src/HOL/Multivariate_Analysis/Derivative.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51478 270b21f3ae0a parent 51363 d4d00c804645 child 51641 cd05e9fcc63d permissions -rw-r--r--
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
```     1 (*  Title:                       HOL/Multivariate_Analysis/Derivative.thy
```
```     2     Author:                      John Harrison
```
```     3     Translation from HOL Light:  Robert Himmelmann, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* Multivariate calculus in Euclidean space. *}
```
```     7
```
```     8 theory Derivative
```
```     9 imports Brouwer_Fixpoint Operator_Norm
```
```    10 begin
```
```    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 filter \<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]:
```
```    27   "(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
```
```    28   unfolding has_derivative_def by auto
```
```    29
```
```    30 lemma netlimit_at_vector:
```
```    31   (* TODO: move *)
```
```    32   fixes a :: "'a::real_normed_vector"
```
```    33   shows "netlimit (at a) = a"
```
```    34 proof (cases "\<exists>x. x \<noteq> a")
```
```    35   case True then obtain x where x: "x \<noteq> a" ..
```
```    36   have "\<not> trivial_limit (at a)"
```
```    37     unfolding trivial_limit_def eventually_at dist_norm
```
```    38     apply clarsimp
```
```    39     apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
```
```    40     apply (simp add: norm_sgn sgn_zero_iff x)
```
```    41     done
```
```    42   thus ?thesis
```
```    43     by (rule netlimit_within [of a UNIV, unfolded within_UNIV])
```
```    44 qed simp
```
```    45
```
```    46 lemma FDERIV_conv_has_derivative:
```
```    47   shows "FDERIV f x :> f' = (f has_derivative f') (at x)"
```
```    48   unfolding fderiv_def has_derivative_def netlimit_at_vector
```
```    49   by (simp add: diff_diff_eq Lim_at_zero [where a=x]
```
```    50     tendsto_norm_zero_iff [where 'b='b, symmetric])
```
```    51
```
```    52 lemma DERIV_conv_has_derivative:
```
```    53   "(DERIV f x :> f') = (f has_derivative op * f') (at x)"
```
```    54   unfolding deriv_fderiv FDERIV_conv_has_derivative
```
```    55   by (subst mult_commute, rule refl)
```
```    56
```
```    57 text {* These are the only cases we'll care about, probably. *}
```
```    58
```
```    59 lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
```
```    60          bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
```
```    61   unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)
```
```    62
```
```    63 lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
```
```    64          bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
```
```    65   using has_derivative_within [of f f' x UNIV] by simp
```
```    66
```
```    67 text {* More explicit epsilon-delta forms. *}
```
```    68
```
```    69 lemma has_derivative_within':
```
```    70   "(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
```
```    71         (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
```
```    72         \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
```
```    73   unfolding has_derivative_within Lim_within dist_norm
```
```    74   unfolding diff_0_right by (simp add: diff_diff_eq)
```
```    75
```
```    76 lemma has_derivative_at':
```
```    77  "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
```
```    78    (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
```
```    79         \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
```
```    80   using has_derivative_within' [of f f' x UNIV] by simp
```
```    81
```
```    82 lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
```
```    83   unfolding has_derivative_within' has_derivative_at' by meson
```
```    84
```
```    85 lemma has_derivative_within_open:
```
```    86   "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
```
```    87   by (simp only: at_within_interior interior_open)
```
```    88
```
```    89 lemma has_derivative_right:
```
```    90   fixes f :: "real \<Rightarrow> real" and y :: "real"
```
```    91   shows "(f has_derivative (op * y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
```
```    92     ((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x <..} \<inter> I))"
```
```    93 proof -
```
```    94   have "((\<lambda>t. (f t - (f x + y * (t - x))) / \<bar>t - x\<bar>) ---> 0) (at x within ({x<..} \<inter> I)) \<longleftrightarrow>
```
```    95     ((\<lambda>t. (f t - f x) / (t - x) - y) ---> 0) (at x within ({x<..} \<inter> I))"
```
```    96     by (intro Lim_cong_within) (auto simp add: diff_divide_distrib add_divide_distrib)
```
```    97   also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f t - f x) / (t - x)) ---> y) (at x within ({x<..} \<inter> I))"
```
```    98     by (simp add: Lim_null[symmetric])
```
```    99   also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x<..} \<inter> I))"
```
```   100     by (intro Lim_cong_within) (simp_all add: field_simps)
```
```   101   finally show ?thesis
```
```   102     by (simp add: bounded_linear_mult_right has_derivative_within)
```
```   103 qed
```
```   104
```
```   105 lemma bounded_linear_imp_linear: "bounded_linear f \<Longrightarrow> linear f" (* TODO: move elsewhere *)
```
```   106 proof -
```
```   107   assume "bounded_linear f"
```
```   108   then interpret f: bounded_linear f .
```
```   109   show "linear f"
```
```   110     by (simp add: f.add f.scaleR linear_def)
```
```   111 qed
```
```   112
```
```   113 lemma derivative_is_linear:
```
```   114   "(f has_derivative f') net \<Longrightarrow> linear f'"
```
```   115   by (rule derivative_linear [THEN bounded_linear_imp_linear])
```
```   116
```
```   117 subsubsection {* Combining theorems. *}
```
```   118
```
```   119 lemma has_derivative_eq_rhs: "(f has_derivative x) F \<Longrightarrow> x = y \<Longrightarrow> (f has_derivative y) F"
```
```   120   by simp
```
```   121
```
```   122 ML {*
```
```   123
```
```   124 structure Has_Derivative_Intros = Named_Thms
```
```   125 (
```
```   126   val name = @{binding has_derivative_intros}
```
```   127   val description = "introduction rules for has_derivative"
```
```   128 )
```
```   129
```
```   130 *}
```
```   131
```
```   132 setup {*
```
```   133   Has_Derivative_Intros.setup #>
```
```   134   Global_Theory.add_thms_dynamic (@{binding has_derivative_eq_intros},
```
```   135     map (fn thm => @{thm has_derivative_eq_rhs} OF [thm]) o Has_Derivative_Intros.get o Context.proof_of);
```
```   136 *}
```
```   137
```
```   138 lemma has_derivative_id[has_derivative_intros]: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
```
```   139   unfolding has_derivative_def
```
```   140   by (simp add: bounded_linear_ident tendsto_const)
```
```   141
```
```   142 lemma has_derivative_const[has_derivative_intros]: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
```
```   143   unfolding has_derivative_def
```
```   144   by (simp add: bounded_linear_zero tendsto_const)
```
```   145
```
```   146 lemma (in bounded_linear) has_derivative'[has_derivative_intros]: "(f has_derivative f) net"
```
```   147   unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
```
```   148   unfolding diff by (simp add: tendsto_const)
```
```   149
```
```   150 lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
```
```   151
```
```   152 lemma (in bounded_linear) has_derivative[has_derivative_intros]:
```
```   153   assumes "((\<lambda>x. g x) has_derivative (\<lambda>h. g' h)) net"
```
```   154   shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>h. f (g' h))) net"
```
```   155   using assms unfolding has_derivative_def
```
```   156   apply safe
```
```   157   apply (erule bounded_linear_compose [OF local.bounded_linear])
```
```   158   apply (drule local.tendsto)
```
```   159   apply (simp add: local.scaleR local.diff local.add local.zero)
```
```   160   done
```
```   161
```
```   162 lemmas scaleR_right_has_derivative[has_derivative_intros] =
```
```   163   bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
```
```   164
```
```   165 lemmas scaleR_left_has_derivative[has_derivative_intros] =
```
```   166   bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
```
```   167
```
```   168 lemmas inner_right_has_derivative[has_derivative_intros] =
```
```   169   bounded_linear.has_derivative [OF bounded_linear_inner_right]
```
```   170
```
```   171 lemmas inner_left_has_derivative[has_derivative_intros] =
```
```   172   bounded_linear.has_derivative [OF bounded_linear_inner_left]
```
```   173
```
```   174 lemmas mult_right_has_derivative[has_derivative_intros] =
```
```   175   bounded_linear.has_derivative [OF bounded_linear_mult_right]
```
```   176
```
```   177 lemmas mult_left_has_derivative[has_derivative_intros] =
```
```   178   bounded_linear.has_derivative [OF bounded_linear_mult_left]
```
```   179
```
```   180 lemma has_derivative_neg[has_derivative_intros]:
```
```   181   assumes "(f has_derivative f') net"
```
```   182   shows "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
```
```   183   using scaleR_right_has_derivative [where r="-1", OF assms] by auto
```
```   184
```
```   185 lemma has_derivative_add[has_derivative_intros]:
```
```   186   assumes "(f has_derivative f') net" and "(g has_derivative g') net"
```
```   187   shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net"
```
```   188 proof-
```
```   189   note as = assms[unfolded has_derivative_def]
```
```   190   show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
```
```   191     using tendsto_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
```
```   192     by (auto simp add: algebra_simps)
```
```   193 qed
```
```   194
```
```   195 lemma has_derivative_add_const:
```
```   196   "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
```
```   197   by (intro has_derivative_eq_intros) auto
```
```   198
```
```   199 lemma has_derivative_sub[has_derivative_intros]:
```
```   200   assumes "(f has_derivative f') net" and "(g has_derivative g') net"
```
```   201   shows "((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
```
```   202   unfolding diff_minus by (intro has_derivative_intros assms)
```
```   203
```
```   204 lemma has_derivative_setsum[has_derivative_intros]:
```
```   205   assumes "finite s" and "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
```
```   206   shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
```
```   207   using assms by (induct, simp_all add: has_derivative_const has_derivative_add)
```
```   208 text {* Somewhat different results for derivative of scalar multiplier. *}
```
```   209
```
```   210 subsubsection {* Limit transformation for derivatives *}
```
```   211
```
```   212 lemma has_derivative_transform_within:
```
```   213   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)"
```
```   214   shows "(g has_derivative f') (at x within s)"
```
```   215   using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
```
```   216   apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
```
```   217   apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
```
```   218
```
```   219 lemma has_derivative_transform_at:
```
```   220   assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
```
```   221   shows "(g has_derivative f') (at x)"
```
```   222   using has_derivative_transform_within [of d x UNIV f g f'] assms by simp
```
```   223
```
```   224 lemma has_derivative_transform_within_open:
```
```   225   assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
```
```   226   shows "(g has_derivative f') (at x)"
```
```   227   using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
```
```   228   apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
```
```   229   apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
```
```   230
```
```   231 subsection {* Differentiability *}
```
```   232
```
```   233 no_notation Deriv.differentiable (infixl "differentiable" 60)
```
```   234
```
```   235 definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" (infixr "differentiable" 30) where
```
```   236   "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
```
```   237
```
```   238 definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
```
```   239   "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
```
```   240
```
```   241 lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
```
```   242   unfolding differentiable_def by auto
```
```   243
```
```   244 lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
```
```   245   unfolding differentiable_def using has_derivative_at_within by blast
```
```   246
```
```   247 lemma differentiable_within_open: (* TODO: delete *)
```
```   248   assumes "a \<in> s" and "open s"
```
```   249   shows "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
```
```   250   using assms by (simp only: at_within_interior interior_open)
```
```   251
```
```   252 lemma differentiable_on_eq_differentiable_at:
```
```   253   "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
```
```   254   unfolding differentiable_on_def
```
```   255   by (auto simp add: at_within_interior interior_open)
```
```   256
```
```   257 lemma differentiable_transform_within:
```
```   258   assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
```
```   259   assumes "f differentiable (at x within s)"
```
```   260   shows "g differentiable (at x within s)"
```
```   261   using assms(4) unfolding differentiable_def
```
```   262   by (auto intro!: has_derivative_transform_within[OF assms(1-3)])
```
```   263
```
```   264 lemma differentiable_transform_at:
```
```   265   assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
```
```   266   shows "g differentiable at x"
```
```   267   using assms(3) unfolding differentiable_def
```
```   268   using has_derivative_transform_at[OF assms(1-2)] by auto
```
```   269
```
```   270 subsection {* Frechet derivative and Jacobian matrix. *}
```
```   271
```
```   272 definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
```
```   273
```
```   274 lemma frechet_derivative_works:
```
```   275  "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
```
```   276   unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
```
```   277
```
```   278 lemma linear_frechet_derivative:
```
```   279   shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
```
```   280   unfolding frechet_derivative_works has_derivative_def
```
```   281   by (auto intro: bounded_linear_imp_linear)
```
```   282
```
```   283 subsection {* Differentiability implies continuity *}
```
```   284
```
```   285 lemma Lim_mul_norm_within:
```
```   286   fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   287   shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
```
```   288   unfolding Lim_within apply(rule,rule)
```
```   289   apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
```
```   290   apply(rule_tac x="min d 1" in exI) apply rule defer
```
```   291   apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
```
```   292   by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
```
```   293
```
```   294 lemma differentiable_imp_continuous_within:
```
```   295   assumes "f differentiable (at x within s)"
```
```   296   shows "continuous (at x within s) f"
```
```   297 proof-
```
```   298   from assms guess f' unfolding differentiable_def has_derivative_within ..
```
```   299   note f'=this
```
```   300   then interpret bounded_linear f' by auto
```
```   301   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"
```
```   302     using zero by auto
```
```   303   have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
```
```   304     apply(rule continuous_within_compose) apply(rule continuous_intros)+
```
```   305     by(rule linear_continuous_within[OF f'[THEN conjunct1]])
```
```   306   show ?thesis unfolding continuous_within
```
```   307     using f'[THEN conjunct2, THEN Lim_mul_norm_within]
```
```   308     apply- apply(drule tendsto_add)
```
```   309     apply(rule **[unfolded continuous_within])
```
```   310     unfolding Lim_within and dist_norm
```
```   311     apply (rule, rule)
```
```   312     apply (erule_tac x=e in allE)
```
```   313     apply (erule impE | assumption)+
```
```   314     apply (erule exE, rule_tac x=d in exI)
```
```   315     by (auto simp add: zero *)
```
```   316 qed
```
```   317
```
```   318 lemma differentiable_imp_continuous_at:
```
```   319   "f differentiable at x \<Longrightarrow> continuous (at x) f"
```
```   320  by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
```
```   321
```
```   322 lemma differentiable_imp_continuous_on:
```
```   323   "f differentiable_on s \<Longrightarrow> continuous_on s f"
```
```   324   unfolding differentiable_on_def continuous_on_eq_continuous_within
```
```   325   using differentiable_imp_continuous_within by blast
```
```   326
```
```   327 lemma has_derivative_within_subset:
```
```   328  "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
```
```   329   unfolding has_derivative_within using Lim_within_subset by blast
```
```   330
```
```   331 lemma differentiable_within_subset:
```
```   332   "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
```
```   333   unfolding differentiable_def using has_derivative_within_subset by blast
```
```   334
```
```   335 lemma differentiable_on_subset:
```
```   336   "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
```
```   337   unfolding differentiable_on_def using differentiable_within_subset by blast
```
```   338
```
```   339 lemma differentiable_on_empty: "f differentiable_on {}"
```
```   340   unfolding differentiable_on_def by auto
```
```   341
```
```   342 text {* Several results are easier using a "multiplied-out" variant.
```
```   343 (I got this idea from Dieudonne's proof of the chain rule). *}
```
```   344
```
```   345 lemma has_derivative_within_alt:
```
```   346  "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
```
```   347   (\<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")
```
```   348 proof
```
```   349   assume ?lhs thus ?rhs
```
```   350     unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
```
```   351     unfolding Lim_within
```
```   352     apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
```
```   353     apply(erule exE,rule_tac x=d in exI)
```
```   354     apply(erule conjE,rule,assumption,rule,rule)
```
```   355   proof-
```
```   356     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>
```
```   357       dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
```
```   358     then interpret bounded_linear f' by auto
```
```   359     show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
```
```   360       case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero)
```
```   361     next
```
```   362       case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
```
```   363         unfolding dist_norm diff_0_right using as(3)
```
```   364         using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded dist_norm]
```
```   365         by (auto simp add: linear_0 linear_sub)
```
```   366       thus ?thesis by(auto simp add:algebra_simps)
```
```   367     qed
```
```   368   qed
```
```   369 next
```
```   370   assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within
```
```   371     apply-apply(erule conjE,rule,assumption)
```
```   372     apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer
```
```   373     apply(erule exE,rule_tac x=d in exI)
```
```   374     apply(erule conjE,rule,assumption,rule,rule)
```
```   375     unfolding dist_norm diff_0_right norm_scaleR
```
```   376     apply(erule_tac x=xa in ballE,erule impE)
```
```   377   proof-
```
```   378     fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
```
```   379         "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
```
```   380     thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
```
```   381       apply(rule_tac le_less_trans[of _ "e/2"])
```
```   382       by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps)
```
```   383   qed auto
```
```   384 qed
```
```   385
```
```   386 lemma has_derivative_at_alt:
```
```   387   "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
```
```   388   (\<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))"
```
```   389   using has_derivative_within_alt[where s=UNIV] by simp
```
```   390
```
```   391 subsection {* The chain rule. *}
```
```   392
```
```   393 lemma diff_chain_within[has_derivative_intros]:
```
```   394   assumes "(f has_derivative f') (at x within s)"
```
```   395   assumes "(g has_derivative g') (at (f x) within (f ` s))"
```
```   396   shows "((g o f) has_derivative (g' o f'))(at x within s)"
```
```   397   unfolding has_derivative_within_alt
```
```   398   apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
```
```   399   apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
```
```   400   apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption)
```
```   401 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)"]
```
```   423       by(auto simp add:algebra_simps)
```
```   424     also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)"
```
```   425       apply(rule add_left_mono) using B1 by(auto simp add:algebra_simps)
```
```   426     also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)"
```
```   427       apply(rule add_right_mono) using d1 d2 d as by auto
```
```   428     also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
```
```   429     also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
```
```   430     finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto
```
```   431
```
```   432     hence "norm (f y - f x) \<le> d * (1 + B1)" apply-
```
```   433       apply(rule order_trans,assumption,rule mult_right_mono)
```
```   434       using as B1 by auto
```
```   435     also have "\<dots> < de" using d B1 by(auto simp add:field_simps)
```
```   436     finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
```
```   437       apply-apply(rule de[THEN conjunct2,rule_format])
```
```   438       using `y\<in>s` using d as by auto
```
```   439     also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto
```
```   440     also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono)
```
```   441       using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
```
```   442     finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
```
```   443
```
```   444     interpret g': bounded_linear g' using assms(2) by auto
```
```   445     interpret f': bounded_linear f' using assms(1) by auto
```
```   446     have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
```
```   447       by(auto simp add:algebra_simps f'.diff g'.diff g'.add)
```
```   448     also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2
```
```   449       by (auto simp add: algebra_simps)
```
```   450     also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))"
```
```   451       apply (rule mult_left_mono) using as d1 d2 d B2 by auto
```
```   452     also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
```
```   453     finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
```
```   454
```
```   455     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)"
```
```   456       using 5 4 by auto
```
```   457     thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)"
```
```   458       unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub)
```
```   459       by assumption
```
```   460   qed
```
```   461 qed
```
```   462
```
```   463 lemma diff_chain_at[has_derivative_intros]:
```
```   464   "(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)"
```
```   465   using diff_chain_within[of f f' x UNIV g g']
```
```   466   using has_derivative_within_subset[of g g' "f x" UNIV "range f"]
```
```   467   by simp
```
```   468
```
```   469 subsection {* Composition rules stated just for differentiability. *}
```
```   470
```
```   471 lemma differentiable_const [intro]:
```
```   472   "(\<lambda>z. c) differentiable (net::'a::real_normed_vector filter)"
```
```   473   unfolding differentiable_def using has_derivative_const by auto
```
```   474
```
```   475 lemma differentiable_id [intro]:
```
```   476   "(\<lambda>z. z) differentiable (net::'a::real_normed_vector filter)"
```
```   477     unfolding differentiable_def using has_derivative_id by auto
```
```   478
```
```   479 lemma differentiable_cmul [intro]:
```
```   480   "f differentiable net \<Longrightarrow>
```
```   481   (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)"
```
```   482   unfolding differentiable_def
```
```   483   apply(erule exE, drule scaleR_right_has_derivative) by auto
```
```   484
```
```   485 lemma differentiable_neg [intro]:
```
```   486   "f differentiable net \<Longrightarrow>
```
```   487   (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector filter)"
```
```   488   unfolding differentiable_def
```
```   489   apply(erule exE, drule has_derivative_neg) by auto
```
```   490
```
```   491 lemma differentiable_add [intro]: "f differentiable net \<Longrightarrow> g differentiable net
```
```   492    \<Longrightarrow> (\<lambda>z. f z + g z) differentiable net"
```
```   493   by (auto intro: has_derivative_eq_intros simp: differentiable_def)
```
```   494
```
```   495 lemma differentiable_sub [intro]: "f differentiable net \<Longrightarrow> g differentiable net
```
```   496   \<Longrightarrow> (\<lambda>z. f z - g z) differentiable net"
```
```   497   by (auto intro: has_derivative_eq_intros simp: differentiable_def)
```
```   498
```
```   499 lemma differentiable_setsum [intro]:
```
```   500   assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
```
```   501   shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net"
```
```   502 proof-
```
```   503   guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
```
```   504   thus ?thesis unfolding differentiable_def apply-
```
```   505     apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto
```
```   506 qed
```
```   507
```
```   508 lemma differentiable_setsum_numseg:
```
```   509   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"
```
```   510   apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto
```
```   511
```
```   512 lemma differentiable_chain_at:
```
```   513   "f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o f) differentiable (at x)"
```
```   514   unfolding differentiable_def by(meson diff_chain_at)
```
```   515
```
```   516 lemma differentiable_chain_within:
```
```   517   "f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s))
```
```   518    \<Longrightarrow> (g o f) differentiable (at x within s)"
```
```   519   unfolding differentiable_def by(meson diff_chain_within)
```
```   520
```
```   521 subsection {* Uniqueness of derivative *}
```
```   522
```
```   523 text {*
```
```   524  The general result is a bit messy because we need approachability of the
```
```   525  limit point from any direction. But OK for nontrivial intervals etc.
```
```   526 *}
```
```   527
```
```   528 lemma frechet_derivative_unique_within:
```
```   529   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   530   assumes "(f has_derivative f') (at x within s)"
```
```   531   assumes "(f has_derivative f'') (at x within s)"
```
```   532   assumes "(\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R i) \<in> s)"
```
```   533   shows "f' = f''"
```
```   534 proof-
```
```   535   note as = assms(1,2)[unfolded has_derivative_def]
```
```   536   then interpret f': bounded_linear f' by auto
```
```   537   from as interpret f'': bounded_linear f'' by auto
```
```   538   have "x islimpt s" unfolding islimpt_approachable
```
```   539   proof(rule,rule)
```
```   540     fix e::real assume "0<e" guess d
```
```   541       using assms(3)[rule_format,OF SOME_Basis `e>0`] ..
```
```   542     thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
```
```   543       apply(rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI)
```
```   544       unfolding dist_norm by (auto simp: SOME_Basis nonzero_Basis)
```
```   545   qed
```
```   546   hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within)
```
```   547     unfolding trivial_limit_within by simp
```
```   548   show ?thesis  apply(rule linear_eq_stdbasis)
```
```   549     unfolding linear_conv_bounded_linear
```
```   550     apply(rule as(1,2)[THEN conjunct1])+
```
```   551   proof(rule,rule ccontr)
```
```   552     fix i :: 'a assume i:"i \<in> Basis" def e \<equiv> "norm (f' i - f'' i)"
```
```   553     assume "f' i \<noteq> f'' i"
```
```   554     hence "e>0" unfolding e_def by auto
```
```   555     guess d using tendsto_diff [OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
```
```   556     guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
```
```   557     have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R i)) + f'' (c *\<^sub>R i)))"
```
```   558       unfolding scaleR_right_distrib by auto
```
```   559     also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
```
```   560       unfolding f'.scaleR f''.scaleR
```
```   561       unfolding scaleR_right_distrib scaleR_minus_right by auto
```
```   562     also have "\<dots> = e" unfolding e_def using c[THEN conjunct1]
```
```   563       using norm_minus_cancel[of "f' i - f'' i"]
```
```   564       by (auto simp add: add.commute ab_diff_minus)
```
```   565     finally show False using c
```
```   566       using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R i"]
```
```   567       unfolding dist_norm
```
```   568       unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
```
```   569         scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
```
```   570       using i by auto
```
```   571   qed
```
```   572 qed
```
```   573
```
```   574 lemma frechet_derivative_unique_at:
```
```   575   shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
```
```   576   unfolding FDERIV_conv_has_derivative [symmetric]
```
```   577   by (rule FDERIV_unique)
```
```   578
```
```   579 lemma frechet_derivative_unique_within_closed_interval:
```
```   580   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   581   assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "x \<in> {a..b}" (is "x\<in>?I")
```
```   582   assumes "(f has_derivative f' ) (at x within {a..b})"
```
```   583   assumes "(f has_derivative f'') (at x within {a..b})"
```
```   584   shows "f' = f''"
```
```   585   apply(rule frechet_derivative_unique_within)
```
```   586   apply(rule assms(3,4))+
```
```   587 proof(rule,rule,rule)
```
```   588   fix e::real and i :: 'a assume "e>0" and i:"i\<in>Basis"
```
```   589   thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> {a..b}"
```
```   590   proof(cases "x\<bullet>i=a\<bullet>i")
```
```   591     case True thus ?thesis
```
```   592       apply(rule_tac x="(min (b\<bullet>i - a\<bullet>i)  e) / 2" in exI)
```
```   593       using assms(1)[THEN bspec[where x=i]] and `e>0` and assms(2)
```
```   594       unfolding mem_interval
```
```   595       using i by (auto simp add: field_simps inner_simps inner_Basis)
```
```   596   next
```
```   597     note * = assms(2)[unfolded mem_interval, THEN bspec, OF i]
```
```   598     case False moreover have "a \<bullet> i < x \<bullet> i" using False * by auto
```
```   599     moreover {
```
```   600       have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> a\<bullet>i *2 + x\<bullet>i - a\<bullet>i"
```
```   601         by auto
```
```   602       also have "\<dots> = a\<bullet>i + x\<bullet>i" by auto
```
```   603       also have "\<dots> \<le> 2 * (x\<bullet>i)" using * by auto
```
```   604       finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2" by auto
```
```   605     }
```
```   606     moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0" using * and `e>0` by auto
```
```   607     hence "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e" using * by auto
```
```   608     ultimately show ?thesis
```
```   609       apply(rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
```
```   610       using assms(1)[THEN bspec, OF i] and `e>0` and assms(2)
```
```   611       unfolding mem_interval
```
```   612       using i by (auto simp add: field_simps inner_simps inner_Basis)
```
```   613   qed
```
```   614 qed
```
```   615
```
```   616 lemma frechet_derivative_unique_within_open_interval:
```
```   617   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   618   assumes "x \<in> {a<..<b}"
```
```   619   assumes "(f has_derivative f' ) (at x within {a<..<b})"
```
```   620   assumes "(f has_derivative f'') (at x within {a<..<b})"
```
```   621   shows "f' = f''"
```
```   622 proof -
```
```   623   from assms(1) have *: "at x within {a<..<b} = at x"
```
```   624     by (simp add: at_within_interior interior_open open_interval)
```
```   625   from assms(2,3) [unfolded *] show "f' = f''"
```
```   626     by (rule frechet_derivative_unique_at)
```
```   627 qed
```
```   628
```
```   629 lemma frechet_derivative_at:
```
```   630   shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
```
```   631   apply(rule frechet_derivative_unique_at[of f],assumption)
```
```   632   unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
```
```   633
```
```   634 lemma frechet_derivative_within_closed_interval:
```
```   635   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```   636   assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" and "x \<in> {a..b}"
```
```   637   assumes "(f has_derivative f') (at x within {a.. b})"
```
```   638   shows "frechet_derivative f (at x within {a.. b}) = f'"
```
```   639   apply(rule frechet_derivative_unique_within_closed_interval[where f=f])
```
```   640   apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
```
```   641   unfolding differentiable_def using assms(3) by auto
```
```   642
```
```   643 subsection {* The traditional Rolle theorem in one dimension. *}
```
```   644
```
```   645 lemma linear_componentwise:
```
```   646   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```   647   assumes lf: "linear f"
```
```   648   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
```
```   649 proof -
```
```   650   have fA: "finite Basis" by simp
```
```   651   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
```
```   652     by (simp add: inner_setsum_left)
```
```   653   then show ?thesis
```
```   654     unfolding linear_setsum_mul[OF lf fA, symmetric]
```
```   655     unfolding euclidean_representation ..
```
```   656 qed
```
```   657
```
```   658 text {* We do not introduce @{text jacobian}, which is defined on matrices, instead we use
```
```   659   the unfolding of it. *}
```
```   660
```
```   661 lemma jacobian_works:
```
```   662   "(f::('a::euclidean_space) \<Rightarrow> ('b::euclidean_space)) differentiable net \<longleftrightarrow>
```
```   663    (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis.
```
```   664       (\<Sum>j\<in>Basis. frechet_derivative f net (j) \<bullet> i * (h \<bullet> j)) *\<^sub>R i)) net"
```
```   665   (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net")
```
```   666 proof
```
```   667   assume *: ?differentiable
```
```   668   { fix h i
```
```   669     have "?SUM h i = frechet_derivative f net h \<bullet> i" using *
```
```   670       by (auto intro!: setsum_cong
```
```   671                simp: linear_componentwise[of _ h i] linear_frechet_derivative) }
```
```   672   with * show "(f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net"
```
```   673     by (simp add: frechet_derivative_works euclidean_representation)
```
```   674 qed (auto intro!: differentiableI)
```
```   675
```
```   676 lemma differential_zero_maxmin_component:
```
```   677   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```   678   assumes k: "k \<in> Basis"
```
```   679     and ball: "0 < e" "((\<forall>y \<in> ball x e. (f y)\<bullet>k \<le> (f x)\<bullet>k) \<or> (\<forall>y\<in>ball x e. (f x)\<bullet>k \<le> (f y)\<bullet>k))"
```
```   680     and diff: "f differentiable (at x)"
```
```   681   shows "(\<Sum>j\<in>Basis. (frechet_derivative f (at x) j \<bullet> k) *\<^sub>R j) = (0::'a)" (is "?D k = 0")
```
```   682 proof (rule ccontr)
```
```   683   assume "?D k \<noteq> 0"
```
```   684   then obtain j where j: "?D k \<bullet> j \<noteq> 0" "j \<in> Basis"
```
```   685     unfolding euclidean_eq_iff[of _ "0::'a"] by auto
```
```   686   hence *: "\<bar>?D k \<bullet> j\<bar> / 2 > 0" by auto
```
```   687   note as = diff[unfolded jacobian_works has_derivative_at_alt]
```
```   688   guess e' using as[THEN conjunct2, rule_format, OF *] .. note e' = this
```
```   689   guess d using real_lbound_gt_zero[OF ball(1) e'[THEN conjunct1]] .. note d = this
```
```   690   { fix c assume "abs c \<le> d"
```
```   691     hence *:"norm (x + c *\<^sub>R j - x) < e'" using norm_Basis[OF j(2)] d by auto
```
```   692     let ?v = "(\<Sum>i\<in>Basis. (\<Sum>l\<in>Basis. ?D i \<bullet> l * ((c *\<^sub>R j :: 'a) \<bullet> l)) *\<^sub>R i)"
```
```   693     have if_dist: "\<And> P a b c. a * (if P then b else c) = (if P then a * b else a * c)" by auto
```
```   694     have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le>
```
```   695         norm (f (x + c *\<^sub>R j) - f x - ?v)" by (rule Basis_le_norm[OF k])
```
```   696     also have "\<dots> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
```
```   697       using e'[THEN conjunct2, rule_format, OF *] and norm_Basis[OF j(2)] j
```
```   698       by simp
```
```   699     finally have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>" by simp
```
```   700     hence "\<bar>f (x + c *\<^sub>R j) \<bullet> k - f x \<bullet> k - c * (?D k \<bullet> j)\<bar> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
```
```   701       using j k
```
```   702       by (simp add: inner_simps field_simps inner_Basis setsum_cases if_dist) }
```
```   703   note * = this
```
```   704   have "x + d *\<^sub>R j \<in> ball x e" "x - d *\<^sub>R j \<in> ball x e"
```
```   705     unfolding mem_ball dist_norm using norm_Basis[OF j(2)] d by auto
```
```   706   hence **:"((f (x - d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<le> (f x)\<bullet>k) \<or>
```
```   707          ((f (x - d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k \<and> (f (x + d *\<^sub>R j))\<bullet>k \<ge> (f x)\<bullet>k)" using ball by auto
```
```   708   have ***: "\<And>y y1 y2 d dx::real.
```
```   709     (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
```
```   710   show False apply(rule ***[OF **, where dx="d * (?D k \<bullet> j)" and d="\<bar>?D k \<bullet> j\<bar> / 2 * \<bar>d\<bar>"])
```
```   711     using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
```
```   712     unfolding mult_minus_left
```
```   713     unfolding abs_mult diff_minus_eq_add scaleR_minus_left
```
```   714     unfolding algebra_simps by (auto intro: mult_pos_pos)
```
```   715 qed
```
```   716
```
```   717 text {* In particular if we have a mapping into @{typ "real"}. *}
```
```   718
```
```   719 lemma differential_zero_maxmin:
```
```   720   fixes f::"'a\<Colon>euclidean_space \<Rightarrow> real"
```
```   721   assumes "x \<in> s" "open s"
```
```   722   and deriv: "(f has_derivative f') (at x)"
```
```   723   and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
```
```   724   shows "f' = (\<lambda>v. 0)"
```
```   725 proof -
```
```   726   obtain e where e:"e>0" "ball x e \<subseteq> s"
```
```   727     using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
```
```   728   with differential_zero_maxmin_component[where 'b=real, of 1 e x f] mono deriv
```
```   729   have "(\<Sum>j\<in>Basis. frechet_derivative f (at x) j *\<^sub>R j) = (0::'a)"
```
```   730     by (auto simp: Basis_real_def differentiable_def)
```
```   731   with frechet_derivative_at[OF deriv, symmetric]
```
```   732   have "\<forall>i\<in>Basis. f' i = 0"
```
```   733     by (simp add: euclidean_eq_iff[of _ "0::'a"] inner_setsum_left_Basis)
```
```   734   with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 1]
```
```   735   show ?thesis by (simp add: fun_eq_iff)
```
```   736 qed
```
```   737
```
```   738 lemma rolle: fixes f::"real\<Rightarrow>real"
```
```   739   assumes "a < b" and "f a = f b" and "continuous_on {a..b} f"
```
```   740   assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
```
```   741   shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)"
```
```   742 proof-
```
```   743   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))"
```
```   744   proof-
```
```   745     have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto
```
```   746     hence *:"{a .. b}\<noteq>{}" by auto
```
```   747     guess d using continuous_attains_sup[OF compact_interval * assms(3)] .. note d=this
```
```   748     guess c using continuous_attains_inf[OF compact_interval * assms(3)] .. note c=this
```
```   749     show ?thesis
```
```   750     proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
```
```   751       case True thus ?thesis
```
```   752         apply(erule_tac disjE) apply(rule_tac x=d in bexI)
```
```   753         apply(rule_tac x=c in bexI)
```
```   754         using d c by auto
```
```   755     next
```
```   756       def e \<equiv> "(a + b) /2"
```
```   757       case False hence "f d = f c" using d c assms(2) by auto
```
```   758       hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d"
```
```   759         using c d apply- apply(erule_tac x=x in ballE)+ by auto
```
```   760       thus ?thesis
```
```   761         apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto
```
```   762     qed
```
```   763   qed
```
```   764   then guess x .. note x=this
```
```   765   hence "f' x = (\<lambda>v. 0)"
```
```   766     apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
```
```   767     defer apply(rule open_interval)
```
```   768     apply(rule assms(4)[unfolded has_derivative_at[THEN sym],THEN bspec[where x=x]],assumption)
```
```   769     unfolding o_def apply(erule disjE,rule disjI2) by auto
```
```   770   thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
```
```   771     apply(drule_tac x=v in fun_cong) using x(1) by auto
```
```   772 qed
```
```   773
```
```   774 subsection {* One-dimensional mean value theorem. *}
```
```   775
```
```   776 lemma mvt: fixes f::"real \<Rightarrow> real"
```
```   777   assumes "a < b" and "continuous_on {a .. b} f"
```
```   778   assumes "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
```
```   779   shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))"
```
```   780 proof-
```
```   781   have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
```
```   782   proof (intro rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"] ballI)
```
```   783     fix x assume x:"x \<in> {a<..<b}"
```
```   784     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)"
```
```   785       by (intro has_derivative_intros assms(3)[rule_format,OF x]
```
```   786         mult_right_has_derivative)
```
```   787   qed (insert assms(1,2), auto intro!: continuous_on_intros simp: field_simps)
```
```   788   then guess x ..
```
```   789   thus ?thesis apply(rule_tac x=x in bexI)
```
```   790     apply(drule fun_cong[of _ _ "b - a"]) by auto
```
```   791 qed
```
```   792
```
```   793 lemma mvt_simple:
```
```   794   fixes f::"real \<Rightarrow> real"
```
```   795   assumes "a<b" and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
```
```   796   shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
```
```   797   apply(rule mvt)
```
```   798   apply(rule assms(1), rule differentiable_imp_continuous_on)
```
```   799   unfolding differentiable_on_def differentiable_def defer
```
```   800 proof
```
```   801   fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)"
```
```   802     unfolding has_derivative_within_open[OF x open_interval,THEN sym]
```
```   803     apply(rule has_derivative_within_subset)
```
```   804     apply(rule assms(2)[rule_format])
```
```   805     using x by auto
```
```   806 qed(insert assms(2), auto)
```
```   807
```
```   808 lemma mvt_very_simple:
```
```   809   fixes f::"real \<Rightarrow> real"
```
```   810   assumes "a \<le> b" and "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
```
```   811   shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
```
```   812 proof (cases "a = b")
```
```   813   interpret bounded_linear "f' b" using assms(2) assms(1) by auto
```
```   814   case True thus ?thesis apply(rule_tac x=a in bexI)
```
```   815     using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
```
```   816     unfolding True using zero by auto next
```
```   817   case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto
```
```   818 qed
```
```   819
```
```   820 text {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
```
```   821
```
```   822 lemma mvt_general:
```
```   823   fixes f::"real\<Rightarrow>'a::euclidean_space"
```
```   824   assumes "a<b" and "continuous_on {a..b} f"
```
```   825   assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
```
```   826   shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))"
```
```   827 proof-
```
```   828   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)"
```
```   829     apply(rule mvt) apply(rule assms(1))
```
```   830     apply(rule continuous_on_inner continuous_on_intros assms(2))+
```
```   831     unfolding o_def apply(rule,rule has_derivative_intros)
```
```   832     using assms(3) by auto
```
```   833   then guess x .. note x=this
```
```   834   show ?thesis proof(cases "f a = f b")
```
```   835     case False
```
```   836     have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2"
```
```   837       by (simp add: power2_eq_square)
```
```   838     also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding power2_norm_eq_inner ..
```
```   839     also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)"
```
```   840       using x unfolding inner_simps by (auto simp add: inner_diff_left)
```
```   841     also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))"
```
```   842       by (rule norm_cauchy_schwarz)
```
```   843     finally show ?thesis using False x(1)
```
```   844       by (auto simp add: real_mult_left_cancel)
```
```   845   next
```
```   846     case True thus ?thesis using assms(1)
```
```   847       apply (rule_tac x="(a + b) /2" in bexI) by auto
```
```   848   qed
```
```   849 qed
```
```   850
```
```   851 text {* Still more general bound theorem. *}
```
```   852
```
```   853 lemma differentiable_bound:
```
```   854   fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```   855   assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)"
```
```   856   assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
```
```   857   shows "norm(f x - f y) \<le> B * norm(x - y)"
```
```   858 proof-
```
```   859   let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
```
```   860   have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
```
```   861     using assms(1)[unfolded convex_alt,rule_format,OF x y]
```
```   862     unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib
```
```   863     by (auto simp add: algebra_simps)
```
```   864   hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply-
```
```   865     apply(rule continuous_on_intros)+
```
```   866     unfolding continuous_on_eq_continuous_within
```
```   867     apply(rule,rule differentiable_imp_continuous_within)
```
```   868     unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
```
```   869     apply(rule has_derivative_within_subset)
```
```   870     apply(rule assms(2)[rule_format]) by auto
```
```   871   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)"
```
```   872   proof rule
```
```   873     case goal1
```
```   874     let ?u = "x + u *\<^sub>R (y - x)"
```
```   875     have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
```
```   876       apply(rule diff_chain_within) apply(rule has_derivative_intros)+
```
```   877       apply(rule has_derivative_within_subset)
```
```   878       apply(rule assms(2)[rule_format]) using goal1 * by auto
```
```   879     thus ?case
```
```   880       unfolding has_derivative_within_open[OF goal1 open_interval] by auto
```
```   881   qed
```
```   882   guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
```
```   883   have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y"
```
```   884   proof-
```
```   885     case goal1
```
```   886     have "norm (f' x y) \<le> onorm (f' x) * norm y"
```
```   887       using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
```
```   888     also have "\<dots> \<le> B * norm y"
```
```   889       apply(rule mult_right_mono)
```
```   890       using assms(3)[rule_format,OF goal1]
```
```   891       by(auto simp add:field_simps)
```
```   892     finally show ?case by simp
```
```   893   qed
```
```   894   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)"
```
```   895     by(auto simp add:norm_minus_commute)
```
```   896   also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
```
```   897   also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
```
```   898   finally show ?thesis by(auto simp add:norm_minus_commute)
```
```   899 qed
```
```   900
```
```   901 lemma differentiable_bound_real:
```
```   902   fixes f::"real \<Rightarrow> real"
```
```   903   assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
```
```   904   assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
```
```   905   shows "norm(f x - f y) \<le> B * norm(x - y)"
```
```   906   using differentiable_bound[of s f f' B x y]
```
```   907   unfolding Ball_def image_iff o_def using assms by auto
```
```   908
```
```   909 text {* In particular. *}
```
```   910
```
```   911 lemma has_derivative_zero_constant:
```
```   912   fixes f::"real\<Rightarrow>real"
```
```   913   assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
```
```   914   shows "\<exists>c. \<forall>x\<in>s. f x = c"
```
```   915 proof(cases "s={}")
```
```   916   case False then obtain x where "x\<in>s" by auto
```
```   917   have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
```
```   918     thus ?case
```
```   919       using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
```
```   920       unfolding onorm_const by auto qed
```
```   921   thus ?thesis apply(rule_tac x="f x" in exI) by auto
```
```   922 qed auto
```
```   923
```
```   924 lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
```
```   925   assumes "convex s" and "a \<in> s" and "f a = c"
```
```   926   assumes "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" and "x\<in>s"
```
```   927   shows "f x = c"
```
```   928   using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
```
```   929
```
```   930 subsection {* Differentiability of inverse function (most basic form). *}
```
```   931
```
```   932 lemma has_derivative_inverse_basic:
```
```   933   fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
```
```   934   assumes "(f has_derivative f') (at (g y))"
```
```   935   assumes "bounded_linear g'" and "g' \<circ> f' = id" and "continuous (at y) g"
```
```   936   assumes "open t" and "y \<in> t" and "\<forall>z\<in>t. f(g z) = z"
```
```   937   shows "(g has_derivative g') (at y)"
```
```   938 proof-
```
```   939   interpret f': bounded_linear f'
```
```   940     using assms unfolding has_derivative_def by auto
```
```   941   interpret g': bounded_linear g' using assms by auto
```
```   942   guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
```
```   943 (*  have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
```
```   944   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)"
```
```   945   proof(rule,rule)
```
```   946     case goal1
```
```   947     have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
```
```   948     guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
```
```   949     guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
```
```   950     guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
```
```   951     guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
```
```   952     thus ?case apply(rule_tac x=d in exI) apply rule defer
```
```   953     proof(rule,rule)
```
```   954       fix z assume as:"norm (z - y) < d" hence "z\<in>t"
```
```   955         using d2 d unfolding dist_norm by auto
```
```   956       have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
```
```   957         unfolding g'.diff f'.diff
```
```   958         unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
```
```   959         unfolding assms(7)[rule_format,OF `z\<in>t`]
```
```   960         apply(subst norm_minus_cancel[THEN sym]) by auto
```
```   961       also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C"
```
```   962         by (rule C [THEN conjunct2, rule_format])
```
```   963       also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
```
```   964         apply(rule mult_right_mono)
```
```   965         apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]])
```
```   966         apply(cases "z=y") defer
```
```   967         apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format])
```
```   968         using as d C d0 by auto
```
```   969       also have "\<dots> \<le> e * norm (g z - g y)"
```
```   970         using C by (auto simp add: field_simps)
```
```   971       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
```
```   972         by simp
```
```   973     qed auto
```
```   974   qed
```
```   975   have *:"(0::real) < 1 / 2" by auto
```
```   976   guess d using lem1[rule_format,OF *] .. note d=this
```
```   977   def B\<equiv>"C*2"
```
```   978   have "B>0" unfolding B_def using C by auto
```
```   979   have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)"
```
```   980   proof(rule,rule) case goal1
```
```   981     have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
```
```   982       by(rule norm_triangle_sub)
```
```   983     also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)"
```
```   984       apply(rule add_left_mono) using d and goal1 by auto
```
```   985     also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
```
```   986       apply(rule add_right_mono) using C by auto
```
```   987     finally show ?case unfolding B_def by(auto simp add:field_simps)
```
```   988   qed
```
```   989   show ?thesis unfolding has_derivative_at_alt
```
```   990   proof(rule,rule assms,rule,rule) case goal1
```
```   991     hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
```
```   992     guess d' using lem1[rule_format,OF *] .. note d'=this
```
```   993     guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
```
```   994     show ?case
```
```   995       apply(rule_tac x=k in exI,rule) defer
```
```   996     proof(rule,rule)
```
```   997       fix z assume as:"norm(z - y) < k"
```
```   998       hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
```
```   999         using d' k by auto
```
```  1000       also have "\<dots> \<le> e * norm(z - y)"
```
```  1001         unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
```
```  1002         using lem2[THEN spec[where x=z]] using k as using `e>0`
```
```  1003         by (auto simp add: field_simps)
```
```  1004       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
```
```  1005         by simp qed(insert k, auto)
```
```  1006   qed
```
```  1007 qed
```
```  1008
```
```  1009 text {* Simply rewrite that based on the domain point x. *}
```
```  1010
```
```  1011 lemma has_derivative_inverse_basic_x:
```
```  1012   fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
```
```  1013   assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
```
```  1014   "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
```
```  1015   shows "(g has_derivative g') (at (f(x)))"
```
```  1016   apply(rule has_derivative_inverse_basic) using assms by auto
```
```  1017
```
```  1018 text {* This is the version in Dieudonne', assuming continuity of f and g. *}
```
```  1019
```
```  1020 lemma has_derivative_inverse_dieudonne:
```
```  1021   fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1022   assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
```
```  1023   (**) "x\<in>s" "(f has_derivative f') (at x)"  "bounded_linear g'" "g' o f' = id"
```
```  1024   shows "(g has_derivative g') (at (f x))"
```
```  1025   apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
```
```  1026   using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)]
```
```  1027     continuous_on_eq_continuous_at[OF assms(2)] by auto
```
```  1028
```
```  1029 text {* Here's the simplest way of not assuming much about g. *}
```
```  1030
```
```  1031 lemma has_derivative_inverse:
```
```  1032   fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1033   assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
```
```  1034   "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
```
```  1035   shows "(g has_derivative g') (at (f x))"
```
```  1036 proof-
```
```  1037   { fix y assume "y\<in>interior (f ` s)"
```
```  1038     then obtain x where "x\<in>s" and *:"y = f x"
```
```  1039       unfolding image_iff using interior_subset by auto
```
```  1040     have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] ..
```
```  1041   } note * = this
```
```  1042   show ?thesis
```
```  1043     apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
```
```  1044     apply(rule continuous_on_interior[OF _ assms(3)])
```
```  1045     apply(rule continuous_on_inv[OF assms(4,1)])
```
```  1046     apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
```
```  1047     by(rule, rule *, assumption)
```
```  1048 qed
```
```  1049
```
```  1050 subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
```
```  1051
```
```  1052 lemma brouwer_surjective:
```
```  1053   fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
```
```  1054   assumes "compact t" "convex t"  "t \<noteq> {}" "continuous_on t f"
```
```  1055   "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
```
```  1056   shows "\<exists>y\<in>t. f y = x"
```
```  1057 proof-
```
```  1058   have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
```
```  1059     by(auto simp add:algebra_simps)
```
```  1060   show ?thesis
```
```  1061     unfolding *
```
```  1062     apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
```
```  1063     apply(rule continuous_on_intros assms)+ using assms(4-6) by auto
```
```  1064 qed
```
```  1065
```
```  1066 lemma brouwer_surjective_cball:
```
```  1067   fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
```
```  1068   assumes "0 < e" "continuous_on (cball a e) f"
```
```  1069   "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
```
```  1070   shows "\<exists>y\<in>cball a e. f y = x"
```
```  1071   apply(rule brouwer_surjective)
```
```  1072   apply(rule compact_cball convex_cball)+
```
```  1073   unfolding cball_eq_empty using assms by auto
```
```  1074
```
```  1075 text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
```
```  1076
```
```  1077 lemma sussmann_open_mapping:
```
```  1078   fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
```
```  1079   assumes "open s" "continuous_on s f" "x \<in> s"
```
```  1080   "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
```
```  1081   "t \<subseteq> s" "x \<in> interior t"
```
```  1082   shows "f x \<in> interior (f ` t)"
```
```  1083 proof-
```
```  1084   interpret f':bounded_linear f'
```
```  1085     using assms unfolding has_derivative_def by auto
```
```  1086   interpret g':bounded_linear g' using assms by auto
```
```  1087   guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this
```
```  1088   hence *:"1/(2*B)>0" by (auto intro!: divide_pos_pos)
```
```  1089   guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
```
```  1090   guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
```
```  1091   have *:"0<e0/B" "0<e1/B"
```
```  1092     apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
```
```  1093   guess e using real_lbound_gt_zero[OF *] .. note e=this
```
```  1094   have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
```
```  1095     apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
```
```  1096     prefer 3 apply(rule,rule)
```
```  1097   proof-
```
```  1098     show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
```
```  1099       unfolding g'.diff
```
```  1100       apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
```
```  1101       apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
```
```  1102       apply(rule continuous_on_subset[OF assms(2)])
```
```  1103       apply(rule,unfold image_iff,erule bexE)
```
```  1104     proof-
```
```  1105       fix y z assume as:"y \<in>cball (f x) e"  "z = x + (g' y - g' (f x))"
```
```  1106       have "dist x z = norm (g' (f x) - g' y)"
```
```  1107         unfolding as(2) and dist_norm by auto
```
```  1108       also have "\<dots> \<le> norm (f x - y) * B"
```
```  1109         unfolding g'.diff[THEN sym] using B by auto
```
```  1110       also have "\<dots> \<le> e * B"
```
```  1111         using as(1)[unfolded mem_cball dist_norm] using B by auto
```
```  1112       also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
```
```  1113       finally have "z\<in>cball x e1" unfolding mem_cball by force
```
```  1114       thus "z \<in> s" using e1 assms(7) by auto
```
```  1115     qed
```
```  1116   next
```
```  1117     fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
```
```  1118     have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
```
```  1119     also have "\<dots> \<le> e * B" apply(rule mult_right_mono)
```
```  1120       using as(2)[unfolded mem_cball dist_norm] and B
```
```  1121       unfolding norm_minus_commute by auto
```
```  1122     also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
```
```  1123     finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
```
```  1124     have **:"f x + f' (x + g' (z - f x) - x) = z"
```
```  1125       using assms(6)[unfolded o_def id_def,THEN cong] by auto
```
```  1126     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)"
```
```  1127       using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"]
```
```  1128       by (auto simp add: algebra_simps)
```
```  1129     also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)"
```
```  1130       using e0[THEN conjunct2,rule_format,OF *]
```
```  1131       unfolding algebra_simps ** by auto
```
```  1132     also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2"
```
```  1133       using as(1)[unfolded mem_cball dist_norm] by auto
```
```  1134     also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2"
```
```  1135       using * and B by (auto simp add: field_simps)
```
```  1136     also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
```
```  1137     also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono)
```
```  1138       using as(2)[unfolded mem_cball dist_norm]
```
```  1139       unfolding norm_minus_commute by auto
```
```  1140     finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
```
```  1141       unfolding mem_cball dist_norm by auto
```
```  1142   qed(insert e, auto) note lem = this
```
```  1143   show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
```
```  1144     apply(rule,rule divide_pos_pos) prefer 3
```
```  1145   proof
```
```  1146     fix y assume "y \<in> ball (f x) (e/2)"
```
```  1147     hence *:"y\<in>cball (f x) (e/2)" by auto
```
```  1148     guess z using lem[rule_format,OF *] .. note z=this
```
```  1149     hence "norm (g' (z - f x)) \<le> norm (z - f x) * B"
```
```  1150       using B by (auto simp add: field_simps)
```
```  1151     also have "\<dots> \<le> e * B"
```
```  1152       apply (rule mult_right_mono) using z(1)
```
```  1153       unfolding mem_cball dist_norm norm_minus_commute using B by auto
```
```  1154     also have "\<dots> \<le> e1"  using e B unfolding less_divide_eq by auto
```
```  1155     finally have "x + g'(z - f x) \<in> t" apply-
```
```  1156       apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
```
```  1157       unfolding mem_cball dist_norm by auto
```
```  1158     thus "y \<in> f ` t" using z by auto
```
```  1159   qed(insert e, auto)
```
```  1160 qed
```
```  1161
```
```  1162 text {* Hence the following eccentric variant of the inverse function theorem.    *)
```
```  1163 (* This has no continuity assumptions, but we do need the inverse function.  *)
```
```  1164 (* We could put f' o g = I but this happens to fit with the minimal linear   *)
```
```  1165 (* algebra theory I've set up so far. *}
```
```  1166
```
```  1167 (* move  before left_inverse_linear in Euclidean_Space*)
```
```  1168
```
```  1169  lemma right_inverse_linear:
```
```  1170    fixes f::"'a::euclidean_space => 'a"
```
```  1171    assumes lf: "linear f" and gf: "f o g = id"
```
```  1172    shows "linear g"
```
```  1173  proof-
```
```  1174    from gf have fi: "surj f" by (auto simp add: surj_def o_def id_def) metis
```
```  1175    from linear_surjective_isomorphism[OF lf fi]
```
```  1176    obtain h:: "'a => 'a" where
```
```  1177      h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
```
```  1178    have "h = g" apply (rule ext) using gf h(2,3)
```
```  1179      by (simp add: o_def id_def fun_eq_iff) metis
```
```  1180    with h(1) show ?thesis by blast
```
```  1181  qed
```
```  1182
```
```  1183 lemma has_derivative_inverse_strong:
```
```  1184   fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
```
```  1185   assumes "open s" and "x \<in> s" and "continuous_on s f"
```
```  1186   assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" and "f' o g' = id"
```
```  1187   shows "(g has_derivative g') (at (f x))"
```
```  1188 proof-
```
```  1189   have linf:"bounded_linear f'"
```
```  1190     using assms(5) unfolding has_derivative_def by auto
```
```  1191   hence ling:"bounded_linear g'"
```
```  1192     unfolding linear_conv_bounded_linear[THEN sym]
```
```  1193     apply- apply(rule right_inverse_linear) using assms(6) by auto
```
```  1194   moreover have "g' \<circ> f' = id" using assms(6) linf ling
```
```  1195     unfolding linear_conv_bounded_linear[THEN sym]
```
```  1196     using linear_inverse_left by auto
```
```  1197   moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)"
```
```  1198     apply(rule,rule,rule,rule sussmann_open_mapping )
```
```  1199     apply(rule assms ling)+ by auto
```
```  1200   have "continuous (at (f x)) g" unfolding continuous_at Lim_at
```
```  1201   proof(rule,rule)
```
```  1202     fix e::real assume "e>0"
```
```  1203     hence "f x \<in> interior (f ` (ball x e \<inter> s))"
```
```  1204       using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
```
```  1205       by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
```
```  1206     then guess d unfolding mem_interior .. note d=this
```
```  1207     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"
```
```  1208       apply(rule_tac x=d in exI)
```
```  1209       apply(rule,rule d[THEN conjunct1])
```
```  1210     proof(rule,rule) case goal1
```
```  1211       hence "g y \<in> g ` f ` (ball x e \<inter> s)"
```
```  1212         using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
```
```  1213         by(auto simp add:dist_commute)
```
```  1214       hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
```
```  1215       thus "dist (g y) (g (f x)) < e"
```
```  1216         using assms(4)[rule_format,OF `x\<in>s`]
```
```  1217         by (auto simp add: dist_commute)
```
```  1218     qed
```
```  1219   qed
```
```  1220   moreover have "f x \<in> interior (f ` s)"
```
```  1221     apply(rule sussmann_open_mapping)
```
```  1222     apply(rule assms ling)+
```
```  1223     using interior_open[OF assms(1)] and `x\<in>s` by auto
```
```  1224   moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y"
```
```  1225   proof- case goal1
```
```  1226     hence "y\<in>f ` s" using interior_subset by auto
```
```  1227     then guess z unfolding image_iff ..
```
```  1228     thus ?case using assms(4) by auto
```
```  1229   qed
```
```  1230   ultimately show ?thesis
```
```  1231     apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)])
```
```  1232     using assms by auto
```
```  1233 qed
```
```  1234
```
```  1235 text {* A rewrite based on the other domain. *}
```
```  1236
```
```  1237 lemma has_derivative_inverse_strong_x:
```
```  1238   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
```
```  1239   assumes "open s" and "g y \<in> s" and "continuous_on s f"
```
```  1240   assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))"
```
```  1241   assumes "f' o g' = id" and "f(g y) = y"
```
```  1242   shows "(g has_derivative g') (at y)"
```
```  1243   using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
```
```  1244
```
```  1245 text {* On a region. *}
```
```  1246
```
```  1247 lemma has_derivative_inverse_on:
```
```  1248   fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
```
```  1249   assumes "open s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
```
```  1250   assumes "\<forall>x\<in>s. g(f x) = x" and "f'(x) o g'(x) = id" and "x\<in>s"
```
```  1251   shows "(g has_derivative g'(x)) (at (f x))"
```
```  1252   apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f])
```
```  1253   apply(rule assms)+
```
```  1254   unfolding continuous_on_eq_continuous_at[OF assms(1)]
```
```  1255   apply(rule,rule differentiable_imp_continuous_at)
```
```  1256   unfolding differentiable_def using assms by auto
```
```  1257
```
```  1258 text {* Invertible derivative continous at a point implies local
```
```  1259 injectivity. It's only for this we need continuity of the derivative,
```
```  1260 except of course if we want the fact that the inverse derivative is
```
```  1261 also continuous. So if we know for some other reason that the inverse
```
```  1262 function exists, it's OK. *}
```
```  1263
```
```  1264 lemma bounded_linear_sub:
```
```  1265   "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
```
```  1266   using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g]
```
```  1267   by (auto simp add: algebra_simps)
```
```  1268
```
```  1269 lemma has_derivative_locally_injective:
```
```  1270   fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  1271   assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
```
```  1272   "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
```
```  1273   "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
```
```  1274   obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)"
```
```  1275 proof-
```
```  1276   interpret bounded_linear g' using assms by auto
```
```  1277   note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
```
```  1278   have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)" defer
```
```  1279     apply(subst euclidean_eq_iff) using f'g' by auto
```
```  1280   hence *:"0 < onorm g'"
```
```  1281     unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastforce
```
```  1282   def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
```
```  1283   guess d1 using assms(6)[rule_format,OF *] .. note d1=this
```
```  1284   from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
```
```  1285   obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
```
```  1286   guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] ..
```
```  1287   note d2=this
```
```  1288   guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] ..
```
```  1289   note d = this
```
```  1290   show ?thesis
```
```  1291   proof
```
```  1292     show "a\<in>ball a d" using d by auto
```
```  1293     show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x"
```
```  1294     proof (intro strip)
```
```  1295       fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
```
```  1296       def ph \<equiv> "\<lambda>w. w - g'(f w - f x)"
```
```  1297       have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
```
```  1298         unfolding ph_def o_def unfolding diff using f'g'
```
```  1299         by (auto simp add: algebra_simps)
```
```  1300       have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
```
```  1301         apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
```
```  1302         apply(rule_tac[!] ballI)
```
```  1303       proof-
```
```  1304         fix u assume u:"u \<in> ball a d"
```
```  1305         hence "u\<in>s" using d d2 by auto
```
```  1306         have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
```
```  1307           unfolding o_def and diff using f'g' by auto
```
```  1308         show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
```
```  1309           unfolding ph' * apply(rule diff_chain_within) defer
```
```  1310           apply(rule bounded_linear.has_derivative'[OF assms(3)])
```
```  1311           apply(rule has_derivative_intros) defer
```
```  1312           apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
```
```  1313           apply(rule has_derivative_at_within)
```
```  1314           using assms(5) and `u\<in>s` `a\<in>s`
```
```  1315           by(auto intro!: has_derivative_intros bounded_linear.has_derivative' derivative_linear)
```
```  1316         have **:"bounded_linear (\<lambda>x. f' u x - f' a x)"
```
```  1317           "bounded_linear (\<lambda>x. f' a x - f' u x)"
```
```  1318           apply(rule_tac[!] bounded_linear_sub)
```
```  1319           apply(rule_tac[!] derivative_linear)
```
```  1320           using assms(5) `u\<in>s` `a\<in>s` by auto
```
```  1321         have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)"
```
```  1322           unfolding * apply(rule onorm_compose)
```
```  1323           unfolding linear_conv_bounded_linear by(rule assms(3) **)+
```
```  1324         also have "\<dots> \<le> onorm g' * k"
```
```  1325           apply(rule mult_left_mono)
```
```  1326           using d1[THEN conjunct2,rule_format,of u]
```
```  1327           using onorm_neg[OF **(1)[unfolded linear_linear]]
```
```  1328           using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]]
```
```  1329           by (auto simp add: algebra_simps)
```
```  1330         also have "\<dots> \<le> 1/2" unfolding k_def by auto
```
```  1331         finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption
```
```  1332       qed
```
```  1333       moreover have "norm (ph y - ph x) = norm (y - x)"
```
```  1334         apply(rule arg_cong[where f=norm])
```
```  1335         unfolding ph_def using diff unfolding as by auto
```
```  1336       ultimately show "x = y" unfolding norm_minus_commute by auto
```
```  1337     qed
```
```  1338   qed auto
```
```  1339 qed
```
```  1340
```
```  1341 subsection {* Uniformly convergent sequence of derivatives. *}
```
```  1342
```
```  1343 lemma has_derivative_sequence_lipschitz_lemma:
```
```  1344   fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
```
```  1345   assumes "convex s"
```
```  1346   assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
```
```  1347   assumes "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
```
```  1348   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)"
```
```  1349 proof (default)+
```
```  1350   fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
```
```  1351   show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
```
```  1352     apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)])
```
```  1353     apply(rule_tac[!] ballI)
```
```  1354   proof-
```
```  1355     fix x assume "x\<in>s"
```
```  1356     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)"
```
```  1357       by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
```
```  1358     { fix h
```
```  1359       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)"
```
```  1360         using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"]
```
```  1361         unfolding norm_minus_commute by (auto simp add: algebra_simps)
```
```  1362       also have "\<dots> \<le> e * norm h+ e * norm h"
```
```  1363         using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h]
```
```  1364         using assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
```
```  1365         by(auto simp add:field_simps)
```
```  1366       finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
```
```  1367     thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
```
```  1368       apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
```
```  1369       unfolding linear_conv_bounded_linear
```
```  1370       using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear]
```
```  1371       by auto
```
```  1372   qed
```
```  1373 qed
```
```  1374
```
```  1375 lemma has_derivative_sequence_lipschitz:
```
```  1376   fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
```
```  1377   assumes "convex s"
```
```  1378   assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
```
```  1379   assumes "\<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)"
```
```  1380   assumes "0 < e"
```
```  1381   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)"
```
```  1382 proof(rule,rule)
```
```  1383   case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
```
```  1384   guess N using assms(3)[rule_format,OF *(2)] ..
```
```  1385   thus ?case
```
```  1386     apply(rule_tac x=N in exI)
```
```  1387     apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
```
```  1388     using assms by auto
```
```  1389 qed
```
```  1390
```
```  1391 lemma has_derivative_sequence:
```
```  1392   fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
```
```  1393   assumes "convex s"
```
```  1394   assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
```
```  1395   assumes "\<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)"
```
```  1396   assumes "x0 \<in> s" and "((\<lambda>n. f n x0) ---> l) sequentially"
```
```  1397   shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and>
```
```  1398     (g has_derivative g'(x)) (at x within s)"
```
```  1399 proof-
```
```  1400   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)"
```
```  1401     apply(rule has_derivative_sequence_lipschitz[where e="42::nat"])
```
```  1402     apply(rule assms)+ by auto
```
```  1403   have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially"
```
```  1404     apply(rule bchoice) unfolding convergent_eq_cauchy
```
```  1405   proof
```
```  1406     fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)"
```
```  1407     proof(cases "x=x0")
```
```  1408       case True thus ?thesis using LIMSEQ_imp_Cauchy[OF assms(5)] by auto
```
```  1409     next
```
```  1410       case False show ?thesis unfolding Cauchy_def
```
```  1411       proof(rule,rule)
```
```  1412         fix e::real assume "e>0"
```
```  1413         hence *:"e/2>0" "e/2/norm(x-x0)>0"
```
```  1414           using False by (auto intro!: divide_pos_pos)
```
```  1415         guess M using LIMSEQ_imp_Cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
```
```  1416         guess N using lem1[rule_format,OF *(2)] .. note N = this
```
```  1417         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
```
```  1418           apply(rule_tac x="max M N" in exI)
```
```  1419         proof(default+)
```
```  1420           fix m n assume as:"max M N \<le>m" "max M N\<le>n"
```
```  1421           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))"
```
```  1422             unfolding dist_norm by(rule norm_triangle_sub)
```
```  1423           also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
```
```  1424             using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False
```
```  1425             by auto
```
```  1426           also have "\<dots> < e / 2 + e / 2"
```
```  1427             apply(rule add_strict_right_mono)
```
```  1428             using as and M[rule_format] unfolding dist_norm by auto
```
```  1429           finally show "dist (f m x) (f n x) < e" by auto
```
```  1430         qed
```
```  1431       qed
```
```  1432     qed
```
```  1433   qed
```
```  1434   then guess g .. note g = this
```
```  1435   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)"
```
```  1436   proof(rule,rule)
```
```  1437     fix e::real assume *:"e>0"
```
```  1438     guess N using lem1[rule_format,OF *] .. note N=this
```
```  1439     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)"
```
```  1440       apply(rule_tac x=N in exI)
```
```  1441     proof(default+)
```
```  1442       fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
```
```  1443       have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
```
```  1444         unfolding eventually_sequentially
```
```  1445         apply(rule_tac x=N in exI)
```
```  1446       proof(rule,rule)
```
```  1447         fix m assume "N\<le>m"
```
```  1448         thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
```
```  1449           using N[rule_format, of n m x y] and as
```
```  1450           by (auto simp add: algebra_simps)
```
```  1451       qed
```
```  1452       thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
```
```  1453         apply-
```
```  1454         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)"])
```
```  1455         apply(rule tendsto_intros g[rule_format] as)+ by assumption
```
```  1456     qed
```
```  1457   qed
```
```  1458   show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
```
```  1459     apply(rule,rule,rule g[rule_format],assumption)
```
```  1460   proof fix x assume "x\<in>s"
```
```  1461     have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
```
```  1462       unfolding LIMSEQ_def
```
```  1463     proof(rule,rule,rule)
```
```  1464       fix u and e::real assume "e>0"
```
```  1465       show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e"
```
```  1466       proof(cases "u=0")
```
```  1467         case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
```
```  1468         show ?thesis apply(rule_tac x=N in exI) unfolding True
```
```  1469           using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto
```
```  1470       next
```
```  1471         case False hence *:"e / 2 / norm u > 0"
```
```  1472           using `e>0` by (auto intro!: divide_pos_pos)
```
```  1473         guess N using assms(3)[rule_format,OF *] .. note N=this
```
```  1474         show ?thesis apply(rule_tac x=N in exI)
```
```  1475         proof(rule,rule) case goal1
```
```  1476           show ?case unfolding dist_norm
```
```  1477             using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
```
```  1478             by (auto simp add:field_simps)
```
```  1479         qed
```
```  1480       qed
```
```  1481     qed
```
```  1482     show "bounded_linear (g' x)"
```
```  1483       unfolding linear_linear linear_def
```
```  1484       apply(rule,rule,rule) defer
```
```  1485     proof(rule,rule)
```
```  1486       fix x' y z::"'m" and c::real
```
```  1487       note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
```
```  1488       show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
```
```  1489         apply(rule tendsto_unique[OF trivial_limit_sequentially])
```
```  1490         apply(rule lem3[rule_format])
```
```  1491         unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
```
```  1492         apply (intro tendsto_intros) by(rule lem3[rule_format])
```
```  1493       show "g' x (y + z) = g' x y + g' x z"
```
```  1494         apply(rule tendsto_unique[OF trivial_limit_sequentially])
```
```  1495         apply(rule lem3[rule_format])
```
```  1496         unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
```
```  1497         apply(rule tendsto_add) by(rule lem3[rule_format])+
```
```  1498     qed
```
```  1499     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)"
```
```  1500     proof(rule,rule) case goal1
```
```  1501       have *:"e/3>0" using goal1 by auto
```
```  1502       guess N1 using assms(3)[rule_format,OF *] .. note N1=this
```
```  1503       guess N2 using lem2[rule_format,OF *] .. note N2=this
```
```  1504       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
```
```  1505       show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1])
```
```  1506       proof(rule,rule)
```
```  1507         fix y assume as:"y \<in> s" "norm (y - x) < d1"
```
```  1508         let ?N ="max N1 N2"
```
```  1509         have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)"
```
```  1510           apply(subst norm_minus_cancel[THEN sym])
```
```  1511           using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto
```
```  1512         moreover
```
```  1513         have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
```
```  1514           using d1 and as by auto
```
```  1515         ultimately
```
```  1516         have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
```
```  1517           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)"]
```
```  1518           by (auto simp add:algebra_simps)
```
```  1519         moreover
```
```  1520         have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
```
```  1521           using N1 `x\<in>s` by auto
```
```  1522         ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
```
```  1523           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)"]
```
```  1524           by(auto simp add:algebra_simps)
```
```  1525       qed
```
```  1526     qed
```
```  1527   qed
```
```  1528 qed
```
```  1529
```
```  1530 text {* Can choose to line up antiderivatives if we want. *}
```
```  1531
```
```  1532 lemma has_antiderivative_sequence:
```
```  1533   fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
```
```  1534   assumes "convex s"
```
```  1535   assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
```
```  1536   assumes "\<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"
```
```  1537   shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
```
```  1538 proof(cases "s={}")
```
```  1539   case False then obtain a where "a\<in>s" by auto
```
```  1540   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
```
```  1541   show ?thesis
```
```  1542     apply(rule *)
```
```  1543     apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
```
```  1544     apply(rule,rule)
```
```  1545     apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
```
```  1546     apply(rule `a\<in>s`) by auto
```
```  1547 qed auto
```
```  1548
```
```  1549 lemma has_antiderivative_limit:
```
```  1550   fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
```
```  1551   assumes "convex s"
```
```  1552   assumes "\<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))"
```
```  1553   shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
```
```  1554 proof-
```
```  1555   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))"
```
```  1556     apply(rule) using assms(2)
```
```  1557     apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
```
```  1558   guess f using *[THEN choice] .. note * = this
```
```  1559   guess f' using *[THEN choice] .. note f=this
```
```  1560   show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer
```
```  1561   proof(rule,rule)
```
```  1562     fix e::real assume "0<e" guess  N using reals_Archimedean[OF `e>0`] .. note N=this
```
```  1563     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"
```
```  1564       apply(rule_tac x=N in exI)
```
```  1565     proof(default+)
```
```  1566       case goal1
```
```  1567       have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
```
```  1568         using goal1(1) by(auto simp add:field_simps)
```
```  1569       show ?case
```
```  1570         using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
```
```  1571         apply(rule order_trans) using N * apply(cases "h=0") by auto
```
```  1572     qed
```
```  1573   qed(insert f,auto)
```
```  1574 qed
```
```  1575
```
```  1576 subsection {* Differentiation of a series. *}
```
```  1577
```
```  1578 definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
```
```  1579 (infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
```
```  1580
```
```  1581 lemma has_derivative_series:
```
```  1582   fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
```
```  1583   assumes "convex s"
```
```  1584   assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
```
```  1585   assumes "\<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)"
```
```  1586   assumes "x\<in>s" and "((\<lambda>n. f n x) sums_seq l) k"
```
```  1587   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)"
```
```  1588   unfolding sums_seq_def
```
```  1589   apply(rule has_derivative_sequence[OF assms(1) _ assms(3)])
```
```  1590   apply(rule,rule)
```
```  1591   apply(rule has_derivative_setsum) defer
```
```  1592   apply(rule,rule assms(2)[rule_format],assumption)
```
```  1593   using assms(4-5) unfolding sums_seq_def by auto
```
```  1594
```
```  1595 subsection {* Derivative with composed bilinear function. *}
```
```  1596
```
```  1597 lemma has_derivative_bilinear_within:
```
```  1598   assumes "(f has_derivative f') (at x within s)"
```
```  1599   assumes "(g has_derivative g') (at x within s)"
```
```  1600   assumes "bounded_bilinear h"
```
```  1601   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)"
```
```  1602 proof-
```
```  1603   have "(g ---> g x) (at x within s)"
```
```  1604     apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
```
```  1605     using assms(2) unfolding differentiable_def by auto
```
```  1606   moreover
```
```  1607   interpret f':bounded_linear f'
```
```  1608     using assms unfolding has_derivative_def by auto
```
```  1609   interpret g':bounded_linear g'
```
```  1610     using assms unfolding has_derivative_def by auto
```
```  1611   interpret h:bounded_bilinear h
```
```  1612     using assms by auto
```
```  1613   have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)"
```
```  1614     unfolding f'.zero[THEN sym]
```
```  1615     using bounded_linear.tendsto [of f' "\<lambda>y. y - x" 0 "at x within s"]
```
```  1616     using tendsto_diff [OF Lim_within_id tendsto_const, of x x s]
```
```  1617     unfolding id_def using assms(1) unfolding has_derivative_def by auto
```
```  1618   hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
```
```  1619     using tendsto_add[OF tendsto_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"]
```
```  1620     by auto
```
```  1621   ultimately
```
```  1622   have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
```
```  1623              + 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)"
```
```  1624     apply-apply(rule tendsto_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)])
```
```  1625     using assms(1-2)  unfolding has_derivative_within by auto
```
```  1626   guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
```
```  1627   guess C using f'.pos_bounded .. note C=this
```
```  1628   guess D using g'.pos_bounded .. note D=this
```
```  1629   have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
```
```  1630   have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)"
```
```  1631     unfolding Lim_within
```
```  1632   proof(rule,rule) case goal1
```
```  1633     hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
```
```  1634     thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule)
```
```  1635     proof(rule,rule,erule conjE)
```
```  1636       fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
```
```  1637       have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
```
```  1638       also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B"
```
```  1639         apply(rule mult_right_mono)
```
```  1640         apply(rule mult_mono) using B C D
```
```  1641         by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
```
```  1642       also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)"
```
```  1643         by (auto simp add: field_simps)
```
```  1644       also have "\<dots> < e * norm (y - x)"
```
```  1645         apply(rule mult_strict_right_mono)
```
```  1646         using as(3)[unfolded dist_norm] and as(2)
```
```  1647         unfolding pos_less_divide_eq[OF bcd] by (auto simp add: field_simps)
```
```  1648       finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
```
```  1649         unfolding dist_norm apply-apply(cases "y = x")
```
```  1650         by(auto simp add: field_simps)
```
```  1651     qed
```
```  1652   qed
```
```  1653   have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))"
```
```  1654     apply (rule bounded_linear_add)
```
```  1655     apply (rule bounded_linear_compose [OF h.bounded_linear_right `bounded_linear g'`])
```
```  1656     apply (rule bounded_linear_compose [OF h.bounded_linear_left `bounded_linear f'`])
```
```  1657     done
```
```  1658   thus ?thesis using tendsto_add[OF * **] unfolding has_derivative_within
```
```  1659     unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
```
```  1660      h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
```
```  1661     scaleR_right_diff_distrib h.zero_right h.zero_left
```
```  1662     by(auto simp add:field_simps)
```
```  1663 qed
```
```  1664
```
```  1665 lemma has_derivative_bilinear_at:
```
```  1666   assumes "(f has_derivative f') (at x)"
```
```  1667   assumes "(g has_derivative g') (at x)"
```
```  1668   assumes "bounded_bilinear h"
```
```  1669   shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
```
```  1670   using has_derivative_bilinear_within[of f f' x UNIV g g' h] assms by simp
```
```  1671
```
```  1672 subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
```
```  1673
```
```  1674 definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real filter \<Rightarrow> bool)"
```
```  1675 (infixl "has'_vector'_derivative" 12) where
```
```  1676  "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
```
```  1677
```
```  1678 definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
```
```  1679
```
```  1680 lemma vector_derivative_works:
```
```  1681   fixes f::"real \<Rightarrow> 'a::real_normed_vector"
```
```  1682   shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
```
```  1683 proof
```
```  1684   assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
```
```  1685   then interpret bounded_linear f' by auto
```
```  1686   show ?r unfolding vector_derivative_def has_vector_derivative_def
```
```  1687     apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
```
```  1688     using f' unfolding scaleR[THEN sym] by auto
```
```  1689 next
```
```  1690   assume ?r thus ?l
```
```  1691     unfolding vector_derivative_def has_vector_derivative_def differentiable_def
```
```  1692     by auto
```
```  1693 qed
```
```  1694
```
```  1695 lemma has_vector_derivative_withinI_DERIV:
```
```  1696   assumes f: "DERIV f x :> y" shows "(f has_vector_derivative y) (at x within s)"
```
```  1697   unfolding has_vector_derivative_def real_scaleR_def
```
```  1698   apply (rule has_derivative_at_within)
```
```  1699   using DERIV_conv_has_derivative[THEN iffD1, OF f]
```
```  1700   apply (subst mult_commute) .
```
```  1701
```
```  1702 lemma vector_derivative_unique_at:
```
```  1703   assumes "(f has_vector_derivative f') (at x)"
```
```  1704   assumes "(f has_vector_derivative f'') (at x)"
```
```  1705   shows "f' = f''"
```
```  1706 proof-
```
```  1707   have "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
```
```  1708     using assms [unfolded has_vector_derivative_def]
```
```  1709     by (rule frechet_derivative_unique_at)
```
```  1710   thus ?thesis unfolding fun_eq_iff by auto
```
```  1711 qed
```
```  1712
```
```  1713 lemma vector_derivative_unique_within_closed_interval:
```
```  1714   fixes f::"real \<Rightarrow> 'n::ordered_euclidean_space"
```
```  1715   assumes "a < b" and "x \<in> {a..b}"
```
```  1716   assumes "(f has_vector_derivative f') (at x within {a..b})"
```
```  1717   assumes "(f has_vector_derivative f'') (at x within {a..b})"
```
```  1718   shows "f' = f''"
```
```  1719 proof-
```
```  1720   have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
```
```  1721     apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
```
```  1722     using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2)
```
```  1723     by (auto simp: Basis_real_def)
```
```  1724   show ?thesis
```
```  1725   proof(rule ccontr)
```
```  1726     assume "f' \<noteq> f''"
```
```  1727     moreover
```
```  1728     hence "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1"
```
```  1729       using * by (auto simp: fun_eq_iff)
```
```  1730     ultimately show False unfolding o_def by auto
```
```  1731   qed
```
```  1732 qed
```
```  1733
```
```  1734 lemma vector_derivative_at:
```
```  1735   shows "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
```
```  1736   apply(rule vector_derivative_unique_at) defer apply assumption
```
```  1737   unfolding vector_derivative_works[THEN sym] differentiable_def
```
```  1738   unfolding has_vector_derivative_def by auto
```
```  1739
```
```  1740 lemma vector_derivative_within_closed_interval:
```
```  1741   fixes f::"real \<Rightarrow> 'a::ordered_euclidean_space"
```
```  1742   assumes "a < b" and "x \<in> {a..b}"
```
```  1743   assumes "(f has_vector_derivative f') (at x within {a..b})"
```
```  1744   shows "vector_derivative f (at x within {a..b}) = f'"
```
```  1745   apply(rule vector_derivative_unique_within_closed_interval)
```
```  1746   using vector_derivative_works[unfolded differentiable_def]
```
```  1747   using assms by(auto simp add:has_vector_derivative_def)
```
```  1748
```
```  1749 lemma has_vector_derivative_within_subset:
```
```  1750  "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
```
```  1751   unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
```
```  1752
```
```  1753 lemma has_vector_derivative_const:
```
```  1754  "((\<lambda>x. c) has_vector_derivative 0) net"
```
```  1755   unfolding has_vector_derivative_def using has_derivative_const by auto
```
```  1756
```
```  1757 lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
```
```  1758   unfolding has_vector_derivative_def using has_derivative_id by auto
```
```  1759
```
```  1760 lemma has_vector_derivative_cmul:
```
```  1761   "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
```
```  1762   unfolding has_vector_derivative_def
```
```  1763   apply (drule scaleR_right_has_derivative)
```
```  1764   by (auto simp add: algebra_simps)
```
```  1765
```
```  1766 lemma has_vector_derivative_cmul_eq:
```
```  1767   assumes "c \<noteq> 0"
```
```  1768   shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
```
```  1769   apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
```
```  1770   apply(rule has_vector_derivative_cmul) using assms by auto
```
```  1771
```
```  1772 lemma has_vector_derivative_neg:
```
```  1773   "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
```
```  1774   unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
```
```  1775
```
```  1776 lemma has_vector_derivative_add:
```
```  1777   assumes "(f has_vector_derivative f') net"
```
```  1778   assumes "(g has_vector_derivative g') net"
```
```  1779   shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
```
```  1780   using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
```
```  1781   unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
```
```  1782
```
```  1783 lemma has_vector_derivative_sub:
```
```  1784   assumes "(f has_vector_derivative f') net"
```
```  1785   assumes "(g has_vector_derivative g') net"
```
```  1786   shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
```
```  1787   using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
```
```  1788   unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
```
```  1789
```
```  1790 lemma has_vector_derivative_bilinear_within:
```
```  1791   assumes "(f has_vector_derivative f') (at x within s)"
```
```  1792   assumes "(g has_vector_derivative g') (at x within s)"
```
```  1793   assumes "bounded_bilinear h"
```
```  1794   shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)"
```
```  1795 proof-
```
```  1796   interpret bounded_bilinear h using assms by auto
```
```  1797   show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
```
```  1798     unfolding o_def has_vector_derivative_def
```
```  1799     using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib
```
```  1800     by auto
```
```  1801 qed
```
```  1802
```
```  1803 lemma has_vector_derivative_bilinear_at:
```
```  1804   assumes "(f has_vector_derivative f') (at x)"
```
```  1805   assumes "(g has_vector_derivative g') (at x)"
```
```  1806   assumes "bounded_bilinear h"
```
```  1807   shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
```
```  1808   using has_vector_derivative_bilinear_within[where s=UNIV] assms by simp
```
```  1809
```
```  1810 lemma has_vector_derivative_at_within:
```
```  1811   "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
```
```  1812   unfolding has_vector_derivative_def
```
```  1813   by (rule has_derivative_at_within)
```
```  1814
```
```  1815 lemma has_vector_derivative_transform_within:
```
```  1816   assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
```
```  1817   assumes "(f has_vector_derivative f') (at x within s)"
```
```  1818   shows "(g has_vector_derivative f') (at x within s)"
```
```  1819   using assms unfolding has_vector_derivative_def
```
```  1820   by (rule has_derivative_transform_within)
```
```  1821
```
```  1822 lemma has_vector_derivative_transform_at:
```
```  1823   assumes "0 < d" and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
```
```  1824   assumes "(f has_vector_derivative f') (at x)"
```
```  1825   shows "(g has_vector_derivative f') (at x)"
```
```  1826   using assms unfolding has_vector_derivative_def
```
```  1827   by (rule has_derivative_transform_at)
```
```  1828
```
```  1829 lemma has_vector_derivative_transform_within_open:
```
```  1830   assumes "open s" and "x \<in> s" and "\<forall>y\<in>s. f y = g y"
```
```  1831   assumes "(f has_vector_derivative f') (at x)"
```
```  1832   shows "(g has_vector_derivative f') (at x)"
```
```  1833   using assms unfolding has_vector_derivative_def
```
```  1834   by (rule has_derivative_transform_within_open)
```
```  1835
```
```  1836 lemma vector_diff_chain_at:
```
```  1837   assumes "(f has_vector_derivative f') (at x)"
```
```  1838   assumes "(g has_vector_derivative g') (at (f x))"
```
```  1839   shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
```
```  1840   using assms(2) unfolding has_vector_derivative_def apply-
```
```  1841   apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
```
```  1842   unfolding o_def real_scaleR_def scaleR_scaleR .
```
```  1843
```
```  1844 lemma vector_diff_chain_within:
```
```  1845   assumes "(f has_vector_derivative f') (at x within s)"
```
```  1846   assumes "(g has_vector_derivative g') (at (f x) within f ` s)"
```
```  1847   shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
```
```  1848   using assms(2) unfolding has_vector_derivative_def apply-
```
```  1849   apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
```
```  1850   unfolding o_def real_scaleR_def scaleR_scaleR .
```
```  1851
```
```  1852 end
```