src/HOL/Multivariate_Analysis/Derivative.thy
author himmelma
Wed, 17 Feb 2010 18:33:45 +0100
changeset 35172 579dd5570f96
parent 35028 108662d50512
child 35290 3707f625314f
permissions -rw-r--r--
Added integration to Multivariate-Analysis (upto FTC)
     1 (*  Title:      HOL/Library/Convex_Euclidean_Space.thy
     2     Author:                     John Harrison
     3     Translation from HOL light: Robert Himmelmann, TU Muenchen *)
     4 
     5 header {* Multivariate calculus in Euclidean space. *}
     6 
     7 theory Derivative
     8   imports Brouwer_Fixpoint RealVector
     9 begin
    10 
    11 
    12 (* Because I do not want to type this all the time *)
    13 lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
    14 
    15 subsection {* Derivatives *}
    16 
    17 text {* The definition is slightly tricky since we make it work over
    18   nets of a particular form. This lets us prove theorems generally and use 
    19   "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}
    20 
    21 definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a net \<Rightarrow> bool)"
    22 (infixl "(has'_derivative)" 12) where
    23  "(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
    24    (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
    25 
    26 lemma derivative_linear[dest]:"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
    27   unfolding has_derivative_def by auto
    28 
    29 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 
    30   assume ?l note as = this[unfolded fderiv_def]
    31   show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
    32     fix e::real assume "e>0"
    33     guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] ..
    34     thus "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
    35       dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e"
    36       apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE)
    37       unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed next
    38   assume ?r note as = this[unfolded has_derivative_def]
    39   show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
    40     fix e::real assume "e>0"
    41     guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] ..
    42     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-
    43       apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
    44       unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed qed
    45 
    46 subsection {* These are the only cases we'll care about, probably. *}
    47 
    48 lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
    49          bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
    50   unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)
    51 
    52 lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
    53          bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
    54   apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto
    55 
    56 subsection {* More explicit epsilon-delta forms. *}
    57 
    58 lemma has_derivative_within':
    59   "(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
    60         (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
    61         \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
    62   unfolding has_derivative_within Lim_within vector_dist_norm
    63   unfolding diff_0_right norm_mul by(simp add: group_simps)
    64 
    65 lemma has_derivative_at':
    66  "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
    67    (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
    68         \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
    69   apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within' by auto
    70 
    71 lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
    72   unfolding has_derivative_within' has_derivative_at' by meson
    73 
    74 lemma has_derivative_within_open:
    75   "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
    76   unfolding has_derivative_within has_derivative_at using Lim_within_open by auto
    77 
    78 subsection {* Derivatives on real = Derivatives on real^1 *}
    79 
    80 lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by(auto simp add:vec1_dest_vec1_simps)
    81 
    82 lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
    83   shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
    84   { assume ?l guess K using linear_bounded[OF `?l`] ..
    85     hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K" apply(rule_tac x=K in exI)
    86       unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) }
    87   thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
    88     unfolding vec1_dest_vec1_simps by auto qed 
    89 
    90 lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
    91   "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
    92   = (f has_derivative f') (at x within s)"
    93   unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
    94   unfolding o_def Lim_within Ball_def unfolding forall_vec1 
    95   unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto  
    96 
    97 lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
    98   "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
    99   using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto
   100 
   101 lemma bounded_linear_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real"
   102   shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
   103   unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
   104   unfolding vec1_dest_vec1_simps by auto
   105 
   106 lemma bounded_linear_dest_vec1: fixes f::"real\<Rightarrow>'a::real_normed_vector"
   107   shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
   108   unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
   109   unfolding vec1_dest_vec1_simps by auto
   110 
   111 lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real" shows
   112   "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
   113   unfolding has_derivative_at unfolding bounded_linear_vec1[unfolded linear_conv_bounded_linear]
   114   unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto
   115 
   116 lemma has_derivative_within_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
   117   "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)"
   118   unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def
   119   unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
   120 
   121 lemma has_derivative_at_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
   122   "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
   123   using has_derivative_within_dest_vec1[where s=UNIV] by(auto simp add:within_UNIV)
   124 
   125 lemma derivative_is_linear: fixes f::"real^'a \<Rightarrow> real^'b" shows
   126   "(f has_derivative f') net \<Longrightarrow> linear f'"
   127   unfolding has_derivative_def and linear_conv_bounded_linear by auto
   128 
   129 
   130 subsection {* Combining theorems. *}
   131 
   132 lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
   133   unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
   134   unfolding diff by(simp add: Lim_const)
   135 
   136 lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
   137   apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp
   138 
   139 lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
   140   unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const)
   141 
   142 lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)" proof
   143   guess K using pos_bounded ..
   144   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
   145     fix x case goal1
   146     hence "abs c * norm (f x) \<le> abs c * (norm x * K)" apply-apply(erule conjE,erule_tac x=x in allE)  
   147       apply(rule mult_left_mono) by auto
   148     thus ?case by(auto simp add:field_simps)
   149   qed qed(auto simp add: scaleR.add_right add scaleR)
   150 
   151 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"
   152   unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
   153   using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]]
   154   unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto 
   155 
   156 lemma has_derivative_cmul_eq: assumes "c \<noteq> 0" 
   157   shows "(((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net \<longleftrightarrow> (f has_derivative f') net)"
   158   apply(rule) defer apply(rule has_derivative_cmul,assumption) 
   159   apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto
   160 
   161 lemma has_derivative_neg:
   162  "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
   163   apply(drule has_derivative_cmul[where c="-1"]) by auto
   164 
   165 lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
   166   apply(rule, drule_tac[!] has_derivative_neg) by auto
   167 
   168 lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net"
   169   shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net" proof-
   170   note as = assms[unfolded has_derivative_def]
   171   show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
   172     using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
   173     by(auto simp add:group_simps scaleR_right_diff_distrib scaleR_right_distrib) qed
   174 
   175 lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
   176   apply(drule has_derivative_add) apply(rule has_derivative_const) by auto
   177 
   178 lemma has_derivative_sub:
   179  "(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"
   180   apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:group_simps)
   181 
   182 lemma has_derivative_setsum: assumes "finite s" "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
   183   shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
   184   apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1)) 
   185 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" 
   186   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"
   187     unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto
   188 qed(auto intro!: has_derivative_const)
   189 
   190 lemma has_derivative_setsum_numseg:
   191   "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
   192   ((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
   193   apply(rule has_derivative_setsum) by auto
   194 
   195 subsection {* somewhat different results for derivative of scalar multiplier. *}
   196 
   197 lemma has_derivative_vmul_component: fixes c::"real^'a \<Rightarrow> real^'b" and v::"real^'c"
   198   assumes "(c has_derivative c') net"
   199   shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net" proof-
   200   have *:"\<And>y. (c y $ k *\<^sub>R v - (c (netlimit net) $ k *\<^sub>R v + c' (y - netlimit net) $ k *\<^sub>R v)) = 
   201         (c y $ k - (c (netlimit net) $ k + c' (y - netlimit net) $ k)) *\<^sub>R v" 
   202     unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto
   203   show ?thesis unfolding has_derivative_def and * and linear_conv_bounded_linear[symmetric]
   204     apply(rule,rule linear_vmul_component[of c' k v, unfolded smult_conv_scaleR]) defer 
   205     apply(subst vector_smult_lzero[THEN sym, of v]) unfolding scaleR_scaleR smult_conv_scaleR apply(rule Lim_vmul)
   206     using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
   207     apply(rule,assumption,rule disjI2,rule,rule) proof-
   208     have *:"\<And>x. x - vec 0 = (x::real^'n)" by auto 
   209     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)
   210     fix e assume "\<not> trivial_limit net" "0 < (e::real)"
   211     then have "eventually (\<lambda>x. dist ((1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e) net"
   212       using assms[unfolded has_derivative_def Lim] by auto
   213     thus "eventually (\<lambda>x. dist (1 / norm (x - netlimit net) * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k))) 0 < e) net"
   214       proof (rule eventually_elim1)
   215       case goal1 thus ?case apply - unfolding vector_dist_norm  apply(rule le_less_trans) prefer 2 apply assumption unfolding * ** and norm_vec1
   216         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
   217       qed
   218       qed(insert assms[unfolded has_derivative_def], auto simp add:linear_conv_bounded_linear) qed 
   219 
   220 lemma has_derivative_vmul_within: fixes c::"real \<Rightarrow> real" and v::"real^'a"
   221   assumes "(c has_derivative c') (at x within s)"
   222   shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x within s)" proof-
   223   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
   224   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]
   225   unfolding * and has_derivative_within_vec1_dest_vec1 unfolding has_derivative_within_dest_vec1 using assms by auto qed
   226 
   227 lemma has_derivative_vmul_at: fixes c::"real \<Rightarrow> real" and v::"real^'a"
   228   assumes "(c has_derivative c') (at x)"
   229   shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x)"
   230   using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV)
   231 
   232 lemma has_derivative_lift_dot:
   233   assumes "(f has_derivative f') net"
   234   shows "((\<lambda>x. inner v (f x)) has_derivative (\<lambda>t. inner v (f' t))) net" proof-
   235   show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule)
   236     apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption)
   237     apply(drule Lim_inner[where a=v]) unfolding o_def
   238     by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed
   239 
   240 lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const
   241    has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul 
   242    bounded_linear.has_derivative has_derivative_lift_dot
   243 
   244 subsection {* limit transformation for derivatives. *}
   245 
   246 lemma has_derivative_transform_within:
   247   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)"
   248   shows "(g has_derivative f') (at x within s)"
   249   using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
   250   apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
   251   apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
   252 
   253 lemma has_derivative_transform_at:
   254   assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
   255   shows "(g has_derivative f') (at x)"
   256   apply(subst within_UNIV[THEN sym]) apply(rule has_derivative_transform_within[OF assms(1)])
   257   using assms(2-3) unfolding within_UNIV by auto
   258 
   259 lemma has_derivative_transform_within_open:
   260   assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
   261   shows "(g has_derivative f') (at x)"
   262   using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
   263   apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
   264   apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
   265 
   266 subsection {* differentiability. *}
   267 
   268 definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
   269   "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
   270 
   271 definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
   272   "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
   273 
   274 lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
   275   unfolding differentiable_def by auto
   276 
   277 lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
   278   unfolding differentiable_def using has_derivative_at_within by blast
   279 
   280 lemma differentiable_within_open: assumes "a \<in> s" "open s" shows 
   281   "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
   282   unfolding differentiable_def has_derivative_within_open[OF assms] by auto
   283 
   284 lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
   285   unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
   286 
   287 lemma differentiable_on_eq_differentiable_at: "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
   288   unfolding differentiable_on_def by(auto simp add: differentiable_within_open)
   289 
   290 lemma differentiable_transform_within:
   291   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)"
   292   shows "g differentiable (at x within s)"
   293   using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)])
   294 
   295 lemma differentiable_transform_at:
   296   assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
   297   shows "g differentiable at x"
   298   using assms(3) unfolding differentiable_def using has_derivative_transform_at[OF assms(1-2)] by auto
   299 
   300 subsection {* Frechet derivative and Jacobian matrix. *}
   301 
   302 definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
   303 
   304 lemma frechet_derivative_works:
   305  "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
   306   unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
   307 
   308 lemma linear_frechet_derivative: fixes f::"real^'a \<Rightarrow> real^'b"
   309   shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
   310   unfolding frechet_derivative_works has_derivative_def unfolding linear_conv_bounded_linear by auto
   311 
   312 definition "jacobian f net = matrix(frechet_derivative f net)"
   313 
   314 lemma jacobian_works: "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
   315   apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer
   316   apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption
   317 
   318 subsection {* Differentiability implies continuity. *}
   319 
   320 lemma Lim_mul_norm_within: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   321   shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
   322   unfolding Lim_within apply(rule,rule) apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
   323   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
   324   by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
   325 
   326 lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)" 
   327   shows "continuous (at x within s) f" proof-
   328   from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this
   329   then interpret bounded_linear f' by auto
   330   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"
   331     using zero by auto
   332   have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
   333     apply(rule continuous_within_compose) apply(rule continuous_intros)+
   334     by(rule linear_continuous_within[OF f'[THEN conjunct1]])
   335   show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within]
   336     apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and vector_dist_norm
   337     apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI)
   338     by(auto simp add:zero * elim!:allE) qed
   339 
   340 lemma differentiable_imp_continuous_at: "f differentiable at x \<Longrightarrow> continuous (at x) f"
   341  by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
   342 
   343 lemma differentiable_imp_continuous_on: "f differentiable_on s \<Longrightarrow> continuous_on s f"
   344   unfolding differentiable_on_def continuous_on_eq_continuous_within
   345   using differentiable_imp_continuous_within by blast
   346 
   347 lemma has_derivative_within_subset:
   348  "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
   349   unfolding has_derivative_within using Lim_within_subset by blast
   350 
   351 lemma differentiable_within_subset:
   352   "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
   353   unfolding differentiable_def using has_derivative_within_subset by blast
   354 
   355 lemma differentiable_on_subset: "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
   356   unfolding differentiable_on_def using differentiable_within_subset by blast
   357 
   358 lemma differentiable_on_empty: "f differentiable_on {}"
   359   unfolding differentiable_on_def by auto
   360 
   361 subsection {* Several results are easier using a "multiplied-out" variant.              *)
   362 (* (I got this idea from Dieudonne's proof of the chain rule). *}
   363 
   364 lemma has_derivative_within_alt:
   365  "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
   366   (\<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")
   367 proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
   368     unfolding Lim_within apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
   369     apply(erule exE,rule_tac x=d in exI) apply(erule conjE,rule,assumption,rule,rule) proof-
   370     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>
   371       dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
   372     then interpret bounded_linear f' by auto
   373     show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
   374       case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next
   375       case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
   376 	unfolding vector_dist_norm diff_0_right norm_mul using as(3)
   377 	using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded vector_dist_norm]
   378 	by(auto simp add:linear_0 linear_sub group_simps)
   379       thus ?thesis by(auto simp add:group_simps) qed qed next
   380   assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within apply-apply(erule conjE,rule,assumption)
   381     apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer apply(erule exE,rule_tac x=d in exI)
   382     apply(erule conjE,rule,assumption,rule,rule) unfolding vector_dist_norm diff_0_right norm_scaleR
   383     apply(erule_tac x=xa in ballE,erule impE) proof-
   384     fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
   385         "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
   386     thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
   387       apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:group_simps) qed auto qed
   388 
   389 lemma has_derivative_at_alt:
   390   "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
   391   (\<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))"
   392   using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto
   393 
   394 subsection {* The chain rule. *}
   395 
   396 lemma diff_chain_within:
   397   assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))"
   398   shows "((g o f) has_derivative (g' o f'))(at x within s)"
   399   unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
   400   apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
   401   apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule)
   402   note assms = assms[unfolded has_derivative_within_alt]
   403   fix e::real assume "0<e"
   404   guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
   405   guess B2 using bounded_linear.pos_bounded[OF assms(2)[THEN conjunct1, rule_format]] .. note B2 = this
   406   have *:"e / 2 / B2 > 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto
   407   guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this
   408   have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto
   409   guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this
   410   guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this
   411 
   412   def d0 \<equiv> "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto
   413   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
   414   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)
   415 
   416   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)
   417     proof(rule,rule `d>0`,rule,rule) 
   418     fix y assume as:"y \<in> s" "norm (y - x) < d"
   419     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
   420 
   421     have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
   422       using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:group_simps)
   423     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)
   424     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
   425     also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
   426     also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
   427     finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto 
   428 
   429     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 
   430     also have "\<dots> < de" using d B1 by(auto simp add:field_simps) 
   431     finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
   432       apply-apply(rule de[THEN conjunct2,rule_format]) using `y\<in>s` using d as by auto 
   433     also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto 
   434     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)
   435     finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
   436     
   437     interpret g': bounded_linear g' using assms(2) by auto
   438     interpret f': bounded_linear f' using assms(1) by auto
   439     have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
   440       by(auto simp add:group_simps f'.diff g'.diff g'.add)
   441     also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:group_simps)
   442     also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto 
   443     also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
   444     finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
   445     
   446     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
   447     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
   448 
   449 lemma diff_chain_at:
   450   "(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)"
   451   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
   452 
   453 subsection {* Composition rules stated just for differentiability. *}
   454 
   455 lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector net)"
   456   unfolding differentiable_def using has_derivative_const by auto
   457 
   458 lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector net)"
   459     unfolding differentiable_def using has_derivative_id by auto
   460 
   461 lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector net)"
   462   unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto
   463 
   464 lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector net)"
   465   unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto
   466 
   467 lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
   468    \<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector net)"
   469     unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
   470     apply(rule has_derivative_add) by auto
   471 
   472 lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
   473   \<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector net)"
   474   unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
   475     apply(rule has_derivative_sub) by auto 
   476 
   477 lemma differentiable_setsum: fixes f::"'a \<Rightarrow> (real^'n \<Rightarrow>real^'n)"
   478   assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
   479   shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net" proof-
   480   guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
   481   thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed
   482 
   483 lemma differentiable_setsum_numseg: fixes f::"_ \<Rightarrow> (real^'n \<Rightarrow>real^'n)"
   484   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"
   485   apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto
   486 
   487 lemma differentiable_chain_at:
   488   "f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o f) differentiable (at x)"
   489   unfolding differentiable_def by(meson diff_chain_at)
   490 
   491 lemma differentiable_chain_within:
   492   "f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s))
   493    \<Longrightarrow> (g o f) differentiable (at x within s)"
   494   unfolding differentiable_def by(meson diff_chain_within)
   495 
   496 subsection {* Uniqueness of derivative.                                                 *)
   497 (*                                                                           *)
   498 (* The general result is a bit messy because we need approachability of the  *)
   499 (* limit point from any direction. But OK for nontrivial intervals etc. *}
   500     
   501 lemma frechet_derivative_unique_within: fixes f::"real^'a \<Rightarrow> real^'b"
   502   assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)"
   503   "(\<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-
   504   note as = assms(1,2)[unfolded has_derivative_def]
   505   then interpret f': bounded_linear f' by auto from as interpret f'': bounded_linear f'' by auto
   506   have "x islimpt s" unfolding islimpt_approachable proof(rule,rule)
   507     guess a using UNIV_witness[where 'a='a] ..
   508     fix e::real assume "0<e" guess d using assms(3)[rule_format,OF`e>0`,of a] ..
   509     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)
   510       using basis_nonzero[of a] norm_basis[of a] unfolding vector_dist_norm by auto qed
   511   hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within) unfolding trivial_limit_within by simp
   512   show ?thesis  apply(rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear
   513     apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule ccontr)
   514     fix i::'a def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
   515     assume "f' (basis i) \<noteq> f'' (basis i)" hence "e>0" unfolding e_def by auto
   516     guess d using Lim_sub[OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
   517     guess c using assms(3)[rule_format,OF d[THEN conjunct1],of i] .. note c=this
   518     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)))"
   519       unfolding scaleR_right_distrib by auto
   520     also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"  
   521       unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto
   522     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)
   523     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 
   524       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
   525 
   526 lemma frechet_derivative_unique_at: fixes f::"real^'a \<Rightarrow> real^'b"
   527   shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
   528   apply(rule frechet_derivative_unique_within[of f f' x UNIV f'']) unfolding within_UNIV apply(assumption)+
   529   apply(rule,rule,rule) apply(rule_tac x="e/2" in exI) by auto
   530  
   531 lemma "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def
   532   unfolding continuous_at Lim_at unfolding dist_nz by auto
   533 
   534 lemma frechet_derivative_unique_within_closed_interval: fixes f::"real^'a \<Rightarrow> real^'b"
   535   assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" (is "x\<in>?I") and
   536   "(f has_derivative f' ) (at x within {a..b})" and
   537   "(f has_derivative f'') (at x within {a..b})"
   538   shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof(rule,rule,rule)
   539   fix e::real and i::'a assume "e>0"
   540   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")
   541     case True thus ?thesis apply(rule_tac x="(min (b$i - a$i)  e) / 2" in exI)
   542       using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
   543       unfolding mem_interval by(auto simp add:field_simps) next
   544     note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
   545     case False moreover have "a $ i < x $ i" using False * by auto
   546     moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
   547     also have "\<dots> = a$i + x$i" by auto also have "\<dots> \<le> 2 * x$i" using * by auto 
   548     finally have "a $ i * 2 + min (x $ i - a $ i) e \<le> x $ i * 2" by auto }
   549     moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
   550     hence "x $ i * 2 \<le> b $ i * 2 + min (x $ i - a $ i) e" using * by auto
   551     ultimately show ?thesis apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
   552       using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
   553       unfolding mem_interval by(auto simp add:field_simps) qed qed
   554 
   555 lemma frechet_derivative_unique_within_open_interval: fixes f::"real^'a \<Rightarrow> real^'b"
   556   assumes "x \<in> {a<..<b}" "(f has_derivative f' ) (at x within {a<..<b})"
   557                          "(f has_derivative f'') (at x within {a<..<b})"
   558   shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(2-3))+ proof(rule,rule,rule)
   559   fix e::real and i::'a assume "e>0"
   560   note * = assms(1)[unfolded mem_interval,THEN spec[where x=i]]
   561   have "a $ i < x $ i" using  * by auto
   562   moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
   563   also have "\<dots> = a$i + x$i" by auto also have "\<dots> < 2 * x$i" using * by auto 
   564   finally have "a $ i * 2 + min (x $ i - a $ i) e < x $ i * 2" by auto }
   565   moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
   566   hence "x $ i * 2 < b $ i * 2 + min (x $ i - a $ i) e" using * by auto
   567   ultimately show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a<..<b}"
   568     apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
   569     using `e>0` and assms(1) unfolding mem_interval by(auto simp add:field_simps) qed
   570 
   571 lemma frechet_derivative_at: fixes f::"real^'a \<Rightarrow> real^'b"
   572   shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
   573   apply(rule frechet_derivative_unique_at[of f],assumption)
   574   unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
   575 
   576 lemma frechet_derivative_within_closed_interval: fixes f::"real^'a \<Rightarrow> real^'b"
   577   assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" "(f has_derivative f') (at x within {a.. b})"
   578   shows "frechet_derivative f (at x within {a.. b}) = f'"
   579   apply(rule frechet_derivative_unique_within_closed_interval[where f=f]) 
   580   apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
   581   unfolding differentiable_def using assms(3) by auto 
   582 
   583 subsection {* Component of the differential must be zero if it exists at a local        *)
   584 (* maximum or minimum for that corresponding component. *}
   585 
   586 lemma differential_zero_maxmin_component: fixes f::"real^'a \<Rightarrow> real^'b"
   587   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))"
   588   "f differentiable (at x)" shows "jacobian f (at x) $ k = 0" proof(rule ccontr)
   589   def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
   590   then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto
   591   hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
   592   note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
   593   guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
   594   guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
   595   { fix c assume "abs c \<le> d" 
   596     hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
   597     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)
   598     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
   599     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
   600     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>"
   601       unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric] 
   602       unfolding dot_rmult dot_basis unfolding smult_conv_scaleR by simp  } note * = this
   603   have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
   604     unfolding mem_ball vector_dist_norm using norm_basis[of j] d by auto
   605   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>
   606          ((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
   607   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
   608   show False apply(rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"]) 
   609     using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
   610     unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding group_simps by (auto intro: mult_pos_pos)
   611 qed
   612 
   613 subsection {* In particular if we have a mapping into R^1. *}
   614 
   615 lemma differential_zero_maxmin: fixes f::"real^'a \<Rightarrow> real"
   616   assumes "x \<in> s" "open s" "(f has_derivative f') (at x)"
   617   "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
   618   shows "f' = (\<lambda>v. 0)" proof-
   619   note deriv = assms(3)[unfolded has_derivative_at_vec1]
   620   obtain e where e:"e>0" "ball x e \<subseteq> s" using assms(2)[unfolded open_contains_ball] and assms(1) by auto
   621   hence **:"(jacobian (vec1 \<circ> f) (at x)) $ 1 = 0" using differential_zero_maxmin_component[of e x "\<lambda>x. vec1 (f x)" 1]
   622     using assms(4) and assms(3)[unfolded has_derivative_at_vec1 o_def]
   623     unfolding differentiable_def o_def by auto 
   624   have *:"jacobian (vec1 \<circ> f) (at x) = matrix (vec1 \<circ> f')" unfolding jacobian_def and frechet_derivative_at[OF deriv] ..
   625   have "vec1 \<circ> f' = (\<lambda>x. 0)" apply(rule) unfolding matrix_works[OF derivative_is_linear[OF deriv],THEN sym]
   626     unfolding Cart_eq matrix_vector_mul_component using **[unfolded *] by auto
   627   thus ?thesis apply-apply(rule,subst vec1_eq[THEN sym]) unfolding o_def apply(drule fun_cong) by auto qed
   628 
   629 subsection {* The traditional Rolle theorem in one dimension. *}
   630 
   631 lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
   632   unfolding vector_le_def by auto
   633 lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
   634   unfolding vector_less_def by auto 
   635 
   636 lemma rolle: fixes f::"real\<Rightarrow>real"
   637   assumes "a < b" "f a = f b" "continuous_on {a..b} f"
   638   "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
   639   shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)" proof-
   640   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-
   641     have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto hence *:"{a .. b}\<noteq>{}" by auto
   642     guess d using continuous_attains_sup[OF compact_real_interval * assms(3)] .. note d=this
   643     guess c using continuous_attains_inf[OF compact_real_interval * assms(3)] .. note c=this
   644     show ?thesis proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
   645       case True thus ?thesis apply(erule_tac disjE) apply(rule_tac x=d in bexI)
   646 	apply(rule_tac[3] x=c in bexI) using d c by auto next def e \<equiv> "(a + b) /2"
   647       case False hence "f d = f c" using d c assms(2) by auto
   648       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
   649       thus ?thesis apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto qed qed
   650   then guess x .. note x=this
   651   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"]) 
   652     unfolding vec1_interval defer apply(rule open_interval) 
   653     apply(rule assms(4)[unfolded has_derivative_at_dest_vec1[THEN sym],THEN bspec[where x=x]],assumption)
   654     unfolding o_def apply(erule disjE,rule disjI2) by(auto simp add: vector_less_def) 
   655   thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule 
   656     apply(drule_tac x="vec1 v" in fun_cong) unfolding vec1_dest_vec1 using x(1) by auto qed
   657 
   658 subsection {* One-dimensional mean value theorem. *}
   659 
   660 lemma mvt: fixes f::"real \<Rightarrow> real"
   661   assumes "a < b" "continuous_on {a .. b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
   662   shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))" proof-
   663   have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
   664     apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"]) defer
   665     apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof
   666     fix x assume x:"x \<in> {a<..<b}"
   667     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)"
   668       by(rule has_derivative_intros assms(3)[rule_format,OF x]
   669         has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+ 
   670   qed(insert assms(1), auto simp add:field_simps)
   671   then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed
   672 
   673 lemma mvt_simple: fixes f::"real \<Rightarrow> real"
   674   assumes "a<b"  "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
   675   shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
   676   apply(rule mvt) apply(rule assms(1), rule differentiable_imp_continuous_on)
   677   unfolding differentiable_on_def differentiable_def defer proof 
   678   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] 
   679     apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using x by auto qed(insert assms(2), auto)
   680 
   681 lemma mvt_very_simple: fixes f::"real \<Rightarrow> real"
   682   assumes "a \<le> b" "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
   683   shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)" proof(cases "a = b")
   684   interpret bounded_linear "f' b" using assms(2) assms(1) by auto
   685   case True thus ?thesis apply(rule_tac x=a in bexI)
   686     using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
   687     unfolding True using zero by auto next
   688   case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed
   689 
   690 subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
   691 
   692 lemma inner_eq_dot: fixes a::"real^'n"
   693   shows "a \<bullet> b = inner a b" unfolding inner_vector_def dot_def by auto
   694 
   695 lemma mvt_general: fixes f::"real\<Rightarrow>real^'n"
   696   assumes "a<b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
   697   shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))" proof-
   698   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)"
   699     apply(rule mvt) apply(rule assms(1))unfolding inner_eq_dot apply(rule continuous_on_inner continuous_on_intros assms(2))+ 
   700     unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
   701   then guess x .. note x=this
   702   show ?thesis proof(cases "f a = f b")
   703     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)
   704     also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding norm_pow_2 ..
   705     also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)" using x by auto
   706     also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz)
   707     finally show ?thesis using False x(1) by(auto simp add: real_mult_left_cancel) next
   708     case True thus ?thesis using assms(1) apply(rule_tac x="(a + b) /2" in bexI) by auto qed qed
   709 
   710 subsection {* Still more general bound theorem. *}
   711 
   712 lemma differentiable_bound: fixes f::"real^'a \<Rightarrow> real^'b"
   713   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"
   714   shows "norm(f x - f y) \<le> B * norm(x - y)" proof-
   715   let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
   716   have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
   717     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)
   718   hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+
   719     unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within)
   720     unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
   721     apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) by auto
   722   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
   723     let ?u = "x + u *\<^sub>R (y - x)"
   724     have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})" 
   725       apply(rule diff_chain_within) apply(rule has_derivative_intros)+ 
   726       apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using goal1 * by auto
   727     thus ?case unfolding has_derivative_within_open[OF goal1 open_interval_real] by auto qed
   728   guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
   729   have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y" proof- case goal1
   730     have "norm (f' x y) \<le> onorm (f' x) * norm y"
   731       using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
   732     also have "\<dots> \<le> B * norm y" apply(rule mult_right_mono)
   733       using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps)
   734     finally show ?case by simp qed
   735   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)"
   736     by(auto simp add:norm_minus_commute) 
   737   also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
   738   also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
   739   finally show ?thesis by(auto simp add:norm_minus_commute) qed 
   740 
   741 (** move this **)
   742 declare norm_vec1[simp]
   743 
   744 lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
   745   shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
   746   have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:Cart_eq)
   747   hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by(auto simp add:norm_vec1)
   748   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
   749   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
   750   have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
   751   show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max norm_vec1) qed
   752 
   753 lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
   754   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"
   755   shows "norm(f x - f y) \<le> B * norm(x - y)" 
   756   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"]
   757   unfolding Ball_def forall_vec1 unfolding has_derivative_within_vec1_dest_vec1 image_iff 
   758   unfolding convex_vec1 unfolding o_def vec1_dest_vec1_simps onorm_vec1 using assms by auto
   759  
   760 subsection {* In particular. *}
   761 
   762 lemma has_derivative_zero_constant: fixes f::"real\<Rightarrow>real"
   763   assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
   764   shows "\<exists>c. \<forall>x\<in>s. f x = c" proof(cases "s={}")
   765   case False then obtain x where "x\<in>s" by auto
   766   have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
   767     thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
   768     unfolding onorm_vec1[of "\<lambda>x. 0", THEN sym] onorm_const norm_vec1 by auto qed
   769   thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto
   770 
   771 lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
   772   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"
   773   shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
   774 
   775 subsection {* Differentiability of inverse function (most basic form). *}
   776 
   777 lemma has_derivative_inverse_basic: fixes f::"real^'b \<Rightarrow> real^'c"
   778   assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \<circ> f' = id" "continuous (at y) g"
   779   "open t" "y \<in> t" "\<forall>z\<in>t. f(g z) = z"
   780   shows "(g has_derivative g') (at y)" proof-
   781   interpret f': bounded_linear f' using assms unfolding has_derivative_def by auto
   782   interpret g': bounded_linear g' using assms by auto
   783   guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
   784 (*  have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
   785   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
   786     have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
   787     guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
   788     guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
   789     guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
   790     guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
   791     thus ?case apply(rule_tac x=d in exI) apply rule defer proof(rule,rule)
   792       fix z assume as:"norm (z - y) < d" hence "z\<in>t" using d2 d unfolding vector_dist_norm by auto
   793       have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
   794         unfolding g'.diff f'.diff unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] 
   795 	unfolding assms(7)[rule_format,OF `z\<in>t`] apply(subst norm_minus_cancel[THEN sym]) by auto
   796       also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format]) 
   797       also have "\<dots> \<le> (e / C) * norm (g z - g y) * C" apply(rule mult_right_mono)
   798 	apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]]) apply(cases "z=y") defer
   799 	apply(rule d1[THEN conjunct2, unfolded vector_dist_norm,rule_format]) using as d C d0 by auto
   800       also have "\<dots> \<le> e * norm (g z - g y)" using C by(auto simp add:field_simps)
   801       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)" by simp qed auto qed
   802   have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\<equiv>"C*2"
   803   have "B>0" unfolding B_def using C by auto
   804   have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)" proof(rule,rule) case goal1
   805     have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" by(rule norm_triangle_sub)
   806     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
   807     also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply(rule add_right_mono) using C by auto
   808     finally show ?case unfolding B_def by(auto simp add:field_simps) qed
   809   show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1
   810     hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
   811     guess d' using lem1[rule_format,OF *] .. note d'=this
   812     guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
   813     show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
   814       hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)" using d' k by auto
   815       also have "\<dots> \<le> e * norm(z - y)" unfolding mult_frac_num pos_divide_le_eq[OF `B>0`]
   816 	using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
   817       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed
   818 
   819 subsection {* Simply rewrite that based on the domain point x. *}
   820 
   821 lemma has_derivative_inverse_basic_x: fixes f::"real^'b \<Rightarrow> real^'c"
   822   assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
   823   "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
   824   shows "(g has_derivative g') (at (f(x)))"
   825   apply(rule has_derivative_inverse_basic) using assms by auto
   826 
   827 subsection {* This is the version in Dieudonne', assuming continuity of f and g. *}
   828 
   829 lemma has_derivative_inverse_dieudonne: fixes f::"real^'a \<Rightarrow> real^'b"
   830   assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
   831   (**) "x\<in>s" "(f has_derivative f') (at x)"  "bounded_linear g'" "g' o f' = id"
   832   shows "(g has_derivative g') (at (f x))"
   833   apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
   834   using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)]  continuous_on_eq_continuous_at[OF assms(2)] by auto
   835 
   836 subsection {* Here's the simplest way of not assuming much about g. *}
   837 
   838 lemma has_derivative_inverse: fixes f::"real^'a \<Rightarrow> real^'b"
   839   assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
   840   "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
   841   shows "(g has_derivative g') (at (f x))" proof-
   842   { fix y assume "y\<in>interior (f ` s)" 
   843     then obtain x where "x\<in>s" and *:"y = f x" unfolding image_iff using interior_subset by auto
   844     have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] .. } note * = this
   845   show ?thesis apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
   846     apply(rule continuous_on_interior[OF _ assms(3)]) apply(rule continuous_on_inverse[OF assms(4,1)])
   847     apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption)  qed
   848 
   849 subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
   850 
   851 lemma brouwer_surjective: fixes f::"real^'n \<Rightarrow> real^'n"
   852   assumes "compact t" "convex t"  "t \<noteq> {}" "continuous_on t f"
   853   "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
   854   shows "\<exists>y\<in>t. f y = x" proof-
   855   have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y" by(auto simp add:group_simps)
   856   show ?thesis  unfolding * apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
   857     apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed
   858 
   859 lemma brouwer_surjective_cball: fixes f::"real^'n \<Rightarrow> real^'n"
   860   assumes "0 < e" "continuous_on (cball a e) f"
   861   "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
   862   shows "\<exists>y\<in>cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+
   863   unfolding cball_eq_empty using assms by auto 
   864 
   865 text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
   866 
   867 lemma sussmann_open_mapping: fixes f::"real^'a \<Rightarrow> real^'b"
   868   assumes "open s" "continuous_on s f" "x \<in> s" 
   869   "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
   870   (**) "t \<subseteq> s" "x \<in> interior t"
   871   shows "f x \<in> interior (f ` t)" proof- 
   872   interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
   873   interpret g':bounded_linear g' using assms by auto
   874   guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos)
   875   guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
   876   guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
   877   have *:"0<e0/B" "0<e1/B" apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
   878   guess e using real_lbound_gt_zero[OF *] .. note e=this
   879   have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
   880     apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
   881     prefer 3 apply(rule,rule) proof- 
   882     show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))" unfolding g'.diff
   883       apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
   884       apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
   885       apply(rule continuous_on_subset[OF assms(2)]) apply(rule,unfold image_iff,erule bexE) proof-
   886       fix y z assume as:"y \<in>cball (f x) e"  "z = x + (g' y - g' (f x))"
   887       have "dist x z = norm (g' (f x) - g' y)" unfolding as(2) and vector_dist_norm by auto
   888       also have "\<dots> \<le> norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto
   889       also have "\<dots> \<le> e * B" using as(1)[unfolded mem_cball vector_dist_norm] using B by auto
   890       also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
   891       finally have "z\<in>cball x e1" unfolding mem_cball by force
   892       thus "z \<in> s" using e1 assms(7) by auto qed next
   893     fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
   894     have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
   895     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
   896     also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
   897     finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
   898     have **:"f x + f' (x + g' (z - f x) - x) = z" using assms(6)[unfolded o_def id_def,THEN cong] by auto
   899     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)"
   900       using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] by(auto simp add:group_simps)
   901     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 
   902     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
   903     also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps)
   904     also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
   905     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
   906     finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e" unfolding mem_cball vector_dist_norm by auto
   907   qed(insert e, auto) note lem = this
   908   show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
   909     apply(rule,rule divide_pos_pos) prefer 3 proof 
   910     fix y assume "y \<in> ball (f x) (e/2)" hence *:"y\<in>cball (f x) (e/2)" by auto
   911     guess z using lem[rule_format,OF *] .. note z=this
   912     hence "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by(auto simp add:field_simps)
   913     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
   914     also have "\<dots> \<le> e1"  using e B unfolding less_divide_eq by auto
   915     finally have "x + g'(z - f x) \<in> t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format]) 
   916       unfolding mem_cball vector_dist_norm by auto
   917     thus "y \<in> f ` t" using z by auto qed(insert e, auto) qed
   918 
   919 text {* Hence the following eccentric variant of the inverse function theorem.    *)
   920 (* This has no continuity assumptions, but we do need the inverse function.  *)
   921 (* We could put f' o g = I but this happens to fit with the minimal linear   *)
   922 (* algebra theory I've set up so far. *}
   923 
   924 lemma has_derivative_inverse_strong: fixes f::"real^'n \<Rightarrow> real^'n"
   925   assumes "open s" "x \<in> s" "continuous_on s f"
   926   "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id"
   927   shows "(g has_derivative g') (at (f x))" proof-
   928   have linf:"bounded_linear f'" using assms(5) unfolding has_derivative_def by auto
   929   hence ling:"bounded_linear g'" unfolding linear_conv_bounded_linear[THEN sym]
   930     apply- apply(rule right_inverse_linear) using assms(6) by auto 
   931   moreover have "g' \<circ> f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym]
   932     using linear_inverse_left by auto
   933   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 )
   934     apply(rule assms ling)+ by auto
   935   have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule)
   936     fix e::real assume "e>0"
   937     hence "f x \<in> interior (f ` (ball x e \<inter> s))" using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
   938       by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
   939     then guess d unfolding mem_interior .. note d=this
   940     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"
   941       apply(rule_tac x=d in exI) apply(rule,rule d[THEN conjunct1]) proof(rule,rule) case goal1
   942       hence "g y \<in> g ` f ` (ball x e \<inter> s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
   943 	by(auto simp add:dist_commute)
   944       hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
   945       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
   946   moreover have "f x \<in> interior (f ` s)" apply(rule sussmann_open_mapping)
   947     apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\<in>s` by auto
   948   moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y" proof- case goal1
   949     hence "y\<in>f ` s" using interior_subset by auto then guess z unfolding image_iff ..
   950     thus ?case using assms(4) by auto qed
   951   ultimately show ?thesis apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)]) using assms by auto qed 
   952 
   953 subsection {* A rewrite based on the other domain. *}
   954 
   955 lemma has_derivative_inverse_strong_x: fixes f::"real^'n \<Rightarrow> real^'n"
   956   assumes "open s" "g y \<in> s" "continuous_on s f"
   957   "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y"
   958   shows "(g has_derivative g') (at y)"
   959   using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
   960 
   961 subsection {* On a region. *}
   962 
   963 lemma has_derivative_inverse_on: fixes f::"real^'n \<Rightarrow> real^'n"
   964   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"
   965   shows "(g has_derivative g'(x)) (at (f x))"
   966   apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f]) apply(rule assms)+
   967   unfolding continuous_on_eq_continuous_at[OF assms(1)]
   968   apply(rule,rule differentiable_imp_continuous_at) unfolding differentiable_def using assms by auto
   969 
   970 subsection {* Invertible derivative continous at a point implies local injectivity.     *)
   971 (* It's only for this we need continuity of the derivative, except of course *)
   972 (* if we want the fact that the inverse derivative is also continuous. So if *)
   973 (* we know for some other reason that the inverse function exists, it's OK. *}
   974 
   975 lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
   976   using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] by(auto simp add:group_simps)
   977 
   978 lemma has_derivative_locally_injective: fixes f::"real^'n \<Rightarrow> real^'m"
   979   assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
   980   "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
   981   "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
   982   obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)" proof-
   983   interpret bounded_linear g' using assms by auto
   984   note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
   985   have "g' (f' a 1) = 1" using f'g' by auto
   986   hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
   987   def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
   988   guess d1 using assms(6)[rule_format,OF *] .. note d1=this
   989   from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
   990   obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
   991   guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] .. note d2=this
   992   guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d = this
   993   show ?thesis proof show "a\<in>ball a d" using d by auto
   994     show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x" proof(intro strip)
   995       fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
   996       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))"
   997 	unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:group_simps)
   998       have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
   999 	apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
  1000 	apply(rule_tac[!] ballI) proof- fix u assume u:"u \<in> ball a d" hence "u\<in>s" using d d2 by auto
  1001 	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
  1002 	show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
  1003 	  unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
  1004 	  apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
  1005 	  apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
  1006 	  by(auto intro!: has_derivative_intros derivative_linear)
  1007 	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)
  1008 	  apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
  1009 	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)
  1010 	  unfolding linear_conv_bounded_linear by(rule assms(3) **)+ 
  1011 	also have "\<dots> \<le> onorm g' * k" apply(rule mult_left_mono) 
  1012 	  using d1[THEN conjunct2,rule_format,of u] using onorm_neg[OF **(1)[unfolded linear_linear]]
  1013 	  using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]] by(auto simp add:group_simps) 
  1014 	also have "\<dots> \<le> 1/2" unfolding k_def by auto
  1015 	finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption qed
  1016       moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm])
  1017 	unfolding ph_def using diff unfolding as by auto
  1018       ultimately show "x = y" unfolding norm_minus_commute by auto qed qed auto qed
  1019 
  1020 subsection {* Uniformly convergent sequence of derivatives. *}
  1021 
  1022 lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \<Rightarrow> real^'m \<Rightarrow> real^'n"
  1023   assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  1024   "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
  1025   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)+ 
  1026   fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
  1027   show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
  1028     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-
  1029     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)"
  1030       by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
  1031     { 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)"
  1032 	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) 
  1033       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]
  1034 	by(auto simp add:field_simps)
  1035       finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
  1036     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)
  1037       unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear] by auto qed qed
  1038 
  1039 lemma has_derivative_sequence_lipschitz: fixes f::"nat \<Rightarrow> real^'m \<Rightarrow> real^'n"
  1040   assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  1041   "\<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"
  1042   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)
  1043   case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
  1044   guess N using assms(3)[rule_format,OF *(2)] ..
  1045   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
  1046 
  1047 lemma has_derivative_sequence: fixes f::"nat\<Rightarrow>real^'m\<Rightarrow>real^'n"
  1048   assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  1049   "\<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)"
  1050   "x0 \<in> s"  "((\<lambda>n. f n x0) ---> l) sequentially"
  1051   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-
  1052   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)"
  1053     apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto
  1054   have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof
  1055     fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)" proof(cases "x=x0")
  1056       case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto next
  1057       case False show ?thesis unfolding Cauchy_def proof(rule,rule)
  1058 	fix e::real assume "e>0" hence *:"e/2>0" "e/2/norm(x-x0)>0" using False by(auto intro!:divide_pos_pos)
  1059 	guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
  1060 	guess N using lem1[rule_format,OF *(2)] .. note N = this
  1061 	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+)
  1062 	  fix m n assume as:"max M N \<le>m" "max M N\<le>n"
  1063 	  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))"
  1064 	    unfolding vector_dist_norm by(rule norm_triangle_sub)
  1065 	  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
  1066 	  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 
  1067 	  finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed
  1068   then guess g .. note g = this
  1069   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)
  1070     fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this
  1071     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+)
  1072       fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
  1073       have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially" 
  1074 	unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule)
  1075 	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)"
  1076 	  using N[rule_format, of n m x y] and as by(auto simp add:group_simps) qed
  1077       thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply-
  1078 	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)"])
  1079 	apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed
  1080   show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
  1081     apply(rule,rule,rule g[rule_format],assumption) proof fix x assume "x\<in>s"
  1082     have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule)
  1083       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")
  1084 	case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
  1085 	show ?thesis apply(rule_tac x=N in exI) unfolding True 
  1086 	  using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto next
  1087 	case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos)
  1088 	guess N using assms(3)[rule_format,OF *] .. note N=this
  1089 	show ?thesis apply(rule_tac x=N in exI) proof(rule,rule) case goal1
  1090 	  show ?case unfolding vector_dist_norm using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
  1091 	    by (auto simp add:field_simps) qed qed qed
  1092     show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
  1093       fix x' y z::"real^'m" and c::real
  1094       note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
  1095       show "g' x (c *s x') = c *s g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially])
  1096 	apply(rule lem3[rule_format]) unfolding smult_conv_scaleR 
  1097         unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
  1098 	apply(rule Lim_cmul) by(rule lem3[rule_format])
  1099       show "g' x (y + z) = g' x y + g' x z" apply(rule Lim_unique[OF trivial_limit_sequentially])
  1100 	apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
  1101         apply(rule Lim_add) by(rule lem3[rule_format])+ qed 
  1102     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
  1103       have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this
  1104       guess N2 using lem2[rule_format,OF *] .. note N2=this
  1105       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
  1106       show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) proof(rule,rule)
  1107 	fix y assume as:"y \<in> s" "norm (y - x) < d1" let ?N ="max N1 N2"
  1108 	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])
  1109 	  using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto moreover
  1110 	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
  1111 	have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)" 
  1112 	  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)"] 
  1113 	  by (auto simp add:group_simps) moreover
  1114 	have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)" using N1 `x\<in>s` by auto
  1115 	ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
  1116 	  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)
  1117 	qed qed qed qed
  1118 
  1119 subsection {* Can choose to line up antiderivatives if we want. *}
  1120 
  1121 lemma has_antiderivative_sequence: fixes f::"nat\<Rightarrow> real^'m \<Rightarrow> real^'n"
  1122   assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  1123   "\<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"
  1124   shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof(cases "s={}")
  1125   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
  1126   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)"])
  1127     apply(rule,rule) apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)  
  1128     apply(rule `a\<in>s`) by(auto intro!: Lim_const) qed auto
  1129 
  1130 lemma has_antiderivative_limit: fixes g'::"real^'m \<Rightarrow> real^'m \<Rightarrow> real^'n"
  1131   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))"
  1132   shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof-
  1133   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))"
  1134     apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
  1135   guess f using *[THEN choice] .. note * = this guess f' using *[THEN choice] .. note f=this
  1136   show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer proof(rule,rule)
  1137     fix e::real assume "0<e" guess  N using reals_Archimedean[OF `e>0`] .. note N=this 
  1138     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
  1139       have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
  1140 	using goal1(1) by(auto simp add:field_simps) 
  1141       show ?case using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]] 
  1142 	apply(rule order_trans) using N * apply(cases "h=0") by auto qed qed(insert f,auto) qed
  1143 
  1144 subsection {* Differentiation of a series. *}
  1145 
  1146 definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
  1147 (infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
  1148 
  1149 lemma has_derivative_series: fixes f::"nat \<Rightarrow> real^'m \<Rightarrow> real^'n"
  1150   assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  1151   "\<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)"
  1152   "x\<in>s" "((\<lambda>n. f n x) sums_seq l) k"
  1153   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)"
  1154   unfolding sums_seq_def apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply(rule,rule)
  1155   apply(rule has_derivative_setsum) defer apply(rule,rule assms(2)[rule_format],assumption)
  1156   using assms(4-5) unfolding sums_seq_def by auto
  1157 
  1158 subsection {* Derivative with composed bilinear function. *}
  1159 
  1160 lemma has_derivative_bilinear_within: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p" and f::"real^'q \<Rightarrow> real^'m"
  1161   assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h"
  1162   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-
  1163   have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
  1164     using assms(2) unfolding differentiable_def by auto moreover
  1165   interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
  1166   interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto
  1167   interpret h:bounded_bilinear h using assms by auto
  1168   have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym]
  1169     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]
  1170     unfolding id_def using assms(1) unfolding has_derivative_def by auto
  1171   hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
  1172     using Lim_add[OF Lim_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately
  1173   have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
  1174              + 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)"
  1175     apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2)  unfolding has_derivative_within by auto
  1176   guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
  1177   guess C using f'.pos_bounded .. note C=this
  1178   guess D using g'.pos_bounded .. note D=this
  1179   have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
  1180   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
  1181     hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
  1182     thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE)
  1183       fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
  1184       have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
  1185       also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
  1186 	apply(rule mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
  1187       also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
  1188       also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
  1189 	using as(3)[unfolded vector_dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
  1190       finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
  1191 	unfolding vector_dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed
  1192   have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))" unfolding linear_linear linear_def
  1193     unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR 
  1194     unfolding h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right by auto
  1195   thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within 
  1196     unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
  1197      h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
  1198     scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed
  1199 
  1200 lemma has_derivative_bilinear_at: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p" and f::"real^'p \<Rightarrow> real^'m"
  1201   assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h"
  1202   shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
  1203   using has_derivative_bilinear_within[of f f' x UNIV g g' h] unfolding within_UNIV using assms by auto
  1204 
  1205 subsection {* Considering derivative R(^1)->R^n as a vector. *}
  1206 
  1207 definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('b) \<Rightarrow> (real net \<Rightarrow> bool)"
  1208 (infixl "has'_vector'_derivative" 12) where
  1209  "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
  1210 
  1211 definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
  1212 
  1213 lemma vector_derivative_works: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  1214   shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
  1215 proof assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
  1216   then interpret bounded_linear f' by auto
  1217   thus ?r unfolding vector_derivative_def has_vector_derivative_def
  1218     apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
  1219     using f' unfolding scaleR[THEN sym] by auto
  1220 next assume ?r thus ?l  unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed
  1221 
  1222 lemma vector_derivative_unique_at: fixes f::"real\<Rightarrow>real^'n"
  1223   assumes "(f has_vector_derivative f') (at x)" "(f has_vector_derivative f'') (at x)" shows "f' = f''" proof-
  1224   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)
  1225     using assms[unfolded has_vector_derivative_def] unfolding has_derivative_at_dest_vec1[THEN sym] by auto
  1226   show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
  1227     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
  1228     ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
  1229 
  1230 lemma vector_derivative_unique_within_closed_interval: fixes f::"real \<Rightarrow> real^'n"
  1231   assumes "a < b" "x \<in> {a..b}"
  1232   "(f has_vector_derivative f') (at x within {a..b})"
  1233   "(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof-
  1234   have *:"(\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1 = (\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1"
  1235     apply(rule frechet_derivative_unique_within_closed_interval[of "vec1 a" "vec1 b"])
  1236     using assms(3-)[unfolded has_vector_derivative_def]
  1237     unfolding has_derivative_within_dest_vec1[THEN sym] vec1_interval using assms(1-2) by auto
  1238   show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
  1239     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
  1240     ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
  1241 
  1242 lemma vector_derivative_at: fixes f::"real \<Rightarrow> real^'a" shows
  1243  "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
  1244   apply(rule vector_derivative_unique_at) defer apply assumption
  1245   unfolding vector_derivative_works[THEN sym] differentiable_def
  1246   unfolding has_vector_derivative_def by auto
  1247 
  1248 lemma vector_derivative_within_closed_interval: fixes f::"real \<Rightarrow> real^'a"
  1249   assumes "a < b" "x \<in> {a..b}" "(f has_vector_derivative f') (at x within {a..b})"
  1250   shows "vector_derivative f (at x within {a..b}) = f'"
  1251   apply(rule vector_derivative_unique_within_closed_interval)
  1252   using vector_derivative_works[unfolded differentiable_def]
  1253   using assms by(auto simp add:has_vector_derivative_def)
  1254 
  1255 lemma has_vector_derivative_within_subset: 
  1256  "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
  1257   unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
  1258 
  1259 lemma has_vector_derivative_const: 
  1260  "((\<lambda>x. c) has_vector_derivative 0) net"
  1261   unfolding has_vector_derivative_def using has_derivative_const by auto
  1262 
  1263 lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
  1264   unfolding has_vector_derivative_def using has_derivative_id by auto
  1265 
  1266 lemma has_vector_derivative_cmul:  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
  1267   unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:group_simps)
  1268 
  1269 lemma has_vector_derivative_cmul_eq: assumes "c \<noteq> 0"
  1270   shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
  1271   apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
  1272   apply(rule has_vector_derivative_cmul) using assms by auto
  1273 
  1274 lemma has_vector_derivative_neg:
  1275  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
  1276   unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
  1277 
  1278 lemma has_vector_derivative_add:
  1279   assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
  1280   shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
  1281   using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
  1282   unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
  1283 
  1284 lemma has_vector_derivative_sub:
  1285   assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
  1286   shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
  1287   using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
  1288   unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
  1289 
  1290 lemma has_vector_derivative_bilinear_within: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p"
  1291   assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h"
  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-
  1293   interpret bounded_bilinear h using assms by auto 
  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]
  1295     unfolding o_def vec1_dest_vec1 has_vector_derivative_def
  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)"]
  1297     using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed
  1298 
  1299 lemma has_vector_derivative_bilinear_at: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p"
  1300   assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h"
  1301   shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
  1302   apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto
  1303 
  1304 lemma has_vector_derivative_at_within: "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
  1305   unfolding has_vector_derivative_def apply(rule has_derivative_at_within) by auto
  1306 
  1307 lemma has_vector_derivative_transform_within:
  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)"
  1309   shows "(g has_vector_derivative f') (at x within s)"
  1310   using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within)
  1311 
  1312 lemma has_vector_derivative_transform_at:
  1313   assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x)"
  1314   shows "(g has_vector_derivative f') (at x)"
  1315   using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at)
  1316 
  1317 lemma has_vector_derivative_transform_within_open:
  1318   assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_vector_derivative f') (at x)"
  1319   shows "(g has_vector_derivative f') (at x)"
  1320   using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within_open)
  1321 
  1322 lemma vector_diff_chain_at:
  1323   assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at (f x))"
  1324   shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
  1325   using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
  1326   unfolding o_def scaleR.scaleR_left by auto
  1327 
  1328 lemma vector_diff_chain_within:
  1329   assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)"
  1330   shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
  1331   using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
  1332   unfolding o_def scaleR.scaleR_left by auto
  1333 
  1334 end