :> f' * g'(f x)" by (metis (full_types) DERIV_chain' mult.commute assms) lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*) assumes "(⋀x. DERIV g x :> g'(x))" and "DERIV f x :> f'" shows "DERIV (λx. g(f x)) x :> f' * g'(f x)" by (metis UNIV_I DERIV_chain_s [of UNIV] assms) text ‹Alternative definition for differentiability› lemma DERIV_LIM_iff: fixes f :: "'a::{real_normed_vector,inverse} ⇒ 'a" shows "((λh. (f (a + h) - f a) / h) ─0→ D) = ((λx. (f x - f a) / (x - a)) ─a→ D)" (is "?lhs = ?rhs") proof assume ?lhs then have "(λx. (f (a + (x + - a)) - f a) / (x + - a)) ─0 - - a→ D" by (rule LIM_offset) then show ?rhs by simp next assume ?rhs then have "(λx. (f (x+a) - f a) / ((x+a) - a)) ─a-a→ D" by (rule LIM_offset) then show ?lhs by (simp add: add.commute) qed lemma has_field_derivative_cong_ev: assumes "x = y" and *: "eventually (λx. x ∈ S ⟶ f x = g x) (nhds x)" and "u = v" "S = t" "x ∈ S" shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative v) (at y within t)" unfolding has_field_derivative_iff proof (rule filterlim_cong) from assms have "f y = g y" by (auto simp: eventually_nhds) with * show "∀⇩_{F}z in at x within S. (f z - f x) / (z - x) = (g z - g y) / (z - y)" unfolding eventually_at_filter by eventually_elim (auto simp: assms ‹f y = g y›) qed (simp_all add: assms) lemma has_field_derivative_cong_eventually: assumes "eventually (λx. f x = g x) (at x within S)" "f x = g x" shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative u) (at x within S)" unfolding has_field_derivative_iff proof (rule tendsto_cong) show "∀⇩_{F}y in at x within S. (f y - f x) / (y - x) = (g y - g x) / (y - x)" using assms by (auto elim: eventually_mono) qed lemma DERIV_cong_ev: "x = y ⟹ eventually (λx. f x = g x) (nhds x) ⟹ u = v ⟹ DERIV f x :> u ⟷ DERIV g y :> v" by (rule has_field_derivative_cong_ev) simp_all lemma DERIV_mirror: "(DERIV f (- x) :> y) ⟷ (DERIV (λx. f (- x)) x :> - y)" for f :: "real ⇒ real" and x y :: real by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right tendsto_minus_cancel_left field_simps conj_commute) lemma DERIV_shift: "(f has_field_derivative y) (at (x + z)) = ((λx. f (x + z)) has_field_derivative y) (at x)" by (simp add: DERIV_def field_simps) lemma DERIV_at_within_shift_lemma: assumes "(f has_field_derivative y) (at (z+x) within (+) z ` S)" shows "(f ∘ (+)z has_field_derivative y) (at x within S)" proof - have "((+)z has_field_derivative 1) (at x within S)" by (rule derivative_eq_intros | simp)+ with assms DERIV_image_chain show ?thesis by (metis mult.right_neutral) qed lemma DERIV_at_within_shift: "(f has_field_derivative y) (at (z+x) within (+) z ` S) ⟷ ((λx. f (z+x)) has_field_derivative y) (at x within S)" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs using DERIV_at_within_shift_lemma unfolding o_def by blast next have [simp]: "(λx. x - z) ` (+) z ` S = S" by force assume R: ?rhs have "(f ∘ (+) z ∘ (+) (- z) has_field_derivative y) (at (z + x) within (+) z ` S)" by (rule DERIV_at_within_shift_lemma) (use R in ‹simp add: o_def›) then show ?lhs by (simp add: o_def) qed lemma floor_has_real_derivative: fixes f :: "real ⇒ 'a::{floor_ceiling,order_topology}" assumes "isCont f x" and "f x ∉ ℤ" shows "((λx. floor (f x)) has_real_derivative 0) (at x)" proof (subst DERIV_cong_ev[OF refl _ refl]) show "((λ_. floor (f x)) has_real_derivative 0) (at x)" by simp have "∀⇩_{F}y in at x. ⌊f y⌋ = ⌊f x⌋" by (rule eventually_floor_eq[OF assms[unfolded continuous_at]]) then show "∀⇩_{F}y in nhds x. real_of_int ⌊f y⌋ = real_of_int ⌊f x⌋" unfolding eventually_at_filter by eventually_elim auto qed lemmas has_derivative_floor[derivative_intros] = floor_has_real_derivative[THEN DERIV_compose_FDERIV] lemma continuous_floor: fixes x::real shows "x ∉ ℤ ⟹ continuous (at x) (real_of_int ∘ floor)" using floor_has_real_derivative [where f=id] by (auto simp: o_def has_field_derivative_def intro: has_derivative_continuous) lemma continuous_frac: fixes x::real assumes "x ∉ ℤ" shows "continuous (at x) frac" proof - have "isCont (λx. real_of_int ⌊x⌋) x" using continuous_floor [OF assms] by (simp add: o_def) then have *: "continuous (at x) (λx. x - real_of_int ⌊x⌋)" by (intro continuous_intros) moreover have "∀⇩_{F}x in nhds x. frac x = x - real_of_int ⌊x⌋" by (simp add: frac_def) ultimately show ?thesis by (simp add: LIM_imp_LIM frac_def isCont_def) qed text ‹Caratheodory formulation of derivative at a point› lemma CARAT_DERIV: "(DERIV f x :> l) ⟷ (∃g. (∀z. f z - f x = g z * (z - x)) ∧ isCont g x ∧ g x = l)" (is "?lhs = ?rhs") proof assume ?lhs show "∃g. (∀z. f z - f x = g z * (z - x)) ∧ isCont g x ∧ g x = l" proof (intro exI conjI) let ?g = "(λz. 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 "isCont ?g x" using ‹?lhs› by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format]) show "?g x = l" by simp qed next assume ?rhs then show ?lhs by (auto simp add: isCont_iff DERIV_def cong: LIM_cong) qed subsection ‹Local extrema› text ‹If \<^term>‹0 < f' x› then \<^term>‹x› is Locally Strictly Increasing At The Right.› lemma has_real_derivative_pos_inc_right: fixes f :: "real ⇒ real" assumes der: "(f has_real_derivative l) (at x within S)" and l: "0 < l" shows "∃d > 0. ∀h > 0. x + h ∈ S ⟶ h < d ⟶ f x < f (x + h)" using assms proof - from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at] obtain s where s: "0 < s" and all: "⋀xa. xa∈S ⟹ xa ≠ x ∧ dist xa x < s ⟶ ¦(f xa - f x) / (xa - x) - l¦ < l" by (auto simp: dist_real_def) then show ?thesis proof (intro exI conjI strip) show "0 < s" by (rule s) next fix h :: real assume "0 < h" "h < s" "x + h ∈ S" with all [of "x + h"] show "f x < f (x+h)" proof (simp add: abs_if dist_real_def pos_less_divide_eq split: if_split_asm) assume "¬ (f (x + h) - f x) / h < l" and h: "0 < h" with l have "0 < (f (x + h) - f x) / h" by arith then show "f x < f (x + h)" by (simp add: pos_less_divide_eq h) qed qed qed lemma DERIV_pos_inc_right: fixes f :: "real ⇒ real" assumes der: "DERIV f x :> l" and l: "0 < l" shows "∃d > 0. ∀h > 0. h < d ⟶ f x < f (x + h)" using has_real_derivative_pos_inc_right[OF assms] by auto lemma has_real_derivative_neg_dec_left: fixes f :: "real ⇒ real" assumes der: "(f has_real_derivative l) (at x within S)" and "l < 0" shows "∃d > 0. ∀h > 0. x - h ∈ S ⟶ h < d ⟶ f x < f (x - h)" proof - from ‹l < 0› have l: "- l > 0" by simp from der [THEN has_field_derivativeD, THEN tendstoD, OF