
huffman@21164: (*  Title       : Deriv.thy
huffman@21164:     Author      : Jacques D. Fleuriot
huffman@21164:     Copyright   : 1998  University of Cambridge
hoelzl@51642:     Author      : Brian Huffman
huffman@21164:     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
huffman@21164:     GMVT by Benjamin Porter, 2005
huffman@21164: *)
huffman@21164: 
huffman@21164: header{* Differentiation *}
huffman@21164: 
huffman@21164: theory Deriv
hoelzl@51526: imports Limits
huffman@21164: begin
huffman@21164: 
hoelzl@56181: subsection {* Frechet derivative *}
hoelzl@56181: 
hoelzl@51642: definition
hoelzl@51642:   has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow>  bool"
hoelzl@56182:   (infix "(has'_derivative)" 50)
hoelzl@51642: where
hoelzl@51642:   "(f has_derivative f') F \<longleftrightarrow>
hoelzl@51642:     (bounded_linear f' \<and>
hoelzl@51642:      ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) ---> 0) F)"
hoelzl@51642: 
hoelzl@56181: text {*
hoelzl@56181:   Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)
hoelzl@56181:   (at x within s)"} means: @{term D} is the derivative of function @{term f} at point @{term x}
hoelzl@56181:   within the set @{term s}. Where @{term s} is used to express left or right sided derivatives. In
hoelzl@56181:   most cases @{term s} is either a variable or @{term UNIV}.
hoelzl@56181: *}
hoelzl@56181: 
hoelzl@56381: lemma has_derivative_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
hoelzl@56381:   by simp
hoelzl@56381: 
hoelzl@56381: definition 
hoelzl@56381:   has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@56381:   (infix "(has'_field'_derivative)" 50)
hoelzl@56381: where
hoelzl@56381:   "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
hoelzl@56381: 
hoelzl@56381: lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"
hoelzl@56381:   by simp
hoelzl@56381: 
hoelzl@56381: definition
hoelzl@56381:   has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
hoelzl@56381:   (infix "has'_vector'_derivative" 50)
hoelzl@56381: where
hoelzl@56381:   "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
hoelzl@56381: 
hoelzl@56381: lemma has_vector_derivative_eq_rhs: "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
hoelzl@56381:   by simp
hoelzl@56381: 
hoelzl@56381: ML {*
hoelzl@56381: 
hoelzl@56381: structure Derivative_Intros = Named_Thms
hoelzl@56381: (
hoelzl@56381:   val name = @{binding derivative_intros}
hoelzl@56381:   val description = "structural introduction rules for derivatives"
hoelzl@56381: )
hoelzl@56381: 
hoelzl@56381: *}
hoelzl@56381: 
hoelzl@56381: setup {*
hoelzl@56381:   let
hoelzl@56381:     val eq_thms = [@{thm has_derivative_eq_rhs}, @{thm DERIV_cong}, @{thm has_vector_derivative_eq_rhs}]
hoelzl@56381:     fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms
hoelzl@56381:   in
hoelzl@56381:     Derivative_Intros.setup #>
hoelzl@56381:     Global_Theory.add_thms_dynamic
hoelzl@56381:       (@{binding derivative_eq_intros}, map_filter eq_rule o Derivative_Intros.get o Context.proof_of)
hoelzl@56381:   end;
hoelzl@56381: *}
hoelzl@56381: 
hoelzl@56181: text {*
hoelzl@56181:   The following syntax is only used as a legacy syntax.
hoelzl@56181: *}
hoelzl@56181: abbreviation (input)
hoelzl@56181:   FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
hoelzl@56181:   ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
hoelzl@56181: where
hoelzl@56181:   "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
hoelzl@56181: 
hoelzl@56181: lemma has_derivative_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
hoelzl@51642:   by (simp add: has_derivative_def)
hoelzl@51642: 
hoelzl@56369: lemma has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"
hoelzl@56369:   using bounded_linear.linear[OF has_derivative_bounded_linear] .
hoelzl@56369: 
hoelzl@56381: lemma has_derivative_ident[derivative_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
hoelzl@51642:   by (simp add: has_derivative_def tendsto_const)
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_const[derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
hoelzl@51642:   by (simp add: has_derivative_def tendsto_const)
hoelzl@51642: 
hoelzl@51642: lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
hoelzl@51642: 
hoelzl@56181: lemma (in bounded_linear) has_derivative:
hoelzl@51642:   "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
hoelzl@51642:   using assms unfolding has_derivative_def
hoelzl@51642:   apply safe
wenzelm@56219:   apply (erule bounded_linear_compose [OF bounded_linear])
wenzelm@56219:   apply (drule tendsto)
wenzelm@56219:   apply (simp add: scaleR diff add zero)
hoelzl@51642:   done
hoelzl@51642: 
hoelzl@56381: lemmas has_derivative_scaleR_right [derivative_intros] =
hoelzl@56181:   bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
hoelzl@51642: 
hoelzl@56381: lemmas has_derivative_scaleR_left [derivative_intros] =
hoelzl@56181:   bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
hoelzl@51642: 
hoelzl@56381: lemmas has_derivative_mult_right [derivative_intros] =
hoelzl@56181:   bounded_linear.has_derivative [OF bounded_linear_mult_right]
hoelzl@51642: 
hoelzl@56381: lemmas has_derivative_mult_left [derivative_intros] =
hoelzl@56181:   bounded_linear.has_derivative [OF bounded_linear_mult_left]
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_add[simp, derivative_intros]:
hoelzl@51642:   assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"
hoelzl@51642:   shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
hoelzl@51642:   unfolding has_derivative_def
hoelzl@51642: proof safe
hoelzl@51642:   let ?x = "Lim F (\<lambda>x. x)"
hoelzl@51642:   let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
hoelzl@51642:   have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"
hoelzl@51642:     using f g by (intro tendsto_add) (auto simp: has_derivative_def)
hoelzl@51642:   then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
hoelzl@51642:     by (simp add: field_simps scaleR_add_right scaleR_diff_right)
hoelzl@56181: qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_setsum[simp, derivative_intros]:
hoelzl@51642:   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"
hoelzl@51642:   shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
hoelzl@51642: proof cases
hoelzl@51642:   assume "finite I" from this f show ?thesis
hoelzl@51642:     by induct (simp_all add: f)
hoelzl@51642: qed simp
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_minus[simp, derivative_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
hoelzl@56181:   using has_derivative_scaleR_right[of f f' F "-1"] by simp
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_diff[simp, derivative_intros]:
hoelzl@56181:   "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
hoelzl@56181:   by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus)
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_at_within:
hoelzl@56181:   "(f has_derivative f') (at x within s) \<longleftrightarrow>
hoelzl@51642:     (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s))"
hoelzl@51642:   by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_iff_norm:
hoelzl@56181:   "(f has_derivative f') (at x within s) \<longleftrightarrow>
hoelzl@51642:     (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ---> 0) (at x within s))"
hoelzl@51642:   using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
hoelzl@56181:   by (simp add: has_derivative_at_within divide_inverse ac_simps)
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_at:
hoelzl@56181:   "(f has_derivative D) (at x) \<longleftrightarrow> (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"
hoelzl@56181:   unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
hoelzl@51642: 
hoelzl@56181: lemma field_has_derivative_at:
hoelzl@51642:   fixes x :: "'a::real_normed_field"
hoelzl@56181:   shows "(f has_derivative op * D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@56181:   apply (unfold has_derivative_at)
hoelzl@56181:   apply (simp add: bounded_linear_mult_right)
hoelzl@51642:   apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
hoelzl@51642:   apply (subst diff_divide_distrib)
hoelzl@51642:   apply (subst times_divide_eq_left [symmetric])
hoelzl@51642:   apply (simp cong: LIM_cong)
hoelzl@51642:   apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)
hoelzl@51642:   done
hoelzl@51642: 
hoelzl@56181: lemma has_derivativeI:
hoelzl@51642:   "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>
hoelzl@56181:   (f has_derivative f') (at x within s)"
hoelzl@56181:   by (simp add: has_derivative_at_within)
hoelzl@51642: 
hoelzl@56181: lemma has_derivativeI_sandwich:
hoelzl@51642:   assumes e: "0 < e" and bounded: "bounded_linear f'"
hoelzl@51642:     and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
hoelzl@51642:     and "(H ---> 0) (at x within s)"
hoelzl@56181:   shows "(f has_derivative f') (at x within s)"
hoelzl@56181:   unfolding has_derivative_iff_norm
hoelzl@51642: proof safe
hoelzl@51642:   show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"
hoelzl@51642:   proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
hoelzl@51642:     show "(H ---> 0) (at x within s)" by fact
hoelzl@51642:     show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
hoelzl@51642:       unfolding eventually_at using e sandwich by auto
hoelzl@51642:   qed (auto simp: le_divide_eq tendsto_const)
hoelzl@51642: qed fact
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_subset: "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
hoelzl@56181:   by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
hoelzl@51642: 
lp15@56261: lemmas has_derivative_within_subset = has_derivative_subset 
lp15@56261: 
lp15@56261: 
hoelzl@51642: subsection {* Continuity *}
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_continuous:
hoelzl@56181:   assumes f: "(f has_derivative f') (at x within s)"
hoelzl@51642:   shows "continuous (at x within s) f"
hoelzl@51642: proof -
hoelzl@56181:   from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
hoelzl@51642:   note F.tendsto[tendsto_intros]
hoelzl@51642:   let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
hoelzl@51642:   have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
hoelzl@56181:     using f unfolding has_derivative_iff_norm by blast
hoelzl@51642:   then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
hoelzl@51642:     by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
hoelzl@51642:   also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
hoelzl@51642:     by (intro filterlim_cong) (simp_all add: eventually_at_filter)
hoelzl@51642:   finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
hoelzl@51642:     by (rule tendsto_norm_zero_cancel)
hoelzl@51642:   then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
hoelzl@51642:     by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
hoelzl@51642:   then have "?L (\<lambda>y. f y - f x)"
hoelzl@51642:     by simp
hoelzl@51642:   from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
hoelzl@51642:     by (simp add: continuous_within)
hoelzl@51642: qed
hoelzl@51642: 
hoelzl@51642: subsection {* Composition *}
hoelzl@51642: 
hoelzl@51642: lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (f ---> y) (inf (nhds x) (principal s))"
hoelzl@51642:   unfolding tendsto_def eventually_inf_principal eventually_at_filter
hoelzl@51642:   by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_in_compose:
hoelzl@56181:   assumes f: "(f has_derivative f') (at x within s)"
hoelzl@56181:   assumes g: "(g has_derivative g') (at (f x) within (f`s))"
hoelzl@56181:   shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
hoelzl@51642: proof -
hoelzl@56181:   from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
hoelzl@56181:   from g interpret G: bounded_linear g' by (rule has_derivative_bounded_linear)
hoelzl@51642:   from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast
hoelzl@51642:   from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
hoelzl@51642:   note G.tendsto[tendsto_intros]
hoelzl@51642: 
hoelzl@51642:   let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
hoelzl@51642:   let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
hoelzl@51642:   let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
hoelzl@51642:   let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
hoelzl@51642:   def Nf \<equiv> "?N f f' x"
hoelzl@51642:   def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"
hoelzl@51642: 
hoelzl@51642:   show ?thesis
hoelzl@56181:   proof (rule has_derivativeI_sandwich[of 1])
hoelzl@51642:     show "bounded_linear (\<lambda>x. g' (f' x))"
hoelzl@56181:       using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
hoelzl@51642:   next
hoelzl@51642:     fix y::'a assume neq: "y \<noteq> x"
hoelzl@51642:     have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
hoelzl@51642:       by (simp add: G.diff G.add field_simps)
hoelzl@51642:     also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
hoelzl@51642:       by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)
hoelzl@51642:     also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"
hoelzl@51642:     proof (intro add_mono mult_left_mono)
hoelzl@51642:       have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
hoelzl@51642:         by simp
hoelzl@51642:       also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
hoelzl@51642:         by (rule norm_triangle_ineq)
hoelzl@51642:       also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
hoelzl@51642:         using kF by (intro add_mono) simp
hoelzl@51642:       finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
hoelzl@51642:         by (simp add: neq Nf_def field_simps)
hoelzl@51642:     qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
hoelzl@51642:     finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
hoelzl@51642:   next
hoelzl@51642:     have [tendsto_intros]: "?L Nf"
hoelzl@56181:       using f unfolding has_derivative_iff_norm Nf_def ..
hoelzl@51642:     from f have "(f ---> f x) (at x within s)"
hoelzl@56181:       by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
hoelzl@51642:     then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
hoelzl@51642:       unfolding filterlim_def
hoelzl@51642:       by (simp add: eventually_filtermap eventually_at_filter le_principal)
hoelzl@51642: 
hoelzl@51642:     have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"
hoelzl@56181:       using g unfolding has_derivative_iff_norm ..
hoelzl@51642:     then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"
hoelzl@51642:       by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
hoelzl@51642: 
hoelzl@51642:     have [tendsto_intros]: "?L Ng"
hoelzl@51642:       unfolding Ng_def by (rule filterlim_compose[OF g' f'])
hoelzl@51642:     show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"
hoelzl@51642:       by (intro tendsto_eq_intros) auto
hoelzl@51642:   qed simp
hoelzl@51642: qed
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_compose:
hoelzl@56181:   "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
hoelzl@56181:   by (blast intro: has_derivative_in_compose has_derivative_subset)
hoelzl@51642: 
hoelzl@51642: lemma (in bounded_bilinear) FDERIV:
hoelzl@56181:   assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
hoelzl@56181:   shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
hoelzl@51642: proof -
hoelzl@56181:   from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
hoelzl@51642:   obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
hoelzl@51642: 
hoelzl@51642:   from pos_bounded obtain K where K: "0 < K" and norm_prod:
hoelzl@51642:     "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
hoelzl@51642:   let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
hoelzl@51642:   let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
hoelzl@51642:   def Ng =="?N g g'" and Nf =="?N f f'"
hoelzl@51642: 
hoelzl@51642:   let ?fun1 = "\<lambda>y. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
hoelzl@51642:   let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
hoelzl@51642:   let ?F = "at x within s"
huffman@21164: 
hoelzl@51642:   show ?thesis
hoelzl@56181:   proof (rule has_derivativeI_sandwich[of 1])
hoelzl@51642:     show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
hoelzl@51642:       by (intro bounded_linear_add
hoelzl@51642:         bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
hoelzl@56181:         has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
hoelzl@51642:   next
hoelzl@51642:     from g have "(g ---> g x) ?F"
hoelzl@56181:       by (intro continuous_within[THEN iffD1] has_derivative_continuous)
hoelzl@51642:     moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"
hoelzl@56181:       by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
hoelzl@51642:     ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
hoelzl@51642:       by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
hoelzl@51642:     then show "(?fun2 ---> 0) ?F"
hoelzl@51642:       by simp
hoelzl@51642:   next
hoelzl@51642:     fix y::'d assume "y \<noteq> x"
hoelzl@51642:     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)"
hoelzl@51642:       by (simp add: diff_left diff_right add_left add_right field_simps)
hoelzl@51642:     also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
hoelzl@51642:         norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
hoelzl@51642:       by (intro divide_right_mono mult_mono'
hoelzl@51642:                 order_trans [OF norm_triangle_ineq add_mono]
hoelzl@51642:                 order_trans [OF norm_prod mult_right_mono]
hoelzl@51642:                 mult_nonneg_nonneg order_refl norm_ge_zero norm_F
hoelzl@51642:                 K [THEN order_less_imp_le])
hoelzl@51642:     also have "\<dots> = ?fun2 y"
hoelzl@51642:       by (simp add: add_divide_distrib Ng_def Nf_def)
hoelzl@51642:     finally show "?fun1 y \<le> ?fun2 y" .
hoelzl@51642:   qed simp
hoelzl@51642: qed
hoelzl@51642: 
hoelzl@56381: lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
hoelzl@56381: lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_setprod[simp, derivative_intros]:
hoelzl@51642:   fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181:   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)"
hoelzl@56181:   shows "((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within s)"
hoelzl@51642: proof cases
hoelzl@51642:   assume "finite I" from this f show ?thesis
hoelzl@51642:   proof induct
hoelzl@51642:     case (insert i I)
hoelzl@51642:     let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
hoelzl@56181:     have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
hoelzl@56181:       using insert by (intro has_derivative_mult) auto
hoelzl@51642:     also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
haftmann@57418:       using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum.cong)
hoelzl@51642:     finally show ?case
hoelzl@51642:       using insert by simp
hoelzl@51642:   qed simp  
hoelzl@51642: qed simp
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_power[simp, derivative_intros]:
hoelzl@51642:   fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181:   assumes f: "(f has_derivative f') (at x within s)"
hoelzl@56181:   shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within s)"
hoelzl@56181:   using has_derivative_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_inverse':
hoelzl@51642:   fixes x :: "'a::real_normed_div_algebra"
hoelzl@51642:   assumes x: "x \<noteq> 0"
hoelzl@56181:   shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
hoelzl@56181:         (is "(?inv has_derivative ?f) _")
hoelzl@56181: proof (rule has_derivativeI_sandwich)
hoelzl@51642:   show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
hoelzl@51642:     apply (rule bounded_linear_minus)
hoelzl@51642:     apply (rule bounded_linear_mult_const)
hoelzl@51642:     apply (rule bounded_linear_const_mult)
hoelzl@51642:     apply (rule bounded_linear_ident)
hoelzl@51642:     done
hoelzl@51642: next
hoelzl@51642:   show "0 < norm x" using x by simp
hoelzl@51642: next
hoelzl@51642:   show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"
hoelzl@51642:     apply (rule tendsto_mult_left_zero)
hoelzl@51642:     apply (rule tendsto_norm_zero)
hoelzl@51642:     apply (rule LIM_zero)
hoelzl@51642:     apply (rule tendsto_inverse)
hoelzl@51642:     apply (rule tendsto_ident_at)
hoelzl@51642:     apply (rule x)
hoelzl@51642:     done
hoelzl@51642: next
hoelzl@51642:   fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
hoelzl@51642:   then have "y \<noteq> 0"
hoelzl@51642:     by (auto simp: norm_conv_dist dist_commute)
hoelzl@51642:   have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
hoelzl@51642:     apply (subst inverse_diff_inverse [OF `y \<noteq> 0` x])
hoelzl@51642:     apply (subst minus_diff_minus)
hoelzl@51642:     apply (subst norm_minus_cancel)
hoelzl@51642:     apply (simp add: left_diff_distrib)
hoelzl@51642:     done
hoelzl@51642:   also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
hoelzl@51642:     apply (rule divide_right_mono [OF _ norm_ge_zero])
hoelzl@51642:     apply (rule order_trans [OF norm_mult_ineq])
hoelzl@51642:     apply (rule mult_right_mono [OF _ norm_ge_zero])
hoelzl@51642:     apply (rule norm_mult_ineq)
hoelzl@51642:     done
hoelzl@51642:   also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
hoelzl@51642:     by simp
hoelzl@51642:   finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
hoelzl@51642:       norm (?inv y - ?inv x) * norm (?inv x)" .
hoelzl@51642: qed
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_inverse[simp, derivative_intros]:
hoelzl@51642:   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181:   assumes x:  "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"
hoelzl@56181:   shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"
hoelzl@56181:   using has_derivative_compose[OF f has_derivative_inverse', OF x] .
hoelzl@51642: 
hoelzl@56381: lemma has_derivative_divide[simp, derivative_intros]:
hoelzl@51642:   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181:   assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" 
lp15@55967:   assumes x: "g x \<noteq> 0"
hoelzl@56181:   shows "((\<lambda>x. f x / g x) has_derivative
hoelzl@56181:                 (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
hoelzl@56181:   using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
hoelzl@56480:   by (simp add: field_simps)
lp15@55967: 
lp15@55967: text{*Conventional form requires mult-AC laws. Types real and complex only.*}
hoelzl@56181: 
hoelzl@56381: lemma has_derivative_divide'[derivative_intros]: 
lp15@55967:   fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
hoelzl@56181:   assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" and x: "g x \<noteq> 0"
hoelzl@56181:   shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within s)"
lp15@55967: proof -
lp15@55967:   { fix h
lp15@55967:     have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
lp15@55967:           (f' h * g x - f x * g' h) / (g x * g x)"
hoelzl@56480:       by (simp add: field_simps x)
lp15@55967:    }
lp15@55967:   then show ?thesis
hoelzl@56181:     using has_derivative_divide [OF f g] x
lp15@55967:     by simp
lp15@55967: qed
hoelzl@51642: 
hoelzl@51642: subsection {* Uniqueness *}
hoelzl@51642: 
hoelzl@51642: text {*
hoelzl@51642: 
hoelzl@56181: This can not generally shown for @{const has_derivative}, as we need to approach the point from
hoelzl@51642: all directions. There is a proof in @{text Multivariate_Analysis} for @{text euclidean_space}.
hoelzl@51642: 
hoelzl@51642: *}
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_zero_unique:
hoelzl@56181:   assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"
hoelzl@51642: proof -
hoelzl@51642:   interpret F: bounded_linear F
hoelzl@56181:     using assms by (rule has_derivative_bounded_linear)
hoelzl@51642:   let ?r = "\<lambda>h. norm (F h) / norm h"
hoelzl@51642:   have *: "?r -- 0 --> 0"
hoelzl@56181:     using assms unfolding has_derivative_at by simp
hoelzl@51642:   show "F = (\<lambda>h. 0)"
hoelzl@51642:   proof
hoelzl@51642:     fix h show "F h = 0"
hoelzl@51642:     proof (rule ccontr)
wenzelm@53374:       assume **: "F h \<noteq> 0"
nipkow@56541:       hence h: "h \<noteq> 0" by (clarsimp simp add: F.zero)
nipkow@56541:       with ** have "0 < ?r h" by simp
hoelzl@51642:       from LIM_D [OF * this] obtain s where s: "0 < s"
hoelzl@51642:         and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
hoelzl@51642:       from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
hoelzl@51642:       let ?x = "scaleR (t / norm h) h"
hoelzl@51642:       have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
hoelzl@51642:       hence "?r ?x < ?r h" by (rule r)
hoelzl@51642:       thus "False" using t h by (simp add: F.scaleR)
hoelzl@51642:     qed
hoelzl@51642:   qed
hoelzl@51642: qed
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_unique:
hoelzl@56181:   assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"
hoelzl@51642: proof -
hoelzl@56181:   have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
hoelzl@56181:     using has_derivative_diff [OF assms] by simp
hoelzl@51642:   hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
hoelzl@56181:     by (rule has_derivative_zero_unique)
hoelzl@51642:   thus "F = F'"
hoelzl@51642:     unfolding fun_eq_iff right_minus_eq .
hoelzl@51642: qed
hoelzl@51642: 
hoelzl@51642: subsection {* Differentiability predicate *}
hoelzl@51642: 
hoelzl@56181: definition
hoelzl@56181:   differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@56182:   (infix "differentiable" 50)
hoelzl@56181: where
hoelzl@56181:   "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
hoelzl@51642: 
hoelzl@56181: lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_subset)
hoelzl@51642: 
lp15@56261: lemmas differentiable_within_subset = differentiable_subset
lp15@56261: 
hoelzl@56381: lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_ident)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_const)
hoelzl@51642: 
hoelzl@51642: lemma differentiable_in_compose:
hoelzl@56181:   "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_in_compose)
hoelzl@51642: 
hoelzl@51642: lemma differentiable_compose:
hoelzl@56181:   "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@51642:   by (blast intro: differentiable_in_compose differentiable_subset)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_sum [simp, derivative_intros]:
hoelzl@56181:   "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_add)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_minus [simp, derivative_intros]:
hoelzl@56181:   "f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_minus)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_diff [simp, derivative_intros]:
hoelzl@56181:   "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_diff)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_mult [simp, derivative_intros]:
hoelzl@51642:   fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
hoelzl@56181:   shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x * g x) differentiable (at x within s)"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_mult)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_inverse [simp, derivative_intros]:
hoelzl@51642:   fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181:   shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable (at x within s)"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_inverse)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_divide [simp, derivative_intros]:
hoelzl@51642:   fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181:   shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
hoelzl@51642:   unfolding divide_inverse using assms by simp
hoelzl@51642: 
hoelzl@56381: lemma differentiable_power [simp, derivative_intros]:
hoelzl@51642:   fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181:   shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_power)
hoelzl@51642: 
hoelzl@56381: lemma differentiable_scaleR [simp, derivative_intros]:
hoelzl@56181:   "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
hoelzl@56181:   unfolding differentiable_def by (blast intro: has_derivative_scaleR)
hoelzl@51642: 
hoelzl@56181: lemma has_derivative_imp_has_field_derivative:
hoelzl@56181:   "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
hoelzl@56181:   unfolding has_field_derivative_def 
haftmann@57512:   by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
hoelzl@56181: 
hoelzl@56181: lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
hoelzl@56181:   by (simp add: has_field_derivative_def)
hoelzl@51642: 
lp15@56261: lemma DERIV_subset: 
lp15@56261:   "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
lp15@56261:    \<Longrightarrow> (f has_field_derivative f') (at x within t)"
lp15@56261:   by (simp add: has_field_derivative_def has_derivative_within_subset)
lp15@56261: 
hoelzl@56181: abbreviation (input)
hoelzl@56381:   DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@56181:   ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
hoelzl@51642: where
hoelzl@56181:   "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
hoelzl@51642: 
hoelzl@56181: abbreviation 
hoelzl@56181:   has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
hoelzl@56182:   (infix "(has'_real'_derivative)" 50)
hoelzl@56181: where
hoelzl@56181:   "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
hoelzl@56181: 
hoelzl@56181: lemma real_differentiable_def:
hoelzl@56181:   "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
hoelzl@51642: proof safe
hoelzl@56181:   assume "f differentiable at x within s"
hoelzl@56181:   then obtain f' where *: "(f has_derivative f') (at x within s)"
hoelzl@56181:     unfolding differentiable_def by auto
hoelzl@56181:   then obtain c where "f' = (op * c)"
haftmann@57512:     by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
hoelzl@56181:   with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181:     unfolding has_field_derivative_def by auto
hoelzl@56181: qed (auto simp: differentiable_def has_field_derivative_def)
hoelzl@51642: 
hoelzl@56181: lemma real_differentiableE [elim?]:
hoelzl@56181:   assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"
hoelzl@56181:   using assms by (auto simp: real_differentiable_def)
hoelzl@51642: 
hoelzl@56181: lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181:   by (auto elim: real_differentiableE)
hoelzl@51642: 
hoelzl@56181: lemma differentiableI: "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
hoelzl@56181:   by (force simp add: real_differentiable_def)
hoelzl@51642: 
hoelzl@56381: lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@56181:   apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right LIM_zero_iff[symmetric, of _ D])
hoelzl@51642:   apply (subst (2) tendsto_norm_zero_iff[symmetric])
hoelzl@51642:   apply (rule filterlim_cong)
hoelzl@51642:   apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
hoelzl@51642:   done
huffman@21164: 
hoelzl@56181: lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
haftmann@57512:   by (simp add: fun_eq_iff mult.commute)
huffman@21164: 
hoelzl@56181: subsection {* Derivatives *}
huffman@21164: 
hoelzl@51642: lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@56381:   by (simp add: DERIV_def)
huffman@21164: 
hoelzl@56381: lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
hoelzl@56181: 
hoelzl@56381: lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
huffman@21164: 
hoelzl@56381: lemma field_differentiable_add[derivative_intros]:
hoelzl@56381:   "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> 
hoelzl@56381:     ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
hoelzl@56381:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
hoelzl@56381:      (auto simp: has_field_derivative_def field_simps mult_commute_abs)
lp15@56261: 
lp15@56261: corollary DERIV_add:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
lp15@56261:   by (rule field_differentiable_add)
lp15@56261: 
hoelzl@56381: lemma field_differentiable_minus[derivative_intros]:
hoelzl@56381:   "(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
hoelzl@56381:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
hoelzl@56381:      (auto simp: has_field_derivative_def field_simps mult_commute_abs)
huffman@21164: 
lp15@56261: corollary DERIV_minus: "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
lp15@56261:   by (rule field_differentiable_minus)
huffman@21164: 
hoelzl@56381: lemma field_differentiable_diff[derivative_intros]:
hoelzl@56381:   "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
hoelzl@56381:   by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
lp15@56261: 
lp15@56261: corollary DERIV_diff:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
lp15@56261:   by (rule field_differentiable_diff)
hoelzl@51642: 
hoelzl@56181: lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
hoelzl@56181:   by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
huffman@21164: 
lp15@56261: corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
lp15@56261:   by (rule DERIV_continuous)
lp15@56261: 
lp15@56261: lemma DERIV_continuous_on:
lp15@56261:   "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative D) (at x)) \<Longrightarrow> continuous_on s f"
lp15@56261:   by (metis DERIV_continuous continuous_at_imp_continuous_on)
hoelzl@51642: 
hoelzl@56181: lemma DERIV_mult':
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181:      (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
huffman@21164: 
hoelzl@56381: lemma DERIV_mult[derivative_intros]:
hoelzl@56181:   "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181:      (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
hoelzl@51642: 
hoelzl@51642: text {* Derivative of linear multiplication *}
huffman@21164: 
hoelzl@51642: lemma DERIV_cmult:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
hoelzl@51642:   by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164: 
lp15@55967: lemma DERIV_cmult_right:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
haftmann@57514:   using DERIV_cmult by (force simp add: ac_simps)
lp15@55967: 
hoelzl@56181: lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
hoelzl@51642:   by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
hoelzl@51642: 
hoelzl@56181: lemma DERIV_cdivide:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
hoelzl@56181:   using DERIV_cmult_right[of f D x s "1 / c"] by simp
huffman@21164: 
huffman@21164: lemma DERIV_unique:
hoelzl@51642:   "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
hoelzl@56381:   unfolding DERIV_def by (rule LIM_unique) 
huffman@21164: 
hoelzl@56381: lemma DERIV_setsum[derivative_intros]:
hoelzl@56181:   "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow> 
hoelzl@56181:     ((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])
hoelzl@56181:      (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
hoelzl@51642: 
hoelzl@56381: lemma DERIV_inverse'[derivative_intros]:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_inverse])
hoelzl@56181:      (auto dest: has_field_derivative_imp_has_derivative)
hoelzl@51642: 
hoelzl@51642: text {* Power of @{text "-1"} *}
hoelzl@51642: 
hoelzl@51642: lemma DERIV_inverse:
hoelzl@56181:   "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
hoelzl@51642:   by (drule DERIV_inverse' [OF DERIV_ident]) simp
hoelzl@51642: 
hoelzl@51642: text {* Derivative of inverse *}
hoelzl@51642: 
hoelzl@51642: lemma DERIV_inverse_fun:
hoelzl@56181:   "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
haftmann@57514:   by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)
hoelzl@51642: 
hoelzl@51642: text {* Derivative of quotient *}
hoelzl@51642: 
hoelzl@56381: lemma DERIV_divide[derivative_intros]:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181:   (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
hoelzl@56480:      (auto dest: has_field_derivative_imp_has_derivative simp: field_simps)
hoelzl@51642: 
hoelzl@51642: lemma DERIV_quotient:
hoelzl@56181:   "(f has_field_derivative d) (at x within s) \<Longrightarrow>
hoelzl@56181:   (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 
hoelzl@56181:   ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
haftmann@57512:   by (drule (2) DERIV_divide) (simp add: mult.commute)
hoelzl@51642: 
hoelzl@51642: lemma DERIV_power_Suc:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181:      (auto simp: has_field_derivative_def)
hoelzl@51642: 
hoelzl@56381: lemma DERIV_power[derivative_intros]:
hoelzl@56181:   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
hoelzl@56181:   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181:      (auto simp: has_field_derivative_def)
hoelzl@31880: 
hoelzl@56181: lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
hoelzl@51642:   apply (cut_tac DERIV_power [OF DERIV_ident])
hoelzl@51642:   apply (simp add: real_of_nat_def)
hoelzl@51642:   done
hoelzl@51642: 
hoelzl@56181: lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 
hoelzl@56181:   ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
hoelzl@56181:   using has_derivative_compose[of f "op * D" x s g "op * E"]
hoelzl@56181:   unfolding has_field_derivative_def mult_commute_abs ac_simps .
hoelzl@51642: 
hoelzl@56181: corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181:   ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
lp15@55967:   by (rule DERIV_chain')
lp15@55967: 
hoelzl@51642: text {* Standard version *}
hoelzl@51642: 
hoelzl@56181: lemma DERIV_chain:
hoelzl@56181:   "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 
hoelzl@56181:   (f o g has_field_derivative Da * Db) (at x within s)"
haftmann@57512:   by (drule (1) DERIV_chain', simp add: o_def mult.commute)
hoelzl@51642: 
lp15@55967: lemma DERIV_image_chain: 
hoelzl@56181:   "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181:   (f o g has_field_derivative Da * Db) (at x within s)"
hoelzl@56181:   using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
hoelzl@56181:   by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
lp15@55967: 
lp15@55967: (*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
lp15@55967: lemma DERIV_chain_s:
lp15@55967:   assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
lp15@55967:       and "DERIV f x :> f'" 
lp15@55967:       and "f x \<in> s"
lp15@55967:     shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
haftmann@57512:   by (metis (full_types) DERIV_chain' mult.commute assms)
lp15@55967: 
lp15@55967: lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
lp15@55967:   assumes "(\<And>x. DERIV g x :> g'(x))"
lp15@55967:       and "DERIV f x :> f'" 
lp15@55967:     shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
lp15@55967:   by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
lp15@55967: 
hoelzl@51642: declare
hoelzl@56381:   DERIV_power[where 'a=real, unfolded real_of_nat_def[symmetric], derivative_intros]
huffman@21164: 
huffman@21164: text{*Alternative definition for differentiability*}
huffman@21164: 
huffman@21164: lemma DERIV_LIM_iff:
huffman@31338:   fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
huffman@21784:      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
huffman@21164:       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
huffman@21164: apply (rule iffI)
huffman@21164: apply (drule_tac k="- a" in LIM_offset)
haftmann@54230: apply simp
huffman@21164: apply (drule_tac k="a" in LIM_offset)
haftmann@57512: apply (simp add: add.commute)
huffman@21164: done
huffman@21164: 
hoelzl@51642: lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
hoelzl@56381:   by (simp add: DERIV_def DERIV_LIM_iff)
huffman@21164: 
hoelzl@51642: lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
hoelzl@51642:     DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
hoelzl@51642:   unfolding DERIV_iff2
hoelzl@51642: proof (rule filterlim_cong)
wenzelm@53374:   assume *: "eventually (\<lambda>x. f x = g x) (nhds x)"
wenzelm@53374:   moreover from * have "f x = g x" by (auto simp: eventually_nhds)
hoelzl@51642:   moreover assume "x = y" "u = v"
hoelzl@51642:   ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
hoelzl@51642:     by (auto simp: eventually_at_filter elim: eventually_elim1)
hoelzl@51642: qed simp_all
huffman@21164: 
hoelzl@51642: lemma DERIV_shift:
hoelzl@51642:   "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
hoelzl@56381:   by (simp add: DERIV_def field_simps)
huffman@21164: 
hoelzl@51642: lemma DERIV_mirror:
hoelzl@51642:   "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
hoelzl@56479:   by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
hoelzl@51642:                 tendsto_minus_cancel_left field_simps conj_commute)
huffman@21164: 
huffman@29975: text {* Caratheodory formulation of derivative at a point *}
huffman@21164: 
lp15@55970: lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
hoelzl@51642:   "(DERIV f x :> l) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
huffman@21164:       (is "?lhs = ?rhs")
huffman@21164: proof
huffman@21164:   assume der: "DERIV f x :> l"
huffman@21784:   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
huffman@21164:   proof (intro exI conjI)
huffman@21784:     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
nipkow@23413:     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
huffman@21164:     show "isCont ?g x" using der
hoelzl@56381:       by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
huffman@21164:     show "?g x = l" by simp
huffman@21164:   qed
huffman@21164: next
huffman@21164:   assume "?rhs"
huffman@21164:   then obtain g where
huffman@21784:     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
huffman@21164:   thus "(DERIV f x :> l)"
hoelzl@56381:      by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
huffman@21164: qed
huffman@21164: 
wenzelm@31899: text {*
wenzelm@31899:  Let's do the standard proof, though theorem
wenzelm@31899:  @{text "LIM_mult2"} follows from a NS proof
wenzelm@31899: *}
huffman@21164: 
huffman@29975: subsection {* Local extrema *}
huffman@29975: 
huffman@21164: text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164: 
paulson@33654: lemma DERIV_pos_inc_right:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   assumes der: "DERIV f x :> l"
huffman@21164:       and l:   "0 < l"
huffman@21164:   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164: proof -
huffman@21164:   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164:   have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
haftmann@54230:     by simp
huffman@21164:   then obtain s
huffman@21164:         where s:   "0 < s"
huffman@21164:           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164:     by auto
huffman@21164:   thus ?thesis
huffman@21164:   proof (intro exI conjI strip)
huffman@23441:     show "0<s" using s .
huffman@21164:     fix h::real
huffman@21164:     assume "0 < h" "h < s"
huffman@21164:     with all [of h] show "f x < f (x+h)"
haftmann@54230:     proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
huffman@21164:       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164:       with l
huffman@21164:       have "0 < (f (x+h) - f x) / h" by arith
huffman@21164:       thus "f x < f (x+h)"
huffman@21164:   by (simp add: pos_less_divide_eq h)
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
paulson@33654: lemma DERIV_neg_dec_left:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   assumes der: "DERIV f x :> l"
huffman@21164:       and l:   "l < 0"
huffman@21164:   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164: proof -
huffman@21164:   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164:   have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
haftmann@54230:     by simp
huffman@21164:   then obtain s
huffman@21164:         where s:   "0 < s"
huffman@21164:           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164:     by auto
huffman@21164:   thus ?thesis
huffman@21164:   proof (intro exI conjI strip)
huffman@23441:     show "0<s" using s .
huffman@21164:     fix h::real
huffman@21164:     assume "0 < h" "h < s"
huffman@21164:     with all [of "-h"] show "f x < f (x-h)"
hoelzl@56479:     proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
hoelzl@56479:       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164:       with l
huffman@21164:       have "0 < (f (x-h) - f x) / h" by arith
huffman@21164:       thus "f x < f (x-h)"
huffman@21164:   by (simp add: pos_less_divide_eq h)
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
paulson@33654: lemma DERIV_pos_inc_left:
paulson@33654:   fixes f :: "real => real"
paulson@33654:   shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
hoelzl@56181:   apply (rule DERIV_neg_dec_left [of "%x. - f x" "-l" x, simplified])
hoelzl@41368:   apply (auto simp add: DERIV_minus)
paulson@33654:   done
paulson@33654: 
paulson@33654: lemma DERIV_neg_dec_right:
paulson@33654:   fixes f :: "real => real"
paulson@33654:   shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
hoelzl@56181:   apply (rule DERIV_pos_inc_right [of "%x. - f x" "-l" x, simplified])
hoelzl@41368:   apply (auto simp add: DERIV_minus)
paulson@33654:   done
paulson@33654: 
huffman@21164: lemma DERIV_local_max:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   assumes der: "DERIV f x :> l"
huffman@21164:       and d:   "0 < d"
huffman@21164:       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164:   shows "l = 0"
huffman@21164: proof (cases rule: linorder_cases [of l 0])
huffman@23441:   case equal thus ?thesis .
huffman@21164: next
huffman@21164:   case less
paulson@33654:   from DERIV_neg_dec_left [OF der less]
huffman@21164:   obtain d' where d': "0 < d'"
huffman@21164:              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164:   from real_lbound_gt_zero [OF d d']
huffman@21164:   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164:   with lt le [THEN spec [where x="x-e"]]
huffman@21164:   show ?thesis by (auto simp add: abs_if)
huffman@21164: next
huffman@21164:   case greater
paulson@33654:   from DERIV_pos_inc_right [OF der greater]
huffman@21164:   obtain d' where d': "0 < d'"
huffman@21164:              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164:   from real_lbound_gt_zero [OF d d']
huffman@21164:   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164:   with lt le [THEN spec [where x="x+e"]]
huffman@21164:   show ?thesis by (auto simp add: abs_if)
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: text{*Similar theorem for a local minimum*}
huffman@21164: lemma DERIV_local_min:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164: by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164: 
huffman@21164: 
huffman@21164: text{*In particular, if a function is locally flat*}
huffman@21164: lemma DERIV_local_const:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164: by (auto dest!: DERIV_local_max)
huffman@21164: 
huffman@29975: 
huffman@29975: subsection {* Rolle's Theorem *}
huffman@29975: 
huffman@21164: text{*Lemma about introducing open ball in open interval*}
huffman@21164: lemma lemma_interval_lt:
huffman@21164:      "[| a < x;  x < b |]
huffman@21164:       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668: 
huffman@22998: apply (simp add: abs_less_iff)
huffman@21164: apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164: apply (rule_tac x = "x-a" in exI)
huffman@21164: apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164:         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164: apply (drule lemma_interval_lt, auto)
huffman@44921: apply force
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Rolle's Theorem.
huffman@21164:    If @{term f} is defined and continuous on the closed interval
huffman@21164:    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164:    and @{term "f(a) = f(b)"},
huffman@21164:    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164: theorem Rolle:
huffman@21164:   assumes lt: "a < b"
huffman@21164:       and eq: "f(a) = f(b)"
huffman@21164:       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181:       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784:   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164: proof -
huffman@21164:   have le: "a \<le> b" using lt by simp
huffman@21164:   from isCont_eq_Ub [OF le con]
huffman@21164:   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164:              and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164:     by blast
huffman@21164:   from isCont_eq_Lb [OF le con]
huffman@21164:   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164:               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164:     by blast
huffman@21164:   show ?thesis
huffman@21164:   proof cases
huffman@21164:     assume axb: "a < x & x < b"
huffman@21164:         --{*@{term f} attains its maximum within the interval*}
chaieb@27668:     hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164:     from lemma_interval [OF ax xb]
huffman@21164:     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164:       by blast
huffman@21164:     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164:       by blast
huffman@21164:     from differentiableD [OF dif [OF axb]]
huffman@21164:     obtain l where der: "DERIV f x :> l" ..
huffman@21164:     have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164:         --{*the derivative at a local maximum is zero*}
huffman@21164:     thus ?thesis using ax xb der by auto
huffman@21164:   next
huffman@21164:     assume notaxb: "~ (a < x & x < b)"
huffman@21164:     hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164:     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164:     show ?thesis
huffman@21164:     proof cases
huffman@21164:       assume ax'b: "a < x' & x' < b"
huffman@21164:         --{*@{term f} attains its minimum within the interval*}
chaieb@27668:       hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164:       from lemma_interval [OF ax' x'b]
huffman@21164:       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164:   by blast
huffman@21164:       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164:   by blast
huffman@21164:       from differentiableD [OF dif [OF ax'b]]
huffman@21164:       obtain l where der: "DERIV f x' :> l" ..
huffman@21164:       have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164:         --{*the derivative at a local minimum is zero*}
huffman@21164:       thus ?thesis using ax' x'b der by auto
huffman@21164:     next
huffman@21164:       assume notax'b: "~ (a < x' & x' < b)"
huffman@21164:         --{*@{term f} is constant througout the interval*}
huffman@21164:       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164:       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164:       from dense [OF lt]
huffman@21164:       obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164:       from lemma_interval [OF ar rb]
huffman@21164:       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164:   by blast
huffman@21164:       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164:       proof (clarify)
huffman@21164:         fix z::real
huffman@21164:         assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164:         show "f z = f b"
huffman@21164:         proof (rule order_antisym)
huffman@21164:           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164:           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164:         qed
huffman@21164:       qed
huffman@21164:       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164:       proof (intro strip)
huffman@21164:         fix y::real
huffman@21164:         assume lt: "\<bar>r-y\<bar> < d"
huffman@21164:         hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164:         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164:       qed
huffman@21164:       from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164:       obtain l where der: "DERIV f r :> l" ..
huffman@21164:       have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164:         --{*the derivative of a constant function is zero*}
huffman@21164:       thus ?thesis using ar rb der by auto
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: subsection{*Mean Value Theorem*}
huffman@21164: 
huffman@21164: lemma lemma_MVT:
huffman@21164:      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
hoelzl@51481:   by (cases "a = b") (simp_all add: field_simps)
huffman@21164: 
huffman@21164: theorem MVT:
huffman@21164:   assumes lt:  "a < b"
huffman@21164:       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181:       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784:   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164:                    (f(b) - f(a) = (b-a) * l)"
huffman@21164: proof -
huffman@21164:   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@44233:   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
hoelzl@56371:     using con by (fast intro: continuous_intros)
hoelzl@56181:   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
huffman@21164:   proof (clarify)
huffman@21164:     fix x::real
huffman@21164:     assume ax: "a < x" and xb: "x < b"
huffman@21164:     from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164:     obtain l where der: "DERIV f x :> l" ..
hoelzl@56181:     show "?F differentiable (at x)"
huffman@21164:       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164:           blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164:   qed
huffman@21164:   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164:   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164:     by blast
huffman@21164:   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164:     by (rule DERIV_cmult_Id)
huffman@21164:   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164:                    :> 0 + (f b - f a) / (b - a)"
huffman@21164:     by (rule DERIV_add [OF der])
huffman@21164:   show ?thesis
huffman@21164:   proof (intro exI conjI)
huffman@23441:     show "a < z" using az .
huffman@23441:     show "z < b" using zb .
huffman@21164:     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164:     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
hoelzl@29803: lemma MVT2:
hoelzl@29803:      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
hoelzl@29803:       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
hoelzl@29803: apply (drule MVT)
hoelzl@29803: apply (blast intro: DERIV_isCont)
hoelzl@56181: apply (force dest: order_less_imp_le simp add: real_differentiable_def)
hoelzl@29803: apply (blast dest: DERIV_unique order_less_imp_le)
hoelzl@29803: done
hoelzl@29803: 
huffman@21164: 
huffman@21164: text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164: 
huffman@21164: lemma DERIV_isconst_end:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| a < b;
huffman@21164:          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164:          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164:         ==> f b = f a"
huffman@21164: apply (drule MVT, assumption)
huffman@21164: apply (blast intro: differentiableI)
huffman@21164: apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_isconst1:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| a < b;
huffman@21164:          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164:          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164:         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164: apply safe
huffman@21164: apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164: apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_isconst2:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| a < b;
huffman@21164:          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164:          \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164:          a \<le> x; x \<le> b |]
huffman@21164:         ==> f x = f a"
huffman@21164: apply (blast dest: DERIV_isconst1)
huffman@21164: done
huffman@21164: 
hoelzl@29803: lemma DERIV_isconst3: fixes a b x y :: real
hoelzl@29803:   assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
hoelzl@29803:   assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803:   shows "f x = f y"
hoelzl@29803: proof (cases "x = y")
hoelzl@29803:   case False
hoelzl@29803:   let ?a = "min x y"
hoelzl@29803:   let ?b = "max x y"
hoelzl@29803:   
hoelzl@29803:   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803:   proof (rule allI, rule impI)
hoelzl@29803:     fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hoelzl@29803:     hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hoelzl@29803:     hence "z \<in> {a<..<b}" by auto
hoelzl@29803:     thus "DERIV f z :> 0" by (rule derivable)
hoelzl@29803:   qed
hoelzl@29803:   hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
hoelzl@29803:     and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
hoelzl@29803: 
hoelzl@29803:   have "?a < ?b" using `x \<noteq> y` by auto
hoelzl@29803:   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803:   show ?thesis by auto
hoelzl@29803: qed auto
hoelzl@29803: 
huffman@21164: lemma DERIV_isconst_all:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164: apply (rule linorder_cases [of x y])
huffman@21164: apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_const_ratio_const:
huffman@21784:   fixes f :: "real => real"
huffman@21784:   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164: apply (rule linorder_cases [of a b], auto)
huffman@21164: apply (drule_tac [!] f = f in MVT)
hoelzl@56181: apply (auto dest: DERIV_isCont DERIV_unique simp add: real_differentiable_def)
haftmann@54230: apply (auto dest: DERIV_unique simp add: ring_distribs)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_const_ratio_const2:
huffman@21784:   fixes f :: "real => real"
huffman@21784:   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
lp15@56217: apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
haftmann@57512: apply (auto dest!: DERIV_const_ratio_const simp add: mult.assoc)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164: by (simp)
huffman@21164: 
huffman@21164: lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164: by (simp)
huffman@21164: 
huffman@21164: text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164: 
huffman@21164: lemma DERIV_const_average:
huffman@21164:   fixes v :: "real => real"
huffman@21164:   assumes neq: "a \<noteq> (b::real)"
huffman@21164:       and der: "\<forall>x. DERIV v x :> k"
huffman@21164:   shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164: proof (cases rule: linorder_cases [of a b])
huffman@21164:   case equal with neq show ?thesis by simp
huffman@21164: next
huffman@21164:   case less
huffman@21164:   have "(v b - v a) / (b - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164:   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164:   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164:   ultimately show ?thesis using neq by force
huffman@21164: next
huffman@21164:   case greater
huffman@21164:   have "(v b - v a) / (b - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164:   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164:   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
haftmann@57512:   ultimately show ?thesis using neq by (force simp add: add.commute)
huffman@21164: qed
huffman@21164: 
paulson@33654: (* A function with positive derivative is increasing. 
paulson@33654:    A simple proof using the MVT, by Jeremy Avigad. And variants.
paulson@33654: *)
lp15@56261: lemma DERIV_pos_imp_increasing_open:
paulson@33654:   fixes a::real and b::real and f::"real => real"
lp15@56261:   assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
lp15@56261:       and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
paulson@33654:   shows "f a < f b"
paulson@33654: proof (rule ccontr)
wenzelm@41550:   assume f: "~ f a < f b"
wenzelm@33690:   have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654:       & f b - f a = (b - a) * l"
wenzelm@33690:     apply (rule MVT)
lp15@56261:       using assms Deriv.differentiableI
lp15@56261:       apply force+
wenzelm@33690:     done
wenzelm@41550:   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
paulson@33654:       and "f b - f a = (b - a) * l"
paulson@33654:     by auto
wenzelm@41550:   with assms f have "~(l > 0)"
huffman@36777:     by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
wenzelm@41550:   with assms z show False
lp15@56261:     by (metis DERIV_unique)
paulson@33654: qed
paulson@33654: 
lp15@56261: lemma DERIV_pos_imp_increasing:
lp15@56261:   fixes a::real and b::real and f::"real => real"
lp15@56261:   assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
lp15@56261:   shows "f a < f b"
lp15@56261: by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)
lp15@56261: 
noschinl@45791: lemma DERIV_nonneg_imp_nondecreasing:
paulson@33654:   fixes a::real and b::real and f::"real => real"
paulson@33654:   assumes "a \<le> b" and
paulson@33654:     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
paulson@33654:   shows "f a \<le> f b"
paulson@33654: proof (rule ccontr, cases "a = b")
wenzelm@41550:   assume "~ f a \<le> f b" and "a = b"
wenzelm@41550:   then show False by auto
haftmann@37891: next
haftmann@37891:   assume A: "~ f a \<le> f b"
haftmann@37891:   assume B: "a ~= b"
paulson@33654:   with assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654:       & f b - f a = (b - a) * l"
wenzelm@33690:     apply -
wenzelm@33690:     apply (rule MVT)
wenzelm@33690:       apply auto
wenzelm@33690:       apply (metis DERIV_isCont)
huffman@36777:      apply (metis differentiableI less_le)
paulson@33654:     done
wenzelm@41550:   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
haftmann@37891:       and C: "f b - f a = (b - a) * l"
paulson@33654:     by auto
haftmann@37891:   with A have "a < b" "f b < f a" by auto
haftmann@37891:   with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
huffman@45051:     (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
wenzelm@41550:   with assms z show False
paulson@33654:     by (metis DERIV_unique order_less_imp_le)
paulson@33654: qed
paulson@33654: 
lp15@56261: lemma DERIV_neg_imp_decreasing_open:
lp15@56261:   fixes a::real and b::real and f::"real => real"
lp15@56261:   assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
lp15@56261:       and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
lp15@56261:   shows "f a > f b"
lp15@56261: proof -
lp15@56261:   have "(%x. -f x) a < (%x. -f x) b"
lp15@56261:     apply (rule DERIV_pos_imp_increasing_open [of a b "%x. -f x"])
lp15@56261:     using assms
lp15@56261:     apply auto
lp15@56261:     apply (metis field_differentiable_minus neg_0_less_iff_less)
lp15@56261:     done
lp15@56261:   thus ?thesis
lp15@56261:     by simp
lp15@56261: qed
lp15@56261: 
paulson@33654: lemma DERIV_neg_imp_decreasing:
paulson@33654:   fixes a::real and b::real and f::"real => real"
paulson@33654:   assumes "a < b" and
paulson@33654:     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
paulson@33654:   shows "f a > f b"
lp15@56261: by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)
paulson@33654: 
paulson@33654: lemma DERIV_nonpos_imp_nonincreasing:
paulson@33654:   fixes a::real and b::real and f::"real => real"
paulson@33654:   assumes "a \<le> b" and
paulson@33654:     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
paulson@33654:   shows "f a \<ge> f b"
paulson@33654: proof -
paulson@33654:   have "(%x. -f x) a \<le> (%x. -f x) b"
noschinl@45791:     apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
wenzelm@33690:     using assms
wenzelm@33690:     apply auto
paulson@33654:     apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654:     done
paulson@33654:   thus ?thesis
paulson@33654:     by simp
paulson@33654: qed
huffman@21164: 
lp15@56289: lemma DERIV_pos_imp_increasing_at_bot:
lp15@56289:   fixes f :: "real => real"
lp15@56289:   assumes "\<And>x. x \<le> b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
lp15@56289:       and lim: "(f ---> flim) at_bot"
lp15@56289:   shows "flim < f b"
lp15@56289: proof -
lp15@56289:   have "flim \<le> f (b - 1)"
lp15@56289:     apply (rule tendsto_ge_const [OF _ lim])
lp15@56289:     apply (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder)
lp15@56289:     apply (rule_tac x="b - 2" in exI)
lp15@56289:     apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289:     done
lp15@56289:   also have "... < f b"
lp15@56289:     by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289:   finally show ?thesis .
lp15@56289: qed
lp15@56289: 
lp15@56289: lemma DERIV_neg_imp_decreasing_at_top:
lp15@56289:   fixes f :: "real => real"
lp15@56289:   assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
lp15@56289:       and lim: "(f ---> flim) at_top"
lp15@56289:   shows "flim < f b"
lp15@56289:   apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
lp15@56289:   apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
lp15@56289:   apply (metis filterlim_at_top_mirror lim)
lp15@56289:   done
lp15@56289: 
huffman@23041: text {* Derivative of inverse function *}
huffman@23041: 
huffman@23041: lemma DERIV_inverse_function:
huffman@23041:   fixes f g :: "real \<Rightarrow> real"
huffman@23041:   assumes der: "DERIV f (g x) :> D"
huffman@23041:   assumes neq: "D \<noteq> 0"
huffman@23044:   assumes a: "a < x" and b: "x < b"
huffman@23044:   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041:   assumes cont: "isCont g x"
huffman@23041:   shows "DERIV g x :> inverse D"
huffman@23041: unfolding DERIV_iff2
huffman@23044: proof (rule LIM_equal2)
huffman@23044:   show "0 < min (x - a) (b - x)"
chaieb@27668:     using a b by arith 
huffman@23044: next
huffman@23041:   fix y
huffman@23044:   assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668:   hence "a < y" and "y < b" 
huffman@23044:     by (simp_all add: abs_less_iff)
huffman@23041:   thus "(g y - g x) / (y - x) =
huffman@23041:         inverse ((f (g y) - x) / (g y - g x))"
huffman@23041:     by (simp add: inj)
huffman@23041: next
huffman@23041:   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041:     by (rule der [unfolded DERIV_iff2])
huffman@23041:   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044:     using inj a b by simp
huffman@23041:   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
wenzelm@56219:   proof (rule exI, safe)
huffman@23044:     show "0 < min (x - a) (b - x)"
huffman@23044:       using a b by simp
huffman@23041:   next
huffman@23041:     fix y
huffman@23044:     assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044:     hence y: "a < y" "y < b"
huffman@23044:       by (simp_all add: abs_less_iff)
huffman@23041:     assume "g y = g x"
huffman@23041:     hence "f (g y) = f (g x)" by simp
huffman@23044:     hence "y = x" using inj y a b by simp
huffman@23041:     also assume "y \<noteq> x"
huffman@23041:     finally show False by simp
huffman@23041:   qed
huffman@23041:   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041:     using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041:   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041:         -- x --> inverse D"
huffman@44568:     using neq by (rule tendsto_inverse)
huffman@23041: qed
huffman@23041: 
huffman@29975: subsection {* Generalized Mean Value Theorem *}
huffman@29975: 
huffman@21164: theorem GMVT:
huffman@21784:   fixes a b :: real
huffman@21164:   assumes alb: "a < b"
wenzelm@41550:     and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
hoelzl@56181:     and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
wenzelm@41550:     and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
hoelzl@56181:     and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
wenzelm@53381:   shows "\<exists>g'c f'c c.
wenzelm@53381:     DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
huffman@21164: proof -
huffman@21164:   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
wenzelm@41550:   from assms have "a < b" by simp
huffman@21164:   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@44233:     using fc gc by simp
hoelzl@56181:   moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
huffman@44233:     using fd gd by simp
huffman@21164:   ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
huffman@21164:   then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164:   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164: 
huffman@21164:   from cdef have cint: "a < c \<and> c < b" by auto
hoelzl@56181:   with gd have "g differentiable (at c)" by simp
huffman@21164:   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164:   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164: 
huffman@21164:   from cdef have "a < c \<and> c < b" by auto
hoelzl@56181:   with fd have "f differentiable (at c)" by simp
huffman@21164:   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164:   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164: 
huffman@21164:   from cdef have "DERIV ?h c :> l" by auto
hoelzl@41368:   moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
hoelzl@56381:     using g'cdef f'cdef by (auto intro!: derivative_eq_intros)
huffman@21164:   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164: 
huffman@21164:   {
huffman@21164:     from cdef have "?h b - ?h a = (b - a) * l" by auto
wenzelm@53374:     also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164:     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164:   }
huffman@21164:   moreover
huffman@21164:   {
huffman@21164:     have "?h b - ?h a =
huffman@21164:          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164:           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667:       by (simp add: algebra_simps)
huffman@21164:     hence "?h b - ?h a = 0" by auto
huffman@21164:   }
huffman@21164:   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164:   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164:   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
haftmann@57514:   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
huffman@21164: 
huffman@21164:   with g'cdef f'cdef cint show ?thesis by auto
huffman@21164: qed
huffman@21164: 
hoelzl@50327: lemma GMVT':
hoelzl@50327:   fixes f g :: "real \<Rightarrow> real"
hoelzl@50327:   assumes "a < b"
hoelzl@50327:   assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
hoelzl@50327:   assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
hoelzl@50327:   assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
hoelzl@50327:   assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
hoelzl@50327:   shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327: proof -
hoelzl@50327:   have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
hoelzl@50327:     a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
hoelzl@56181:     using assms by (intro GMVT) (force simp: real_differentiable_def)+
hoelzl@50327:   then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327:     using DERIV_f DERIV_g by (force dest: DERIV_unique)
hoelzl@50327:   then show ?thesis
hoelzl@50327:     by auto
hoelzl@50327: qed
hoelzl@50327: 
hoelzl@51529: 
hoelzl@51529: subsection {* L'Hopitals rule *}
hoelzl@51529: 
hoelzl@51641: lemma isCont_If_ge:
hoelzl@51641:   fixes a :: "'a :: linorder_topology"
hoelzl@51641:   shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
hoelzl@51641:   unfolding isCont_def continuous_within
hoelzl@51641:   apply (intro filterlim_split_at)
hoelzl@51641:   apply (subst filterlim_cong[OF refl refl, where g=g])
hoelzl@51641:   apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641:   apply (subst filterlim_cong[OF refl refl, where g=f])
hoelzl@51641:   apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641:   done
hoelzl@51641: 
hoelzl@50327: lemma lhopital_right_0:
hoelzl@50329:   fixes f0 g0 :: "real \<Rightarrow> real"
hoelzl@50329:   assumes f_0: "(f0 ---> 0) (at_right 0)"
hoelzl@50329:   assumes g_0: "(g0 ---> 0) (at_right 0)"
hoelzl@50327:   assumes ev:
hoelzl@50329:     "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
hoelzl@50327:     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50329:     "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
hoelzl@50329:     "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
hoelzl@50327:   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50329:   shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
hoelzl@50327: proof -
hoelzl@50329:   def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
hoelzl@50329:   then have "f 0 = 0" by simp
hoelzl@50329: 
hoelzl@50329:   def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
hoelzl@50329:   then have "g 0 = 0" by simp
hoelzl@50329: 
hoelzl@50329:   have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
hoelzl@50329:       DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
hoelzl@50329:     using ev by eventually_elim auto
hoelzl@50329:   then obtain a where [arith]: "0 < a"
hoelzl@50329:     and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
hoelzl@50327:     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50329:     and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
hoelzl@50329:     and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
wenzelm@56219:     unfolding eventually_at by (auto simp: dist_real_def)
hoelzl@50327: 
hoelzl@50329:   have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
hoelzl@50329:     using g0_neq_0 by (simp add: g_def)
hoelzl@50329: 
hoelzl@50329:   { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
hoelzl@50329:       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
hoelzl@50329:          (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329:   note f = this
hoelzl@50329: 
hoelzl@50329:   { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
hoelzl@50329:       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
hoelzl@50329:          (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329:   note g = this
hoelzl@50329: 
hoelzl@50329:   have "isCont f 0"
hoelzl@51641:     unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
hoelzl@51641: 
hoelzl@50329:   have "isCont g 0"
hoelzl@51641:     unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
hoelzl@50329: 
hoelzl@50327:   have "\<exists>\<zeta>. \<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
hoelzl@50327:   proof (rule bchoice, rule)
hoelzl@50327:     fix x assume "x \<in> {0 <..< a}"
hoelzl@50327:     then have x[arith]: "0 < x" "x < a" by auto
hoelzl@50327:     with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
hoelzl@50327:       by auto
hoelzl@50328:     have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
hoelzl@50328:       using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328:     moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
hoelzl@50328:       using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328:     ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
hoelzl@50328:       using f g `x < a` by (intro GMVT') auto
wenzelm@53374:     then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
wenzelm@53374:       by blast
hoelzl@50327:     moreover
wenzelm@53374:     from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
hoelzl@50327:       by (simp add: field_simps)
hoelzl@50327:     ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
hoelzl@50327:       using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
hoelzl@50327:   qed
wenzelm@53381:   then obtain \<zeta> where "\<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)" ..
hoelzl@50327:   then have \<zeta>: "eventually (\<lambda>x. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)) (at_right 0)"
hoelzl@51641:     unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
hoelzl@50327:   moreover
hoelzl@50327:   from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
hoelzl@50327:     by eventually_elim auto
hoelzl@50327:   then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
hoelzl@50327:     by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
hoelzl@51641:        (auto intro: tendsto_const tendsto_ident_at)
hoelzl@50327:   then have "(\<zeta> ---> 0) (at_right 0)"
hoelzl@50327:     by (rule tendsto_norm_zero_cancel)
hoelzl@50327:   with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
hoelzl@51641:     by (auto elim!: eventually_elim1 simp: filterlim_at)
hoelzl@50327:   from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
hoelzl@50327:     by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
hoelzl@50329:   ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
hoelzl@50328:     by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
hoelzl@50328:        (auto elim: eventually_elim1)
hoelzl@50329:   also have "?P \<longleftrightarrow> ?thesis"
hoelzl@51641:     by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
hoelzl@50329:   finally show ?thesis .
hoelzl@50327: qed
hoelzl@50327: 
hoelzl@50330: lemma lhopital_right:
hoelzl@50330:   "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330:   ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330:   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330:   by (rule lhopital_right_0)
hoelzl@50330: 
hoelzl@50330: lemma lhopital_left:
hoelzl@50330:   "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330:   ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330:   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@56479:   by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330: 
hoelzl@50330: lemma lhopital:
hoelzl@50330:   "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330:   ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330:   unfolding eventually_at_split filterlim_at_split
hoelzl@50330:   by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
hoelzl@50330: 
hoelzl@50327: lemma lhopital_right_0_at_top:
hoelzl@50327:   fixes f g :: "real \<Rightarrow> real"
hoelzl@50327:   assumes g_0: "LIM x at_right 0. g x :> at_top"
hoelzl@50327:   assumes ev:
hoelzl@50327:     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50327:     "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
hoelzl@50327:     "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
hoelzl@50327:   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50327:   shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
hoelzl@50327:   unfolding tendsto_iff
hoelzl@50327: proof safe
hoelzl@50327:   fix e :: real assume "0 < e"
hoelzl@50327: 
hoelzl@50327:   with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
hoelzl@50327:   have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
hoelzl@50327:   from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
hoelzl@50327:   obtain a where [arith]: "0 < a"
hoelzl@50327:     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50327:     and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
hoelzl@50327:     and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
hoelzl@50327:     and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
hoelzl@51641:     unfolding eventually_at_le by (auto simp: dist_real_def)
hoelzl@51641:     
hoelzl@50327: 
hoelzl@50327:   from Df have
hoelzl@50328:     "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
hoelzl@51641:     unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
hoelzl@50327: 
hoelzl@50327:   moreover
hoelzl@50328:   have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
hoelzl@50346:     using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
hoelzl@50327: 
hoelzl@50327:   moreover
hoelzl@50327:   have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
hoelzl@50327:     using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
hoelzl@50327:     by (rule filterlim_compose)
hoelzl@50327:   then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
hoelzl@50327:     by (intro tendsto_intros)
hoelzl@50327:   then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
hoelzl@50327:     by (simp add: inverse_eq_divide)
hoelzl@50327:   from this[unfolded tendsto_iff, rule_format, of 1]
hoelzl@50327:   have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
hoelzl@50327:     by (auto elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50327: 
hoelzl@50327:   moreover
hoelzl@50327:   from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)"
hoelzl@50327:     by (intro tendsto_intros)
hoelzl@50327:   then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
hoelzl@50327:     by (simp add: inverse_eq_divide)
hoelzl@50327:   from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
hoelzl@50327:   have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
hoelzl@50327:     by (auto simp: dist_real_def)
hoelzl@50327: 
hoelzl@50327:   ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
hoelzl@50327:   proof eventually_elim
hoelzl@50327:     fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
hoelzl@50327:     assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
hoelzl@50327: 
hoelzl@50327:     have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
hoelzl@50327:       using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
wenzelm@53381:     then obtain y where [arith]: "t < y" "y < a"
wenzelm@53381:       and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
wenzelm@53381:       by blast
wenzelm@53381:     from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
hoelzl@50327:       using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
hoelzl@50327: 
hoelzl@50327:     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"
hoelzl@50327:       by (simp add: field_simps)
hoelzl@50327:     have "norm (f t / g t - x) \<le>
hoelzl@50327:         norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
hoelzl@50327:       unfolding * by (rule norm_triangle_ineq)
hoelzl@50327:     also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
hoelzl@50327:       by (simp add: abs_mult D_eq dist_real_def)
hoelzl@50327:     also have "\<dots> < (e / 4) * 2 + e / 2"
hoelzl@50327:       using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
hoelzl@50327:     finally show "dist (f t / g t) x < e"
hoelzl@50327:       by (simp add: dist_real_def)
hoelzl@50327:   qed
hoelzl@50327: qed
hoelzl@50327: 
hoelzl@50330: lemma lhopital_right_at_top:
hoelzl@50330:   "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330:   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330:   by (rule lhopital_right_0_at_top)
hoelzl@50330: 
hoelzl@50330: lemma lhopital_left_at_top:
hoelzl@50330:   "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330:   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@56479:   by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330: 
hoelzl@50330: lemma lhopital_at_top:
hoelzl@50330:   "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330:     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330:     ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330:   unfolding eventually_at_split filterlim_at_split
hoelzl@50330:   by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
hoelzl@50330: 
hoelzl@50347: lemma lhospital_at_top_at_top:
hoelzl@50347:   fixes f g :: "real \<Rightarrow> real"
hoelzl@50347:   assumes g_0: "LIM x at_top. g x :> at_top"
hoelzl@50347:   assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
hoelzl@50347:   assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
hoelzl@50347:   assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
hoelzl@50347:   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
hoelzl@50347:   shows "((\<lambda> x. f x / g x) ---> x) at_top"
hoelzl@50347:   unfolding filterlim_at_top_to_right
hoelzl@50347: proof (rule lhopital_right_0_at_top)
hoelzl@50347:   let ?F = "\<lambda>x. f (inverse x)"
hoelzl@50347:   let ?G = "\<lambda>x. g (inverse x)"
hoelzl@50347:   let ?R = "at_right (0::real)"
hoelzl@50347:   let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
hoelzl@50347: 
hoelzl@50347:   show "LIM x ?R. ?G x :> at_top"
hoelzl@50347:     using g_0 unfolding filterlim_at_top_to_right .
hoelzl@50347: 
hoelzl@50347:   show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
hoelzl@50347:     unfolding eventually_at_right_to_top
hoelzl@50347:     using Dg eventually_ge_at_top[where c="1::real"]
hoelzl@50347:     apply eventually_elim
hoelzl@50347:     apply (rule DERIV_cong)
hoelzl@50347:     apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347:     apply (auto intro!:  DERIV_inverse)
hoelzl@50347:     done
hoelzl@50347: 
hoelzl@50347:   show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
hoelzl@50347:     unfolding eventually_at_right_to_top
hoelzl@50347:     using Df eventually_ge_at_top[where c="1::real"]
hoelzl@50347:     apply eventually_elim
hoelzl@50347:     apply (rule DERIV_cong)
hoelzl@50347:     apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347:     apply (auto intro!:  DERIV_inverse)
hoelzl@50347:     done
hoelzl@50347: 
hoelzl@50347:   show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
hoelzl@50347:     unfolding eventually_at_right_to_top
hoelzl@50347:     using g' eventually_ge_at_top[where c="1::real"]
hoelzl@50347:     by eventually_elim auto
hoelzl@50347:     
hoelzl@50347:   show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
hoelzl@50347:     unfolding filterlim_at_right_to_top
hoelzl@50347:     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
hoelzl@50347:     using eventually_ge_at_top[where c="1::real"]
hoelzl@56479:     by eventually_elim simp
hoelzl@50347: qed
hoelzl@50347: 
huffman@21164: end




