# HG changeset patch
# User wenzelm
# Date 1379795622 -7200
# Node ID 1e86d0b668667cf4420e35dd1b65d850096e2a28
# Parent  ef62204a126b011da364d2a7b05138d9bb407d75
tuned proofs;

diff -r ef62204a126b -r 1e86d0b66866 src/HOL/Multivariate_Analysis/Derivative.thy
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Sat Sep 21 20:58:32 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Sat Sep 21 22:33:42 2013 +0200
@@ -1,19 +1,21 @@
-(*  Title:                       HOL/Multivariate_Analysis/Derivative.thy
-    Author:                      John Harrison
-    Translation from HOL Light:  Robert Himmelmann, TU Muenchen
+(*  Title:      HOL/Multivariate_Analysis/Derivative.thy
+    Author:     John Harrison
+    Author:     Robert Himmelmann, TU Muenchen (translation from HOL Light)
 *)
 
-header {* Multivariate calculus in Euclidean space. *}
+header {* Multivariate calculus in Euclidean space *}
 
 theory Derivative
 imports Brouwer_Fixpoint Operator_Norm
 begin
 
-lemma bounded_linear_imp_linear: "bounded_linear f \<Longrightarrow> linear f" (* TODO: move elsewhere *)
+lemma bounded_linear_imp_linear: (* TODO: move elsewhere *)
+  assumes "bounded_linear f"
+  shows "linear f"
 proof -
-  assume "bounded_linear f"
-  then interpret f: bounded_linear f .
-  show "linear f"
+  interpret f: bounded_linear f
+    using assms .
+  show ?thesis
     by (simp add: f.add f.scaleR linear_iff)
 qed
 
@@ -28,25 +30,23 @@
     apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
     apply (simp add: norm_sgn sgn_zero_iff x)
     done
-  thus ?thesis
+  then show ?thesis
     by (rule netlimit_within [of a UNIV])
 qed simp
 
 (* Because I do not want to type this all the time *)
-lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
+lemmas linear_linear = linear_conv_bounded_linear[symmetric]
 
-lemma derivative_linear[dest]:
-  "(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
+lemma derivative_linear[dest]: "(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
   unfolding has_derivative_def by auto
 
-lemma derivative_is_linear:
-  "(f has_derivative f') net \<Longrightarrow> linear f'"
+lemma derivative_is_linear: "(f has_derivative f') net \<Longrightarrow> linear f'"
   by (rule derivative_linear [THEN bounded_linear_imp_linear])
 
-lemma DERIV_conv_has_derivative:
-  "(DERIV f x :> f') = (f has_derivative op * f') (at x)"
+lemma DERIV_conv_has_derivative: "(DERIV f x :> f') \<longleftrightarrow> (f has_derivative op * f') (at x)"
   using deriv_fderiv[of f x UNIV f'] by (subst (asm) mult_commute)
 
+
 subsection {* Derivatives *}
 
 subsubsection {* Combining theorems. *}
@@ -68,56 +68,66 @@
   "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
   by (intro FDERIV_eq_intros) auto
 
+
 subsection {* Derivative with composed bilinear function. *}
 
 lemma has_derivative_bilinear_within:
   assumes "(f has_derivative f') (at x within s)"
-  assumes "(g has_derivative g') (at x within s)"
-  assumes "bounded_bilinear h"
+    and "(g has_derivative g') (at x within s)"
+    and "bounded_bilinear h"
   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)"
   using bounded_bilinear.FDERIV[OF assms(3,1,2)] .
 
 lemma has_derivative_bilinear_at:
   assumes "(f has_derivative f') (at x)"
-  assumes "(g has_derivative g') (at x)"
-  assumes "bounded_bilinear h"
+    and "(g has_derivative g') (at x)"
+    and "bounded_bilinear h"
   shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
   using has_derivative_bilinear_within[of f f' x UNIV g g' h] assms by simp
 
 text {* These are the only cases we'll care about, probably. *}
 
 lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
-         bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
-  unfolding has_derivative_def Lim by (auto simp add: netlimit_within inverse_eq_divide field_simps)
+    bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
+  unfolding has_derivative_def Lim
+  by (auto simp add: netlimit_within inverse_eq_divide field_simps)
 
 lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
-         bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
-  using has_derivative_within [of f f' x UNIV] by simp
+    bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
+  using has_derivative_within [of f f' x UNIV]
+  by simp
 
 text {* More explicit epsilon-delta forms. *}
 
 lemma has_derivative_within':
-  "(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
-        (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
-        \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
+  "(f has_derivative f')(at x within s) \<longleftrightarrow>
+    bounded_linear f' \<and>
+    (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
+      norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
   unfolding has_derivative_within Lim_within dist_norm
-  unfolding diff_0_right by (simp add: diff_diff_eq)
+  unfolding diff_0_right
+  by (simp add: diff_diff_eq)
 
 lemma has_derivative_at':
- "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
-   (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
-        \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
-  using has_derivative_within' [of f f' x UNIV] by simp
+  "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
+    (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
+      norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
+  using has_derivative_within' [of f f' x UNIV]
+  by simp
 
-lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
-  unfolding has_derivative_within' has_derivative_at' by blast
+lemma has_derivative_at_within:
+  "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
+  unfolding has_derivative_within' has_derivative_at'
+  by blast
 
 lemma has_derivative_within_open:
-  "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
+  "a \<in> s \<Longrightarrow> open s \<Longrightarrow>
+    (f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a)"
   by (simp only: at_within_interior interior_open)
 
 lemma has_derivative_right:
-  fixes f :: "real \<Rightarrow> real" and y :: "real"
+  fixes f :: "real \<Rightarrow> real"
+    and y :: "real"
   shows "(f has_derivative (op * y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
     ((\<lambda>t. (f x - f t) / (x - t)) ---> y) (at x within ({x <..} \<inter> I))"
 proof -
@@ -132,91 +142,154 @@
     by (simp add: bounded_linear_mult_right has_derivative_within)
 qed
 
+
 subsubsection {* Limit transformation for derivatives *}
 
 lemma has_derivative_transform_within:
-  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)"
+  assumes "0 < d"
+    and "x \<in> s"
+    and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+    and "(f has_derivative f') (at x within s)"
   shows "(g has_derivative f') (at x within s)"
-  using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
-  apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
-  apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
+  using assms(4)
+  unfolding has_derivative_within
+  apply -
+  apply (erule conjE)
+  apply rule
+  apply assumption
+  apply (rule Lim_transform_within[OF assms(1)])
+  defer
+  apply assumption
+  apply rule
+  apply rule
+  apply (drule assms(3)[rule_format])
+  using assms(3)[rule_format, OF assms(2)]
+  apply auto
+  done
 
 lemma has_derivative_transform_at:
-  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
+  assumes "0 < d"
+    and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
+    and "(f has_derivative f') (at x)"
   shows "(g has_derivative f') (at x)"
-  using has_derivative_transform_within [of d x UNIV f g f'] assms by simp
+  using has_derivative_transform_within [of d x UNIV f g f'] assms
+  by simp
 
 lemma has_derivative_transform_within_open:
-  assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
+  assumes "open s"
+    and "x \<in> s"
+    and "\<forall>y\<in>s. f y = g y"
+    and "(f has_derivative f') (at x)"
   shows "(g has_derivative f') (at x)"
-  using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
-  apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
-  apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
+  using assms(4)
+  unfolding has_derivative_at
+  apply -
+  apply (erule conjE)
+  apply rule
+  apply assumption
+  apply (rule Lim_transform_within_open[OF assms(1,2)])
+  defer
+  apply assumption
+  apply rule
+  apply rule
+  apply (drule assms(3)[rule_format])
+  using assms(3)[rule_format, OF assms(2)]
+  apply auto
+  done
 
 subsection {* Differentiability *}
 
 no_notation Deriv.differentiable (infixl "differentiable" 60)
 
-abbreviation differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" (infixr "differentiable" 30) where
-  "f differentiable net \<equiv> isDiff net f"
+abbreviation
+  differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
+    (infixr "differentiable" 30)
+  where "f differentiable net \<equiv> isDiff net f"
 
-definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
-  "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
+definition
+  differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool"
+    (infixr "differentiable'_on" 30)
+  where "f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable (at x within s))"
 
 lemmas differentiable_def = isDiff_def
 
 lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
-  unfolding differentiable_def by auto
+  unfolding differentiable_def
+  by auto
 
 lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
-  unfolding differentiable_def using has_derivative_at_within by blast
+  unfolding differentiable_def
+  using has_derivative_at_within
+  by blast
 
 lemma differentiable_within_open: (* TODO: delete *)
-  assumes "a \<in> s" and "open s"
-  shows "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
-  using assms by (simp only: at_within_interior interior_open)
+  assumes "a \<in> s"
+    and "open s"
+  shows "f differentiable (at a within s) \<longleftrightarrow> f differentiable (at a)"
+  using assms
+  by (simp only: at_within_interior interior_open)
 
 lemma differentiable_on_eq_differentiable_at:
-  "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
+  "open s \<Longrightarrow> f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x)"
   unfolding differentiable_on_def
   by (metis at_within_interior interior_open)
 
 lemma differentiable_transform_within:
-  assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+  assumes "0 < d"
+    and "x \<in> s"
+    and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
   assumes "f differentiable (at x within s)"
   shows "g differentiable (at x within s)"
-  using assms(4) unfolding differentiable_def
+  using assms(4)
+  unfolding differentiable_def
   by (auto intro!: has_derivative_transform_within[OF assms(1-3)])
 
 lemma differentiable_transform_at:
-  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
+  assumes "0 < d"
+    and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
+    and "f differentiable at x"
   shows "g differentiable at x"
-  using assms(3) unfolding differentiable_def
-  using has_derivative_transform_at[OF assms(1-2)] by auto
+  using assms(3)
+  unfolding differentiable_def
+  using has_derivative_transform_at[OF assms(1-2)]
+  by auto
 
-subsection {* Frechet derivative and Jacobian matrix. *}
+
+subsection {* Frechet derivative and Jacobian matrix *}
 
 definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
 
 lemma frechet_derivative_works:
- "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
-  unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
+  "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
+  unfolding frechet_derivative_def differentiable_def
+  unfolding some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
 
-lemma linear_frechet_derivative:
-  shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
+lemma linear_frechet_derivative: "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
   unfolding frechet_derivative_works has_derivative_def
   by (auto intro: bounded_linear_imp_linear)
 
+
 subsection {* Differentiability implies continuity *}
 
 lemma Lim_mul_norm_within:
-  fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
-  shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
-  unfolding Lim_within apply(rule,rule)
-  apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
-  apply(rule_tac x="min d 1" in exI) apply rule defer
-  apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
-  by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
+  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
+  shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f x) ---> 0) (at a within s)"
+  unfolding Lim_within
+  apply rule
+  apply rule
+  apply (erule_tac x=e in allE)
+  apply (erule impE)
+  apply assumption
+  apply (erule exE)
+  apply (erule conjE)
+  apply (rule_tac x="min d 1" in exI)
+  apply rule
+  defer
+  apply rule
+  apply (erule_tac x=x in ballE)
+  unfolding dist_norm diff_0_right
+  apply (auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
+  done
 
 lemma differentiable_imp_continuous_within:
   "f differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
@@ -228,91 +301,162 @@
   using differentiable_imp_continuous_within by blast
 
 lemma has_derivative_within_subset:
- "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
-  unfolding has_derivative_within using tendsto_within_subset by blast
+  "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
+    (f has_derivative f') (at x within t)"
+  unfolding has_derivative_within
+  using tendsto_within_subset
+  by blast
 
 lemma differentiable_within_subset:
-  "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
-  unfolding differentiable_def using has_derivative_within_subset by blast
+  "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow>
+    f differentiable (at x within s)"
+  unfolding differentiable_def
+  using has_derivative_within_subset
+  by blast
 
 lemma differentiable_on_subset:
   "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
-  unfolding differentiable_on_def using differentiable_within_subset by blast
+  unfolding differentiable_on_def
+  using differentiable_within_subset
+  by blast
 
 lemma differentiable_on_empty: "f differentiable_on {}"
-  unfolding differentiable_on_def by auto
+  unfolding differentiable_on_def
+  by auto
 
 text {* Several results are easier using a "multiplied-out" variant.
 (I got this idea from Dieudonne's proof of the chain rule). *}
 
 lemma has_derivative_within_alt:
- "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
-  (\<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")
+  "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
+    (\<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")
 proof
-  assume ?lhs thus ?rhs
-    unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
+  assume ?lhs
+  then show ?rhs
+    unfolding has_derivative_within
+    apply -
+    apply (erule conjE)
+    apply rule
+    apply assumption
     unfolding Lim_within
-    apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
-    apply(erule exE,rule_tac x=d in exI)
-    apply(erule conjE,rule,assumption,rule,rule)
+    apply rule
+    apply (erule_tac x=e in allE)
+    apply rule
+    apply (erule impE)
+    apply assumption
+    apply (erule exE)
+    apply (rule_tac x=d in exI)
+    apply (erule conjE)
+    apply rule
+    apply assumption
+    apply rule
+    apply rule
   proof-
-    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>
-      dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
-    then interpret bounded_linear f' by auto
-    show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
-      case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero)
+    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>
+        dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e"
+      "y \<in> s"
+      "bounded_linear f'"
+    then interpret bounded_linear f'
+      by auto
+    show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)"
+    proof (cases "y = x")
+      case True
+      then show ?thesis
+        using `bounded_linear f'` by (auto simp add: zero)
     next
-      case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
-        unfolding dist_norm diff_0_right using as(3)
+      case False
+        then have "norm (f y - (f x + f' (y - x))) < e * norm (y - x)"
+        using as(4)[rule_format, OF `y \<in> s`]
+        unfolding dist_norm diff_0_right
+        using as(3)
         using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded dist_norm]
         by (auto simp add: linear_0 linear_sub)
-      thus ?thesis by(auto simp add:algebra_simps)
+      then show ?thesis
+        by (auto simp add: algebra_simps)
     qed
   qed
 next
-  assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within
-    apply-apply(erule conjE,rule,assumption)
-    apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer
-    apply(erule exE,rule_tac x=d in exI)
-    apply(erule conjE,rule,assumption,rule,rule)
+  assume ?rhs
+  then show ?lhs
+    unfolding has_derivative_within Lim_within
+    apply -
+    apply (erule conjE)
+    apply rule
+    apply assumption
+    apply rule
+    apply (erule_tac x="e/2" in allE)
+    apply rule
+    apply (erule impE)
+    defer
+    apply (erule exE)
+    apply (rule_tac x=d in exI)
+    apply (erule conjE)
+    apply rule
+    apply assumption
+    apply rule
+    apply rule
     unfolding dist_norm diff_0_right norm_scaleR
-    apply(erule_tac x=xa in ballE,erule impE)
-  proof-
-    fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
-        "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
-    thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
-      apply(rule_tac le_less_trans[of _ "e/2"])
-      by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps)
+    apply (erule_tac x=xa in ballE)
+    apply (erule impE)
+  proof -
+    fix e d y
+    assume "bounded_linear f'"
+      and "0 < e"
+      and "0 < d"
+      and "y \<in> s"
+      and "0 < norm (y - x) \<and> norm (y - x) < d"
+      and "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
+    then show "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
+      apply (rule_tac le_less_trans[of _ "e/2"])
+      apply (auto intro!: mult_imp_div_pos_le simp add: algebra_simps)
+      done
   qed auto
 qed
 
 lemma has_derivative_at_alt:
-  "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
-  (\<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))"
-  using has_derivative_within_alt[where s=UNIV] by simp
+  "(f has_derivative f') (at x) \<longleftrightarrow>
+    bounded_linear f' \<and>
+    (\<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))"
+  using has_derivative_within_alt[where s=UNIV]
+  by simp
 
-subsection {* The chain rule. *}
+
+subsection {* The chain rule *}
 
 lemma diff_chain_within[FDERIV_intros]:
   assumes "(f has_derivative f') (at x within s)"
-  assumes "(g has_derivative g') (at (f x) within (f ` s))"
-  shows "((g o f) has_derivative (g' o f'))(at x within s)"
-  using FDERIV_in_compose[OF assms] by (simp add: comp_def)
+    and "(g has_derivative g') (at (f x) within (f ` s))"
+  shows "((g \<circ> f) has_derivative (g' \<circ> f'))(at x within s)"
+  using FDERIV_in_compose[OF assms]
+  by (simp add: comp_def)
 
 lemma diff_chain_at[FDERIV_intros]:
-  "(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)"
-  using FDERIV_compose[of f f' x UNIV g g'] by (simp add: comp_def)
+  "(f has_derivative f') (at x) \<Longrightarrow>
+    (g has_derivative g') (at (f x)) \<Longrightarrow> ((g \<circ> f) has_derivative (g' \<circ> f')) (at x)"
+  using FDERIV_compose[of f f' x UNIV g g']
+  by (simp add: comp_def)
 
 
-subsection {* Composition rules stated just for differentiability. *}
+subsection {* Composition rules stated just for differentiability *}
 
 lemma differentiable_chain_at:
-  "f differentiable (at x) \<Longrightarrow> g differentiable (at (f x)) \<Longrightarrow> (g o f) differentiable (at x)"
-  unfolding differentiable_def by(meson diff_chain_at)
+  "f differentiable (at x) \<Longrightarrow>
+    g differentiable (at (f x)) \<Longrightarrow> (g \<circ> f) differentiable (at x)"
+  unfolding differentiable_def
+  by (meson diff_chain_at)
 
 lemma differentiable_chain_within:
-  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s)) \<Longrightarrow> (g o f) differentiable (at x within s)"
-  unfolding differentiable_def by(meson diff_chain_within)
+  "f differentiable (at x within s) \<Longrightarrow>
+    g differentiable (at(f x) within (f ` s)) \<Longrightarrow> (g \<circ> f) differentiable (at x within s)"
+  unfolding differentiable_def
+  by (meson diff_chain_within)
+
 
 subsection {* Uniqueness of derivative *}
 
@@ -324,95 +468,127 @@
 lemma frechet_derivative_unique_within:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_derivative f') (at x within s)"
-  assumes "(f has_derivative f'') (at x within s)"
-  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)"
+    and "(f has_derivative f'') (at x within s)"
+    and "\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < abs d \<and> abs d < e \<and> (x + d *\<^sub>R i) \<in> s"
   shows "f' = f''"
-proof-
+proof -
   note as = assms(1,2)[unfolded has_derivative_def]
   then interpret f': bounded_linear f' by auto
   from as interpret f'': bounded_linear f'' by auto
   have "x islimpt s" unfolding islimpt_approachable
-  proof(rule,rule)
-    fix e::real assume "0<e" guess d
-      using assms(3)[rule_format,OF SOME_Basis `e>0`] ..
-    thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
-      apply(rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI)
-      unfolding dist_norm by (auto simp: SOME_Basis nonzero_Basis)
+  proof (rule, rule)
+    fix e :: real
+    assume "e > 0"
+    guess d using assms(3)[rule_format,OF SOME_Basis `e>0`] ..
+    then show "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
+      apply (rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI)
+      unfolding dist_norm
+      apply (auto simp: SOME_Basis nonzero_Basis)
+      done
   qed
-  hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within)
-    unfolding trivial_limit_within by simp
-  show ?thesis  apply(rule linear_eq_stdbasis)
+  then have *: "netlimit (at x within s) = x"
+    apply -
+    apply (rule netlimit_within)
+    unfolding trivial_limit_within
+    apply simp
+    done
+  show ?thesis
+    apply (rule linear_eq_stdbasis)
     unfolding linear_conv_bounded_linear
-    apply(rule as(1,2)[THEN conjunct1])+
-  proof(rule,rule ccontr)
-    fix i :: 'a assume i:"i \<in> Basis" def e \<equiv> "norm (f' i - f'' i)"
+    apply (rule as(1,2)[THEN conjunct1])+
+  proof (rule, rule ccontr)
+    fix i :: 'a
+    assume i: "i \<in> Basis"
+    def e \<equiv> "norm (f' i - f'' i)"
     assume "f' i \<noteq> f'' i"
-    hence "e>0" unfolding e_def by auto
+    then have "e > 0"
+      unfolding e_def by auto
     guess d using tendsto_diff [OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
     guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
-    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)))"
+    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)))"
       unfolding scaleR_right_distrib by auto
-    also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"  
+    also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
       unfolding f'.scaleR f''.scaleR
-      unfolding scaleR_right_distrib scaleR_minus_right by auto
-    also have "\<dots> = e" unfolding e_def using c[THEN conjunct1]
+      unfolding scaleR_right_distrib scaleR_minus_right
+      by auto
+    also have "\<dots> = e"
+      unfolding e_def
+      using c[THEN conjunct1]
       using norm_minus_cancel[of "f' i - f'' i"]
       by (auto simp add: add.commute ab_diff_minus)
-    finally show False using c
+    finally show False
+      using c
       using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R i"]
       unfolding dist_norm
       unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
         scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
-      using i by (auto simp: inverse_eq_divide)
+      using i
+      by (auto simp: inverse_eq_divide)
   qed
 qed
 
 lemma frechet_derivative_unique_at:
-  shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
+  "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
   by (rule FDERIV_unique)
 
 lemma frechet_derivative_unique_within_closed_interval:
   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "x \<in> {a..b}" (is "x\<in>?I")
-  assumes "(f has_derivative f' ) (at x within {a..b})"
-  assumes "(f has_derivative f'') (at x within {a..b})"
+  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
+    and "x \<in> {a..b}"
+    and "(f has_derivative f' ) (at x within {a..b})"
+    and "(f has_derivative f'') (at x within {a..b})"
   shows "f' = f''"
   apply(rule frechet_derivative_unique_within)
   apply(rule assms(3,4))+
-proof(rule,rule,rule)
-  fix e::real and i :: 'a assume "e>0" and i:"i\<in>Basis"
-  thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> {a..b}"
-  proof(cases "x\<bullet>i=a\<bullet>i")
-    case True thus ?thesis
-      apply(rule_tac x="(min (b\<bullet>i - a\<bullet>i)  e) / 2" in exI)
+proof (rule, rule, rule)
+  fix e :: real
+  fix i :: 'a
+  assume "e > 0" and i: "i \<in> Basis"
+  then show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> {a..b}"
+  proof (cases "x\<bullet>i = a\<bullet>i")
+    case True
+    then show ?thesis
+      apply (rule_tac x="(min (b\<bullet>i - a\<bullet>i)  e) / 2" in exI)
       using assms(1)[THEN bspec[where x=i]] and `e>0` and assms(2)
       unfolding mem_interval
-      using i by (auto simp add: field_simps inner_simps inner_Basis)
-  next 
+      using i
+      apply (auto simp add: field_simps inner_simps inner_Basis)
+      done
+  next
     note * = assms(2)[unfolded mem_interval, THEN bspec, OF i]
-    case False moreover have "a \<bullet> i < x \<bullet> i" using False * by auto
+    case False
+    moreover have "a \<bullet> i < x \<bullet> i"
+      using False * by auto
     moreover {
       have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> a\<bullet>i *2 + x\<bullet>i - a\<bullet>i"
         by auto
-      also have "\<dots> = a\<bullet>i + x\<bullet>i" by auto
-      also have "\<dots> \<le> 2 * (x\<bullet>i)" using * by auto
-      finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2" by auto
+      also have "\<dots> = a\<bullet>i + x\<bullet>i"
+        by auto
+      also have "\<dots> \<le> 2 * (x\<bullet>i)"
+        using * by auto
+      finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2"
+        by auto
     }
-    moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0" using * and `e>0` by auto
-    hence "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e" using * by auto
+    moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0"
+      using * and `e>0` by auto
+    then have "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e"
+      using * by auto
     ultimately show ?thesis
-      apply(rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
+      apply (rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
       using assms(1)[THEN bspec, OF i] and `e>0` and assms(2)
       unfolding mem_interval
-      using i by (auto simp add: field_simps inner_simps inner_Basis)
+      using i
+      apply (auto simp add: field_simps inner_simps inner_Basis)
+      done
   qed
 qed
 
 lemma frechet_derivative_unique_within_open_interval:
   fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "x \<in> {a<..<b}"
-  assumes "(f has_derivative f' ) (at x within {a<..<b})"
-  assumes "(f has_derivative f'') (at x within {a<..<b})"
+    and "(f has_derivative f' ) (at x within {a<..<b})"
+    and "(f has_derivative f'') (at x within {a<..<b})"
   shows "f' = f''"
 proof -
   from assms(1) have *: "at x within {a<..<b} = at x"
@@ -422,27 +598,38 @@
 qed
 
 lemma frechet_derivative_at:
-  shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
-  apply(rule frechet_derivative_unique_at[of f],assumption)
-  unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
+  "(f has_derivative f') (at x) \<Longrightarrow> f' = frechet_derivative f (at x)"
+  apply (rule frechet_derivative_unique_at[of f])
+  apply assumption
+  unfolding frechet_derivative_works[symmetric]
+  using differentiable_def
+  apply auto
+  done
 
 lemma frechet_derivative_within_closed_interval:
-  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" and "x \<in> {a..b}"
-  assumes "(f has_derivative f') (at x within {a.. b})"
-  shows "frechet_derivative f (at x within {a.. b}) = f'"
-  apply(rule frechet_derivative_unique_within_closed_interval[where f=f]) 
-  apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
-  unfolding differentiable_def using assms(3) by auto 
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
+  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
+    and "x \<in> {a..b}"
+    and "(f has_derivative f') (at x within {a..b})"
+  shows "frechet_derivative f (at x within {a..b}) = f'"
+  apply (rule frechet_derivative_unique_within_closed_interval[where f=f])
+  apply (rule assms(1,2))+
+  unfolding frechet_derivative_works[symmetric]
+  unfolding differentiable_def
+  using assms(3)
+  apply auto
+  done
 
-subsection {* The traditional Rolle theorem in one dimension. *}
+
+subsection {* The traditional Rolle theorem in one dimension *}
 
 lemma linear_componentwise:
   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes lf: "linear f"
   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
 proof -
-  have fA: "finite Basis" by simp
+  have fA: "finite Basis"
+    by simp
   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
     by (simp add: inner_setsum_left)
   then show ?thesis
@@ -454,16 +641,19 @@
   the unfolding of it. *}
 
 lemma jacobian_works:
-  "(f::('a::euclidean_space) \<Rightarrow> ('b::euclidean_space)) differentiable net \<longleftrightarrow>
-   (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis.
-      (\<Sum>j\<in>Basis. frechet_derivative f net (j) \<bullet> i * (h \<bullet> j)) *\<^sub>R i)) net"
-  (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net")
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  shows "f differentiable net \<longleftrightarrow>
+    (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis.
+      (\<Sum>j\<in>Basis. frechet_derivative f net j \<bullet> i * (h \<bullet> j)) *\<^sub>R i)) net"
+    (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net")
 proof
   assume *: ?differentiable
-  { fix h i
-    have "?SUM h i = frechet_derivative f net h \<bullet> i" using *
-      by (auto intro!: setsum_cong
-               simp: linear_componentwise[of _ h i] linear_frechet_derivative) }
+  {
+    fix h i
+    have "?SUM h i = frechet_derivative f net h \<bullet> i"
+      using *
+      by (auto intro!: setsum_cong simp: linear_componentwise[of _ h i] linear_frechet_derivative)
+  }
   with * show "(f has_derivative (\<lambda>h. \<Sum>i\<in>Basis. ?SUM h i *\<^sub>R i)) net"
     by (simp add: frechet_derivative_works euclidean_representation)
 qed (auto intro!: differentiableI)
@@ -471,54 +661,69 @@
 lemma differential_zero_maxmin_component:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes k: "k \<in> Basis"
-    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))"
+    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)"
     and diff: "f differentiable (at x)"
   shows "(\<Sum>j\<in>Basis. (frechet_derivative f (at x) j \<bullet> k) *\<^sub>R j) = (0::'a)" (is "?D k = 0")
 proof (rule ccontr)
-  assume "?D k \<noteq> 0"
+  assume "\<not> ?thesis"
   then obtain j where j: "?D k \<bullet> j \<noteq> 0" "j \<in> Basis"
     unfolding euclidean_eq_iff[of _ "0::'a"] by auto
-  hence *: "\<bar>?D k \<bullet> j\<bar> / 2 > 0" by auto
+  then have *: "\<bar>?D k \<bullet> j\<bar> / 2 > 0"
+    by auto
   note as = diff[unfolded jacobian_works has_derivative_at_alt]
   guess e' using as[THEN conjunct2, rule_format, OF *] .. note e' = this
   guess d using real_lbound_gt_zero[OF ball(1) e'[THEN conjunct1]] .. note d = this
-  { fix c assume "abs c \<le> d"
-    hence *:"norm (x + c *\<^sub>R j - x) < e'" using norm_Basis[OF j(2)] d by auto
+  {
+    fix c
+    assume "abs c \<le> d"
+    then have *: "norm (x + c *\<^sub>R j - x) < e'"
+      using norm_Basis[OF j(2)] d by auto
     let ?v = "(\<Sum>i\<in>Basis. (\<Sum>l\<in>Basis. ?D i \<bullet> l * ((c *\<^sub>R j :: 'a) \<bullet> l)) *\<^sub>R i)"
-    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
-    have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le>
-        norm (f (x + c *\<^sub>R j) - f x - ?v)" by (rule Basis_le_norm[OF k])
+    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
+    have "\<bar>(f (x + c *\<^sub>R j) - f x - ?v) \<bullet> k\<bar> \<le> norm (f (x + c *\<^sub>R j) - f x - ?v)"
+      by (rule Basis_le_norm[OF k])
     also have "\<dots> \<le> \<bar>?D k \<bullet> j\<bar> / 2 * \<bar>c\<bar>"
       using e'[THEN conjunct2, rule_format, OF *] and norm_Basis[OF j(2)] j
       by simp
-    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
-    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>"
+    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
+    then have "\<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>"
       using j k
-      by (simp add: inner_simps field_simps inner_Basis setsum_cases if_dist) }
+      by (simp add: inner_simps field_simps inner_Basis setsum_cases if_dist)
+  }
   note * = this
   have "x + d *\<^sub>R j \<in> ball x e" "x - d *\<^sub>R j \<in> ball x e"
-    unfolding mem_ball dist_norm using norm_Basis[OF j(2)] d by auto
-  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>
-         ((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
-  have ***: "\<And>y y1 y2 d dx::real.
-    (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
-  show False apply(rule ***[OF **, where dx="d * (?D k \<bullet> j)" and d="\<bar>?D k \<bullet> j\<bar> / 2 * \<bar>d\<bar>"])
+    unfolding mem_ball dist_norm
+    using norm_Basis[OF j(2)] d
+    by auto
+  then have **: "((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>
+      ((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
+  have ***: "\<And>y y1 y2 d dx :: real. y1 \<le> y \<and> y2 \<le> y \<or> y \<le> y1 \<and> y \<le> y2 \<Longrightarrow>
+      d < abs dx \<Longrightarrow> abs (y1 - y - - dx) \<le> d \<Longrightarrow> abs (y2 - y - dx) \<le> d \<Longrightarrow> False"
+    by arith
+  show False
+    apply (rule ***[OF **, where dx="d * (?D k \<bullet> j)" and d="\<bar>?D k \<bullet> j\<bar> / 2 * \<bar>d\<bar>"])
     using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
     unfolding mult_minus_left
     unfolding abs_mult diff_minus_eq_add scaleR_minus_left
-    unfolding algebra_simps by (auto intro: mult_pos_pos)
+    unfolding algebra_simps
+    apply (auto intro: mult_pos_pos)
+    done
 qed
 
 text {* In particular if we have a mapping into @{typ "real"}. *}
 
 lemma differential_zero_maxmin:
-  fixes f::"'a\<Colon>euclidean_space \<Rightarrow> real"
-  assumes "x \<in> s" "open s"
-  and deriv: "(f has_derivative f') (at x)"
-  and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
+  fixes f::"'a::euclidean_space \<Rightarrow> real"
+  assumes "x \<in> s"
+    and "open s"
+    and deriv: "(f has_derivative f') (at x)"
+    and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
   shows "f' = (\<lambda>v. 0)"
 proof -
-  obtain e where e:"e>0" "ball x e \<subseteq> s"
+  obtain e where e: "e > 0" "ball x e \<subseteq> s"
     using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
   with differential_zero_maxmin_component[where 'b=real, of 1 e x f] mono deriv
   have "(\<Sum>j\<in>Basis. frechet_derivative f (at x) j *\<^sub>R j) = (0::'a)"
@@ -527,529 +732,856 @@
   have "\<forall>i\<in>Basis. f' i = 0"
     by (simp add: euclidean_eq_iff[of _ "0::'a"] inner_setsum_left_Basis)
   with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 1]
-  show ?thesis by (simp add: fun_eq_iff)
+  show ?thesis
+    by (simp add: fun_eq_iff)
 qed
 
 lemma rolle:
-  fixes f::"real\<Rightarrow>real"
-  assumes "a < b" and "f a = f b" and "continuous_on {a..b} f"
-  assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+  fixes f :: "real \<Rightarrow> real"
+  assumes "a < b"
+    and "f a = f b"
+    and "continuous_on {a..b} f"
+    and "\<forall>x\<in>{a<..<b}. (f has_derivative f' x) (at x)"
   shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)"
-proof-
-  have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))"
-  proof-
-    have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto
-    hence *:"{a .. b}\<noteq>{}" by auto
+proof -
+  have "\<exists>x\<in>{a<..<b}. (\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x)"
+  proof -
+    have "(a + b) / 2 \<in> {a .. b}"
+      using assms(1) by auto
+    then have *: "{a..b} \<noteq> {}"
+      by auto
     guess d using continuous_attains_sup[OF compact_interval * assms(3)] .. note d=this
     guess c using continuous_attains_inf[OF compact_interval * assms(3)] .. note c=this
     show ?thesis
-    proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
-      case True thus ?thesis
-        apply(erule_tac disjE) apply(rule_tac x=d in bexI)
-        apply(rule_tac[3] x=c in bexI)
-        using d c by auto
+    proof (cases "d \<in> {a<..<b} \<or> c \<in> {a<..<b}")
+      case True
+      then show ?thesis
+        apply (erule_tac disjE)
+        apply (rule_tac x=d in bexI)
+        apply (rule_tac[3] x=c in bexI)
+        using d c
+        apply auto
+        done
     next
       def e \<equiv> "(a + b) /2"
-      case False hence "f d = f c" using d c assms(2) by auto
-      hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d"
-        using c d apply- apply(erule_tac x=x in ballE)+ by auto
-      thus ?thesis
-        apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto
+      case False
+      then have "f d = f c"
+        using d c assms(2) by auto
+      then have "\<And>x. x \<in> {a..b} \<Longrightarrow> f x = f d"
+        using c d
+        apply -
+        apply (erule_tac x=x in ballE)+
+        apply auto
+        done
+      then show ?thesis
+        apply (rule_tac x=e in bexI)
+        unfolding e_def
+        using assms(1)
+        apply auto
+        done
     qed
   qed
   then guess x .. note x=this
-  hence "f' x = (\<lambda>v. 0)"
-    apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
-    defer apply(rule open_interval)
-    apply(rule assms(4)[unfolded has_derivative_at[THEN sym],THEN bspec[where x=x]],assumption)
-    unfolding o_def apply(erule disjE,rule disjI2) by auto
-  thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
-    apply(drule_tac x=v in fun_cong) using x(1) by auto
+  then have "f' x = (\<lambda>v. 0)"
+    apply (rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
+    defer
+    apply (rule open_interval)
+    apply (rule assms(4)[unfolded has_derivative_at[symmetric],THEN bspec[where x=x]])
+    apply assumption
+    unfolding o_def
+    apply (erule disjE)
+    apply (rule disjI2)
+    apply auto
+    done
+  then show ?thesis
+    apply (rule_tac x=x in bexI)
+    unfolding o_def
+    apply rule
+    apply (drule_tac x=v in fun_cong)
+    using x(1)
+    apply auto
+    done
 qed
 
-subsection {* One-dimensional mean value theorem. *}
+
+subsection {* One-dimensional mean value theorem *}
 
-lemma mvt: fixes f::"real \<Rightarrow> real"
-  assumes "a < b" and "continuous_on {a .. b} f"
+lemma mvt:
+  fixes f :: "real \<Rightarrow> real"
+  assumes "a < b"
+    and "continuous_on {a..b} f"
   assumes "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
-  shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))"
-proof-
+  shows "\<exists>x\<in>{a<..<b}. f b - f a = (f' x) (b - a)"
+proof -
   have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
   proof (intro rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"] ballI)
-    fix x assume x:"x \<in> {a<..<b}"
-    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)"
+    fix x
+    assume x: "x \<in> {a<..<b}"
+    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)"
       by (intro FDERIV_intros assms(3)[rule_format,OF x] mult_right_has_derivative)
   qed (insert assms(1,2), auto intro!: continuous_on_intros simp: field_simps)
   then guess x ..
-  thus ?thesis apply(rule_tac x=x in bexI)
-    apply(drule fun_cong[of _ _ "b - a"]) by auto
+  then show ?thesis
+    apply (rule_tac x=x in bexI)
+    apply (drule fun_cong[of _ _ "b - a"])
+    apply auto
+    done
 qed
 
 lemma mvt_simple:
-  fixes f::"real \<Rightarrow> real"
-  assumes "a<b" and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
+  fixes f :: "real \<Rightarrow> real"
+  assumes "a < b"
+    and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
   shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
-  apply(rule mvt)
-  apply(rule assms(1), rule differentiable_imp_continuous_on)
-  unfolding differentiable_on_def differentiable_def defer
+  apply (rule mvt)
+  apply (rule assms(1))
+  apply (rule differentiable_imp_continuous_on)
+  unfolding differentiable_on_def differentiable_def
+  defer
 proof
-  fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)"
-    unfolding has_derivative_within_open[OF x open_interval,THEN sym] 
-    apply(rule has_derivative_within_subset)
-    apply(rule assms(2)[rule_format])
-    using x by auto
-qed(insert assms(2), auto)
+  fix x
+  assume x: "x \<in> {a<..<b}"
+  show "(f has_derivative f' x) (at x)"
+    unfolding has_derivative_within_open[OF x open_interval,symmetric]
+    apply (rule has_derivative_within_subset)
+    apply (rule assms(2)[rule_format])
+    using x
+    apply auto
+    done
+qed (insert assms(2), auto)
 
 lemma mvt_very_simple:
-  fixes f::"real \<Rightarrow> real"
-  assumes "a \<le> b" and "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
+  fixes f :: "real \<Rightarrow> real"
+  assumes "a \<le> b"
+    and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
   shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
 proof (cases "a = b")
-  interpret bounded_linear "f' b" using assms(2) assms(1) by auto
-  case True thus ?thesis apply(rule_tac x=a in bexI)
-    using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
-    unfolding True using zero by auto next
-  case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto
+  interpret bounded_linear "f' b"
+    using assms(2) assms(1) by auto
+  case True
+  then show ?thesis
+    apply (rule_tac x=a in bexI)
+    using assms(2)[THEN bspec[where x=a]]
+    unfolding has_derivative_def
+    unfolding True
+    using zero
+    apply auto
+    done
+next
+  case False
+  then show ?thesis
+    using mvt_simple[OF _ assms(2)]
+    using assms(1)
+    by auto
 qed
 
 text {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
 
 lemma mvt_general:
-  fixes f::"real\<Rightarrow>'a::euclidean_space"
-  assumes "a<b" and "continuous_on {a..b} f"
-  assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
-  shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))"
-proof-
+  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+  assumes "a < b"
+    and "continuous_on {a..b} f"
+    and "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+  shows "\<exists>x\<in>{a<..<b}. norm (f b - f a) \<le> norm (f' x (b - a))"
+proof -
   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)"
-    apply(rule mvt) apply(rule assms(1))
-    apply(rule continuous_on_inner continuous_on_intros assms(2) ballI)+
+    apply (rule mvt)
+    apply (rule assms(1))
+    apply (rule continuous_on_inner continuous_on_intros assms(2) ballI)+
     unfolding o_def
-    apply(rule FDERIV_inner_right)
-    using assms(3) by auto
+    apply (rule FDERIV_inner_right)
+    using assms(3)
+    apply auto
+    done
   then guess x .. note x=this
-  show ?thesis proof(cases "f a = f b")
+  show ?thesis
+  proof (cases "f a = f b")
     case False
     have "norm (f b - f a) * norm (f b - f a) = (norm (f b - f a))\<^sup>2"
       by (simp add: power2_eq_square)
-    also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding power2_norm_eq_inner ..
+    also have "\<dots> = (f b - f a) \<bullet> (f b - f a)"
+      unfolding power2_norm_eq_inner ..
     also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)"
-      using x unfolding inner_simps by (auto simp add: inner_diff_left)
+      using x
+      unfolding inner_simps
+      by (auto simp add: inner_diff_left)
     also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))"
       by (rule norm_cauchy_schwarz)
-    finally show ?thesis using False x(1)
+    finally show ?thesis
+      using False x(1)
       by (auto simp add: real_mult_left_cancel)
   next
-    case True thus ?thesis using assms(1)
-      apply (rule_tac x="(a + b) /2" in bexI) by auto
+    case True
+    then show ?thesis
+      using assms(1)
+      apply (rule_tac x="(a + b) /2" in bexI)
+      apply auto
+      done
   qed
 qed
 
 text {* Still more general bound theorem. *}
 
 lemma differentiable_bound:
-  fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)"
-  assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
-  shows "norm(f x - f y) \<le> B * norm(x - y)"
-proof-
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "convex s"
+    and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
+    and "\<forall>x\<in>s. onorm (f' x) \<le> B"
+    and x: "x \<in> s"
+    and y: "y \<in> s"
+  shows "norm (f x - f y) \<le> B * norm (x - y)"
+proof -
   let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
-  have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
+  have *: "\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
     using assms(1)[unfolded convex_alt,rule_format,OF x y]
     unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib
     by (auto simp add: algebra_simps)
-  hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply-
-    apply(rule continuous_on_intros)+
+  then have 1: "continuous_on {0..1} (f \<circ> ?p)"
+    apply -
+    apply (rule continuous_on_intros)+
     unfolding continuous_on_eq_continuous_within
-    apply(rule,rule differentiable_imp_continuous_within)
-    unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
-    apply(rule has_derivative_within_subset)
-    apply(rule assms(2)[rule_format]) by auto
-  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)"
+    apply rule
+    apply (rule differentiable_imp_continuous_within)
+    unfolding differentiable_def
+    apply (rule_tac x="f' xa" in exI)
+    apply (rule has_derivative_within_subset)
+    apply (rule assms(2)[rule_format])
+    apply auto
+    done
+  have 2: "\<forall>u\<in>{0<..<1}.
+    ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)"
   proof rule
     case goal1
     let ?u = "x + u *\<^sub>R (y - x)"
-    have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})" 
-      apply(rule diff_chain_within) apply(rule FDERIV_intros)+ 
-      apply(rule has_derivative_within_subset)
-      apply(rule assms(2)[rule_format]) using goal1 * by auto
-    thus ?case
-      unfolding has_derivative_within_open[OF goal1 open_interval] by auto
+    have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
+      apply (rule diff_chain_within)
+      apply (rule FDERIV_intros)+
+      apply (rule has_derivative_within_subset)
+      apply (rule assms(2)[rule_format])
+      using goal1 *
+      apply auto
+      done
+    then show ?case
+      unfolding has_derivative_within_open[OF goal1 open_interval]
+      by auto
   qed
   guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
-  have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y"
-  proof-
+  have **: "\<And>x y. x \<in> s \<Longrightarrow> norm (f' x y) \<le> B * norm y"
+  proof -
     case goal1
     have "norm (f' x y) \<le> onorm (f' x) * norm y"
-      using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
+      by (rule onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]])
     also have "\<dots> \<le> B * norm y"
-      apply(rule mult_right_mono)
+      apply (rule mult_right_mono)
       using assms(3)[rule_format,OF goal1]
-      by(auto simp add:field_simps)
-    finally show ?case by simp
+      apply (auto simp add: field_simps)
+      done
+    finally show ?case
+      by simp
   qed
   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)"
-    by(auto simp add:norm_minus_commute) 
-  also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
-  also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
-  finally show ?thesis by(auto simp add:norm_minus_commute)
+    by (auto simp add: norm_minus_commute)
+  also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))"
+    using u by auto
+  also have "\<dots> \<le> B * norm(y - x)"
+    apply (rule **)
+    using * and u
+    apply auto
+    done
+  finally show ?thesis
+    by (auto simp add: norm_minus_commute)
 qed
 
 lemma differentiable_bound_real:
-  fixes f::"real \<Rightarrow> real"
-  assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
-  assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
-  shows "norm(f x - f y) \<le> B * norm(x - y)"
+  fixes f :: "real \<Rightarrow> real"
+  assumes "convex s"
+    and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
+    and "\<forall>x\<in>s. onorm (f' x) \<le> B"
+    and x: "x \<in> s"
+    and y: "y \<in> s"
+  shows "norm (f x - f y) \<le> B * norm (x - y)"
   using differentiable_bound[of s f f' B x y]
-  unfolding Ball_def image_iff o_def using assms by auto
+  unfolding Ball_def image_iff o_def
+  using assms
+  by auto
 
 text {* In particular. *}
 
 lemma has_derivative_zero_constant:
-  fixes f::"real\<Rightarrow>real"
-  assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
+  fixes f :: "real \<Rightarrow> real"
+  assumes "convex s"
+    and "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
   shows "\<exists>c. \<forall>x\<in>s. f x = c"
-proof(cases "s={}")
-  case False then obtain x where "x\<in>s" by auto
-  have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
-    thus ?case
-      using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
-      unfolding onorm_const by auto qed
-  thus ?thesis apply(rule_tac x="f x" in exI) by auto
-qed auto
+proof (cases "s={}")
+  case False
+  then obtain x where "x \<in> s"
+    by auto
+  have "\<And>y. y \<in> s \<Longrightarrow> f x = f y"
+  proof -
+    case goal1
+    then show ?case
+      using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x \<in> s`
+      unfolding onorm_const
+      by auto
+  qed
+  then show ?thesis
+    apply (rule_tac x="f x" in exI)
+    apply auto
+    done
+next
+  case True
+  then show ?thesis by auto
+qed
 
-lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
-  assumes "convex s" and "a \<in> s" and "f a = c"
-  assumes "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" and "x\<in>s"
+lemma has_derivative_zero_unique:
+  fixes f :: "real \<Rightarrow> real"
+  assumes "convex s"
+    and "a \<in> s"
+    and "f a = c"
+    and "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
+    and "x \<in> s"
   shows "f x = c"
-  using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
+  using has_derivative_zero_constant[OF assms(1,4)]
+  using assms(2-3,5)
+  by auto
 
-subsection {* Differentiability of inverse function (most basic form). *}
+
+subsection {* Differentiability of inverse function (most basic form) *}
 
 lemma has_derivative_inverse_basic:
-  fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
+  fixes f :: "'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
   assumes "(f has_derivative f') (at (g y))"
-  assumes "bounded_linear g'" and "g' \<circ> f' = id" and "continuous (at y) g"
-  assumes "open t" and "y \<in> t" and "\<forall>z\<in>t. f(g z) = z"
+    and "bounded_linear g'"
+    and "g' \<circ> f' = id"
+    and "continuous (at y) g"
+    and "open t"
+    and "y \<in> t"
+    and "\<forall>z\<in>t. f (g z) = z"
   shows "(g has_derivative g') (at y)"
-proof-
+proof -
   interpret f': bounded_linear f'
     using assms unfolding has_derivative_def by auto
-  interpret g': bounded_linear g' using assms by auto
+  interpret g': bounded_linear g'
+    using assms by auto
   guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
-(*  have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
-  have lem1:"\<forall>e>0. \<exists>d>0. \<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y - g'(z - y)) \<le> e * norm(g z - g y)"
-  proof(rule,rule)
+  have lem1: "\<forall>e>0. \<exists>d>0. \<forall>z.
+    norm (z - y) < d \<longrightarrow> norm (g z - g y - g'(z - y)) \<le> e * norm (g z - g y)"
+  proof (rule, rule)
     case goal1
-    have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
+    have *: "e / C > 0"
+      apply (rule divide_pos_pos)
+      using `e > 0` C
+      apply auto
+      done
     guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
     guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
     guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
     guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
-    thus ?case apply(rule_tac x=d in exI) apply rule defer
-    proof(rule,rule)
-      fix z assume as:"norm (z - y) < d" hence "z\<in>t"
+    then show ?case
+      apply (rule_tac x=d in exI)
+      apply rule
+      defer
+      apply rule
+      apply rule
+    proof -
+      fix z
+      assume as: "norm (z - y) < d"
+      then have "z \<in> t"
         using d2 d unfolding dist_norm by auto
       have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
         unfolding g'.diff f'.diff
-        unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] 
+        unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
         unfolding assms(7)[rule_format,OF `z\<in>t`]
-        apply(subst norm_minus_cancel[THEN sym]) by auto
-      also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C"
+        apply (subst norm_minus_cancel[symmetric])
+        apply auto
+        done
+      also have "\<dots> \<le> norm (f (g z) - y - f' (g z - g y)) * C"
         by (rule C [THEN conjunct2, rule_format])
       also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
-        apply(rule mult_right_mono)
-        apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]])
-        apply(cases "z=y") defer
-        apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format])
-        using as d C d0 by auto
+        apply (rule mult_right_mono)
+        apply (rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]])
+        apply (cases "z = y")
+        defer
+        apply (rule d1[THEN conjunct2, unfolded dist_norm,rule_format])
+        using as d C d0
+        apply auto
+        done
       also have "\<dots> \<le> e * norm (g z - g y)"
         using C by (auto simp add: field_simps)
       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
         by simp
     qed auto
   qed
-  have *:"(0::real) < 1 / 2" by auto
+  have *: "(0::real) < 1 / 2"
+    by auto
   guess d using lem1[rule_format,OF *] .. note d=this
-  def B\<equiv>"C*2"
-  have "B>0" unfolding B_def using C by auto
-  have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)"
-  proof(rule,rule) case goal1
+  def B\<equiv>"C * 2"
+  have "B > 0"
+    unfolding B_def using C by auto
+  have lem2: "\<forall>z. norm(z - y) < d \<longrightarrow> norm (g z - g y) \<le> B * norm (z - y)"
+  proof (rule, rule)
+    case goal1
     have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
-      by(rule norm_triangle_sub)
-    also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)"
-      apply(rule add_left_mono) using d and goal1 by auto
+      by (rule norm_triangle_sub)
+    also have "\<dots> \<le> norm (g' (z - y)) + 1 / 2 * norm (g z - g y)"
+      apply (rule add_left_mono)
+      using d and goal1
+      apply auto
+      done
     also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
-      apply(rule add_right_mono) using C by auto
-    finally show ?case unfolding B_def by(auto simp add:field_simps)
+      apply (rule add_right_mono)
+      using C
+      apply auto
+      done
+    finally show ?case
+      unfolding B_def
+      by (auto simp add: field_simps)
   qed
-  show ?thesis unfolding has_derivative_at_alt
-  proof(rule,rule assms,rule,rule) case goal1
-    hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
+  show ?thesis
+    unfolding has_derivative_at_alt
+    apply rule
+    apply (rule assms)
+    apply rule
+    apply rule
+  proof -
+    case goal1
+    then have *: "e / B >0"
+      apply -
+      apply (rule divide_pos_pos)
+      using `B > 0`
+      apply auto
+      done
     guess d' using lem1[rule_format,OF *] .. note d'=this
     guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
     show ?case
-      apply(rule_tac x=k in exI,rule) defer
-    proof(rule,rule)
-      fix z assume as:"norm(z - y) < k"
-      hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
+      apply (rule_tac x=k in exI)
+      apply rule
+      defer
+      apply rule
+      apply rule
+    proof -
+      fix z
+      assume as: "norm (z - y) < k"
+      then have "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
         using d' k by auto
-      also have "\<dots> \<le> e * norm(z - y)"
+      also have "\<dots> \<le> e * norm (z - y)"
         unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
-        using lem2[THEN spec[where x=z]] using k as using `e>0`
+        using lem2[THEN spec[where x=z]]
+        using k as using `e > 0`
         by (auto simp add: field_simps)
       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
-        by simp qed(insert k, auto)
+        by simp
+    qed(insert k, auto)
   qed
 qed
 
 text {* Simply rewrite that based on the domain point x. *}
 
 lemma has_derivative_inverse_basic_x:
-  fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
-  assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
-  "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
-  shows "(g has_derivative g') (at (f(x)))"
-  apply(rule has_derivative_inverse_basic) using assms by auto
+  fixes f :: "'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
+  assumes "(f has_derivative f') (at x)"
+    and "bounded_linear g'"
+    and "g' \<circ> f' = id"
+    and "continuous (at (f x)) g"
+    and "g (f x) = x"
+    and "open t"
+    and "f x \<in> t"
+    and "\<forall>y\<in>t. f (g y) = y"
+  shows "(g has_derivative g') (at (f x))"
+  apply (rule has_derivative_inverse_basic)
+  using assms
+  apply auto
+  done
 
 text {* This is the version in Dieudonne', assuming continuity of f and g. *}
 
 lemma has_derivative_inverse_dieudonne:
-  fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
-  (**) "x\<in>s" "(f has_derivative f') (at x)"  "bounded_linear g'" "g' o f' = id"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "open s"
+    and "open (f ` s)"
+    and "continuous_on s f"
+    and "continuous_on (f ` s) g"
+    and "\<forall>x\<in>s. g (f x) = x"
+    and "x \<in> s"
+    and "(f has_derivative f') (at x)"
+    and "bounded_linear g'"
+    and "g' \<circ> f' = id"
   shows "(g has_derivative g') (at (f x))"
-  apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
-  using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)]
-    continuous_on_eq_continuous_at[OF assms(2)] by auto
+  apply (rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
+  using assms(3-6)
+  unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)]
+  apply auto
+  done
 
 text {* Here's the simplest way of not assuming much about g. *}
 
 lemma has_derivative_inverse:
-  fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
-  "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "compact s"
+    and "x \<in> s"
+    and "f x \<in> interior (f ` s)"
+    and "continuous_on s f"
+    and "\<forall>y\<in>s. g (f y) = y"
+    and "(f has_derivative f') (at x)"
+    and "bounded_linear g'"
+    and "g' \<circ> f' = id"
   shows "(g has_derivative g') (at (f x))"
-proof-
-  { fix y assume "y\<in>interior (f ` s)" 
-    then obtain x where "x\<in>s" and *:"y = f x"
-      unfolding image_iff using interior_subset by auto
-    have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] ..
+proof -
+  {
+    fix y
+    assume "y \<in> interior (f ` s)"
+    then obtain x where "x \<in> s" and *: "y = f x"
+      unfolding image_iff
+      using interior_subset
+      by auto
+    have "f (g y) = y"
+      unfolding * and assms(5)[rule_format,OF `x\<in>s`] ..
   } note * = this
   show ?thesis
-    apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
-    apply(rule continuous_on_interior[OF _ assms(3)])
-    apply(rule continuous_on_inv[OF assms(4,1)])
-    apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
-    by(rule, rule *, assumption)
+    apply (rule has_derivative_inverse_basic_x[OF assms(6-8)])
+    apply (rule continuous_on_interior[OF _ assms(3)])
+    apply (rule continuous_on_inv[OF assms(4,1)])
+    apply (rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
+    apply rule
+    apply (rule *)
+    apply assumption
+    done
 qed
 
-subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
+
+subsection {* Proving surjectivity via Brouwer fixpoint theorem *}
 
 lemma brouwer_surjective:
-  fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
-  assumes "compact t" "convex t"  "t \<noteq> {}" "continuous_on t f"
-  "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+  assumes "compact t"
+    and "convex t"
+    and "t \<noteq> {}"
+    and "continuous_on t f"
+    and "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t"
+    and "x \<in> s"
   shows "\<exists>y\<in>t. f y = x"
-proof-
-  have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
-    by(auto simp add:algebra_simps)
+proof -
+  have *: "\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
+    by (auto simp add: algebra_simps)
   show ?thesis
     unfolding *
-    apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
-    apply(rule continuous_on_intros assms)+ using assms(4-6) by auto
+    apply (rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
+    apply (rule continuous_on_intros assms)+
+    using assms(4-6)
+    apply auto
+    done
 qed
 
 lemma brouwer_surjective_cball:
-  fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
-  assumes "0 < e" "continuous_on (cball a e) f"
-  "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+  assumes "e > 0"
+    and "continuous_on (cball a e) f"
+    and "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e"
+    and "x \<in> s"
   shows "\<exists>y\<in>cball a e. f y = x"
-  apply(rule brouwer_surjective)
-  apply(rule compact_cball convex_cball)+
-  unfolding cball_eq_empty using assms by auto
+  apply (rule brouwer_surjective)
+  apply (rule compact_cball convex_cball)+
+  unfolding cball_eq_empty
+  using assms
+  apply auto
+  done
 
 text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
 
 lemma sussmann_open_mapping:
-  fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
-  assumes "open s" "continuous_on s f" "x \<in> s" 
-  "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
-  "t \<subseteq> s" "x \<in> interior t"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
+  assumes "open s"
+    and "continuous_on s f"
+    and "x \<in> s"
+    and "(f has_derivative f') (at x)"
+    and "bounded_linear g'" "f' \<circ> g' = id"
+    and "t \<subseteq> s"
+    and "x \<in> interior t"
   shows "f x \<in> interior (f ` t)"
-proof- 
-  interpret f':bounded_linear f'
-    using assms unfolding has_derivative_def by auto
-  interpret g':bounded_linear g' using assms by auto
+proof -
+  interpret f': bounded_linear f'
+    using assms
+    unfolding has_derivative_def
+    by auto
+  interpret g': bounded_linear g'
+    using assms
+    by auto
   guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this
-  hence *:"1/(2*B)>0" by (auto intro!: divide_pos_pos)
+  then have *: "1 / (2 * B) > 0"
+    by (auto intro!: divide_pos_pos)
   guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
   guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
-  have *:"0<e0/B" "0<e1/B"
-    apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
+  have *: "0 < e0 / B" "0 < e1 / B"
+    apply (rule_tac[!] divide_pos_pos)
+    using e0 e1 B
+    apply auto
+    done
   guess e using real_lbound_gt_zero[OF *] .. note e=this
-  have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
-    apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
-    prefer 3 apply(rule,rule)
+  have "\<forall>z\<in>cball (f x) (e / 2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
+    apply rule
+    apply (rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
+    prefer 3
+    apply rule
+    apply rule
   proof-
     show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
       unfolding g'.diff
-      apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
-      apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
-      apply(rule continuous_on_subset[OF assms(2)])
-      apply(rule,unfold image_iff,erule bexE)
+      apply (rule continuous_on_compose[of _ _ f, unfolded o_def])
+      apply (rule continuous_on_intros linear_continuous_on[OF assms(5)])+
+      apply (rule continuous_on_subset[OF assms(2)])
+      apply rule
+      apply (unfold image_iff)
+      apply (erule bexE)
     proof-
-      fix y z assume as:"y \<in>cball (f x) e"  "z = x + (g' y - g' (f x))"
+      fix y z
+      assume as: "y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
       have "dist x z = norm (g' (f x) - g' y)"
         unfolding as(2) and dist_norm by auto
       also have "\<dots> \<le> norm (f x - y) * B"
-        unfolding g'.diff[THEN sym] using B by auto
+        unfolding g'.diff[symmetric]
+        using B
+        by auto
       also have "\<dots> \<le> e * B"
-        using as(1)[unfolded mem_cball dist_norm] using B by auto
-      also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
-      finally have "z\<in>cball x e1" unfolding mem_cball by force
-      thus "z \<in> s" using e1 assms(7) by auto
+        using as(1)[unfolded mem_cball dist_norm]
+        using B
+        by auto
+      also have "\<dots> \<le> e1"
+        using e
+        unfolding less_divide_eq
+        using B
+        by auto
+      finally have "z \<in> cball x e1"
+        unfolding mem_cball
+        by force
+      then show "z \<in> s"
+        using e1 assms(7) by auto
     qed
   next
-    fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
-    have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
-    also have "\<dots> \<le> e * B" apply(rule mult_right_mono)
+    fix y z
+    assume as: "y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
+    have "norm (g' (z - f x)) \<le> norm (z - f x) * B"
+      using B by auto
+    also have "\<dots> \<le> e * B"
+      apply (rule mult_right_mono)
       using as(2)[unfolded mem_cball dist_norm] and B
-      unfolding norm_minus_commute by auto
-    also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
-    finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
-    have **:"f x + f' (x + g' (z - f x) - x) = z"
-      using assms(6)[unfolded o_def id_def,THEN cong] by auto
-    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)"
+      unfolding norm_minus_commute
+      apply auto
+      done
+    also have "\<dots> < e0"
+      using e and B
+      unfolding less_divide_eq
+      by auto
+    finally have *: "norm (x + g' (z - f x) - x) < e0"
+      by auto
+    have **: "f x + f' (x + g' (z - f x) - x) = z"
+      using assms(6)[unfolded o_def id_def,THEN cong]
+      by auto
+    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)"
       using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"]
       by (auto simp add: algebra_simps)
     also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)"
       using e0[THEN conjunct2,rule_format,OF *]
-      unfolding algebra_simps ** by auto
+      unfolding algebra_simps **
+      by auto
     also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2"
-      using as(1)[unfolded mem_cball dist_norm] by auto
+      using as(1)[unfolded mem_cball dist_norm]
+      by auto
     also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2"
-      using * and B by (auto simp add: field_simps)
-    also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
-    also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono)
+      using * and B
+      by (auto simp add: field_simps)
+    also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2"
+      by auto
+    also have "\<dots> \<le> e/2 + e/2"
+      apply (rule add_right_mono)
       using as(2)[unfolded mem_cball dist_norm]
-      unfolding norm_minus_commute by auto
+      unfolding norm_minus_commute
+      apply auto
+      done
     finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
-      unfolding mem_cball dist_norm by auto
-  qed(insert e, auto) note lem = this
-  show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
-    apply(rule,rule divide_pos_pos) prefer 3
+      unfolding mem_cball dist_norm
+      by auto
+  qed (insert e, auto) note lem = this
+  show ?thesis
+    unfolding mem_interior
+    apply (rule_tac x="e/2" in exI)
+    apply rule
+    apply (rule divide_pos_pos)
+    prefer 3
   proof
-    fix y assume "y \<in> ball (f x) (e/2)"
-    hence *:"y\<in>cball (f x) (e/2)" by auto
+    fix y
+    assume "y \<in> ball (f x) (e / 2)"
+    then have *: "y \<in> cball (f x) (e / 2)"
+      by auto
     guess z using lem[rule_format,OF *] .. note z=this
-    hence "norm (g' (z - f x)) \<le> norm (z - f x) * B"
-      using B by (auto simp add: field_simps)
+    then have "norm (g' (z - f x)) \<le> norm (z - f x) * B"
+      using B
+      by (auto simp add: field_simps)
     also have "\<dots> \<le> e * B"
-      apply (rule mult_right_mono) using z(1)
-      unfolding mem_cball dist_norm norm_minus_commute using B by auto
-    also have "\<dots> \<le> e1"  using e B unfolding less_divide_eq by auto
-    finally have "x + g'(z - f x) \<in> t" apply-
-      apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
-      unfolding mem_cball dist_norm by auto
-    thus "y \<in> f ` t" using z by auto
-  qed(insert e, auto)
+      apply (rule mult_right_mono)
+      using z(1)
+      unfolding mem_cball dist_norm norm_minus_commute
+      using B
+      apply auto
+      done
+    also have "\<dots> \<le> e1"
+      using e B unfolding less_divide_eq by auto
+    finally have "x + g'(z - f x) \<in> t"
+      apply -
+      apply (rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
+      unfolding mem_cball dist_norm
+      apply auto
+      done
+    then show "y \<in> f ` t"
+      using z by auto
+  qed (insert e, auto)
 qed
 
 text {* Hence the following eccentric variant of the inverse function theorem.    *)
 (* This has no continuity assumptions, but we do need the inverse function.  *)
-(* We could put f' o g = I but this happens to fit with the minimal linear   *)
+(* We could put f' \<circ> g = I but this happens to fit with the minimal linear   *)
 (* algebra theory I've set up so far. *}
 
 (* move  before left_inverse_linear in Euclidean_Space*)
 
- lemma right_inverse_linear:
-   fixes f::"'a::euclidean_space => 'a"
-   assumes lf: "linear f" and gf: "f o g = id"
-   shows "linear g"
- proof-
-   from gf have fi: "surj f" by (auto simp add: surj_def o_def id_def) metis
-   from linear_surjective_isomorphism[OF lf fi]
-   obtain h:: "'a => 'a" where
-     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
-   have "h = g" apply (rule ext) using gf h(2,3)
-     by (simp add: o_def id_def fun_eq_iff) metis
-   with h(1) show ?thesis by blast
- qed
- 
+lemma right_inverse_linear:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
+  assumes lf: "linear f"
+    and gf: "f \<circ> g = id"
+  shows "linear g"
+proof -
+  from gf have fi: "surj f"
+    by (auto simp add: surj_def o_def id_def) metis
+  from linear_surjective_isomorphism[OF lf fi]
+  obtain h:: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
+    by blast
+  have "h = g"
+    apply (rule ext)
+    using gf h(2,3)
+    apply (simp add: o_def id_def fun_eq_iff)
+    apply metis
+    done
+  with h(1) show ?thesis by blast
+qed
+
 lemma has_derivative_inverse_strong:
-  fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
-  assumes "open s" and "x \<in> s" and "continuous_on s f"
-  assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" and "f' o g' = id"
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+  assumes "open s"
+    and "x \<in> s"
+    and "continuous_on s f"
+    and "\<forall>x\<in>s. g (f x) = x"
+    and "(f has_derivative f') (at x)"
+    and "f' \<circ> g' = id"
   shows "(g has_derivative g') (at (f x))"
-proof-
-  have linf:"bounded_linear f'"
+proof -
+  have linf: "bounded_linear f'"
     using assms(5) unfolding has_derivative_def by auto
-  hence ling:"bounded_linear g'"
-    unfolding linear_conv_bounded_linear[THEN sym]
-    apply- apply(rule right_inverse_linear) using assms(6) by auto
-  moreover have "g' \<circ> f' = id" using assms(6) linf ling
-    unfolding linear_conv_bounded_linear[THEN sym]
-    using linear_inverse_left by auto
-  moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)"
-    apply(rule,rule,rule,rule sussmann_open_mapping )
-    apply(rule assms ling)+ by auto
-  have "continuous (at (f x)) g" unfolding continuous_at Lim_at
-  proof(rule,rule)
-    fix e::real assume "e>0"
-    hence "f x \<in> interior (f ` (ball x e \<inter> s))"
-      using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
-      by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
+  then have ling: "bounded_linear g'"
+    unfolding linear_conv_bounded_linear[symmetric]
+    apply -
+    apply (rule right_inverse_linear)
+    using assms(6)
+    apply auto
+    done
+  moreover have "g' \<circ> f' = id"
+    using assms(6) linf ling
+    unfolding linear_conv_bounded_linear[symmetric]
+    using linear_inverse_left
+    by auto
+  moreover have *:"\<forall>t\<subseteq>s. x \<in> interior t \<longrightarrow> f x \<in> interior (f ` t)"
+    apply rule
+    apply rule
+    apply rule
+    apply (rule sussmann_open_mapping)
+    apply (rule assms ling)+
+    apply auto
+    done
+  have "continuous (at (f x)) g"
+    unfolding continuous_at Lim_at
+  proof (rule, rule)
+    fix e :: real
+    assume "e > 0"
+    then have "f x \<in> interior (f ` (ball x e \<inter> s))"
+      using *[rule_format,of "ball x e \<inter> s"] `x \<in> s`
+      by (auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
     then guess d unfolding mem_interior .. note d=this
     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"
-      apply(rule_tac x=d in exI)
-      apply(rule,rule d[THEN conjunct1])
-    proof(rule,rule) case goal1
-      hence "g y \<in> g ` f ` (ball x e \<inter> s)"
+      apply (rule_tac x=d in exI)
+      apply rule
+      apply (rule d[THEN conjunct1])
+      apply rule
+      apply rule
+    proof -
+      case goal1
+      then have "g y \<in> g ` f ` (ball x e \<inter> s)"
         using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
-        by(auto simp add:dist_commute)
-      hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
-      thus "dist (g y) (g (f x)) < e"
-        using assms(4)[rule_format,OF `x\<in>s`]
+        by (auto simp add: dist_commute)
+      then have "g y \<in> ball x e \<inter> s"
+        using assms(4) by auto
+      then show "dist (g y) (g (f x)) < e"
+        using assms(4)[rule_format,OF `x \<in> s`]
         by (auto simp add: dist_commute)
     qed
   qed
   moreover have "f x \<in> interior (f ` s)"
-    apply(rule sussmann_open_mapping)
-    apply(rule assms ling)+
-    using interior_open[OF assms(1)] and `x\<in>s` by auto
+    apply (rule sussmann_open_mapping)
+    apply (rule assms ling)+
+    using interior_open[OF assms(1)] and `x \<in> s`
+    apply auto
+    done
   moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y"
-  proof- case goal1
-    hence "y\<in>f ` s" using interior_subset by auto
+  proof -
+    case goal1
+    then have "y \<in> f ` s"
+      using interior_subset by auto
     then guess z unfolding image_iff ..
-    thus ?case using assms(4) by auto
+    then show ?case
+      using assms(4) by auto
   qed
   ultimately show ?thesis
-    apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)])
-    using assms by auto
+    apply -
+    apply (rule has_derivative_inverse_basic_x[OF assms(5)])
+    using assms
+    apply auto
+    done
 qed
 
 text {* A rewrite based on the other domain. *}
 
 lemma has_derivative_inverse_strong_x:
-  fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
-  assumes "open s" and "g y \<in> s" and "continuous_on s f"
-  assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))"
-  assumes "f' o g' = id" and "f(g y) = y"
+  fixes f :: "'a::ordered_euclidean_space \<Rightarrow> 'a"
+  assumes "open s"
+    and "g y \<in> s"
+    and "continuous_on s f"
+    and "\<forall>x\<in>s. g (f x) = x"
+    and "(f has_derivative f') (at (g y))"
+    and "f' \<circ> g' = id"
+    and "f (g y) = y"
   shows "(g has_derivative g') (at y)"
-  using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
+  using has_derivative_inverse_strong[OF assms(1-6)]
+  unfolding assms(7)
+  by simp
 
 text {* On a region. *}
 
 lemma has_derivative_inverse_on:
-  fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
-  assumes "open s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
-  assumes "\<forall>x\<in>s. g(f x) = x" and "f'(x) o g'(x) = id" and "x\<in>s"
+  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'n"
+  assumes "open s"
+    and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
+    and "\<forall>x\<in>s. g (f x) = x"
+    and "f' x \<circ> g' x = id"
+    and "x \<in> s"
   shows "(g has_derivative g'(x)) (at (f x))"
-  apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f])
-  apply(rule assms)+
+  apply (rule has_derivative_inverse_strong[where g'="g' x" and f=f])
+  apply (rule assms)+
   unfolding continuous_on_eq_continuous_at[OF assms(1)]
-  apply(rule,rule differentiable_imp_continuous_within)
-  unfolding differentiable_def using assms by auto
+  apply rule
+  apply (rule differentiable_imp_continuous_within)
+  unfolding differentiable_def
+  using assms
+  apply auto
+  done
 
 text {* Invertible derivative continous at a point implies local
 injectivity. It's only for this we need continuity of the derivative,
@@ -1057,269 +1589,381 @@
 also continuous. So if we know for some other reason that the inverse
 function exists, it's OK. *}
 
-lemma bounded_linear_sub:
-  "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
+lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_linear (\<lambda>x. f x - g x)"
   using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g]
   by (auto simp add: algebra_simps)
 
 lemma has_derivative_locally_injective:
-  fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
-  assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
-  "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
-  "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
-  obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)"
-proof-
-  interpret bounded_linear g' using assms by auto
+  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
+  assumes "a \<in> s"
+    and "open s"
+    and "bounded_linear g'"
+    and "g' \<circ> f' a = id"
+    and "\<forall>x\<in>s. (f has_derivative f' x) (at x)"
+    and "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e"
+  obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. f x' = f x \<longrightarrow> x' = x"
+proof -
+  interpret bounded_linear g'
+    using assms by auto
   note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
-  have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)" defer 
-    apply(subst euclidean_eq_iff) using f'g' by auto
-  hence *:"0 < onorm g'"
-    unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastforce
-  def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
+  have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)"
+    defer
+    apply (subst euclidean_eq_iff)
+    using f'g'
+    apply auto
+    done
+  then have *: "0 < onorm g'"
+    unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]]
+    by fastforce
+  def k \<equiv> "1 / onorm g' / 2"
+  have *: "k > 0"
+    unfolding k_def using * by auto
   guess d1 using assms(6)[rule_format,OF *] .. note d1=this
-  from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
-  obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
+  from `open s` obtain d2 where "d2 > 0" "ball a d2 \<subseteq> s"
+    using `a\<in>s` ..
+  obtain d2 where "d2 > 0" "ball a d2 \<subseteq> s"
+    using assms(2,1) ..
   guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] ..
   note d2=this
   guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] ..
   note d = this
   show ?thesis
   proof
-    show "a\<in>ball a d" using d by auto
+    show "a \<in> ball a d"
+      using d by auto
     show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x"
     proof (intro strip)
-      fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
-      def ph \<equiv> "\<lambda>w. w - g'(f w - f x)"
+      fix x y
+      assume as: "x \<in> ball a d" "y \<in> ball a d" "f x = f y"
+      def ph \<equiv> "\<lambda>w. w - g' (f w - f x)"
       have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
-        unfolding ph_def o_def unfolding diff using f'g'
+        unfolding ph_def o_def
+        unfolding diff
+        using f'g'
         by (auto simp add: algebra_simps)
-      have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
-        apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
-        apply(rule_tac[!] ballI)
-      proof-
-        fix u assume u:"u \<in> ball a d"
-        hence "u\<in>s" using d d2 by auto
-        have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
-          unfolding o_def and diff using f'g' by auto
+      have "norm (ph x - ph y) \<le> (1 / 2) * norm (x - y)"
+        apply (rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
+        apply (rule_tac[!] ballI)
+      proof -
+        fix u
+        assume u: "u \<in> ball a d"
+        then have "u \<in> s"
+          using d d2 by auto
+        have *: "(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
+          unfolding o_def and diff
+          using f'g' by auto
         show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
           unfolding ph' *
-          apply(simp add: comp_def)
-          apply(rule bounded_linear.FDERIV[OF assms(3)])
-          apply(rule FDERIV_intros) defer
-          apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
-          apply(rule has_derivative_at_within)
-          using assms(5) and `u\<in>s` `a\<in>s`
+          apply (simp add: comp_def)
+          apply (rule bounded_linear.FDERIV[OF assms(3)])
+          apply (rule FDERIV_intros)
+          defer
+          apply (rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
+          apply (rule has_derivative_at_within)
+          using assms(5) and `u \<in> s` `a \<in> s`
           apply (auto intro!: FDERIV_intros bounded_linear.FDERIV[of _ "\<lambda>x. x"] derivative_linear)
           done
-        have **:"bounded_linear (\<lambda>x. f' u x - f' a x)"
-          "bounded_linear (\<lambda>x. f' a x - f' u x)"
-          apply(rule_tac[!] bounded_linear_sub)
-          apply(rule_tac[!] derivative_linear)
-          using assms(5) `u\<in>s` `a\<in>s` by auto
+        have **: "bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)"
+          apply (rule_tac[!] bounded_linear_sub)
+          apply (rule_tac[!] derivative_linear)
+          using assms(5) `u \<in> s` `a \<in> s`
+          apply auto
+          done
         have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)"
-          unfolding * apply(rule onorm_compose)
-          unfolding linear_conv_bounded_linear by(rule assms(3) **)+
+          unfolding *
+          apply (rule onorm_compose)
+          unfolding linear_conv_bounded_linear
+          apply (rule assms(3) **)+
+          done
         also have "\<dots> \<le> onorm g' * k"
-          apply(rule mult_left_mono) 
+          apply (rule mult_left_mono)
           using d1[THEN conjunct2,rule_format,of u]
           using onorm_neg[OF **(1)[unfolded linear_linear]]
           using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]]
-          by (auto simp add: algebra_simps)
-        also have "\<dots> \<le> 1/2" unfolding k_def by auto
-        finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption
+          apply (auto simp add: algebra_simps)
+          done
+        also have "\<dots> \<le> 1 / 2"
+          unfolding k_def by auto
+        finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" .
       qed
       moreover have "norm (ph y - ph x) = norm (y - x)"
-        apply(rule arg_cong[where f=norm])
-        unfolding ph_def using diff unfolding as by auto
-      ultimately show "x = y" unfolding norm_minus_commute by auto
+        apply (rule arg_cong[where f=norm])
+        unfolding ph_def
+        using diff
+        unfolding as
+        apply auto
+        done
+      ultimately show "x = y"
+        unfolding norm_minus_commute by auto
     qed
   qed auto
 qed
 
-subsection {* Uniformly convergent sequence of derivatives. *}
+
+subsection {* Uniformly convergent sequence of derivatives *}
 
 lemma has_derivative_sequence_lipschitz_lemma:
-  fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+  fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "convex s"
-  assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-  assumes "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
-  shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
-proof (default)+
-  fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
-  show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
-    apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)])
-    apply(rule_tac[!] ballI)
-  proof-
-    fix x assume "x\<in>s"
+    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+    and "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
+  shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm ((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm (x - y)"
+proof rule+
+  fix m n x y
+  assume as: "N \<le> m" "N \<le> n" "x \<in> s" "y \<in> s"
+  show "norm ((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm (x - y)"
+    apply (rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)])
+    apply (rule_tac[!] ballI)
+  proof -
+    fix x
+    assume "x \<in> s"
     show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within s)"
-      by(rule FDERIV_intros assms(2)[rule_format] `x\<in>s`)+
-    { fix h
+      by (rule FDERIV_intros assms(2)[rule_format] `x\<in>s`)+
+    {
+      fix h
       have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
         using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"]
-        unfolding norm_minus_commute by (auto simp add: algebra_simps)
-      also have "\<dots> \<le> e * norm h+ e * norm h"
-        using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h]
-        using assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
-        by(auto simp add:field_simps)
-      finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
-    thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
-      apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
+        unfolding norm_minus_commute
+        by (auto simp add: algebra_simps)
+      also have "\<dots> \<le> e * norm h + e * norm h"
+        using assms(3)[rule_format,OF `N \<le> m` `x \<in> s`, of h]
+        using assms(3)[rule_format,OF `N \<le> n` `x \<in> s`, of h]
+        by (auto simp add: field_simps)
+      finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h"
+        by auto
+    }
+    then show "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
+      apply -
+      apply (rule onorm(2))
+      apply (rule linear_compose_sub)
       unfolding linear_conv_bounded_linear
-      using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear]
-      by auto
+      using assms(2)[rule_format,OF `x \<in> s`, THEN derivative_linear]
+      apply auto
+      done
   qed
 qed
 
 lemma has_derivative_sequence_lipschitz:
-  fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+  fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "convex s"
-  assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-  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)"
-  assumes "0 < e"
-  shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
-proof(rule,rule)
-  case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
+    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+    and "\<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"
+    and "e > 0"
+  shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s.
+    norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
+proof (rule, rule)
+  case goal1 have *: "2 * (1/2* e) = e" "1/2 * e >0"
+    using `e > 0` by auto
   guess N using assms(3)[rule_format,OF *(2)] ..
-  thus ?case
-    apply(rule_tac x=N in exI)
-    apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
-    using assms by auto
+  then show ?case
+    apply (rule_tac x=N in exI)
+    apply (rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
+    using assms
+    apply auto
+    done
 qed
 
 lemma has_derivative_sequence:
   fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "convex s"
-  assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-  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)"
-  assumes "x0 \<in> s" and "((\<lambda>n. f n x0) ---> l) sequentially"
-  shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and>
-    (g has_derivative g'(x)) (at x within s)"
-proof-
-  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)"
-    apply(rule has_derivative_sequence_lipschitz[where e="42::nat"])
-    apply(rule assms)+ by auto
+    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+    and "\<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"
+    and "x0 \<in> s"
+    and "((\<lambda>n. f n x0) ---> l) sequentially"
+  shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)"
+proof -
+  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)"
+    apply (rule has_derivative_sequence_lipschitz[where e="42::nat"])
+    apply (rule assms)+
+    apply auto
+    done
   have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially"
-    apply(rule bchoice) unfolding convergent_eq_cauchy
+    apply (rule bchoice)
+    unfolding convergent_eq_cauchy
   proof
-    fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)"
-    proof(cases "x=x0")
-      case True thus ?thesis using LIMSEQ_imp_Cauchy[OF assms(5)] by auto
+    fix x
+    assume "x \<in> s"
+    show "Cauchy (\<lambda>n. f n x)"
+    proof (cases "x = x0")
+      case True
+      then show ?thesis
+        using LIMSEQ_imp_Cauchy[OF assms(5)] by auto
     next
-      case False show ?thesis unfolding Cauchy_def
-      proof(rule,rule)
-        fix e::real assume "e>0"
-        hence *:"e/2>0" "e/2/norm(x-x0)>0"
+      case False
+      show ?thesis
+        unfolding Cauchy_def
+      proof (rule, rule)
+        fix e :: real
+        assume "e > 0"
+        then have *: "e / 2 > 0" "e / 2 / norm (x - x0) > 0"
           using False by (auto intro!: divide_pos_pos)
         guess M using LIMSEQ_imp_Cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
         guess N using lem1[rule_format,OF *(2)] .. note N = this
         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
-          apply(rule_tac x="max M N" in exI)
-        proof(default+)
-          fix m n assume as:"max M N \<le>m" "max M N\<le>n"
-          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))"
-            unfolding dist_norm by(rule norm_triangle_sub)
+          apply (rule_tac x="max M N" in exI)
+        proof rule+
+          fix m n
+          assume as: "max M N \<le>m" "max M N\<le>n"
+          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))"
+            unfolding dist_norm
+            by (rule norm_triangle_sub)
           also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
             using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False
             by auto
           also have "\<dots> < e / 2 + e / 2"
-            apply(rule add_strict_right_mono)
-            using as and M[rule_format] unfolding dist_norm by auto
-          finally show "dist (f m x) (f n x) < e" by auto
+            apply (rule add_strict_right_mono)
+            using as and M[rule_format]
+            unfolding dist_norm
+            apply auto
+            done
+          finally show "dist (f m x) (f n x) < e"
+            by auto
         qed
       qed
     qed
   qed
   then guess g .. note g = this
-  have lem2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f n x - f n y) - (g x - g y)) \<le> e * norm(x - y)"
-  proof(rule,rule)
-    fix e::real assume *:"e>0"
+  have lem2: "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm ((f n x - f n y) - (g x - g y)) \<le> e * norm (x - y)"
+  proof (rule, rule)
+    fix e :: real
+    assume *: "e > 0"
     guess N using lem1[rule_format,OF *] .. note N=this
     show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
-      apply(rule_tac x=N in exI)
-    proof(default+)
-      fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
+      apply (rule_tac x=N in exI)
+    proof rule+
+      fix n x y
+      assume as: "N \<le> n" "x \<in> s" "y \<in> s"
       have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
         unfolding eventually_sequentially
-        apply(rule_tac x=N in exI)
-      proof(rule,rule)
-        fix m assume "N\<le>m"
-        thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
+        apply (rule_tac x=N in exI)
+        apply rule
+        apply rule
+      proof -
+        fix m
+        assume "N \<le> m"
+        then show "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
           using N[rule_format, of n m x y] and as
           by (auto simp add: algebra_simps)
       qed
-      thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
-        apply-
-        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)"])
-        apply(rule tendsto_intros g[rule_format] as)+ by assumption
+      then show "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
+        apply -
+        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)"])
+        apply (rule tendsto_intros g[rule_format] as)+
+        apply assumption
+        done
     qed
   qed
-  show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
-    apply(rule,rule,rule g[rule_format],assumption)
-  proof fix x assume "x\<in>s"
-    have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
+  show ?thesis
+    unfolding has_derivative_within_alt
+    apply (rule_tac x=g in exI)
+    apply rule
+    apply rule
+    apply (rule g[rule_format])
+    apply assumption
+  proof
+    fix x
+    assume "x \<in> s"
+    have lem3: "\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
       unfolding LIMSEQ_def
-    proof(rule,rule,rule)
-      fix u and e::real assume "e>0"
+    proof (rule, rule, rule)
+      fix u
+      fix e :: real
+      assume "e > 0"
       show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e"
-      proof(cases "u=0")
-        case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
-        show ?thesis apply(rule_tac x=N in exI) unfolding True 
-          using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto
+      proof (cases "u = 0")
+        case True
+        guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
+        show ?thesis
+          apply (rule_tac x=N in exI)
+          unfolding True
+          using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0`
+          apply auto
+          done
       next
-        case False hence *:"e / 2 / norm u > 0"
-          using `e>0` by (auto intro!: divide_pos_pos)
+        case False
+        then have *: "e / 2 / norm u > 0"
+          using `e > 0`
+          by (auto intro!: divide_pos_pos)
         guess N using assms(3)[rule_format,OF *] .. note N=this
-        show ?thesis apply(rule_tac x=N in exI)
-        proof(rule,rule) case goal1
-          show ?case unfolding dist_norm
+        show ?thesis
+          apply (rule_tac x=N in exI)
+          apply rule
+          apply rule
+        proof -
+          case goal1
+          show ?case
+            unfolding dist_norm
             using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
-            by (auto simp add:field_simps)
+            by (auto simp add: field_simps)
         qed
       qed
     qed
     show "bounded_linear (g' x)"
       unfolding linear_linear linear_iff
-      apply(rule,rule,rule) defer
-    proof(rule,rule)
-      fix x' y z::"'m" and c::real
+      apply rule
+      apply rule
+      apply rule
+      defer
+      apply rule
+      apply rule
+    proof -
+      fix x' y z :: 'm
+      fix c :: real
       note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
       show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
-        apply(rule tendsto_unique[OF trivial_limit_sequentially])
-        apply(rule lem3[rule_format])
+        apply (rule tendsto_unique[OF trivial_limit_sequentially])
+        apply (rule lem3[rule_format])
         unfolding lin[THEN bounded_linear_imp_linear, THEN linear_cmul]
-        apply (intro tendsto_intros) by(rule lem3[rule_format])
+        apply (intro tendsto_intros)
+        apply (rule lem3[rule_format])
+        done
       show "g' x (y + z) = g' x y + g' x z"
-        apply(rule tendsto_unique[OF trivial_limit_sequentially])
-        apply(rule lem3[rule_format])
+        apply (rule tendsto_unique[OF trivial_limit_sequentially])
+        apply (rule lem3[rule_format])
         unfolding lin[THEN bounded_linear_imp_linear, THEN linear_add]
-        apply(rule tendsto_add) by(rule lem3[rule_format])+
+        apply (rule tendsto_add)
+        apply (rule lem3[rule_format])+
+        done
     qed
     show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
-    proof(rule,rule) case goal1
-      have *:"e/3>0" using goal1 by auto
+    proof (rule, rule)
+      case goal1
+      have *: "e / 3 > 0"
+        using goal1 by auto
       guess N1 using assms(3)[rule_format,OF *] .. note N1=this
       guess N2 using lem2[rule_format,OF *] .. note N2=this
       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
-      show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1])
-      proof(rule,rule)
-        fix y assume as:"y \<in> s" "norm (y - x) < d1"
-        let ?N ="max N1 N2"
+      show ?case
+        apply (rule_tac x=d1 in exI)
+        apply rule
+        apply (rule d1[THEN conjunct1])
+        apply rule
+        apply rule
+      proof -
+        fix y
+        assume as: "y \<in> s" "norm (y - x) < d1"
+        let ?N = "max N1 N2"
         have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)"
-          apply(subst norm_minus_cancel[THEN sym])
-          using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto
+          apply (subst norm_minus_cancel[symmetric])
+          using N2[rule_format, OF _ `y \<in> s` `x \<in> s`, of ?N]
+          apply auto
+          done
         moreover
         have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
-          using d1 and as by auto
+          using d1 and as
+          by auto
         ultimately
-        have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)" 
+        have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
           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)"]
-          by (auto simp add:algebra_simps)
+          by (auto simp add: algebra_simps)
         moreover
         have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
-          using N1 `x\<in>s` by auto
+          using N1 `x \<in> s` by auto
         ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
           using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"]
-          by(auto simp add:algebra_simps)
+          by (auto simp add: algebra_simps)
       qed
     qed
   qed
@@ -1328,122 +1972,172 @@
 text {* Can choose to line up antiderivatives if we want. *}
 
 lemma has_antiderivative_sequence:
-  fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+  fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "convex s"
-  assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-  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"
-  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
-proof(cases "s={}")
-  case False then obtain a where "a\<in>s" by auto
-  have *:"\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x" by auto
+    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+    and "\<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"
+  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g' x) (at x within s)"
+proof (cases "s = {}")
+  case False
+  then obtain a where "a \<in> s"
+    by auto
+  have *: "\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x"
+    by auto
   show ?thesis
-    apply(rule *)
-    apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
-    apply(rule,rule)
-    apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)  
-    apply(rule `a\<in>s`) by auto
+    apply (rule *)
+    apply (rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
+    apply rule
+    apply rule
+    apply (rule has_derivative_add_const, rule assms(2)[rule_format])
+    apply assumption
+    apply (rule `a \<in> s`)
+    apply auto
+    done
 qed auto
 
 lemma has_antiderivative_limit:
-  fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
+  fixes g' :: "'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
   assumes "convex s"
-  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))"
-  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
-proof-
-  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))"
-    apply(rule) using assms(2)
-    apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
+    and "\<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)"
+  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g' x) (at x within s)"
+proof -
+  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)"
+    apply rule
+    using assms(2)
+    apply (erule_tac x="inverse (real (Suc n))" in allE)
+    apply auto
+    done
   guess f using *[THEN choice] .. note * = this
-  guess f' using *[THEN choice] .. note f=this
-  show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer
-  proof(rule,rule)
-    fix e::real assume "0<e" guess  N using reals_Archimedean[OF `e>0`] .. note N=this 
+  guess f' using *[THEN choice] .. note f = this
+  show ?thesis
+    apply (rule has_antiderivative_sequence[OF assms(1), of f f'])
+    defer
+    apply rule
+    apply rule
+  proof -
+    fix e :: real
+    assume "e > 0"
+    guess  N using reals_Archimedean[OF `e>0`] .. note N=this
     show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
-      apply(rule_tac x=N in exI)
-    proof(default+)
+      apply (rule_tac x=N in exI)
+    proof rule+
       case goal1
-      have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
-        using goal1(1) by(auto simp add:field_simps) 
+      have *: "inverse (real (Suc n)) \<le> e"
+        apply (rule order_trans[OF _ N[THEN less_imp_le]])
+        using goal1(1)
+        apply (auto simp add: field_simps)
+        done
       show ?case
-        using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]] 
-        apply(rule order_trans) using N * apply(cases "h=0") by auto
+        using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
+        apply (rule order_trans)
+        using N *
+        apply (cases "h = 0")
+        apply auto
+        done
     qed
-  qed(insert f,auto)
+  qed (insert f, auto)
 qed
 
-subsection {* Differentiation of a series. *}
+
+subsection {* Differentiation of a series *}
 
-definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
-(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
+definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> nat set \<Rightarrow> bool"
+    (infixl "sums'_seq" 12)
+  where "(f sums_seq l) s \<longleftrightarrow> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
 
 lemma has_derivative_series:
-  fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+  fixes f :: "nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "convex s"
-  assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-  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)"
-  assumes "x\<in>s" and "((\<lambda>n. f n x) sums_seq l) k"
-  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)"
+    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+    and "\<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"
+    and "x \<in> s"
+    and "((\<lambda>n. f n x) sums_seq l) k"
+  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)"
   unfolding sums_seq_def
-  apply(rule has_derivative_sequence[OF assms(1) _ assms(3)])
-  apply(rule, rule)
-  apply(rule has_derivative_setsum)
-  apply(rule assms(2)[rule_format])
+  apply (rule has_derivative_sequence[OF assms(1) _ assms(3)])
+  apply rule
+  apply rule
+  apply (rule has_derivative_setsum)
+  apply (rule assms(2)[rule_format])
   apply assumption
-  using assms(4-5) unfolding sums_seq_def by auto
+  using assms(4-5)
+  unfolding sums_seq_def
+  apply auto
+  done
 
-subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
+
+text {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
 
-definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real filter \<Rightarrow> bool)"
-(infixl "has'_vector'_derivative" 12) where
- "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
+definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
+    (infixl "has'_vector'_derivative" 12)
+  where "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
 
-definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
+definition "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"
 
 lemma vector_derivative_works:
-  fixes f::"real \<Rightarrow> 'a::real_normed_vector"
-  shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
+  fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
+  shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net"
+    (is "?l = ?r")
 proof
-  assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
-  then interpret bounded_linear f' by auto
-  show ?r unfolding vector_derivative_def has_vector_derivative_def
-    apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
-    using f' unfolding scaleR[THEN sym] by auto
+  assume ?l
+  guess f' using `?l`[unfolded differentiable_def] .. note f' = this
+  then interpret bounded_linear f'
+    by auto
+  show ?r
+    unfolding vector_derivative_def has_vector_derivative_def
+    apply -
+    apply (rule someI_ex,rule_tac x="f' 1" in exI)
+    using f'
+    unfolding scaleR[symmetric]
+    apply auto
+    done
 next
-  assume ?r thus ?l
+  assume ?r
+  then show ?l
     unfolding vector_derivative_def has_vector_derivative_def differentiable_def
     by auto
 qed
 
 lemma has_vector_derivative_withinI_DERIV:
-  assumes f: "DERIV f x :> y" shows "(f has_vector_derivative y) (at x within s)"
+  assumes f: "DERIV f x :> y"
+  shows "(f has_vector_derivative y) (at x within s)"
   unfolding has_vector_derivative_def real_scaleR_def
   apply (rule has_derivative_at_within)
   using DERIV_conv_has_derivative[THEN iffD1, OF f]
-  apply (subst mult_commute) .
+  apply (subst mult_commute)
+  apply assumption
+  done
 
 lemma vector_derivative_unique_at:
   assumes "(f has_vector_derivative f') (at x)"
-  assumes "(f has_vector_derivative f'') (at x)"
+    and "(f has_vector_derivative f'') (at x)"
   shows "f' = f''"
-proof-
+proof -
   have "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
     using assms [unfolded has_vector_derivative_def]
     by (rule frechet_derivative_unique_at)
-  thus ?thesis unfolding fun_eq_iff by auto
+  then show ?thesis
+    unfolding fun_eq_iff by auto
 qed
 
 lemma vector_derivative_unique_within_closed_interval:
-  assumes "a < b" and "x \<in> {a..b}"
+  assumes "a < b"
+    and "x \<in> {a..b}"
   assumes "(f has_vector_derivative f') (at x within {a..b})"
   assumes "(f has_vector_derivative f'') (at x within {a..b})"
   shows "f' = f''"
-proof-
-  have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
-    apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
-    using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2)
-    by auto
+proof -
+  have *: "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
+    apply (rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
+    using assms(3-)[unfolded has_vector_derivative_def]
+    using assms(1-2)
+    apply auto
+    done
   show ?thesis
-  proof(rule ccontr)
+  proof (rule ccontr)
     assume **: "f' \<noteq> f''"
     with * have "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1"
       by (auto simp: fun_eq_iff)
@@ -1453,76 +2147,106 @@
 qed
 
 lemma vector_derivative_at:
-  shows "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
-  apply(rule vector_derivative_unique_at) defer apply assumption
-  unfolding vector_derivative_works[THEN sym] differentiable_def
-  unfolding has_vector_derivative_def by auto
+  "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
+  apply (rule vector_derivative_unique_at)
+  defer
+  apply assumption
+  unfolding vector_derivative_works[symmetric] differentiable_def
+  unfolding has_vector_derivative_def
+  apply auto
+  done
 
 lemma vector_derivative_within_closed_interval:
-  assumes "a < b" and "x \<in> {a..b}"
+  assumes "a < b"
+    and "x \<in> {a..b}"
   assumes "(f has_vector_derivative f') (at x within {a..b})"
   shows "vector_derivative f (at x within {a..b}) = f'"
-  apply(rule vector_derivative_unique_within_closed_interval)
+  apply (rule vector_derivative_unique_within_closed_interval)
   using vector_derivative_works[unfolded differentiable_def]
-  using assms by(auto simp add:has_vector_derivative_def)
+  using assms
+  apply (auto simp add:has_vector_derivative_def)
+  done
 
-lemma has_vector_derivative_within_subset: 
- "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
-  unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
+lemma has_vector_derivative_within_subset:
+  "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
+    (f has_vector_derivative f') (at x within t)"
+  unfolding has_vector_derivative_def
+  apply (rule has_derivative_within_subset)
+  apply auto
+  done
 
-lemma has_vector_derivative_const: 
- "((\<lambda>x. c) has_vector_derivative 0) net"
-  unfolding has_vector_derivative_def using has_derivative_const by auto
+lemma has_vector_derivative_const: "((\<lambda>x. c) has_vector_derivative 0) net"
+  unfolding has_vector_derivative_def
+  using has_derivative_const
+  by auto
 
 lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
-  unfolding has_vector_derivative_def using has_derivative_id by auto
+  unfolding has_vector_derivative_def
+  using has_derivative_id
+  by auto
 
 lemma has_vector_derivative_cmul:
-  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
+  "(f has_vector_derivative f') net \<Longrightarrow>
+    ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
   unfolding has_vector_derivative_def
   apply (drule scaleR_right_has_derivative)
-  by (auto simp add: algebra_simps)
+  apply (auto simp add: algebra_simps)
+  done
 
 lemma has_vector_derivative_cmul_eq:
   assumes "c \<noteq> 0"
   shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
-  apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
-  apply(rule has_vector_derivative_cmul) using assms by auto
+  apply rule
+  apply (drule has_vector_derivative_cmul[where c="1/c"])
+  defer
+  apply (rule has_vector_derivative_cmul)
+  using assms
+  apply auto
+  done
 
 lemma has_vector_derivative_neg:
-  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
-  unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
+  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. - f x) has_vector_derivative (- f')) net"
+  unfolding has_vector_derivative_def
+  apply (drule has_derivative_neg)
+  apply auto
+  done
 
 lemma has_vector_derivative_add:
   assumes "(f has_vector_derivative f') net"
-  assumes "(g has_vector_derivative g') net"
-  shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
+    and "(g has_vector_derivative g') net"
+  shows "((\<lambda>x. f x + g x) has_vector_derivative (f' + g')) net"
   using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
-  unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
+  unfolding has_vector_derivative_def
+  unfolding scaleR_right_distrib
+  by auto
 
 lemma has_vector_derivative_sub:
   assumes "(f has_vector_derivative f') net"
-  assumes "(g has_vector_derivative g') net"
-  shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
+    and "(g has_vector_derivative g') net"
+  shows "((\<lambda>x. f x - g x) has_vector_derivative (f' - g')) net"
   using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
-  unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
+  unfolding has_vector_derivative_def scaleR_right_diff_distrib
+  by auto
 
 lemma has_vector_derivative_bilinear_within:
   assumes "(f has_vector_derivative f') (at x within s)"
-  assumes "(g has_vector_derivative g') (at x within s)"
+    and "(g has_vector_derivative g') (at x within s)"
   assumes "bounded_bilinear h"
   shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)"
-proof-
-  interpret bounded_bilinear h using assms by auto 
-  show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
+proof -
+  interpret bounded_bilinear h
+    using assms by auto
+  show ?thesis
+    using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
     unfolding o_def has_vector_derivative_def
-    using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib
+    using assms(3)
+    unfolding scaleR_right scaleR_left scaleR_right_distrib
     by auto
 qed
 
 lemma has_vector_derivative_bilinear_at:
   assumes "(f has_vector_derivative f') (at x)"
-  assumes "(g has_vector_derivative g') (at x)"
+    and "(g has_vector_derivative g') (at x)"
   assumes "bounded_bilinear h"
   shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
   using has_vector_derivative_bilinear_within[OF assms] .
@@ -1533,40 +2257,54 @@
   by (rule has_derivative_at_within)
 
 lemma has_vector_derivative_transform_within:
-  assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+  assumes "0 < d"
+    and "x \<in> s"
+    and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
   assumes "(f has_vector_derivative f') (at x within s)"
   shows "(g has_vector_derivative f') (at x within s)"
-  using assms unfolding has_vector_derivative_def
+  using assms
+  unfolding has_vector_derivative_def
   by (rule has_derivative_transform_within)
 
 lemma has_vector_derivative_transform_at:
-  assumes "0 < d" and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
-  assumes "(f has_vector_derivative f') (at x)"
+  assumes "0 < d"
+    and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
+    and "(f has_vector_derivative f') (at x)"
   shows "(g has_vector_derivative f') (at x)"
-  using assms unfolding has_vector_derivative_def
+  using assms
+  unfolding has_vector_derivative_def
   by (rule has_derivative_transform_at)
 
 lemma has_vector_derivative_transform_within_open:
-  assumes "open s" and "x \<in> s" and "\<forall>y\<in>s. f y = g y"
-  assumes "(f has_vector_derivative f') (at x)"
+  assumes "open s"
+    and "x \<in> s"
+    and "\<forall>y\<in>s. f y = g y"
+    and "(f has_vector_derivative f') (at x)"
   shows "(g has_vector_derivative f') (at x)"
-  using assms unfolding has_vector_derivative_def
+  using assms
+  unfolding has_vector_derivative_def
   by (rule has_derivative_transform_within_open)
 
 lemma vector_diff_chain_at:
   assumes "(f has_vector_derivative f') (at x)"
-  assumes "(g has_vector_derivative g') (at (f x))"
+    and "(g has_vector_derivative g') (at (f x))"
   shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
-  using assms(2) unfolding has_vector_derivative_def apply-
-  apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
-  unfolding o_def real_scaleR_def scaleR_scaleR .
+  using assms(2)
+  unfolding has_vector_derivative_def
+  apply -
+  apply (drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
+  apply (simp only: o_def real_scaleR_def scaleR_scaleR)
+  done
 
 lemma vector_diff_chain_within:
   assumes "(f has_vector_derivative f') (at x within s)"
-  assumes "(g has_vector_derivative g') (at (f x) within f ` s)"
-  shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
-  using assms(2) unfolding has_vector_derivative_def apply-
-  apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
-  unfolding o_def real_scaleR_def scaleR_scaleR .
+    and "(g has_vector_derivative g') (at (f x) within f ` s)"
+  shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
+  using assms(2)
+  unfolding has_vector_derivative_def
+  apply -
+  apply (drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
+  apply (simp only: o_def real_scaleR_def scaleR_scaleR)
+  done
 
 end