1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy Tue Nov 17 18:52:30 2009 +0100
1.3 @@ -0,0 +1,1332 @@
1.4 +(* Title: HOL/Library/Convex_Euclidean_Space.thy
1.5 + Author: John Harrison
1.6 + Translated to from HOL light: Robert Himmelmann, TU Muenchen *)
1.7 +
1.8 +header {* Multivariate calculus in Euclidean space. *}
1.9 +
1.10 +theory Derivative
1.11 + imports Brouwer_Fixpoint RealVector
1.12 +begin
1.13 +
1.14 +
1.15 +(* Because I do not want to type this all the time *)
1.16 +lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
1.17 +
1.18 +subsection {* Derivatives *}
1.19 +
1.20 +text {* The definition is slightly tricky since we make it work over
1.21 + nets of a particular form. This lets us prove theorems generally and use
1.22 + "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}
1.23 +
1.24 +definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a net \<Rightarrow> bool)"
1.25 +(infixl "(has'_derivative)" 12) where
1.26 + "(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
1.27 + (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
1.28 +
1.29 +lemma derivative_linear[dest]:"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
1.30 + unfolding has_derivative_def by auto
1.31 +
1.32 +lemma FDERIV_conv_has_derivative:"FDERIV f (x::'a::{real_normed_vector,perfect_space}) :> f' = (f has_derivative f') (at x)" (is "?l = ?r") proof
1.33 + assume ?l note as = this[unfolded fderiv_def]
1.34 + show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
1.35 + fix e::real assume "e>0"
1.36 + guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] ..
1.37 + thus "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
1.38 + dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e"
1.39 + apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE)
1.40 + unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed next
1.41 + assume ?r note as = this[unfolded has_derivative_def]
1.42 + show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
1.43 + fix e::real assume "e>0"
1.44 + guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] ..
1.45 + thus "\<exists>s>0. \<forall>xa. xa \<noteq> 0 \<and> dist xa 0 < s \<longrightarrow> dist (norm (f (x + xa) - f x - f' xa) / norm xa) 0 < e" apply-
1.46 + apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
1.47 + unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed qed
1.48 +
1.49 +subsection {* These are the only cases we'll care about, probably. *}
1.50 +
1.51 +lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
1.52 + bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
1.53 + unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)
1.54 +
1.55 +lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
1.56 + bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
1.57 + apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto
1.58 +
1.59 +subsection {* More explicit epsilon-delta forms. *}
1.60 +
1.61 +lemma has_derivative_within':
1.62 + "(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
1.63 + (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
1.64 + \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
1.65 + unfolding has_derivative_within Lim_within vector_dist_norm
1.66 + unfolding diff_0_right norm_mul by(simp add: group_simps)
1.67 +
1.68 +lemma has_derivative_at':
1.69 + "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
1.70 + (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
1.71 + \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
1.72 + apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within' by auto
1.73 +
1.74 +lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
1.75 + unfolding has_derivative_within' has_derivative_at' by meson
1.76 +
1.77 +lemma has_derivative_within_open:
1.78 + "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
1.79 + unfolding has_derivative_within has_derivative_at using Lim_within_open by auto
1.80 +
1.81 +subsection {* Derivatives on real = Derivatives on real^1 *}
1.82 +
1.83 +lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by(auto simp add:vec1_dest_vec1_simps)
1.84 +
1.85 +lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
1.86 + shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
1.87 + { assume ?l guess K using linear_bounded[OF `?l`] ..
1.88 + hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K" apply(rule_tac x=K in exI)
1.89 + unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) }
1.90 + thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
1.91 + unfolding vec1_dest_vec1_simps by auto qed
1.92 +
1.93 +lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
1.94 + "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
1.95 + = (f has_derivative f') (at x within s)"
1.96 + unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
1.97 + unfolding o_def Lim_within Ball_def unfolding forall_vec1
1.98 + unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
1.99 +
1.100 +lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
1.101 + "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
1.102 + using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto
1.103 +
1.104 +lemma bounded_linear_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real"
1.105 + shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
1.106 + unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
1.107 + unfolding vec1_dest_vec1_simps by auto
1.108 +
1.109 +lemma bounded_linear_dest_vec1: fixes f::"real\<Rightarrow>'a::real_normed_vector"
1.110 + shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
1.111 + unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
1.112 + unfolding vec1_dest_vec1_simps by auto
1.113 +
1.114 +lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real" shows
1.115 + "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
1.116 + unfolding has_derivative_at unfolding bounded_linear_vec1[unfolded linear_conv_bounded_linear]
1.117 + unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto
1.118 +
1.119 +lemma has_derivative_within_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
1.120 + "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)"
1.121 + unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def
1.122 + unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
1.123 +
1.124 +lemma has_derivative_at_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
1.125 + "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
1.126 + using has_derivative_within_dest_vec1[where s=UNIV] by(auto simp add:within_UNIV)
1.127 +
1.128 +lemma derivative_is_linear: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite" shows
1.129 + "(f has_derivative f') net \<Longrightarrow> linear f'"
1.130 + unfolding has_derivative_def and linear_conv_bounded_linear by auto
1.131 +
1.132 +
1.133 +subsection {* Combining theorems. *}
1.134 +
1.135 +lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
1.136 + unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
1.137 + unfolding diff by(simp add: Lim_const)
1.138 +
1.139 +lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
1.140 + apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp
1.141 +
1.142 +lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
1.143 + unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const)
1.144 +
1.145 +lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)" proof
1.146 + guess K using pos_bounded ..
1.147 + thus "\<exists>K. \<forall>x. norm ((c::real) *\<^sub>R f x) \<le> norm x * K" apply(rule_tac x="abs c * K" in exI) proof
1.148 + fix x case goal1
1.149 + hence "abs c * norm (f x) \<le> abs c * (norm x * K)" apply-apply(erule conjE,erule_tac x=x in allE)
1.150 + apply(rule mult_left_mono) by auto
1.151 + thus ?case by(auto simp add:field_simps)
1.152 + qed qed(auto simp add: scaleR.add_right add scaleR)
1.153 +
1.154 +lemma has_derivative_cmul: assumes "(f has_derivative f') net" shows "((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net"
1.155 + unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
1.156 + using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]]
1.157 + unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto
1.158 +
1.159 +lemma has_derivative_cmul_eq: assumes "c \<noteq> 0"
1.160 + shows "(((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net \<longleftrightarrow> (f has_derivative f') net)"
1.161 + apply(rule) defer apply(rule has_derivative_cmul,assumption)
1.162 + apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto
1.163 +
1.164 +lemma has_derivative_neg:
1.165 + "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
1.166 + apply(drule has_derivative_cmul[where c="-1"]) by auto
1.167 +
1.168 +lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
1.169 + apply(rule, drule_tac[!] has_derivative_neg) by auto
1.170 +
1.171 +lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net"
1.172 + shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net" proof-
1.173 + note as = assms[unfolded has_derivative_def]
1.174 + show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
1.175 + using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
1.176 + by(auto simp add:group_simps scaleR_right_diff_distrib scaleR_right_distrib) qed
1.177 +
1.178 +lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
1.179 + apply(drule has_derivative_add) apply(rule has_derivative_const) by auto
1.180 +
1.181 +lemma has_derivative_sub:
1.182 + "(f has_derivative f') net \<Longrightarrow> (g has_derivative g') net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
1.183 + apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:group_simps)
1.184 +
1.185 +lemma has_derivative_setsum: assumes "finite s" "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
1.186 + shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
1.187 + apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1))
1.188 +proof- fix x F assume as:"finite F" "x \<notin> F" "x\<in>s" "((\<lambda>x. \<Sum>a\<in>F. f a x) has_derivative (\<lambda>h. \<Sum>a\<in>F. f' a h)) net"
1.189 + thus "((\<lambda>xa. \<Sum>a\<in>insert x F. f a xa) has_derivative (\<lambda>h. \<Sum>a\<in>insert x F. f' a h)) net"
1.190 + unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto
1.191 +qed(auto intro!: has_derivative_const)
1.192 +
1.193 +lemma has_derivative_setsum_numseg:
1.194 + "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
1.195 + ((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
1.196 + apply(rule has_derivative_setsum) by auto
1.197 +
1.198 +subsection {* somewhat different results for derivative of scalar multiplier. *}
1.199 +
1.200 +lemma has_derivative_vmul_component: fixes c::"real^'a::finite \<Rightarrow> real^'b::finite" and v::"real^'c::finite"
1.201 + assumes "(c has_derivative c') net"
1.202 + shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net" proof-
1.203 + have *:"\<And>y. (c y $ k *\<^sub>R v - (c (netlimit net) $ k *\<^sub>R v + c' (y - netlimit net) $ k *\<^sub>R v)) =
1.204 + (c y $ k - (c (netlimit net) $ k + c' (y - netlimit net) $ k)) *\<^sub>R v"
1.205 + unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto
1.206 + show ?thesis unfolding has_derivative_def and * and linear_conv_bounded_linear[symmetric]
1.207 + apply(rule,rule linear_vmul_component[of c' k v, unfolded smult_conv_scaleR]) defer
1.208 + apply(subst vector_smult_lzero[THEN sym, of v]) unfolding scaleR_scaleR smult_conv_scaleR apply(rule Lim_vmul)
1.209 + using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
1.210 + apply(rule,assumption,rule disjI2,rule,rule) proof-
1.211 + have *:"\<And>x. x - vec 0 = (x::real^'n)" by auto
1.212 + have **:"\<And>d x. d * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k)) = (d *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net) ))) $k" by(auto simp add:field_simps)
1.213 + fix e assume "\<not> trivial_limit net" "0 < (e::real)"
1.214 + then obtain A where A:"A\<in>Rep_net net" "\<forall>x\<in>A. dist ((1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e"
1.215 + using assms[unfolded has_derivative_def Lim] unfolding eventually_def by auto
1.216 + show "eventually (\<lambda>x. dist (1 / norm (x - netlimit net) * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k))) 0 < e) net"
1.217 + unfolding eventually_def apply(rule_tac x=A in bexI) apply rule proof-
1.218 + case goal1 thus ?case apply -apply(drule A(2)[rule_format]) unfolding vector_dist_norm vec1_vec apply(rule le_less_trans) prefer 2 apply assumption unfolding * ** and norm_vec1[unfolded vec1_vec]
1.219 + using component_le_norm[of "(1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net))) - 0" k] by auto
1.220 + qed(insert A, auto) qed(insert assms[unfolded has_derivative_def], auto simp add:linear_conv_bounded_linear) qed
1.221 +
1.222 +lemma has_derivative_vmul_within: fixes c::"real \<Rightarrow> real" and v::"real^'a::finite"
1.223 + assumes "(c has_derivative c') (at x within s)"
1.224 + shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x within s)" proof-
1.225 + have *:"\<And>c. (\<lambda>x. (vec1 \<circ> c \<circ> dest_vec1) x $ 1 *\<^sub>R v) = (\<lambda>x. (c x) *\<^sub>R v) \<circ> dest_vec1" unfolding o_def by auto
1.226 + show ?thesis using has_derivative_vmul_component[of "vec1 \<circ> c \<circ> dest_vec1" "vec1 \<circ> c' \<circ> dest_vec1" "at (vec1 x) within vec1 ` s" 1 v]
1.227 + unfolding * and has_derivative_within_vec1_dest_vec1 unfolding has_derivative_within_dest_vec1 using assms by auto qed
1.228 +
1.229 +lemma has_derivative_vmul_at: fixes c::"real \<Rightarrow> real" and v::"real^'a::finite"
1.230 + assumes "(c has_derivative c') (at x)"
1.231 + shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x)"
1.232 + using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV)
1.233 +
1.234 +lemma has_derivative_lift_dot:
1.235 + assumes "(f has_derivative f') net"
1.236 + shows "((\<lambda>x. inner v (f x)) has_derivative (\<lambda>t. inner v (f' t))) net" proof-
1.237 + show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule)
1.238 + apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption)
1.239 + apply(drule Lim_inner[where a=v]) unfolding o_def
1.240 + by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed
1.241 +
1.242 +lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const
1.243 + has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul
1.244 + bounded_linear.has_derivative has_derivative_lift_dot
1.245 +
1.246 +subsection {* limit transformation for derivatives. *}
1.247 +
1.248 +lemma has_derivative_transform_within:
1.249 + assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x within s)"
1.250 + shows "(g has_derivative f') (at x within s)"
1.251 + using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
1.252 + apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
1.253 + apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
1.254 +
1.255 +lemma has_derivative_transform_at:
1.256 + assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
1.257 + shows "(g has_derivative f') (at x)"
1.258 + apply(subst within_UNIV[THEN sym]) apply(rule has_derivative_transform_within[OF assms(1)])
1.259 + using assms(2-3) unfolding within_UNIV by auto
1.260 +
1.261 +lemma has_derivative_transform_within_open:
1.262 + assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
1.263 + shows "(g has_derivative f') (at x)"
1.264 + using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
1.265 + apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
1.266 + apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
1.267 +
1.268 +subsection {* differentiability. *}
1.269 +
1.270 +definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
1.271 + "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
1.272 +
1.273 +definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
1.274 + "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
1.275 +
1.276 +lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
1.277 + unfolding differentiable_def by auto
1.278 +
1.279 +lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
1.280 + unfolding differentiable_def using has_derivative_at_within by blast
1.281 +
1.282 +lemma differentiable_within_open: assumes "a \<in> s" "open s" shows
1.283 + "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
1.284 + unfolding differentiable_def has_derivative_within_open[OF assms] by auto
1.285 +
1.286 +lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n::finite) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
1.287 + unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
1.288 +
1.289 +lemma differentiable_on_eq_differentiable_at: "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
1.290 + unfolding differentiable_on_def by(auto simp add: differentiable_within_open)
1.291 +
1.292 +lemma differentiable_transform_within:
1.293 + assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable (at x within s)"
1.294 + shows "g differentiable (at x within s)"
1.295 + using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)])
1.296 +
1.297 +lemma differentiable_transform_at:
1.298 + assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
1.299 + shows "g differentiable at x"
1.300 + using assms(3) unfolding differentiable_def using has_derivative_transform_at[OF assms(1-2)] by auto
1.301 +
1.302 +subsection {* Frechet derivative and Jacobian matrix. *}
1.303 +
1.304 +definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
1.305 +
1.306 +lemma frechet_derivative_works:
1.307 + "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
1.308 + unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
1.309 +
1.310 +lemma linear_frechet_derivative: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.311 + shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
1.312 + unfolding frechet_derivative_works has_derivative_def unfolding linear_conv_bounded_linear by auto
1.313 +
1.314 +definition "jacobian f net = matrix(frechet_derivative f net)"
1.315 +
1.316 +lemma jacobian_works: "(f::(real^'a::finite) \<Rightarrow> (real^'b::finite)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
1.317 + apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer
1.318 + apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption
1.319 +
1.320 +subsection {* Differentiability implies continuity. *}
1.321 +
1.322 +lemma Lim_mul_norm_within: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
1.323 + shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
1.324 + unfolding Lim_within apply(rule,rule) apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
1.325 + apply(rule_tac x="min d 1" in exI) apply rule defer apply(rule,erule_tac x=x in ballE) unfolding vector_dist_norm diff_0_right norm_mul
1.326 + by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
1.327 +
1.328 +lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)"
1.329 + shows "continuous (at x within s) f" proof-
1.330 + from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this
1.331 + then interpret bounded_linear f' by auto
1.332 + have *:"\<And>xa. x\<noteq>xa \<Longrightarrow> (f' \<circ> (\<lambda>y. y - x)) xa + norm (xa - x) *\<^sub>R ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) - ((f' \<circ> (\<lambda>y. y - x)) x + 0) = f xa - f x"
1.333 + using zero by auto
1.334 + have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
1.335 + apply(rule continuous_within_compose) apply(rule continuous_intros)+
1.336 + by(rule linear_continuous_within[OF f'[THEN conjunct1]])
1.337 + show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within]
1.338 + apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and vector_dist_norm
1.339 + apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI)
1.340 + by(auto simp add:zero * elim!:allE) qed
1.341 +
1.342 +lemma differentiable_imp_continuous_at: "f differentiable at x \<Longrightarrow> continuous (at x) f"
1.343 + by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
1.344 +
1.345 +lemma differentiable_imp_continuous_on: "f differentiable_on s \<Longrightarrow> continuous_on s f"
1.346 + unfolding differentiable_on_def continuous_on_eq_continuous_within
1.347 + using differentiable_imp_continuous_within by blast
1.348 +
1.349 +lemma has_derivative_within_subset:
1.350 + "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
1.351 + unfolding has_derivative_within using Lim_within_subset by blast
1.352 +
1.353 +lemma differentiable_within_subset:
1.354 + "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
1.355 + unfolding differentiable_def using has_derivative_within_subset by blast
1.356 +
1.357 +lemma differentiable_on_subset: "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
1.358 + unfolding differentiable_on_def using differentiable_within_subset by blast
1.359 +
1.360 +lemma differentiable_on_empty: "f differentiable_on {}"
1.361 + unfolding differentiable_on_def by auto
1.362 +
1.363 +subsection {* Several results are easier using a "multiplied-out" variant. *)
1.364 +(* (I got this idea from Dieudonne's proof of the chain rule). *}
1.365 +
1.366 +lemma has_derivative_within_alt:
1.367 + "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
1.368 + (\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm(f(y) - f(x) - f'(y - x)) \<le> e * norm(y - x))" (is "?lhs \<longleftrightarrow> ?rhs")
1.369 +proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
1.370 + unfolding Lim_within apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
1.371 + apply(erule exE,rule_tac x=d in exI) apply(erule conjE,rule,assumption,rule,rule) proof-
1.372 + fix x y e d assume as:"0 < e" "0 < d" "norm (y - x) < d" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
1.373 + dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
1.374 + then interpret bounded_linear f' by auto
1.375 + show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
1.376 + case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next
1.377 + case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
1.378 + unfolding vector_dist_norm diff_0_right norm_mul using as(3)
1.379 + using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded vector_dist_norm]
1.380 + by(auto simp add:linear_0 linear_sub group_simps)
1.381 + thus ?thesis by(auto simp add:group_simps) qed qed next
1.382 + assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within apply-apply(erule conjE,rule,assumption)
1.383 + apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer apply(erule exE,rule_tac x=d in exI)
1.384 + apply(erule conjE,rule,assumption,rule,rule) unfolding vector_dist_norm diff_0_right norm_scaleR
1.385 + apply(erule_tac x=xa in ballE,erule impE) proof-
1.386 + fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
1.387 + "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
1.388 + thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
1.389 + apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:group_simps) qed auto qed
1.390 +
1.391 +lemma has_derivative_at_alt:
1.392 + "(f has_derivative f') (at (x::real^'n::finite)) \<longleftrightarrow> bounded_linear f' \<and>
1.393 + (\<forall>e>0. \<exists>d>0. \<forall>y. norm(y - x) < d \<longrightarrow> norm(f y - f x - f'(y - x)) \<le> e * norm(y - x))"
1.394 + using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto
1.395 +
1.396 +subsection {* The chain rule. *}
1.397 +
1.398 +lemma diff_chain_within:
1.399 + assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))"
1.400 + shows "((g o f) has_derivative (g' o f'))(at x within s)"
1.401 + unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
1.402 + apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
1.403 + apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule)
1.404 + note assms = assms[unfolded has_derivative_within_alt]
1.405 + fix e::real assume "0<e"
1.406 + guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
1.407 + guess B2 using bounded_linear.pos_bounded[OF assms(2)[THEN conjunct1, rule_format]] .. note B2 = this
1.408 + have *:"e / 2 / B2 > 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto
1.409 + guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this
1.410 + have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto
1.411 + guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this
1.412 + guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this
1.413 +
1.414 + def d0 \<equiv> "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto
1.415 + def d \<equiv> "(min d0 (de / (B1 + 1))) / 2" have "de * 2 / (B1 + 1) > de / (B1 + 1)" apply(rule divide_strict_right_mono) using B1 de by auto
1.416 + hence d:"d>0" "d < d1" "d < d2" "d < (de / (B1 + 1))" unfolding d_def using d0 d1 d2 de B1 by(auto intro!: divide_pos_pos simp add:min_less_iff_disj not_less)
1.417 +
1.418 + show "\<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" apply(rule_tac x=d in exI)
1.419 + proof(rule,rule `d>0`,rule,rule)
1.420 + fix y assume as:"y \<in> s" "norm (y - x) < d"
1.421 + hence 1:"norm (f y - f x - f' (y - x)) \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x))" using d1 d2 d by auto
1.422 +
1.423 + have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
1.424 + using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:group_simps)
1.425 + also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)" apply(rule add_left_mono) using B1 by(auto simp add:group_simps)
1.426 + also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)" apply(rule add_right_mono) using d1 d2 d as by auto
1.427 + also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
1.428 + also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
1.429 + finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto
1.430 +
1.431 + hence "norm (f y - f x) \<le> d * (1 + B1)" apply- apply(rule order_trans,assumption,rule mult_right_mono) using as B1 by auto
1.432 + also have "\<dots> < de" using d B1 by(auto simp add:field_simps)
1.433 + finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
1.434 + apply-apply(rule de[THEN conjunct2,rule_format]) using `y\<in>s` using d as by auto
1.435 + also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto
1.436 + also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono) using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
1.437 + finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
1.438 +
1.439 + interpret g': bounded_linear g' using assms(2) by auto
1.440 + interpret f': bounded_linear f' using assms(1) by auto
1.441 + have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
1.442 + by(auto simp add:group_simps f'.diff g'.diff g'.add)
1.443 + also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:group_simps)
1.444 + also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto
1.445 + also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
1.446 + finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
1.447 +
1.448 + have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \<le> e * norm (y - x)" using 5 4 by auto
1.449 + thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub) by assumption qed qed
1.450 +
1.451 +lemma diff_chain_at:
1.452 + "(f has_derivative f') (at x) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow> ((g o f) has_derivative (g' o f')) (at x)"
1.453 + using diff_chain_within[of f f' x UNIV g g'] using has_derivative_within_subset[of g g' "f x" UNIV "range f"] unfolding within_UNIV by auto
1.454 +
1.455 +subsection {* Composition rules stated just for differentiability. *}
1.456 +
1.457 +lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector net)"
1.458 + unfolding differentiable_def using has_derivative_const by auto
1.459 +
1.460 +lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector net)"
1.461 + unfolding differentiable_def using has_derivative_id by auto
1.462 +
1.463 +lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector net)"
1.464 + unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto
1.465 +
1.466 +lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector net)"
1.467 + unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto
1.468 +
1.469 +lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
1.470 + \<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector net)"
1.471 + unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
1.472 + apply(rule has_derivative_add) by auto
1.473 +
1.474 +lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
1.475 + \<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector net)"
1.476 + unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
1.477 + apply(rule has_derivative_sub) by auto
1.478 +
1.479 +lemma differentiable_setsum: fixes f::"'a \<Rightarrow> (real^'n::finite \<Rightarrow>real^'n)"
1.480 + assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
1.481 + shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net" proof-
1.482 + guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
1.483 + thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed
1.484 +
1.485 +lemma differentiable_setsum_numseg: fixes f::"_ \<Rightarrow> (real^'n::finite \<Rightarrow>real^'n)"
1.486 + shows "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> (f i) differentiable net \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) {m::nat..n}) differentiable net"
1.487 + apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto
1.488 +
1.489 +lemma differentiable_chain_at:
1.490 + "f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o f) differentiable (at x)"
1.491 + unfolding differentiable_def by(meson diff_chain_at)
1.492 +
1.493 +lemma differentiable_chain_within:
1.494 + "f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s))
1.495 + \<Longrightarrow> (g o f) differentiable (at x within s)"
1.496 + unfolding differentiable_def by(meson diff_chain_within)
1.497 +
1.498 +subsection {* Uniqueness of derivative. *)
1.499 +(* *)
1.500 +(* The general result is a bit messy because we need approachability of the *)
1.501 +(* limit point from any direction. But OK for nontrivial intervals etc. *}
1.502 +
1.503 +lemma frechet_derivative_unique_within: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.504 + assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)"
1.505 + "(\<forall>i::'a::finite. \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis i) \<in> s)" shows "f' = f''" proof-
1.506 + note as = assms(1,2)[unfolded has_derivative_def]
1.507 + then interpret f': bounded_linear f' by auto from as interpret f'': bounded_linear f'' by auto
1.508 + have "x islimpt s" unfolding islimpt_approachable proof(rule,rule)
1.509 + guess a using UNIV_witness[where 'a='a] ..
1.510 + fix e::real assume "0<e" guess d using assms(3)[rule_format,OF`e>0`,of a] ..
1.511 + thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x="x + d*\<^sub>R basis a" in bexI)
1.512 + using basis_nonzero[of a] norm_basis[of a] unfolding vector_dist_norm by auto qed
1.513 + hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within) unfolding trivial_limit_within by simp
1.514 + show ?thesis apply(rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear
1.515 + apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule ccontr)
1.516 + fix i::'a def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
1.517 + assume "f' (basis i) \<noteq> f'' (basis i)" hence "e>0" unfolding e_def by auto
1.518 + guess d using Lim_sub[OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
1.519 + guess c using assms(3)[rule_format,OF d[THEN conjunct1],of i] .. note c=this
1.520 + have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))"
1.521 + unfolding scaleR_right_distrib by auto
1.522 + also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"
1.523 + unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto
1.524 + also have "\<dots> = e" unfolding e_def norm_mul using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by(auto simp add:group_simps)
1.525 + finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"] using norm_basis[of i] unfolding vector_dist_norm
1.526 + unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib by auto qed qed
1.527 +
1.528 +lemma frechet_derivative_unique_at: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.529 + shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
1.530 + apply(rule frechet_derivative_unique_within[of f f' x UNIV f'']) unfolding within_UNIV apply(assumption)+
1.531 + apply(rule,rule,rule) apply(rule_tac x="e/2" in exI) by auto
1.532 +
1.533 +lemma "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def
1.534 + unfolding continuous_at Lim_at unfolding dist_nz by auto
1.535 +
1.536 +lemma frechet_derivative_unique_within_closed_interval: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.537 + assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" (is "x\<in>?I") and
1.538 + "(f has_derivative f' ) (at x within {a..b})" and
1.539 + "(f has_derivative f'') (at x within {a..b})"
1.540 + shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof(rule,rule,rule)
1.541 + fix e::real and i::'a assume "e>0"
1.542 + thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}" proof(cases "x$i=a$i")
1.543 + case True thus ?thesis apply(rule_tac x="(min (b$i - a$i) e) / 2" in exI)
1.544 + using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
1.545 + unfolding mem_interval by(auto simp add:field_simps) next
1.546 + note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
1.547 + case False moreover have "a $ i < x $ i" using False * by auto
1.548 + moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
1.549 + also have "\<dots> = a$i + x$i" by auto also have "\<dots> \<le> 2 * x$i" using * by auto
1.550 + finally have "a $ i * 2 + min (x $ i - a $ i) e \<le> x $ i * 2" by auto }
1.551 + moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
1.552 + hence "x $ i * 2 \<le> b $ i * 2 + min (x $ i - a $ i) e" using * by auto
1.553 + ultimately show ?thesis apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
1.554 + using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
1.555 + unfolding mem_interval by(auto simp add:field_simps) qed qed
1.556 +
1.557 +lemma frechet_derivative_unique_within_open_interval: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.558 + assumes "x \<in> {a<..<b}" "(f has_derivative f' ) (at x within {a<..<b})"
1.559 + "(f has_derivative f'') (at x within {a<..<b})"
1.560 + shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(2-3))+ proof(rule,rule,rule)
1.561 + fix e::real and i::'a assume "e>0"
1.562 + note * = assms(1)[unfolded mem_interval,THEN spec[where x=i]]
1.563 + have "a $ i < x $ i" using * by auto
1.564 + moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
1.565 + also have "\<dots> = a$i + x$i" by auto also have "\<dots> < 2 * x$i" using * by auto
1.566 + finally have "a $ i * 2 + min (x $ i - a $ i) e < x $ i * 2" by auto }
1.567 + moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
1.568 + hence "x $ i * 2 < b $ i * 2 + min (x $ i - a $ i) e" using * by auto
1.569 + ultimately show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a<..<b}"
1.570 + apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
1.571 + using `e>0` and assms(1) unfolding mem_interval by(auto simp add:field_simps) qed
1.572 +
1.573 +lemma frechet_derivative_at: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.574 + shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
1.575 + apply(rule frechet_derivative_unique_at[of f],assumption)
1.576 + unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
1.577 +
1.578 +lemma frechet_derivative_within_closed_interval: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.579 + assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" "(f has_derivative f') (at x within {a.. b})"
1.580 + shows "frechet_derivative f (at x within {a.. b}) = f'"
1.581 + apply(rule frechet_derivative_unique_within_closed_interval[where f=f])
1.582 + apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
1.583 + unfolding differentiable_def using assms(3) by auto
1.584 +
1.585 +subsection {* Component of the differential must be zero if it exists at a local *)
1.586 +(* maximum or minimum for that corresponding component. *}
1.587 +
1.588 +lemma differential_zero_maxmin_component: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.589 + assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
1.590 + "f differentiable (at x)" shows "jacobian f (at x) $ k = 0" proof(rule ccontr)
1.591 + def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
1.592 + then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto
1.593 + hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
1.594 + note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
1.595 + guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
1.596 + guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
1.597 + { fix c assume "abs c \<le> d"
1.598 + hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
1.599 + have "\<bar>(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\<bar> \<le> norm (f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j))" by(rule component_le_norm)
1.600 + also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto
1.601 + finally have "\<bar>(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
1.602 + hence "\<bar>f (x + c *\<^sub>R basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
1.603 + unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric]
1.604 + unfolding dot_rmult dot_basis unfolding smult_conv_scaleR by simp } note * = this
1.605 + have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
1.606 + unfolding mem_ball vector_dist_norm using norm_basis[of j] d by auto
1.607 + hence **:"((f (x - d *\<^sub>R basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R basis j))$k \<le> (f x)$k) \<or>
1.608 + ((f (x - d *\<^sub>R basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R basis j))$k \<ge> (f x)$k)" using assms(2) by auto
1.609 + have ***:"\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
1.610 + show False apply(rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"])
1.611 + using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
1.612 + unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding group_simps by auto qed
1.613 +
1.614 +subsection {* In particular if we have a mapping into R^1. *}
1.615 +
1.616 +lemma differential_zero_maxmin: fixes f::"real^'a::finite \<Rightarrow> real"
1.617 + assumes "x \<in> s" "open s" "(f has_derivative f') (at x)"
1.618 + "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
1.619 + shows "f' = (\<lambda>v. 0)" proof-
1.620 + note deriv = assms(3)[unfolded has_derivative_at_vec1]
1.621 + obtain e where e:"e>0" "ball x e \<subseteq> s" using assms(2)[unfolded open_contains_ball] and assms(1) by auto
1.622 + hence **:"(jacobian (vec1 \<circ> f) (at x)) $ 1 = 0" using differential_zero_maxmin_component[of e x "\<lambda>x. vec1 (f x)" 1]
1.623 + unfolding dest_vec1_def[THEN sym] vec1_dest_vec1 using assms(4) and assms(3)[unfolded has_derivative_at_vec1 o_def]
1.624 + unfolding differentiable_def o_def by auto
1.625 + have *:"jacobian (vec1 \<circ> f) (at x) = matrix (vec1 \<circ> f')" unfolding jacobian_def and frechet_derivative_at[OF deriv] ..
1.626 + have "vec1 \<circ> f' = (\<lambda>x. 0)" apply(rule) unfolding matrix_works[OF derivative_is_linear[OF deriv],THEN sym]
1.627 + unfolding Cart_eq matrix_vector_mul_component using **[unfolded *] by auto
1.628 + thus ?thesis apply-apply(rule,subst vec1_eq[THEN sym]) unfolding o_def apply(drule fun_cong) by auto qed
1.629 +
1.630 +subsection {* The traditional Rolle theorem in one dimension. *}
1.631 +
1.632 +lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
1.633 + unfolding vector_less_eq_def by auto
1.634 +lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
1.635 + unfolding vector_less_def by auto
1.636 +
1.637 +lemma rolle: fixes f::"real\<Rightarrow>real"
1.638 + assumes "a < b" "f a = f b" "continuous_on {a..b} f"
1.639 + "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
1.640 + shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)" proof-
1.641 + have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))" proof-
1.642 + have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto hence *:"{a .. b}\<noteq>{}" by auto
1.643 + guess d using continuous_attains_sup[OF compact_real_interval * assms(3)] .. note d=this
1.644 + guess c using continuous_attains_inf[OF compact_real_interval * assms(3)] .. note c=this
1.645 + show ?thesis proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
1.646 + case True thus ?thesis apply(erule_tac disjE) apply(rule_tac x=d in bexI)
1.647 + apply(rule_tac[3] x=c in bexI) using d c by auto next def e \<equiv> "(a + b) /2"
1.648 + case False hence "f d = f c" using d c assms(2) by auto
1.649 + hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d" using c d apply- apply(erule_tac x=x in ballE)+ by auto
1.650 + thus ?thesis apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto qed qed
1.651 + then guess x .. note x=this
1.652 + hence "f' x \<circ> dest_vec1 = (\<lambda>v. 0)" apply(rule_tac differential_zero_maxmin[of "vec1 x" "vec1 ` {a<..<b}" "f \<circ> dest_vec1" "(f' x) \<circ> dest_vec1"])
1.653 + unfolding vec1_interval defer apply(rule open_interval)
1.654 + apply(rule assms(4)[unfolded has_derivative_at_dest_vec1[THEN sym],THEN bspec[where x=x]],assumption)
1.655 + unfolding o_def apply(erule disjE,rule disjI2) by(auto simp add: vector_less_def dest_vec1_def)
1.656 + thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
1.657 + apply(drule_tac x="vec1 v" in fun_cong) unfolding vec1_dest_vec1 using x(1) by auto qed
1.658 +
1.659 +subsection {* One-dimensional mean value theorem. *}
1.660 +
1.661 +lemma mvt: fixes f::"real \<Rightarrow> real"
1.662 + assumes "a < b" "continuous_on {a .. b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
1.663 + shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))" proof-
1.664 + have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
1.665 + apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"]) defer
1.666 + apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof
1.667 + fix x assume x:"x \<in> {a<..<b}"
1.668 + show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) has_derivative (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)"
1.669 + by(rule has_derivative_intros assms(3)[rule_format,OF x]
1.670 + has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+
1.671 + qed(insert assms(1), auto simp add:field_simps)
1.672 + then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed
1.673 +
1.674 +lemma mvt_simple: fixes f::"real \<Rightarrow> real"
1.675 + assumes "a<b" "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
1.676 + shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
1.677 + apply(rule mvt) apply(rule assms(1), rule differentiable_imp_continuous_on)
1.678 + unfolding differentiable_on_def differentiable_def defer proof
1.679 + fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)" unfolding has_derivative_within_open[OF x open_interval_real,THEN sym]
1.680 + apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using x by auto qed(insert assms(2), auto)
1.681 +
1.682 +lemma mvt_very_simple: fixes f::"real \<Rightarrow> real"
1.683 + assumes "a \<le> b" "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
1.684 + shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)" proof(cases "a = b")
1.685 + interpret bounded_linear "f' b" using assms(2) assms(1) by auto
1.686 + case True thus ?thesis apply(rule_tac x=a in bexI)
1.687 + using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
1.688 + unfolding True using zero by auto next
1.689 + case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed
1.690 +
1.691 +subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
1.692 +
1.693 +lemma inner_eq_dot: fixes a::"real^'n::finite"
1.694 + shows "a \<bullet> b = inner a b" unfolding inner_vector_def dot_def by auto
1.695 +
1.696 +lemma mvt_general: fixes f::"real\<Rightarrow>real^'n::finite"
1.697 + assumes "a<b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
1.698 + shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))" proof-
1.699 + have "\<exists>x\<in>{a<..<b}. (op \<bullet> (f b - f a) \<circ> f) b - (op \<bullet> (f b - f a) \<circ> f) a = (f b - f a) \<bullet> f' x (b - a)"
1.700 + apply(rule mvt) apply(rule assms(1))unfolding inner_eq_dot apply(rule continuous_on_inner continuous_on_intros assms(2))+
1.701 + unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
1.702 + then guess x .. note x=this
1.703 + show ?thesis proof(cases "f a = f b")
1.704 + case False have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" by(simp add:class_semiring.semiring_rules)
1.705 + also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding norm_pow_2 ..
1.706 + also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)" using x by auto
1.707 + also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz)
1.708 + finally show ?thesis using False x(1) by(auto simp add: real_mult_left_cancel) next
1.709 + case True thus ?thesis using assms(1) apply(rule_tac x="(a + b) /2" in bexI) by auto qed qed
1.710 +
1.711 +subsection {* Still more general bound theorem. *}
1.712 +
1.713 +lemma differentiable_bound: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.714 + assumes "convex s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
1.715 + shows "norm(f x - f y) \<le> B * norm(x - y)" proof-
1.716 + let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
1.717 + have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
1.718 + using assms(1)[unfolded convex_alt,rule_format,OF x y] unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib by(auto simp add:group_simps)
1.719 + hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+
1.720 + unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within)
1.721 + unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
1.722 + apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) by auto
1.723 + have 2:"\<forall>u\<in>{0<..<1}. ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)" proof rule case goal1
1.724 + let ?u = "x + u *\<^sub>R (y - x)"
1.725 + have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
1.726 + apply(rule diff_chain_within) apply(rule has_derivative_intros)+
1.727 + apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using goal1 * by auto
1.728 + thus ?case unfolding has_derivative_within_open[OF goal1 open_interval_real] by auto qed
1.729 + guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
1.730 + have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y" proof- case goal1
1.731 + have "norm (f' x y) \<le> onorm (f' x) * norm y"
1.732 + using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
1.733 + also have "\<dots> \<le> B * norm y" apply(rule mult_right_mono)
1.734 + using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps)
1.735 + finally show ?case by simp qed
1.736 + have "norm (f x - f y) = norm ((f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 1 - (f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 0)"
1.737 + by(auto simp add:norm_minus_commute)
1.738 + also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
1.739 + also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
1.740 + finally show ?thesis by(auto simp add:norm_minus_commute) qed
1.741 +
1.742 +lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
1.743 + shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
1.744 + have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 unfolding norm_vec1 by auto
1.745 + hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by(auto simp add:norm_vec1)
1.746 + have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto
1.747 +
1.748 + have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
1.749 + have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
1.750 + show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max norm_vec1) qed
1.751 +
1.752 +lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
1.753 + assumes "convex s" "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
1.754 + shows "norm(f x - f y) \<le> B * norm(x - y)"
1.755 + using differentiable_bound[of "vec1 ` s" "vec1 \<circ> f \<circ> dest_vec1" "\<lambda>x. vec1 \<circ> (f' (dest_vec1 x)) \<circ> dest_vec1" B "vec1 x" "vec1 y"]
1.756 + unfolding Ball_def forall_vec1 unfolding has_derivative_within_vec1_dest_vec1 image_iff
1.757 + unfolding convex_vec1 unfolding o_def vec1_dest_vec1_simps onorm_vec1 using assms by auto
1.758 +
1.759 +subsection {* In particular. *}
1.760 +
1.761 +lemma has_derivative_zero_constant: fixes f::"real\<Rightarrow>real"
1.762 + assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
1.763 + shows "\<exists>c. \<forall>x\<in>s. f x = c" proof(cases "s={}")
1.764 + case False then obtain x where "x\<in>s" by auto
1.765 + have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
1.766 + thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
1.767 + unfolding onorm_vec1[of "\<lambda>x. 0", THEN sym] onorm_const norm_vec1 by auto qed
1.768 + thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto
1.769 +
1.770 +lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
1.771 + assumes "convex s" "a \<in> s" "f a = c" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
1.772 + shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
1.773 +
1.774 +subsection {* Differentiability of inverse function (most basic form). *}
1.775 +
1.776 +lemma has_derivative_inverse_basic: fixes f::"real^'b::finite \<Rightarrow> real^'c::finite"
1.777 + assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \<circ> f' = id" "continuous (at y) g"
1.778 + "open t" "y \<in> t" "\<forall>z\<in>t. f(g z) = z"
1.779 + shows "(g has_derivative g') (at y)" proof-
1.780 + interpret f': bounded_linear f' using assms unfolding has_derivative_def by auto
1.781 + interpret g': bounded_linear g' using assms by auto
1.782 + guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
1.783 +(* have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
1.784 + have lem1:"\<forall>e>0. \<exists>d>0. \<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y - g'(z - y)) \<le> e * norm(g z - g y)" proof(rule,rule) case goal1
1.785 + have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
1.786 + guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
1.787 + guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
1.788 + guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
1.789 + guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
1.790 + thus ?case apply(rule_tac x=d in exI) apply rule defer proof(rule,rule)
1.791 + fix z assume as:"norm (z - y) < d" hence "z\<in>t" using d2 d unfolding vector_dist_norm by auto
1.792 + have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
1.793 + unfolding g'.diff f'.diff unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
1.794 + unfolding assms(7)[rule_format,OF `z\<in>t`] apply(subst norm_minus_cancel[THEN sym]) by auto
1.795 + also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format])
1.796 + also have "\<dots> \<le> (e / C) * norm (g z - g y) * C" apply(rule mult_right_mono)
1.797 + apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]]) apply(cases "z=y") defer
1.798 + apply(rule d1[THEN conjunct2, unfolded vector_dist_norm,rule_format]) using as d C d0 by auto
1.799 + also have "\<dots> \<le> e * norm (g z - g y)" using C by(auto simp add:field_simps)
1.800 + finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)" by simp qed auto qed
1.801 + have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\<equiv>"C*2"
1.802 + have "B>0" unfolding B_def using C by auto
1.803 + have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)" proof(rule,rule) case goal1
1.804 + have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" by(rule norm_triangle_sub)
1.805 + also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)" apply(rule add_left_mono) using d and goal1 by auto
1.806 + also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply(rule add_right_mono) using C by auto
1.807 + finally show ?case unfolding B_def by(auto simp add:field_simps) qed
1.808 + show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1
1.809 + hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
1.810 + guess d' using lem1[rule_format,OF *] .. note d'=this
1.811 + guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
1.812 + show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
1.813 + hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)" using d' k by auto
1.814 + also have "\<dots> \<le> e * norm(z - y)" unfolding mult_frac_num pos_divide_le_eq[OF `B>0`]
1.815 + using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
1.816 + finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed
1.817 +
1.818 +subsection {* Simply rewrite that based on the domain point x. *}
1.819 +
1.820 +lemma has_derivative_inverse_basic_x: fixes f::"real^'b::finite \<Rightarrow> real^'c::finite"
1.821 + assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
1.822 + "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
1.823 + shows "(g has_derivative g') (at (f(x)))"
1.824 + apply(rule has_derivative_inverse_basic) using assms by auto
1.825 +
1.826 +subsection {* This is the version in Dieudonne', assuming continuity of f and g. *}
1.827 +
1.828 +lemma has_derivative_inverse_dieudonne: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.829 + assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
1.830 + (**) "x\<in>s" "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
1.831 + shows "(g has_derivative g') (at (f x))"
1.832 + apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
1.833 + using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)] by auto
1.834 +
1.835 +subsection {* Here's the simplest way of not assuming much about g. *}
1.836 +
1.837 +lemma has_derivative_inverse: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.838 + assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
1.839 + "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
1.840 + shows "(g has_derivative g') (at (f x))" proof-
1.841 + { fix y assume "y\<in>interior (f ` s)"
1.842 + then obtain x where "x\<in>s" and *:"y = f x" unfolding image_iff using interior_subset by auto
1.843 + have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] .. } note * = this
1.844 + show ?thesis apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
1.845 + apply(rule continuous_on_interior[OF _ assms(3)]) apply(rule continuous_on_inverse[OF assms(4,1)])
1.846 + apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption) qed
1.847 +
1.848 +subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
1.849 +
1.850 +lemma brouwer_surjective: fixes f::"real^'n::finite \<Rightarrow> real^'n"
1.851 + assumes "compact t" "convex t" "t \<noteq> {}" "continuous_on t f"
1.852 + "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
1.853 + shows "\<exists>y\<in>t. f y = x" proof-
1.854 + have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y" by(auto simp add:group_simps)
1.855 + show ?thesis unfolding * apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
1.856 + apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed
1.857 +
1.858 +lemma brouwer_surjective_cball: fixes f::"real^'n::finite \<Rightarrow> real^'n"
1.859 + assumes "0 < e" "continuous_on (cball a e) f"
1.860 + "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
1.861 + shows "\<exists>y\<in>cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+
1.862 + unfolding cball_eq_empty using assms by auto
1.863 +
1.864 +text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
1.865 +
1.866 +lemma sussmann_open_mapping: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
1.867 + assumes "open s" "continuous_on s f" "x \<in> s"
1.868 + "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
1.869 + (**) "t \<subseteq> s" "x \<in> interior t"
1.870 + shows "f x \<in> interior (f ` t)" proof-
1.871 + interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
1.872 + interpret g':bounded_linear g' using assms by auto
1.873 + guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos)
1.874 + guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
1.875 + guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
1.876 + have *:"0<e0/B" "0<e1/B" apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
1.877 + guess e using real_lbound_gt_zero[OF *] .. note e=this
1.878 + have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
1.879 + apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
1.880 + prefer 3 apply(rule,rule) proof-
1.881 + show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))" unfolding g'.diff
1.882 + apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
1.883 + apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
1.884 + apply(rule continuous_on_subset[OF assms(2)]) apply(rule,unfold image_iff,erule bexE) proof-
1.885 + fix y z assume as:"y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
1.886 + have "dist x z = norm (g' (f x) - g' y)" unfolding as(2) and vector_dist_norm by auto
1.887 + also have "\<dots> \<le> norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto
1.888 + also have "\<dots> \<le> e * B" using as(1)[unfolded mem_cball vector_dist_norm] using B by auto
1.889 + also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
1.890 + finally have "z\<in>cball x e1" unfolding mem_cball by force
1.891 + thus "z \<in> s" using e1 assms(7) by auto qed next
1.892 + fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
1.893 + have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
1.894 + also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using as(2)[unfolded mem_cball vector_dist_norm] and B unfolding norm_minus_commute by auto
1.895 + also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
1.896 + finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
1.897 + have **:"f x + f' (x + g' (z - f x) - x) = z" using assms(6)[unfolded o_def id_def,THEN cong] by auto
1.898 + have "norm (f x - (y + (z - f (x + g' (z - f x))))) \<le> norm (f (x + g' (z - f x)) - z) + norm (f x - y)"
1.899 + using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] by(auto simp add:group_simps)
1.900 + also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding group_simps ** by auto
1.901 + also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball vector_dist_norm] by auto
1.902 + also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps)
1.903 + also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
1.904 + also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono) using as(2)[unfolded mem_cball vector_dist_norm] unfolding norm_minus_commute by auto
1.905 + finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e" unfolding mem_cball vector_dist_norm by auto
1.906 + qed(insert e, auto) note lem = this
1.907 + show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
1.908 + apply(rule,rule divide_pos_pos) prefer 3 proof
1.909 + fix y assume "y \<in> ball (f x) (e/2)" hence *:"y\<in>cball (f x) (e/2)" by auto
1.910 + guess z using lem[rule_format,OF *] .. note z=this
1.911 + hence "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by(auto simp add:field_simps)
1.912 + also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using z(1) unfolding mem_cball vector_dist_norm norm_minus_commute using B by auto
1.913 + also have "\<dots> \<le> e1" using e B unfolding less_divide_eq by auto
1.914 + finally have "x + g'(z - f x) \<in> t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
1.915 + unfolding mem_cball vector_dist_norm by auto
1.916 + thus "y \<in> f ` t" using z by auto qed(insert e, auto) qed
1.917 +
1.918 +text {* Hence the following eccentric variant of the inverse function theorem. *)
1.919 +(* This has no continuity assumptions, but we do need the inverse function. *)
1.920 +(* We could put f' o g = I but this happens to fit with the minimal linear *)
1.921 +(* algebra theory I've set up so far. *}
1.922 +
1.923 +lemma has_derivative_inverse_strong: fixes f::"real^'n::finite \<Rightarrow> real^'n"
1.924 + assumes "open s" "x \<in> s" "continuous_on s f"
1.925 + "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id"
1.926 + shows "(g has_derivative g') (at (f x))" proof-
1.927 + have linf:"bounded_linear f'" using assms(5) unfolding has_derivative_def by auto
1.928 + hence ling:"bounded_linear g'" unfolding linear_conv_bounded_linear[THEN sym]
1.929 + apply- apply(rule right_inverse_linear) using assms(6) by auto
1.930 + moreover have "g' \<circ> f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym]
1.931 + using linear_inverse_left by auto
1.932 + moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)" apply(rule,rule,rule,rule sussmann_open_mapping )
1.933 + apply(rule assms ling)+ by auto
1.934 + have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule)
1.935 + fix e::real assume "e>0"
1.936 + hence "f x \<in> interior (f ` (ball x e \<inter> s))" using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
1.937 + by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
1.938 + then guess d unfolding mem_interior .. note d=this
1.939 + show "\<exists>d>0. \<forall>y. 0 < dist y (f x) \<and> dist y (f x) < d \<longrightarrow> dist (g y) (g (f x)) < e"
1.940 + apply(rule_tac x=d in exI) apply(rule,rule d[THEN conjunct1]) proof(rule,rule) case goal1
1.941 + hence "g y \<in> g ` f ` (ball x e \<inter> s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
1.942 + by(auto simp add:dist_commute)
1.943 + hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
1.944 + thus "dist (g y) (g (f x)) < e" using assms(4)[rule_format,OF `x\<in>s`] by(auto simp add:dist_commute) qed qed
1.945 + moreover have "f x \<in> interior (f ` s)" apply(rule sussmann_open_mapping)
1.946 + apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\<in>s` by auto
1.947 + moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y" proof- case goal1
1.948 + hence "y\<in>f ` s" using interior_subset by auto then guess z unfolding image_iff ..
1.949 + thus ?case using assms(4) by auto qed
1.950 + ultimately show ?thesis apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)]) using assms by auto qed
1.951 +
1.952 +subsection {* A rewrite based on the other domain. *}
1.953 +
1.954 +lemma has_derivative_inverse_strong_x: fixes f::"real^'n::finite \<Rightarrow> real^'n"
1.955 + assumes "open s" "g y \<in> s" "continuous_on s f"
1.956 + "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y"
1.957 + shows "(g has_derivative g') (at y)"
1.958 + using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
1.959 +
1.960 +subsection {* On a region. *}
1.961 +
1.962 +lemma has_derivative_inverse_on: fixes f::"real^'n::finite \<Rightarrow> real^'n"
1.963 + assumes "open s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)" "\<forall>x\<in>s. g(f x) = x" "f'(x) o g'(x) = id" "x\<in>s"
1.964 + shows "(g has_derivative g'(x)) (at (f x))"
1.965 + apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f]) apply(rule assms)+
1.966 + unfolding continuous_on_eq_continuous_at[OF assms(1)]
1.967 + apply(rule,rule differentiable_imp_continuous_at) unfolding differentiable_def using assms by auto
1.968 +
1.969 +subsection {* Invertible derivative continous at a point implies local injectivity. *)
1.970 +(* It's only for this we need continuity of the derivative, except of course *)
1.971 +(* if we want the fact that the inverse derivative is also continuous. So if *)
1.972 +(* we know for some other reason that the inverse function exists, it's OK. *}
1.973 +
1.974 +lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
1.975 + using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] by(auto simp add:group_simps)
1.976 +
1.977 +lemma has_derivative_locally_injective: fixes f::"real^'n::finite \<Rightarrow> real^'m::finite"
1.978 + assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
1.979 + "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
1.980 + "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
1.981 + obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)" proof-
1.982 + interpret bounded_linear g' using assms by auto
1.983 + note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
1.984 + have "g' (f' a 1) = 1" using f'g' by auto
1.985 + hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
1.986 + def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
1.987 + guess d1 using assms(6)[rule_format,OF *] .. note d1=this
1.988 + from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
1.989 + obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
1.990 + guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] .. note d2=this
1.991 + guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d = this
1.992 + show ?thesis proof show "a\<in>ball a d" using d by auto
1.993 + show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x" proof(intro strip)
1.994 + fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
1.995 + def ph \<equiv> "\<lambda>w. w - g'(f w - f x)" have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
1.996 + unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:group_simps)
1.997 + have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
1.998 + apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
1.999 + apply(rule_tac[!] ballI) proof- fix u assume u:"u \<in> ball a d" hence "u\<in>s" using d d2 by auto
1.1000 + have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)" unfolding o_def and diff using f'g' by auto
1.1001 + show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
1.1002 + unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
1.1003 + apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
1.1004 + apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
1.1005 + by(auto intro!: has_derivative_intros derivative_linear)
1.1006 + have **:"bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub)
1.1007 + apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
1.1008 + have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)" unfolding * apply(rule onorm_compose)
1.1009 + unfolding linear_conv_bounded_linear by(rule assms(3) **)+
1.1010 + also have "\<dots> \<le> onorm g' * k" apply(rule mult_left_mono)
1.1011 + using d1[THEN conjunct2,rule_format,of u] using onorm_neg[OF **(1)[unfolded linear_linear]]
1.1012 + using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]] by(auto simp add:group_simps)
1.1013 + also have "\<dots> \<le> 1/2" unfolding k_def by auto
1.1014 + finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption qed
1.1015 + moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm])
1.1016 + unfolding ph_def using diff unfolding as by auto
1.1017 + ultimately show "x = y" unfolding norm_minus_commute by auto qed qed auto qed
1.1018 +
1.1019 +subsection {* Uniformly convergent sequence of derivatives. *}
1.1020 +
1.1021 +lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
1.1022 + assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
1.1023 + "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
1.1024 + shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)" proof(default)+
1.1025 + fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
1.1026 + show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
1.1027 + apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)]) apply(rule_tac[!] ballI) proof-
1.1028 + fix x assume "x\<in>s" show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within s)"
1.1029 + by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
1.1030 + { fix h have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
1.1031 + using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"] unfolding norm_minus_commute by(auto simp add:group_simps)
1.1032 + also have "\<dots> \<le> e * norm h+ e * norm h" using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h] assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
1.1033 + by(auto simp add:field_simps)
1.1034 + finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
1.1035 + thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e" apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
1.1036 + unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear] by auto qed qed
1.1037 +
1.1038 +lemma has_derivative_sequence_lipschitz: fixes f::"nat \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
1.1039 + assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
1.1040 + "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)" "0 < e"
1.1041 + shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)" proof(rule,rule)
1.1042 + case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
1.1043 + guess N using assms(3)[rule_format,OF *(2)] ..
1.1044 + thus ?case apply(rule_tac x=N in exI) apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *]) using assms by auto qed
1.1045 +
1.1046 +lemma has_derivative_sequence: fixes f::"nat\<Rightarrow>real^'m::finite\<Rightarrow>real^'n::finite"
1.1047 + assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
1.1048 + "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
1.1049 + "x0 \<in> s" "((\<lambda>n. f n x0) ---> l) sequentially"
1.1050 + shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)" proof-
1.1051 + have lem1:"\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
1.1052 + apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto
1.1053 + have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof
1.1054 + fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)" proof(cases "x=x0")
1.1055 + case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto next
1.1056 + case False show ?thesis unfolding Cauchy_def proof(rule,rule)
1.1057 + fix e::real assume "e>0" hence *:"e/2>0" "e/2/norm(x-x0)>0" using False by(auto intro!:divide_pos_pos)
1.1058 + guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
1.1059 + guess N using lem1[rule_format,OF *(2)] .. note N = this
1.1060 + show " \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e" apply(rule_tac x="max M N" in exI) proof(default+)
1.1061 + fix m n assume as:"max M N \<le>m" "max M N\<le>n"
1.1062 + have "dist (f m x) (f n x) \<le> norm (f m x0 - f n x0) + norm (f m x - f n x - (f m x0 - f n x0))"
1.1063 + unfolding vector_dist_norm by(rule norm_triangle_sub)
1.1064 + also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2" using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False by auto
1.1065 + also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_right_mono) using as and M[rule_format] unfolding vector_dist_norm by auto
1.1066 + finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed
1.1067 + then guess g .. note g = this
1.1068 + have lem2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f n x - f n y) - (g x - g y)) \<le> e * norm(x - y)" proof(rule,rule)
1.1069 + fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this
1.1070 + show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply(rule_tac x=N in exI) proof(default+)
1.1071 + fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
1.1072 + have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
1.1073 + unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule)
1.1074 + fix m assume "N\<le>m" thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
1.1075 + using N[rule_format, of n m x y] and as by(auto simp add:group_simps) qed
1.1076 + thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply-
1.1077 + apply(rule Lim_norm_ubound[OF trivial_limit_sequentially, where f="\<lambda>m. (f n x - f n y) - (f m x - f m y)"])
1.1078 + apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed
1.1079 + show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
1.1080 + apply(rule,rule,rule g[rule_format],assumption) proof fix x assume "x\<in>s"
1.1081 + have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule)
1.1082 + fix u and e::real assume "e>0" show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e" proof(cases "u=0")
1.1083 + case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
1.1084 + show ?thesis apply(rule_tac x=N in exI) unfolding True
1.1085 + using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto next
1.1086 + case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos)
1.1087 + guess N using assms(3)[rule_format,OF *] .. note N=this
1.1088 + show ?thesis apply(rule_tac x=N in exI) proof(rule,rule) case goal1
1.1089 + show ?case unfolding vector_dist_norm using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
1.1090 + by (auto simp add:field_simps) qed qed qed
1.1091 + show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
1.1092 + fix x' y z::"real^'m" and c::real
1.1093 + note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
1.1094 + show "g' x (c *s x') = c *s g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially])
1.1095 + apply(rule lem3[rule_format]) unfolding smult_conv_scaleR
1.1096 + unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
1.1097 + apply(rule Lim_cmul) by(rule lem3[rule_format])
1.1098 + show "g' x (y + z) = g' x y + g' x z" apply(rule Lim_unique[OF trivial_limit_sequentially])
1.1099 + apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
1.1100 + apply(rule Lim_add) by(rule lem3[rule_format])+ qed
1.1101 + show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)" proof(rule,rule) case goal1
1.1102 + have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this
1.1103 + guess N2 using lem2[rule_format,OF *] .. note N2=this
1.1104 + guess d1 using assms(2)[unfolded has_derivative_within_alt, rule_format,OF `x\<in>s`, of "max N1 N2",THEN conjunct2,rule_format,OF *] .. note d1=this
1.1105 + show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) proof(rule,rule)
1.1106 + fix y assume as:"y \<in> s" "norm (y - x) < d1" let ?N ="max N1 N2"
1.1107 + have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)" apply(subst norm_minus_cancel[THEN sym])
1.1108 + using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto moreover
1.1109 + have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)" using d1 and as by auto ultimately
1.1110 + have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
1.1111 + using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
1.1112 + by (auto simp add:group_simps) moreover
1.1113 + have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)" using N1 `x\<in>s` by auto
1.1114 + ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
1.1115 + using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"] by(auto simp add:group_simps)
1.1116 + qed qed qed qed
1.1117 +
1.1118 +subsection {* Can choose to line up antiderivatives if we want. *}
1.1119 +
1.1120 +lemma has_antiderivative_sequence: fixes f::"nat\<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
1.1121 + assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
1.1122 + "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm h"
1.1123 + shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof(cases "s={}")
1.1124 + case False then obtain a where "a\<in>s" by auto have *:"\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x" by auto
1.1125 + show ?thesis apply(rule *) apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
1.1126 + apply(rule,rule) apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
1.1127 + apply(rule `a\<in>s`) by(auto intro!: Lim_const) qed auto
1.1128 +
1.1129 +lemma has_antiderivative_limit: fixes g'::"real^'m::finite \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
1.1130 + assumes "convex s" "\<forall>e>0. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> e * norm(h))"
1.1131 + shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof-
1.1132 + have *:"\<forall>n. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm(h))"
1.1133 + apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
1.1134 + guess f using *[THEN choice] .. note * = this guess f' using *[THEN choice] .. note f=this
1.1135 + show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer proof(rule,rule)
1.1136 + fix e::real assume "0<e" guess N using reals_Archimedean[OF `e>0`] .. note N=this
1.1137 + show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h" apply(rule_tac x=N in exI) proof(default+) case goal1
1.1138 + have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
1.1139 + using goal1(1) by(auto simp add:field_simps)
1.1140 + show ?case using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
1.1141 + apply(rule order_trans) using N * apply(cases "h=0") by auto qed qed(insert f,auto) qed
1.1142 +
1.1143 +subsection {* Differentiation of a series. *}
1.1144 +
1.1145 +definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
1.1146 +(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
1.1147 +
1.1148 +lemma has_derivative_series: fixes f::"nat \<Rightarrow> real^'m::finite \<Rightarrow> real^'n::finite"
1.1149 + assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
1.1150 + "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(setsum (\<lambda>i. f' i x h) (k \<inter> {0..n}) - g' x h) \<le> e * norm(h)"
1.1151 + "x\<in>s" "((\<lambda>n. f n x) sums_seq l) k"
1.1152 + shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) sums_seq (g x)) k \<and> (g has_derivative g'(x)) (at x within s)"
1.1153 + unfolding sums_seq_def apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply(rule,rule)
1.1154 + apply(rule has_derivative_setsum) defer apply(rule,rule assms(2)[rule_format],assumption)
1.1155 + using assms(4-5) unfolding sums_seq_def by auto
1.1156 +
1.1157 +subsection {* Derivative with composed bilinear function. *}
1.1158 +
1.1159 +lemma has_derivative_bilinear_within: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite" and f::"real^'q::finite \<Rightarrow> real^'m"
1.1160 + assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h"
1.1161 + shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" proof-
1.1162 + have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
1.1163 + using assms(2) unfolding differentiable_def by auto moreover
1.1164 + interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
1.1165 + interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto
1.1166 + interpret h:bounded_bilinear h using assms by auto
1.1167 + have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym]
1.1168 + apply(rule Lim_linear[of "\<lambda>y. y - x" 0 "at x within s" f']) using Lim_sub[OF Lim_within_id Lim_const, of x x s]
1.1169 + unfolding id_def using assms(1) unfolding has_derivative_def by auto
1.1170 + hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
1.1171 + using Lim_add[OF Lim_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately
1.1172 + have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
1.1173 + + h ((1/ (norm (x' - x))) *\<^sub>R (f x' - (f x + f' (x' - x)))) (g x')) ---> h (f x) 0 + h 0 (g x)) (at x within s)"
1.1174 + apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2) unfolding has_derivative_within by auto
1.1175 + guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
1.1176 + guess C using f'.pos_bounded .. note C=this
1.1177 + guess D using g'.pos_bounded .. note D=this
1.1178 + have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
1.1179 + have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)" unfolding Lim_within proof(rule,rule) case goal1
1.1180 + hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
1.1181 + thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE)
1.1182 + fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
1.1183 + have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
1.1184 + also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
1.1185 + apply(rule pordered_semiring_class.mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
1.1186 + also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
1.1187 + also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
1.1188 + using as(3)[unfolded vector_dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
1.1189 + finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
1.1190 + unfolding vector_dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed
1.1191 + have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))" unfolding linear_linear linear_def
1.1192 + unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR
1.1193 + unfolding h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right by auto
1.1194 + thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within
1.1195 + unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
1.1196 + h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
1.1197 + scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed
1.1198 +
1.1199 +lemma has_derivative_bilinear_at: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite" and f::"real^'p::finite \<Rightarrow> real^'m"
1.1200 + assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h"
1.1201 + shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
1.1202 + using has_derivative_bilinear_within[of f f' x UNIV g g' h] unfolding within_UNIV using assms by auto
1.1203 +
1.1204 +subsection {* Considering derivative R(^1)->R^n as a vector. *}
1.1205 +
1.1206 +definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('b) \<Rightarrow> (real net \<Rightarrow> bool)"
1.1207 +(infixl "has'_vector'_derivative" 12) where
1.1208 + "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
1.1209 +
1.1210 +definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
1.1211 +
1.1212 +lemma vector_derivative_works: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
1.1213 + shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
1.1214 +proof assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
1.1215 + then interpret bounded_linear f' by auto
1.1216 + thus ?r unfolding vector_derivative_def has_vector_derivative_def
1.1217 + apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
1.1218 + using f' unfolding scaleR[THEN sym] by auto
1.1219 +next assume ?r thus ?l unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed
1.1220 +
1.1221 +lemma vector_derivative_unique_at: fixes f::"real\<Rightarrow>real^'n::finite"
1.1222 + assumes "(f has_vector_derivative f') (at x)" "(f has_vector_derivative f'') (at x)" shows "f' = f''" proof-
1.1223 + have *:"(\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1 = (\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1" apply(rule frechet_derivative_unique_at)
1.1224 + using assms[unfolded has_vector_derivative_def] unfolding has_derivative_at_dest_vec1[THEN sym] by auto
1.1225 + show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
1.1226 + hence "((\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1) (vec1 1) = ((\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1) (vec1 1)" using * by auto
1.1227 + ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
1.1228 +
1.1229 +lemma vector_derivative_unique_within_closed_interval: fixes f::"real \<Rightarrow> real^'n::finite"
1.1230 + assumes "a < b" "x \<in> {a..b}"
1.1231 + "(f has_vector_derivative f') (at x within {a..b})"
1.1232 + "(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof-
1.1233 + have *:"(\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1 = (\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1"
1.1234 + apply(rule frechet_derivative_unique_within_closed_interval[of "vec1 a" "vec1 b"])
1.1235 + using assms(3-)[unfolded has_vector_derivative_def]
1.1236 + unfolding has_derivative_within_dest_vec1[THEN sym] vec1_interval using assms(1-2) by auto
1.1237 + show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
1.1238 + hence "((\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1) (vec1 1) = ((\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1) (vec1 1)" using * by auto
1.1239 + ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
1.1240 +
1.1241 +lemma vector_derivative_at: fixes f::"real \<Rightarrow> real^'a::finite" shows
1.1242 + "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
1.1243 + apply(rule vector_derivative_unique_at) defer apply assumption
1.1244 + unfolding vector_derivative_works[THEN sym] differentiable_def
1.1245 + unfolding has_vector_derivative_def by auto
1.1246 +
1.1247 +lemma vector_derivative_within_closed_interval: fixes f::"real \<Rightarrow> real^'a::finite"
1.1248 + assumes "a < b" "x \<in> {a..b}" "(f has_vector_derivative f') (at x within {a..b})"
1.1249 + shows "vector_derivative f (at x within {a..b}) = f'"
1.1250 + apply(rule vector_derivative_unique_within_closed_interval)
1.1251 + using vector_derivative_works[unfolded differentiable_def]
1.1252 + using assms by(auto simp add:has_vector_derivative_def)
1.1253 +
1.1254 +lemma has_vector_derivative_within_subset: fixes f::"real \<Rightarrow> real^'a::finite" shows
1.1255 + "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
1.1256 + unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
1.1257 +
1.1258 +lemma has_vector_derivative_const: fixes c::"real^'n::finite" shows
1.1259 + "((\<lambda>x. c) has_vector_derivative 0) net"
1.1260 + unfolding has_vector_derivative_def using has_derivative_const by auto
1.1261 +
1.1262 +lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
1.1263 + unfolding has_vector_derivative_def using has_derivative_id by auto
1.1264 +
1.1265 +lemma has_vector_derivative_cmul: fixes f::"real \<Rightarrow> real^'a::finite"
1.1266 + shows "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
1.1267 + unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:group_simps)
1.1268 +
1.1269 +lemma has_vector_derivative_cmul_eq: fixes f::"real \<Rightarrow> real^'a::finite" assumes "c \<noteq> 0"
1.1270 + shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
1.1271 + apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
1.1272 + apply(rule has_vector_derivative_cmul) using assms by auto
1.1273 +
1.1274 +lemma has_vector_derivative_neg:
1.1275 + "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
1.1276 + unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
1.1277 +
1.1278 +lemma has_vector_derivative_add:
1.1279 + assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
1.1280 + shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
1.1281 + using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
1.1282 + unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
1.1283 +
1.1284 +lemma has_vector_derivative_sub:
1.1285 + assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
1.1286 + shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
1.1287 + using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
1.1288 + unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
1.1289 +
1.1290 +lemma has_vector_derivative_bilinear_within: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite"
1.1291 + assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h"
1.1292 + shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)" proof-
1.1293 + interpret bounded_bilinear h using assms by auto
1.1294 + show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def has_derivative_within_dest_vec1[THEN sym]], where h=h]
1.1295 + unfolding o_def vec1_dest_vec1 has_vector_derivative_def
1.1296 + unfolding has_derivative_within_dest_vec1[unfolded o_def, where f="\<lambda>x. h (f x) (g x)" and f'="\<lambda>d. h (f x) (d *\<^sub>R g') + h (d *\<^sub>R f') (g x)"]
1.1297 + using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed
1.1298 +
1.1299 +lemma has_vector_derivative_bilinear_at: fixes h::"real^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real^'p::finite"
1.1300 + assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h"
1.1301 + shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
1.1302 + apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto
1.1303 +
1.1304 +lemma has_vector_derivative_at_within: "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
1.1305 + unfolding has_vector_derivative_def apply(rule has_derivative_at_within) by auto
1.1306 +
1.1307 +lemma has_vector_derivative_transform_within:
1.1308 + assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x within s)"
1.1309 + shows "(g has_vector_derivative f') (at x within s)"
1.1310 + using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within)
1.1311 +
1.1312 +lemma has_vector_derivative_transform_at:
1.1313 + assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x)"
1.1314 + shows "(g has_vector_derivative f') (at x)"
1.1315 + using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at)
1.1316 +
1.1317 +lemma has_vector_derivative_transform_within_open:
1.1318 + assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_vector_derivative f') (at x)"
1.1319 + shows "(g has_vector_derivative f') (at x)"
1.1320 + using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within_open)
1.1321 +
1.1322 +lemma vector_diff_chain_at:
1.1323 + assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at (f x))"
1.1324 + shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
1.1325 + using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
1.1326 + unfolding o_def scaleR.scaleR_left by auto
1.1327 +
1.1328 +lemma vector_diff_chain_within:
1.1329 + assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)"
1.1330 + shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
1.1331 + using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
1.1332 + unfolding o_def scaleR.scaleR_left by auto
1.1333 +
1.1334 +end
1.1335 +