(* 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))" by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at) 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)" unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp 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) lemmas has_derivative_within_subset = has_derivative_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_within_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) 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 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_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_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_within_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)" apply (subst tendsto_norm_zero_iff[symmetric], rule filterlim_cong) apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide) done 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 .. 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› lemma has_field_derivative_iff_has_vector_derivative: "(f has_field_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)" unfolding has_field_derivative_def by (rule has_derivative_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_field_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_field_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]) 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_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_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_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_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 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 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 pos_less_divide_eq dist_real_def 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_neg_dec_left: fixes f :: "real ⇒ real" assumes der: "DERIV f x :> l" and l: "l < 0" shows "∃d > 0. ∀h > 0. h < d ⟶ f x < f (x - h)" using has_real_derivative_neg_dec_left[OF assms] by auto lemma has_real_derivative_pos_inc_left: fixes f :: "real ⇒ real" shows "(f has_real_derivative l) (at x within S) ⟹ 0 < l ⟹ ∃d>0. ∀h>0. x - h ∈ S ⟶ h < d ⟶ f (x - h) < f x" by (rule has_real_derivative_neg_dec_left [of "λx. - f x" "-l" x S, simplified]) (auto simp add: DERIV_minus) lemma DERIV_pos_inc_left: fixes f :: "real ⇒ real" shows "DERIV f x :> l ⟹ 0 < l ⟹ ∃d > 0. ∀h > 0. h < d ⟶ f (x - h) < f x" using has_real_derivative_pos_inc_left by blast lemma has_real_derivative_neg_dec_right: fixes f :: "real ⇒ real" shows "(f has_real_derivative l) (at x within S) ⟹ l < 0 ⟹ ∃d > 0. ∀h > 0. x + h ∈ S ⟶ h < d ⟶ f x > f (x + h)" by (rule has_real_derivative_pos_inc_right [of "λx. - f x" "-l" x S, simplified]) (auto simp add: DERIV_minus) lemma DERIV_neg_dec_right: fixes f :: "real ⇒ real" shows "DERIV f x :> l ⟹ l < 0 ⟹ ∃d > 0. ∀h > 0. h < d ⟶ f x > f (x + h)" using has_real_derivative_neg_dec_right by blast lemma DERIV_local_max: fixes f :: "real ⇒ real" assumes der: "DERIV f x :> l" and d: "0 < d" and le: "∀y. ¦x - y¦ < d ⟶ f y ≤ f x" shows "l = 0" proof (cases rule: linorder_cases [of l 0]) case equal then show ?thesis . next case less from DERIV_neg_dec_left [OF der less] obtain d' where d': "0 < d'" and lt: "∀h > 0. h < d' ⟶ f x < f (x - h)" by blast obtain e where "0 < e ∧ e < d ∧ e < d'" using field_lbound_gt_zero [OF d d'] .. with lt le [THEN spec [where x="x - e"]] show ?thesis by (auto simp add: abs_if) next case greater from DERIV_pos_inc_right [OF der greater] obtain d' where d': "0 < d'" and lt: "∀h > 0. h < d' ⟶ f x < f (x + h)" by blast obtain e where "0 < e ∧ e < d ∧ e < d'" using field_lbound_gt_zero [OF d d'] .. with lt le [THEN spec [where x="x + e"]] show ?thesis by (auto simp add: abs_if) qed text ‹Similar theorem for a local minimum› lemma DERIV_local_min: fixes f :: "real ⇒ real" shows "DERIV f x :> l ⟹ 0 < d ⟹ ∀y. ¦x - y¦ < d ⟶ f x ≤ f y ⟹ l = 0" by (drule DERIV_minus [THEN DERIV_local_max]) auto text‹In particular, if a function is locally flat› lemma DERIV_local_const: fixes f :: "real ⇒ real" shows "DERIV f x :> l ⟹ 0 < d ⟹ ∀y. ¦x - y¦ < d ⟶ f x = f y ⟹ l = 0" by (auto dest!: DERIV_local_max) subsection ‹Rolle's Theorem› text ‹Lemma about introducing open ball in open interval› lemma lemma_interval_lt: fixes a b x :: real assumes "a < x" "x < b" shows "∃d. 0 < d ∧ (∀y. ¦x - y¦ < d ⟶ a < y ∧ y < b)" using linorder_linear [of "x - a" "b - x"] proof assume "x - a ≤ b - x" with assms show ?thesis by (rule_tac x = "x - a" in exI) auto next assume "b - x ≤ x - a" with assms show ?thesis by (rule_tac x = "b - x" in exI) auto qed lemma lemma_interval: "a < x ⟹ x < b ⟹ ∃d. 0 < d ∧ (∀y. ¦x - y¦ < d ⟶ a ≤ y ∧ y ≤ b)" for a b x :: real by (force dest: lemma_interval_lt) text ‹Rolle's Theorem. If \<^term>‹f› is defined and continuous on the closed interval ‹[a,b]› and differentiable on the open interval ‹(a,b)›, and \<^term>‹f a = f b›, then there exists ‹x0 ∈ (a,b)› such that \<^term>‹f' x0 = 0›› theorem Rolle_deriv: fixes f :: "real ⇒ real" assumes "a < b" and fab: "f a = f b" and contf: "continuous_on {a..b} f" and derf: "⋀x. ⟦a < x; x < b⟧ ⟹ (f has_derivative f' x) (at x)" shows "∃z. a < z ∧ z < b ∧ f' z = (λv. 0)" proof - have le: "a ≤ b" using ‹a < b› by simp have "(a + b) / 2 ∈ {a..b}" using assms(1) by auto then have *: "{a..b} ≠ {}" by auto obtain x where x_max: "∀z. a ≤ z ∧ z ≤ b ⟶ f z ≤ f x" and "a ≤ x" "x ≤ b" using continuous_attains_sup[OF compact_Icc * contf] by (meson atLeastAtMost_iff) obtain x' where x'_min: "∀z. a ≤ z ∧ z ≤ b ⟶ f x' ≤ f z" and "a ≤ x'" "x' ≤ b" using continuous_attains_inf[OF compact_Icc * contf] by (meson atLeastAtMost_iff) consider "a < x" "x < b" | "x = a ∨ x = b" using ‹a ≤ x› ‹x ≤ b› by arith then show ?thesis proof cases case 1 ― ‹\<^term>‹f› attains its maximum within the interval› then obtain l where der: "DERIV f x :> l" using derf differentiable_def real_differentiable_def by blast obtain d where d: "0 < d" and bound: "∀y. ¦x - y¦ < d ⟶ a ≤ y ∧ y ≤ b" using lemma_interval [OF 1] by blast then have bound': "∀y. ¦x - y¦ < d ⟶ f y ≤ f x" using x_max by blast ― ‹the derivative at a local maximum is zero› have "l = 0" by (rule DERIV_local_max [OF der d bound']) with 1 der derf [of x] show ?thesis by (metis has_derivative_unique has_field_derivative_def mult_zero_left) next case 2 then have fx: "f b = f x" by (auto simp add: fab) consider "a < x'" "x' < b" | "x' = a ∨ x' = b" using ‹a ≤ x'› ‹x' ≤ b› by arith then show ?thesis proof cases case 1 ― ‹\<^term>‹f› attains its minimum within the interval› then obtain l where der: "DERIV f x' :> l" using derf differentiable_def real_differentiable_def by blast from lemma_interval [OF 1] obtain d where d: "0<d" and bound: "∀y. ¦x'-y¦ < d ⟶ a ≤ y ∧ y ≤ b" by blast then have bound': "∀y. ¦x' - y¦ < d ⟶ f x' ≤ f y" using x'_min by blast have "l = 0" by (rule DERIV_local_min [OF der d bound']) ― ‹the derivative at a local minimum is zero› then show ?thesis using 1 der derf [of x'] by (metis has_derivative_unique has_field_derivative_def mult_zero_left) next case 2 ― ‹\<^term>‹f› is constant throughout the interval› then have fx': "f b = f x'" by (auto simp: fab) from dense [OF ‹a < b›] obtain r where r: "a < r" "r < b" by blast obtain d where d: "0 < d" and bound: "∀y. ¦r - y¦ < d ⟶ a ≤ y ∧ y ≤ b" using lemma_interval [OF r] by blast have eq_fb: "f z = f b" if "a ≤ z" and "z ≤ b" for z proof (rule order_antisym) show "f z ≤ f b" by (simp add: fx x_max that) show "f b ≤ f z" by (simp add: fx' x'_min that) qed have bound': "∀y. ¦r - y¦ < d ⟶ f r = f y" proof (intro strip) fix y :: real assume lt: "¦r - y¦ < d" then have "f y = f b" by (simp add: eq_fb bound) then show "f r = f y" by (simp add: eq_fb r order_less_imp_le) qed obtain l where der: "DERIV f r :> l" using derf differentiable_def r(1) r(2) real_differentiable_def by blast have "l = 0" by (rule DERIV_local_const [OF der d bound']) ― ‹the derivative of a constant function is zero› with r der derf [of r] show ?thesis by (metis has_derivative_unique has_field_derivative_def mult_zero_left) qed qed qed corollary Rolle: fixes a b :: real assumes ab: "a < b" "f a = f b" "continuous_on {a..b} f" and dif [rule_format]: "⋀x. ⟦a < x; x < b⟧ ⟹ f differentiable (at x)" shows "∃z. a < z ∧ z < b ∧ DERIV f z :> 0" proof - obtain f' where f': "⋀x. ⟦a < x; x < b⟧ ⟹ (f has_derivative f' x) (at x)" using dif unfolding differentiable_def by metis then have "∃z. a < z ∧ z < b ∧ f' z = (λv. 0)" by (metis Rolle_deriv [OF ab]) then show ?thesis using f' has_derivative_imp_has_field_derivative by fastforce qed subsection ‹Mean Value Theorem› theorem mvt: fixes f :: "real ⇒ real" assumes "a < b" and contf: "continuous_on {a..b} f" and derf: "⋀x. ⟦a < x; x < b⟧ ⟹ (f has_derivative f' x) (at x)" obtains ξ where "a < ξ" "ξ < b" "f b - f a = (f' ξ) (b - a)" proof - have "∃x. a < x ∧ x < b ∧ (λy. f' x y - (f b - f a) / (b - a) * y) = (λv. 0)" proof (intro Rolle_deriv[OF ‹a < b›]) fix x assume x: "a < x" "x < b" show "((λx. f x - (f b - f a) / (b - a) * x) has_derivative (λy. f' x y - (f b - f a) / (b - a) * y)) (at x)" by (intro derivative_intros derf[OF x]) qed (use assms in ‹auto intro!: continuous_intros simp: field_simps›) then obtain ξ where "a < ξ" "ξ < b" "(λy. f' ξ y - (f b - f a) / (b - a) * y) = (λv. 0)" by metis then show ?thesis by (metis (no_types, hide_lams) that add.right_neutral add_diff_cancel_left' add_diff_eq ‹a < b› less_irrefl nonzero_eq_divide_eq) qed theorem MVT: fixes a b :: real assumes lt: "a < b" and contf: "continuous_on {a..b} f" and dif: "⋀x. ⟦a < x; x < b⟧ ⟹ f differentiable (at x)" shows "∃l z. a < z ∧ z < b ∧ DERIV f z :> l ∧ f b - f a = (b - a) * l" proof - obtain f' :: "real ⇒ real ⇒ real" where derf: "⋀x. a < x ⟹ x < b ⟹ (f has_derivative f' x) (at x)" using dif unfolding differentiable_def by metis then obtain z where "a < z" "z < b" "f b - f a = (f' z) (b - a)" using mvt [OF lt contf] by blast then show ?thesis by (simp add: ac_simps) (metis derf dif has_derivative_unique has_field_derivative_imp_has_derivative real_differentiable_def) qed corollary MVT2: assumes "a < b" and der: "⋀x. ⟦a ≤ x; x ≤ b⟧ ⟹ DERIV f x :> f' x" shows "∃z::real. a < z ∧ z < b ∧ (f b - f a = (b - a) * f' z)" proof - have "∃l z. a < z ∧ z < b ∧ (f has_real_derivative l) (at z) ∧ f b - f a = (b - a) * l" proof (rule MVT [OF ‹a < b›]) show "continuous_on {a..b} f" by (meson DERIV_continuous atLeastAtMost_iff continuous_at_imp_continuous_on der) show "⋀x. ⟦a < x; x < b⟧ ⟹ f differentiable (at x)" using assms by (force dest: order_less_imp_le simp add: real_differentiable_def) qed with assms show ?thesis by (blast dest: DERIV_unique order_less_imp_le) qed lemma pos_deriv_imp_strict_mono: assumes "⋀x. (f has_real_derivative f' x) (at x)" assumes "⋀x. f' x > 0" shows "strict_mono f" proof (rule strict_monoI) fix x y :: real assume xy: "x < y" from assms and xy have "∃z>x. z < y ∧ f y - f x = (y - x) * f' z" by (intro MVT2) (auto dest: connectedD_interval) then obtain z where z: "z > x" "z < y" "f y - f x = (y - x) * f' z" by blast note ‹f y - f x = (y - x) * f' z› also have "(y - x) * f' z > 0" using xy assms by (intro mult_pos_pos) auto finally show "f x < f y" by simp qed proposition deriv_nonneg_imp_mono: assumes deriv: "⋀x. x ∈ {a..b} ⟹ (g has_real_derivative g' x) (at x)" assumes nonneg: "⋀x. x ∈ {a..b} ⟹ g' x ≥ 0" assumes ab: "a ≤ b" shows "g a ≤ g b" proof (cases "a < b") assume "a < b" from deriv have "⋀x. ⟦x ≥ a; x ≤ b⟧ ⟹ (g has_real_derivative g' x) (at x)" by simp with MVT2[OF ‹a < b›] and deriv obtain ξ where ξ_ab: "ξ > a" "ξ < b" and g_ab: "g b - g a = (b - a) * g' ξ" by blast from ξ_ab ab nonneg have "(b - a) * g' ξ ≥ 0" by simp with g_ab show ?thesis by simp qed (insert ab, simp) subsubsection ‹A function is constant if its derivative is 0 over an interval.› lemma DERIV_isconst_end: fixes f :: "real ⇒ real" assumes "a < b" and contf: "continuous_on {a..b} f" and 0: "⋀x. ⟦a < x; x < b⟧ ⟹ DERIV f x :> 0" shows "f b = f a" using MVT [OF ‹a < b›] "0" DERIV_unique contf real_differentiable_def by (fastforce simp: algebra_simps) lemma DERIV_isconst2: fixes f :: "real ⇒ real" assumes "a < b" and contf: "continuous_on {a..b} f" and derf: "⋀x. ⟦a < x; x < b⟧ ⟹ DERIV f x :> 0" and "a ≤ x" "x ≤ b" shows "f x = f a" proof (cases "a < x") case True have *: "continuous_on {a..x} f" using ‹x ≤ b› contf continuous_on_subset by fastforce show ?thesis by (rule DERIV_isconst_end [OF True *]) (use ‹x ≤ b› derf in auto) qed (use ‹a ≤ x› in auto) lemma DERIV_isconst3: fixes a b x y :: real assumes "a < b" and "x ∈ {a <..< b}" and "y ∈ {a <..< b}" and derivable: "⋀x. x ∈ {a <..< b} ⟹ DERIV f x :> 0" shows "f x = f y" proof (cases "x = y") case False let ?a = "min x y" let ?b = "max x y" have *: "DERIV f z :> 0" if "?a ≤ z" "z ≤ ?b" for z proof - have "a < z" and "z < b" using that ‹x ∈ {a <..< b}› and ‹y ∈ {a <..< b}› by auto then have "z ∈ {a<..<b}" by auto then show "DERIV f z :> 0" by (rule derivable) qed have isCont: "continuous_on {?a..?b} f" by (meson * DERIV_continuous_on atLeastAtMost_iff has_field_derivative_at_within) have DERIV: "⋀z. ⟦?a < z; z < ?b⟧ ⟹ DERIV f z :> 0" using * by auto have "?a < ?b" using ‹x ≠ y› by auto from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y] show ?thesis by auto qed auto lemma DERIV_isconst_all: fixes f :: "real ⇒ real" shows "∀x. DERIV f x :> 0 ⟹ f x = f y" apply (rule linorder_cases [of x y]) apply (metis DERIV_continuous DERIV_isconst_end continuous_at_imp_continuous_on)+ done lemma DERIV_const_ratio_const: fixes f :: "real ⇒ real" assumes "a ≠ b" and df: "⋀x. DERIV f x :> k" shows "f b - f a = (b - a) * k" proof (cases a b rule: linorder_cases) case less show ?thesis using MVT [OF less] df by (metis DERIV_continuous DERIV_unique continuous_at_imp_continuous_on real_differentiable_def) next case greater have "f a - f b = (a - b) * k" using MVT [OF greater] df by (metis DERIV_continuous DERIV_unique continuous_at_imp_continuous_on real_differentiable_def) then show ?thesis by (simp add: algebra_simps) qed auto lemma DERIV_const_ratio_const2: fixes f :: "real ⇒ real" assumes "a ≠ b" and df: "⋀x. DERIV f x :> k" shows "(f b - f a) / (b - a) = k" using DERIV_const_ratio_const [OF assms] ‹a ≠ b› by auto lemma real_average_minus_first [simp]: "(a + b) / 2 - a = (b - a) / 2" for a b :: real by simp lemma real_average_minus_second [simp]: "(b + a) / 2 - a = (b - a) / 2" for a b :: real by simp text ‹Gallileo's "trick": average velocity = av. of end velocities.› lemma DERIV_const_average: fixes v :: "real ⇒ real" and a b :: real assumes neq: "a ≠ b" and der: "⋀x. DERIV v x :> k" shows "v ((a + b) / 2) = (v a + v b) / 2" proof (cases rule: linorder_cases [of a b]) case equal with neq show ?thesis by simp next case less have "(v b - v a) / (b - a) = k" by (rule DERIV_const_ratio_const2 [OF neq der]) then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k" by simp moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k" by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq) ultimately show ?thesis using neq by force next case greater have "(v b - v a) / (b - a) = k" by (rule DERIV_const_ratio_const2 [OF neq der]) then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k" by simp moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k" by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq) ultimately show ?thesis using neq by (force simp add: add.commute) qed subsubsection‹A function with positive derivative is increasing› text ‹A simple proof using the MVT, by Jeremy Avigad. And variants.› lemma DERIV_pos_imp_increasing_open: fixes a b :: real and f :: "real ⇒ real" assumes "a < b" and "⋀x. a < x ⟹ x < b ⟹ (∃y. DERIV f x :> y ∧ y > 0)" and con: "continuous_on {a..b} f" shows "f a < f b" proof (rule ccontr) assume f: "¬ ?thesis" have "∃l z. a < z ∧ z < b ∧ DERIV f z :> l ∧ f b - f a = (b - a) * l" by (rule MVT) (use assms real_differentiable_def in ‹force+›) then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" and "f b - f a = (b - a) * l" by auto with assms f have "¬ l > 0" by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le) with assms z show False by (metis DERIV_unique) qed lemma DERIV_pos_imp_increasing: fixes a b :: real and f :: "real ⇒ real" assumes "a < b" and der: "⋀x. ⟦a ≤ x; x ≤ b⟧ ⟹ ∃y. DERIV f x :> y ∧ y > 0" shows "f a < f b" by (metis less_le_not_le DERIV_atLeastAtMost_imp_continuous_on DERIV_pos_imp_increasing_open [OF ‹a < b›] der) lemma DERIV_nonneg_imp_nondecreasing: fixes a b :: real and f :: "real ⇒ real" assumes "a ≤ b" and "⋀x. ⟦a ≤ x; x ≤ b⟧ ⟹ ∃y. DERIV f x :> y ∧ y ≥ 0" shows "f a ≤ f b" proof (rule ccontr, cases "a = b") assume "¬ ?thesis" and "a = b" then show False by auto next assume *: "¬ ?thesis" assume "a ≠ b" with ‹a ≤ b› have "a < b" by linarith moreover have "continuous_on {a..b} f" by (meson DERIV_isCont assms(2) atLeastAtMost_iff continuous_at_imp_continuous_on) ultimately have "∃l z. a < z ∧ z < b ∧ DERIV f z :> l ∧ f b - f a = (b - a) * l" using assms MVT [OF ‹a < b›, of f] real_differentiable_def less_eq_real_def by blast then obtain l z where lz: "a < z" "z < b" "DERIV f z :> l" and **: "f b - f a = (b - a) * l" by auto with * have "a < b" "f b < f a" by auto with ** have "¬ l ≥ 0" by (auto simp add: not_le algebra_simps) (metis * add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl) with assms lz show False by (metis DERIV_unique order_less_imp_le) qed lemma DERIV_neg_imp_decreasing_open: fixes a b :: real and f :: "real ⇒ real" assumes "a < b" and "⋀x. a < x ⟹ x < b ⟹ ∃y. DERIV f x :> y ∧ y < 0" and con: "continuous_on {a..b} f" shows "f a > f b" proof - have "(λx. -f x) a < (λx. -f x) b" proof (rule DERIV_pos_imp_increasing_open [of a b]) show "⋀x. ⟦a < x; x < b⟧ ⟹ ∃y. ((λx. - f x) has_real_derivative y) (at x) ∧ 0 < y" using assms by simp (metis field_differentiable_minus neg_0_less_iff_less) show "continuous_on {a..b} (λx. - f x)" using con continuous_on_minus by blast qed (use assms in auto) then show ?thesis by simp qed lemma DERIV_neg_imp_decreasing: fixes a b :: real and f :: "real ⇒ real" assumes "a < b" and der: "⋀x. ⟦a ≤ x; x ≤ b⟧ ⟹ ∃y. DERIV f x :> y ∧ y < 0" shows "f a > f b" by (metis less_le_not_le DERIV_atLeastAtMost_imp_continuous_on DERIV_neg_imp_decreasing_open [OF ‹a < b›] der) lemma DERIV_nonpos_imp_nonincreasing: fixes a b :: real and f :: "real ⇒ real" assumes "a ≤ b" and "⋀x. ⟦a ≤ x; x ≤ b⟧ ⟹ ∃y. DERIV f x :> y ∧ y ≤ 0" shows "f a ≥ f b" proof - have "(λx. -f x) a ≤ (λx. -f x) b" using DERIV_nonneg_imp_nondecreasing [of a b "λx. -f x"] assms DERIV_minus by fastforce then show ?thesis by simp qed lemma DERIV_pos_imp_increasing_at_bot: fixes f :: "real ⇒ real" assumes "⋀x. x ≤ b ⟹ (∃y. DERIV f x :> y ∧ y > 0)" and lim: "(f ⤏ flim) at_bot" shows "flim < f b" proof - have "∃N. ∀n≤N. f n ≤ f (b - 1)" by (rule_tac x="b - 2" in exI) (force intro: order.strict_implies_order DERIV_pos_imp_increasing assms) then have "flim ≤ f (b - 1)" by (auto simp: eventually_at_bot_linorder tendsto_upperbound [OF lim]) also have "… < f b" by (force intro: DERIV_pos_imp_increasing [where f=f] assms) finally show ?thesis . qed lemma DERIV_neg_imp_decreasing_at_top: fixes f :: "real ⇒ real" assumes der: "⋀x. x ≥ b ⟹ ∃y. DERIV f x :> y ∧ y < 0" and lim: "(f ⤏ flim) at_top" shows "flim < f b" apply (rule DERIV_pos_imp_increasing_at_bot [where f = "λi. f (-i)" and b = "-b", simplified]) apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less) apply (metis filterlim_at_top_mirror lim) done text ‹Derivative of inverse function› lemma DERIV_inverse_function: fixes f g :: "real ⇒ real" assumes der: "DERIV f (g x) :> D" and neq: "D ≠ 0" and x: "a < x" "x < b" and inj: "⋀y. ⟦a < y; y < b⟧ ⟹ f (g y) = y" and cont: "isCont g x" shows "DERIV g x :> inverse D" unfolding has_field_derivative_iff proof (rule LIM_equal2) show "0 < min (x - a) (b - x)" using x by arith next fix y assume "norm (y - x) < min (x - a) (b - x)" then have "a < y" and "y < b" by (simp_all add: abs_less_iff) then show "(g y - g x) / (y - x) = inverse ((f (g y) - x) / (g y - g x))" by (simp add: inj) next have "(λz. (f z - f (g x)) / (z - g x)) ─g x→ D" by (rule der [unfolded has_field_derivative_iff]) then have 1: "(λz. (f z - x) / (z - g x)) ─g x→ D" using inj x by simp have 2: "∃d>0. ∀y. y ≠ x ∧ norm (y - x) < d ⟶ g y ≠ g x" proof (rule exI, safe) show "0 < min (x - a) (b - x)" using x by simp next fix y assume "norm (y - x) < min (x - a) (b - x)" then have y: "a < y" "y < b" by (simp_all add: abs_less_iff) assume "g y = g x" then have "f (g y) = f (g x)" by simp then have "y = x" using inj y x by simp also assume "y ≠ x" finally show False by simp qed have "(λy. (f (g y) - x) / (g y - g x)) ─x→ D" using cont 1 2 by (rule isCont_LIM_compose2) then show "(λy. inverse ((f (g y) - x) / (g y - g x))) ─x→ inverse D" using neq by (rule tendsto_inverse) qed subsection ‹Generalized Mean Value Theorem› theorem GMVT: fixes a b :: real assumes alb: "a < b" and fc: "∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x" and fd: "∀x. a < x ∧ x < b ⟶ f differentiable (at x)" and gc: "∀x. a ≤ x ∧ x ≤ b ⟶ isCont g x" and gd: "∀x. a < x ∧ x < b ⟶ g differentiable (at x)" shows "∃g'c f'c c. DERIV g c :> g'c ∧ DERIV f c :> f'c ∧ a < c ∧ c < b ∧ (f b - f a) * g'c = (g b - g a) * f'c" proof - let ?h = "λx. (f b - f a) * g x - (g b - g a) * f x" have "∃l z. a < z ∧ z < b ∧ DERIV ?h z :> l ∧ ?h b - ?h a = (b - a) * l" proof (rule MVT) from assms show "a < b" by simp show "continuous_on {a..b} ?h" by (simp add: continuous_at_imp_continuous_on fc gc) show "⋀x. ⟦a < x; x < b⟧ ⟹ ?h differentiable (at x)" using fd gd by simp qed then obtain l where l: "∃z. a < z ∧ z < b ∧ DERIV ?h z :> l ∧ ?h b - ?h a = (b - a) * l" .. then obtain c where c: "a < c ∧ c < b ∧ DERIV ?h c :> l ∧ ?h b - ?h a = (b - a) * l" .. from c have cint: "a < c ∧ c < b" by auto then obtain g'c where g'c: "DERIV g c :> g'c" using gd real_differentiable_def by blast from c have "a < c ∧ c < b" by auto then obtain f'c where f'c: "DERIV f c :> f'c" using fd real_differentiable_def by blast from c have "DERIV ?h c :> l" by auto moreover have "DERIV ?h c :> g'c * (f b - f a) - f'c * (g b - g a)" using g'c f'c by (auto intro!: derivative_eq_intros) ultimately have leq: "l = g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique) have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" proof - from c have "?h b - ?h a = (b - a) * l" by auto also from leq have "… = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp finally show ?thesis by simp qed moreover have "?h b - ?h a = 0" proof - have "?h b - ?h a = ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) - ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))" by (simp add: algebra_simps) then show ?thesis by auto qed ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp then have "g'c * (f b - f a) = f'c * (g b - g a)" by simp then have "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps) with g'c f'c cint show ?thesis by auto qed lemma GMVT': fixes f g :: "real ⇒ real" assumes "a < b" and isCont_f: "⋀z. a ≤ z ⟹ z ≤ b ⟹ isCont f z" and isCont_g: "⋀z. a ≤ z ⟹ z ≤ b ⟹ isCont g z" and DERIV_g: "⋀z. a < z ⟹ z < b ⟹ DERIV g z :> (g' z)" and DERIV_f: "⋀z. a < z ⟹ z < b ⟹ DERIV f z :> (f' z)" shows "∃c. a < c ∧ c < b ∧ (f b - f a) * g' c = (g b - g a) * f' c" proof - have "∃g'c f'c c. DERIV g c :> g'c ∧ DERIV f c :> f'c ∧ a < c ∧ c < b ∧ (f b - f a) * g'c = (g b - g a) * f'c" using assms by (intro GMVT) (force simp: real_differentiable_def)+ then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c" using DERIV_f DERIV_g by (force dest: DERIV_unique) then show ?thesis by auto qed subsection ‹L'Hopitals rule› lemma isCont_If_ge: fixes a :: "'a :: linorder_topology" assumes "continuous (at_left a) g" and f: "(f ⤏ g a) (at_right a)" shows "isCont (λx. if x ≤ a then g x else f x) a" (is "isCont ?gf a") proof - have g: "(g ⤏ g a) (at_left a)" using assms continuous_within by blast show ?thesis unfolding isCont_def continuous_within proof (intro filterlim_split_at; simp) show "(?gf ⤏ g a) (at_left a)" by (subst filterlim_cong[OF refl refl, where g=g]) (simp_all add: eventually_at_filter less_le g) show "(?gf ⤏ g a) (at_right a)" by (subst filterlim_cong[OF refl refl, where g=f]) (simp_all add: eventually_at_filter less_le f) qed qed lemma lhopital_right_0: fixes f0 g0 :: "real ⇒ real" assumes f_0: "(f0 ⤏ 0) (at_right 0)" and g_0: "(g0 ⤏ 0) (at_right 0)" and ev: "eventually (λx. g0 x ≠ 0) (at_right 0)" "eventually (λx. g' x ≠ 0) (at_right 0)" "eventually (λx. DERIV f0 x :> f' x) (at_right 0)" "eventually (λx. DERIV g0 x :> g' x) (at_right 0)" and lim: "filterlim (λ x. (f' x / g' x)) F (at_right 0)" shows "filterlim (λ x. f0 x / g0 x) F (at_right 0)" proof - define f where [abs_def]: "f x = (if x ≤ 0 then 0 else f0 x)" for x then have "f 0 = 0" by simp define g where [abs_def]: "g x = (if x ≤ 0 then 0 else g0 x)" for x then have "g 0 = 0" by simp have "eventually (λx. g0 x ≠ 0 ∧ g' x ≠ 0 ∧ DERIV f0 x :> (f' x) ∧ DERIV g0 x :> (g' x)) (at_right 0)" using ev by eventually_elim auto then obtain a where [arith]: "0 < a" and g0_neq_0: "⋀x. 0 < x ⟹ x < a ⟹ g0 x ≠ 0" and g'_neq_0: "⋀x. 0 < x ⟹ x < a ⟹ g' x ≠ 0" and f0: "⋀x. 0 < x ⟹ x < a ⟹ DERIV f0 x :> (f' x)" and g0: "⋀x. 0 < x ⟹ x < a ⟹ DERIV g0 x :> (g' x)" unfolding eventually_at by (auto simp: dist_real_def) have g_neq_0: "⋀x. 0 < x ⟹ x < a ⟹ g x ≠ 0" using g0_neq_0 by (simp add: g_def) have f: "DERIV f x :> (f' x)" if x: "0 < x" "x < a" for x using that by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]]) (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) have g: "DERIV g x :> (g' x)" if x: "0 < x" "x < a" for x using that by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]]) (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) have "isCont f 0" unfolding f_def by (intro isCont_If_ge f_0 continuous_const) have "isCont g 0" unfolding g_def by (intro isCont_If_ge g_0 continuous_const) have "∃ζ. ∀x∈{0 <..< a}. 0 < ζ x ∧ ζ x < x ∧ f x / g x = f' (ζ x) / g' (ζ x)" proof (rule bchoice, rule ballI) fix x assume "x ∈ {0 <..< a}" then have x[arith]: "0 < x" "x < a" by auto with g'_neq_0 g_neq_0 ‹g 0 = 0› have g': "⋀x. 0 < x ⟹ x < a ⟹ 0 ≠ g' x" "g 0 ≠ g x" by auto have "⋀x. 0 ≤ x ⟹ x < a ⟹ isCont f x" using ‹isCont f 0› f by (auto intro: DERIV_isCont simp: le_less) moreover have "⋀x. 0 ≤ x ⟹ x < a ⟹ isCont g x" using ‹isCont g 0› g by (auto intro: DERIV_isCont simp: le_less) ultimately have "∃c. 0 < c ∧ c < x ∧ (f x - f 0) * g' c = (g x - g 0) * f' c" using f g ‹x < a› by (intro GMVT') auto then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c" by blast moreover from * g'(1)[of c] g'(2) have "(f x - f 0) / (g x - g 0) = f' c / g' c" by (simp add: field_simps) ultimately show "∃y. 0 < y ∧ y < x ∧ f x / g x = f' y / g' y" using ‹f 0 = 0› ‹g 0 = 0› by (auto intro!: exI[of _ c]) qed then obtain ζ where "∀x∈{0 <..< a}. 0 < ζ x ∧ ζ x < x ∧ f x / g x = f' (ζ x) / g' (ζ x)" .. then have ζ: "eventually (λx. 0 < ζ x ∧ ζ x < x ∧ f x / g x = f' (ζ x) / g' (ζ x)) (at_right 0)" unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def) moreover from ζ have "eventually (λx. norm (ζ x) ≤ x) (at_right 0)" by eventually_elim auto then have "((λx. norm (ζ x)) ⤏ 0) (at_right 0)" by (rule_tac real_tendsto_sandwich[where f="λx. 0" and h="λx. x"]) auto then have "(ζ ⤏ 0) (at_right 0)" by (rule tendsto_norm_zero_cancel) with ζ have "filterlim ζ (at_right 0) (at_right 0)" by (auto elim!: eventually_mono simp: filterlim_at) from this lim have "filterlim (λt. f' (ζ t) / g' (ζ t)) F (at_right 0)" by (rule_tac filterlim_compose[of _ _ _ ζ]) ultimately have "filterlim (λt. f t / g t) F (at_right 0)" (is ?P) by (rule_tac filterlim_cong[THEN iffD1, OF refl refl]) (auto elim: eventually_mono) also have "?P ⟷ ?thesis" by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter) finally show ?thesis . qed lemma lhopital_right: "(f ⤏ 0) (at_right x) ⟹ (g ⤏ 0) (at_right x) ⟹ eventually (λx. g x ≠ 0) (at_right x) ⟹ eventually (λx. g' x ≠ 0) (at_right x) ⟹ eventually (λx. DERIV f x :> f' x) (at_right x) ⟹ eventually (λx. DERIV g x :> g' x) (at_right x) ⟹ filterlim (λ x. (f' x / g' x)) F (at_right x) ⟹ filterlim (λ x. f x / g x) F (at_right x)" for x :: real unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift by (rule lhopital_right_0) lemma lhopital_left: "(f ⤏ 0) (at_left x) ⟹ (g ⤏ 0) (at_left x) ⟹ eventually (λx. g x ≠ 0) (at_left x) ⟹ eventually (λx. g' x ≠ 0) (at_left x) ⟹ eventually (λx. DERIV f x :> f' x) (at_left x) ⟹ eventually (λx. DERIV g x :> g' x) (at_left x) ⟹ filterlim (λ x. (f' x / g' x)) F (at_left x) ⟹ filterlim (λ x. f x / g x) F (at_left x)" for x :: real unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror by (rule lhopital_right[where f'="λx. - f' (- x)"]) (auto simp: DERIV_mirror) lemma lhopital: "(f ⤏ 0) (at x) ⟹ (g ⤏ 0) (at x) ⟹ eventually (λx. g x ≠ 0) (at x) ⟹ eventually (λx. g' x ≠ 0) (at x) ⟹ eventually (λx. DERIV f x :> f' x) (at x) ⟹ eventually (λx. DERIV g x :> g' x) (at x) ⟹ filterlim (λ x. (f' x / g' x)) F (at x) ⟹ filterlim (λ x. f x / g x) F (at x)" for x :: real unfolding eventually_at_split filterlim_at_split by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f']) lemma lhopital_right_0_at_top: fixes f g :: "real ⇒ real" assumes g_0: "LIM x at_right 0. g x :> at_top" and ev: "eventually (λx. g' x ≠ 0) (at_right 0)" "eventually (λx. DERIV f x :> f' x) (at_right 0)" "eventually (λx. DERIV g x :> g' x) (at_right 0)" and lim: "((λ x. (f' x / g' x)) ⤏ x) (at_right 0)" shows "((λ x. f x / g x) ⤏ x) (at_right 0)" unfolding tendsto_iff proof safe fix e :: real assume "0 < e" with lim[unfolded tendsto_iff, rule_format, of "e / 4"] have "eventually (λt. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]] obtain a where [arith]: "0 < a" and g'_neq_0: "⋀x. 0 < x ⟹ x < a ⟹ g' x ≠ 0" and f0: "⋀x. 0 < x ⟹ x ≤ a ⟹ DERIV f x :> (f' x)" and g0: "⋀x. 0 < x ⟹ x ≤ a ⟹ DERIV g x :> (g' x)" and Df: "⋀t. 0 < t ⟹ t < a ⟹ dist (f' t / g' t) x < e / 4" unfolding eventually_at_le by (auto simp: dist_real_def) from Df have "eventually (λt. t < a) (at_right 0)" "eventually (λt::real. 0 < t) (at_right 0)" unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def) moreover have "eventually (λt. 0 < g t) (at_right 0)" "eventually (λt. g a < g t) (at_right 0)" using g_0 by (auto elim: eventually_mono simp: filterlim_at_top_dense) moreover have inv_g: "((λx. inverse (g x)) ⤏ 0) (at_right 0)" using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl] by (rule filterlim_compose) then have "((λx. norm (1 - g a * inverse (g x))) ⤏ norm (1 - g a * 0)) (at_right 0)" by (intro tendsto_intros) then have "((λx. norm (1 - g a / g x)) ⤏ 1) (at_right 0)" by (simp add: inverse_eq_divide) from this[unfolded tendsto_iff, rule_format, of 1] have "eventually (λx. norm (1 - g a / g x) < 2) (at_right 0)" by (auto elim!: eventually_mono simp: dist_real_def) moreover from inv_g have "((λt. norm ((f a - x * g a) * inverse (g t))) ⤏ norm ((f a - x * g a) * 0)) (at_right 0)" by (intro tendsto_intros) then have "((λt. norm (f a - x * g a) / norm (g t)) ⤏ 0) (at_right 0)" by (simp add: inverse_eq_divide) from this[unfolded tendsto_iff, rule_format, of "e / 2"] ‹0 < e› have "eventually (λt. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)" by (auto simp: dist_real_def) ultimately show "eventually (λt. dist (f t / g t) x < e) (at_right 0)" proof eventually_elim fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t" assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2" have "∃y. t < y ∧ y < a ∧ (g a - g t) * f' y = (f a - f t) * g' y" using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+ then obtain y where [arith]: "t < y" "y < a" and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y" by blast from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y" using ‹g a < g t› g'_neq_0[of y] by (auto simp add: field_simps) have *: "f t / g t - x = ((f t - f a) / (g t - g a) - x) * (1 - g a / g t) + (f a - x * g a) / g t" by (simp add: field_simps) have "norm (f t / g t - x) ≤ norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)" unfolding * by (rule norm_triangle_ineq) also have "… = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)" by (simp add: abs_mult D_eq dist_real_def) also have "… < (e / 4) * 2 + e / 2" using ineq Df[of y] ‹0 < e› by (intro add_le_less_mono mult_mono) auto finally show "dist (f t / g t) x < e" by (simp add: dist_real_def) qed qed lemma lhopital_right_at_top: "LIM x at_right x. (g::real ⇒ real) x :> at_top ⟹ eventually (λx. g' x ≠ 0) (at_right x) ⟹ eventually (λx. DERIV f x :> f' x) (at_right x) ⟹ eventually (λx. DERIV g x :> g' x) (at_right x) ⟹ ((λ x. (f' x / g' x)) ⤏ y) (at_right x) ⟹ ((λ x. f x / g x) ⤏ y) (at_right x)" unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift by (rule lhopital_right_0_at_top) lemma lhopital_left_at_top: "LIM x at_left x. g x :> at_top ⟹ eventually (λx. g' x ≠ 0) (at_left x) ⟹ eventually (λx. DERIV f x :> f' x) (at_left x) ⟹ eventually (λx. DERIV g x :> g' x) (at_left x) ⟹ ((λ x. (f' x / g' x)) ⤏ y) (at_left x) ⟹ ((λ x. f x / g x) ⤏ y) (at_left x)" for x :: real unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror by (rule lhopital_right_at_top[where f'="λx. - f' (- x)"]) (auto simp: DERIV_mirror) lemma lhopital_at_top: "LIM x at x. (g::real ⇒ real) x :> at_top ⟹ eventually (λx. g' x ≠ 0) (at x) ⟹ eventually (λx. DERIV f x :> f' x) (at x) ⟹ eventually (λx. DERIV g x :> g' x) (at x) ⟹ ((λ x. (f' x / g' x)) ⤏ y) (at x) ⟹ ((λ x. f x / g x) ⤏ y) (at x)" unfolding eventually_at_split filterlim_at_split by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f']) lemma lhospital_at_top_at_top: fixes f g :: "real ⇒ real" assumes g_0: "LIM x at_top. g x :> at_top" and g': "eventually (λx. g' x ≠ 0) at_top" and Df: "eventually (λx. DERIV f x :> f' x) at_top" and Dg: "eventually (λx. DERIV g x :> g' x) at_top" and lim: "((λ x. (f' x / g' x)) ⤏ x) at_top" shows "((λ x. f x / g x) ⤏ x) at_top" unfolding filterlim_at_top_to_right proof (rule lhopital_right_0_at_top) let ?F = "λx. f (inverse x)" let ?G = "λx. g (inverse x)" let ?R = "at_right (0::real)" let ?D = "λf' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))" show "LIM x ?R. ?G x :> at_top" using g_0 unfolding filterlim_at_top_to_right . show "eventually (λx. DERIV ?G x :> ?D g' x) ?R" unfolding eventually_at_right_to_top using Dg eventually_ge_at_top[where c=1] by eventually_elim (rule derivative_eq_intros DERIV_chain'[where f=inverse] | simp)+ show "eventually (λx. DERIV ?F x :> ?D f' x) ?R" unfolding eventually_at_right_to_top using Df eventually_ge_at_top[where c=1] by eventually_elim (rule derivative_eq_intros DERIV_chain'[where f=inverse] | simp)+ show "eventually (λx. ?D g' x ≠ 0) ?R" unfolding eventually_at_right_to_top using g' eventually_ge_at_top[where c=1] by eventually_elim auto show "((λx. ?D f' x / ?D g' x) ⤏ x) ?R" unfolding filterlim_at_right_to_top apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim]) using eventually_ge_at_top[where c=1] by eventually_elim simp qed lemma lhopital_right_at_top_at_top: fixes f g :: "real ⇒ real" assumes f_0: "LIM x at_right a. f x :> at_top" assumes g_0: "LIM x at_right a. g x :> at_top" and ev: "eventually (λx. DERIV f x :> f' x) (at_right a)" "eventually (λx. DERIV g x :> g' x) (at_right a)" and lim: "filterlim (λ x. (f' x / g' x)) at_top (at_right a)" shows "filterlim (λ x. f x / g x) at_top (at_right a)" proof - from lim have pos: "eventually (λx. f' x / g' x > 0) (at_right a)" unfolding filterlim_at_top_dense by blast have "((λx. g x / f x) ⤏ 0) (at_right a)" proof (rule lhopital_right_at_top) from pos show "eventually (λx. f' x ≠ 0) (at_right a)" by eventually_elim auto from tendsto_inverse_0_at_top[OF lim] show "((λx. g' x / f' x) ⤏ 0) (at_right a)" by simp qed fact+ moreover from f_0 g_0 have "eventually (λx. f x > 0) (at_right a)" "eventually (λx. g x > 0) (at_right a)" unfolding filterlim_at_top_dense by blast+ hence "eventually (λx. g x / f x > 0) (at_right a)" by eventually_elim simp ultimately have "filterlim (λx. inverse (g x / f x)) at_top (at_right a)" by (rule filterlim_inverse_at_top) thus ?thesis by simp qed lemma lhopital_right_at_top_at_bot: fixes f g :: "real ⇒ real" assumes f_0: "LIM x at_right a. f x :> at_top" assumes g_0: "LIM x at_right a. g x :> at_bot" and ev: "eventually (λx. DERIV f x :> f' x) (at_right a)" "eventually (λx. DERIV g x :> g' x) (at_right a)" and lim: "filterlim (λ x. (f' x / g' x)) at_bot (at_right a)" shows "filterlim (λ x. f x / g x) at_bot (at_right a)" proof - from ev(2) have ev': "eventually (λx. DERIV (λx. -g x) x :> -g' x) (at_right a)" by eventually_elim (auto intro: derivative_intros) have "filterlim (λx. f x / (-g x)) at_top (at_right a)" by (rule lhopital_right_at_top_at_top[where f' = f' and g' = "λx. -g' x"]) (insert assms ev', auto simp: filterlim_uminus_at_bot) hence "filterlim (λx. -(f x / g x)) at_top (at_right a)" by simp thus ?thesis by (simp add: filterlim_uminus_at_bot) qed lemma lhopital_left_at_top_at_top: fixes f g :: "real ⇒ real" assumes f_0: "LIM x at_left a. f x :> at_top" assumes g_0: "LIM x at_left a. g x :> at_top" and ev: "eventually (λx. DERIV f x :> f' x) (at_left a)" "eventually (λx. DERIV g x :> g' x) (at_left a)" and lim: "filterlim (λ x. (f' x / g' x)) at_top (at_left a)" shows "filterlim (λ x. f x / g x) at_top (at_left a)" by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror, rule lhopital_right_at_top_at_top[where f'="λx. - f' (- x)"]) (insert assms, auto simp: DERIV_mirror) lemma lhopital_left_at_top_at_bot: fixes f g :: "real ⇒ real" assumes f_0: "LIM x at_left a. f x :> at_top" assumes g_0: "LIM x at_left a. g x :> at_bot" and ev: "eventually (λx. DERIV f x :> f' x) (at_left a)" "eventually (λx. DERIV g x :> g' x) (at_left a)" and lim: "filterlim (λ x. (f' x / g' x)) at_bot (at_left a)" shows "filterlim (λ x. f x / g x) at_bot (at_left a)" by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror, rule lhopital_right_at_top_at_bot[where f'="λx. - f' (- x)"]) (insert assms, auto simp: DERIV_mirror) lemma lhopital_at_top_at_top: fixes f g :: "real ⇒ real" assumes f_0: "LIM x at a. f x :> at_top" assumes g_0: "LIM x at a. g x :> at_top" and ev: "eventually (λx. DERIV f x :> f' x) (at a)" "eventually (λx. DERIV g x :> g' x) (at a)" and lim: "filterlim (λ x. (f' x / g' x)) at_top (at a)" shows "filterlim (λ x. f x / g x) at_top (at a)" using assms unfolding eventually_at_split filterlim_at_split by (auto intro!: lhopital_right_at_top_at_top[of f a g f' g'] lhopital_left_at_top_at_top[of f a g f' g']) lemma lhopital_at_top_at_bot: fixes f g :: "real ⇒ real" assumes f_0: "LIM x at a. f x :> at_top" assumes g_0: "LIM x at a. g x :> at_bot" and ev: "eventually (λx. DERIV f x :> f' x) (at a)" "eventually (λx. DERIV g x :> g' x) (at a)" and lim: "filterlim (λ x. (f' x / g' x)) at_bot (at a)" shows "filterlim (λ x. f x / g x) at_bot (at a)" using assms unfolding eventually_at_split filterlim_at_split by (auto intro!: lhopital_right_at_top_at_bot[of f a g f' g'] lhopital_left_at_top_at_bot[of f a g f' g']) end