misc tuning and modernization;
authorwenzelm
Thu, 28 Jul 2016 20:39:46 +0200
changeset 63558 0aa33085c8b1
parent 63557 e506baad44fa
child 63559 113cee845044
misc tuning and modernization;
src/HOL/Deriv.thy
src/HOL/NthRoot.thy
src/HOL/Transcendental.thy
--- a/src/HOL/Deriv.thy	Wed Jul 27 10:24:50 2016 +0200
+++ b/src/HOL/Deriv.thy	Thu Jul 28 20:39:46 2016 +0200
@@ -1,26 +1,23 @@
-(*  Title       : Deriv.thy
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998  University of Cambridge
-    Author      : Brian Huffman
-    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
-    GMVT by Benjamin Porter, 2005
+(*  Title:      HOL/Deriv.thy
+    Author:     Jacques D. Fleuriot, University of Cambridge, 1998
+    Author:     Brian Huffman
+    Author:     Lawrence C Paulson, 2004
+    Author:     Benjamin Porter, 2005
 *)
 
-section\<open>Differentiation\<close>
+section \<open>Differentiation\<close>
 
 theory Deriv
-imports Limits
+  imports Limits
 begin
 
 subsection \<open>Frechet derivative\<close>
 
-definition
-  has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow>  bool"
-  (infix "(has'_derivative)" 50)
-where
-  "(f has_derivative f') F \<longleftrightarrow>
-    (bounded_linear f' \<and>
-     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> 0) F)"
+definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow>
+    ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool"  (infix "(has'_derivative)" 50)
+  where "(f has_derivative f') F \<longleftrightarrow>
+    bounded_linear f' \<and>
+    ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> 0) F"
 
 text \<open>
   Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)
@@ -32,22 +29,19 @@
 lemma has_derivative_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
   by simp
 
-definition 
-  has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
-  (infix "(has'_field'_derivative)" 50)
-where
-  "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
+definition has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
+    (infix "(has'_field'_derivative)" 50)
+  where "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
 
 lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"
   by simp
 
-definition
-  has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
-  (infix "has'_vector'_derivative" 50)
-where
-  "(f has_vector_derivative f') net \<longleftrightarrow> (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"
+    (infix "has'_vector'_derivative" 50)
+  where "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
 
-lemma has_vector_derivative_eq_rhs: "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
+lemma has_vector_derivative_eq_rhs:
+  "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
   by simp
 
 named_theorems derivative_intros "structural introduction rules for derivatives"
@@ -70,8 +64,7 @@
 abbreviation (input)
   FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
   ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
-where
-  "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
+  where "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
 
 lemma has_derivative_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
   by (simp add: has_derivative_def)
@@ -94,7 +87,7 @@
   "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
   unfolding has_derivative_def
   apply safe
-  apply (erule bounded_linear_compose [OF bounded_linear])
+   apply (erule bounded_linear_compose [OF bounded_linear])
   apply (drule tendsto)
   apply (simp add: scaleR diff add zero)
   done
@@ -112,7 +105,8 @@
   bounded_linear.has_derivative [OF bounded_linear_mult_left]
 
 lemma has_derivative_add[simp, derivative_intros]:
-  assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"
+  assumes f: "(f has_derivative f') F"
+    and g: "(g has_derivative g') F"
   shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
   unfolding has_derivative_def
 proof safe
@@ -127,16 +121,22 @@
 lemma has_derivative_setsum[simp, derivative_intros]:
   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"
   shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
-proof cases
-  assume "finite I" from this f show ?thesis
+proof (cases "finite I")
+  case True
+  from this f show ?thesis
     by induct (simp_all add: f)
-qed simp
+next
+  case False
+  then show ?thesis by simp
+qed
 
-lemma has_derivative_minus[simp, derivative_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
+lemma has_derivative_minus[simp, derivative_intros]:
+  "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
   using has_derivative_scaleR_right[of f f' F "-1"] by simp
 
 lemma has_derivative_diff[simp, derivative_intros]:
-  "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
+  "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow>
+    ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
   by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus)
 
 lemma has_derivative_at_within:
@@ -146,12 +146,13 @@
 
 lemma has_derivative_iff_norm:
   "(f has_derivative f') (at x within s) \<longleftrightarrow>
-    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
+    bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
   using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
   by (simp add: has_derivative_at_within divide_inverse ac_simps)
 
 lemma has_derivative_at:
-  "(f has_derivative D) (at x) \<longleftrightarrow> (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) \<midarrow>0\<rightarrow> 0)"
+  "(f has_derivative D) (at x) \<longleftrightarrow>
+    (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) \<midarrow>0\<rightarrow> 0)"
   unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
 
 lemma field_has_derivative_at:
@@ -167,13 +168,16 @@
   done
 
 lemma has_derivativeI:
-  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s) \<Longrightarrow>
-  (f has_derivative f') (at x within s)"
+  "bounded_linear f' \<Longrightarrow>
+    ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s) \<Longrightarrow>
+    (f has_derivative f') (at x within s)"
   by (simp add: has_derivative_at_within)
 
 lemma has_derivativeI_sandwich:
-  assumes e: "0 < e" and bounded: "bounded_linear f'"
-    and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
+  assumes e: "0 < e"
+    and bounded: "bounded_linear f'"
+    and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow>
+      norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
     and "(H \<longlongrightarrow> 0) (at x within s)"
   shows "(f has_derivative f') (at x within s)"
   unfolding has_derivative_iff_norm
@@ -186,10 +190,11 @@
   qed (auto simp: le_divide_eq)
 qed fact
 
-lemma has_derivative_subset: "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
+lemma has_derivative_subset:
+  "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
   by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
 
-lemmas has_derivative_within_subset = has_derivative_subset 
+lemmas has_derivative_within_subset = has_derivative_subset
 
 
 subsection \<open>Continuity\<close>
@@ -198,7 +203,8 @@
   assumes f: "(f has_derivative f') (at x within s)"
   shows "continuous (at x within s) f"
 proof -
-  from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
+  from f interpret F: bounded_linear f'
+    by (rule has_derivative_bounded_linear)
   note F.tendsto[tendsto_intros]
   let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
   have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
@@ -217,21 +223,27 @@
     by (simp add: continuous_within)
 qed
 
+
 subsection \<open>Composition\<close>
 
-lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f \<longlongrightarrow> y) (at x within s) \<longleftrightarrow> (f \<longlongrightarrow> y) (inf (nhds x) (principal s))"
+lemma tendsto_at_iff_tendsto_nhds_within:
+  "f x = y \<Longrightarrow> (f \<longlongrightarrow> y) (at x within s) \<longleftrightarrow> (f \<longlongrightarrow> y) (inf (nhds x) (principal s))"
   unfolding tendsto_def eventually_inf_principal eventually_at_filter
   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
 
 lemma has_derivative_in_compose:
   assumes f: "(f has_derivative f') (at x within s)"
-  assumes g: "(g has_derivative g') (at (f x) within (f`s))"
+    and g: "(g has_derivative g') (at (f x) within (f`s))"
   shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
 proof -
-  from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
-  from g interpret G: bounded_linear g' by (rule has_derivative_bounded_linear)
-  from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast
-  from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
+  from f interpret F: bounded_linear f'
+    by (rule has_derivative_bounded_linear)
+  from g interpret G: bounded_linear g'
+    by (rule has_derivative_bounded_linear)
+  from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF"
+    by fast
+  from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG"
+    by fast
   note G.tendsto[tendsto_intros]
 
   let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
@@ -246,7 +258,8 @@
     show "bounded_linear (\<lambda>x. g' (f' x))"
       using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
   next
-    fix y::'a assume neq: "y \<noteq> x"
+    fix y :: 'a
+    assume neq: "y \<noteq> x"
     have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
       by (simp add: G.diff G.add field_simps)
     also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
@@ -261,7 +274,7 @@
         using kF by (intro add_mono) simp
       finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
         by (simp add: neq Nf_def field_simps)
-    qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
+    qed (use kG in \<open>simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps\<close>)
     finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
   next
     have [tendsto_intros]: "?L Nf"
@@ -296,8 +309,9 @@
   from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
   obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
 
-  from pos_bounded obtain K where K: "0 < K" and norm_prod:
-    "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
+  from pos_bounded obtain K
+    where K: "0 < K" and norm_prod: "\<And>a b. norm (a ** b) \<le> norm a * norm b * K"
+    by fast
   let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
   let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
   define Ng where "Ng = ?N g g'"
@@ -323,8 +337,10 @@
     then show "(?fun2 \<longlongrightarrow> 0) ?F"
       by simp
   next
-    fix y::'d assume "y \<noteq> x"
-    have "?fun1 y = norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
+    fix y :: 'd
+    assume "y \<noteq> x"
+    have "?fun1 y =
+        norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
       by (simp add: diff_left diff_right add_left add_right field_simps)
     also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
         norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
@@ -343,22 +359,30 @@
 lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
 
 lemma has_derivative_setprod[simp, derivative_intros]:
-  fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
+  fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
   assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)"
   shows "((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within s)"
-proof cases
-  assume "finite I" from this f show ?thesis
+proof (cases "finite I")
+  case True
+  from this f show ?thesis
   proof induct
+    case empty
+    then show ?case by simp
+  next
     case (insert i I)
     let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
     have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
       using insert by (intro has_derivative_mult) auto
     also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
-      using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum.cong)
+      using insert(1,2)
+      by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum.cong)
     finally show ?case
       using insert by simp
-  qed simp  
-qed simp
+  qed
+next
+  case False
+  then show ?thesis by simp
+qed
 
 lemma has_derivative_power[simp, derivative_intros]:
   fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
@@ -370,7 +394,7 @@
   fixes x :: "'a::real_normed_div_algebra"
   assumes x: "x \<noteq> 0"
   shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
-        (is "(?inv has_derivative ?f) _")
+    (is "(?inv has_derivative ?f) _")
 proof (rule has_derivativeI_sandwich)
   show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
     apply (rule bounded_linear_minus)
@@ -378,21 +402,21 @@
     apply (rule bounded_linear_const_mult)
     apply (rule bounded_linear_ident)
     done
-next
   show "0 < norm x" using x by simp
-next
   show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) \<longlongrightarrow> 0) (at x within s)"
     apply (rule tendsto_mult_left_zero)
     apply (rule tendsto_norm_zero)
     apply (rule LIM_zero)
     apply (rule tendsto_inverse)
-    apply (rule tendsto_ident_at)
+     apply (rule tendsto_ident_at)
     apply (rule x)
     done
 next
-  fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
+  fix y :: 'a
+  assume h: "y \<noteq> x" "dist y x < norm x"
   then have "y \<noteq> 0" by auto
-  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
+  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) =
+      norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
     apply (subst inverse_diff_inverse [OF \<open>y \<noteq> 0\<close> x])
     apply (subst minus_diff_minus)
     apply (subst norm_minus_cancel)
@@ -407,52 +431,56 @@
   also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
     by simp
   finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
-      norm (?inv y - ?inv x) * norm (?inv x)" .
+    norm (?inv y - ?inv x) * norm (?inv x)" .
 qed
 
 lemma has_derivative_inverse[simp, derivative_intros]:
   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
-  assumes x:  "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"
-  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"
+  assumes x:  "f x \<noteq> 0"
+    and f: "(f has_derivative f') (at x within s)"
+  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x))))
+    (at x within s)"
   using has_derivative_compose[OF f has_derivative_inverse', OF x] .
 
 lemma has_derivative_divide[simp, derivative_intros]:
   fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
-  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" 
+  assumes f: "(f has_derivative f') (at x within s)"
+    and g: "(g has_derivative g') (at x within s)"
   assumes x: "g x \<noteq> 0"
   shows "((\<lambda>x. f x / g x) has_derivative
                 (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
   using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
   by (simp add: field_simps)
 
-text\<open>Conventional form requires mult-AC laws. Types real and complex only.\<close>
+
+text \<open>Conventional form requires mult-AC laws. Types real and complex only.\<close>
 
-lemma has_derivative_divide'[derivative_intros]: 
+lemma has_derivative_divide'[derivative_intros]:
   fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
-  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" and x: "g x \<noteq> 0"
+  assumes f: "(f has_derivative f') (at x within s)"
+    and g: "(g has_derivative g') (at x within s)"
+    and x: "g x \<noteq> 0"
   shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within s)"
 proof -
-  { fix h
-    have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
-          (f' h * g x - f x * g' h) / (g x * g x)"
-      by (simp add: field_simps x)
-   }
+  have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
+      (f' h * g x - f x * g' h) / (g x * g x)" for h
+    by (simp add: field_simps x)
   then show ?thesis
     using has_derivative_divide [OF f g] x
     by simp
 qed
 
+
 subsection \<open>Uniqueness\<close>
 
 text \<open>
-
 This can not generally shown for @{const has_derivative}, as we need to approach the point from
 all directions. There is a proof in \<open>Multivariate_Analysis\<close> for \<open>euclidean_space\<close>.
-
 \<close>
 
 lemma has_derivative_zero_unique:
-  assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"
+  assumes "((\<lambda>x. 0) has_derivative F) (at x)"
+  shows "F = (\<lambda>h. 0)"
 proof -
   interpret F: bounded_linear F
     using assms by (rule has_derivative_bounded_linear)
@@ -461,42 +489,50 @@
     using assms unfolding has_derivative_at by simp
   show "F = (\<lambda>h. 0)"
   proof
-    fix h show "F h = 0"
+    show "F h = 0" for h
     proof (rule ccontr)
-      assume **: "F h \<noteq> 0"
-      hence h: "h \<noteq> 0" by (clarsimp simp add: F.zero)
-      with ** have "0 < ?r h" by simp
-      from LIM_D [OF * this] obtain s where s: "0 < s"
-        and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
+      assume **: "\<not> ?thesis"
+      then have h: "h \<noteq> 0"
+        by (auto simp add: F.zero)
+      with ** have "0 < ?r h"
+        by simp
+      from LIM_D [OF * this] obtain s
+        where s: "0 < s" and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h"
+        by auto
       from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
       let ?x = "scaleR (t / norm h) h"
-      have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
-      hence "?r ?x < ?r h" by (rule r)
-      thus "False" using t h by (simp add: F.scaleR)
+      have "?x \<noteq> 0" and "norm ?x < s"
+        using t h by simp_all
+      then have "?r ?x < ?r h"
+        by (rule r)
+      then show False
+        using t h by (simp add: F.scaleR)
     qed
   qed
 qed
 
 lemma has_derivative_unique:
-  assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"
+  assumes "(f has_derivative F) (at x)"
+    and "(f has_derivative F') (at x)"
+  shows "F = F'"
 proof -
   have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
     using has_derivative_diff [OF assms] by simp
-  hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
+  then have "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
     by (rule has_derivative_zero_unique)
-  thus "F = F'"
+  then show "F = F'"
     unfolding fun_eq_iff right_minus_eq .
 qed
 
+
 subsection \<open>Differentiability predicate\<close>
 
-definition
-  differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
-  (infix "differentiable" 50)
-where
-  "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
+definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
+    (infix "differentiable" 50)
+  where "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
 
-lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
+lemma differentiable_subset:
+  "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
   unfolding differentiable_def by (blast intro: has_derivative_subset)
 
 lemmas differentiable_within_subset = differentiable_subset
@@ -508,11 +544,13 @@
   unfolding differentiable_def by (blast intro: has_derivative_const)
 
 lemma differentiable_in_compose:
-  "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
+  "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
+    (\<lambda>x. f (g x)) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_in_compose)
 
 lemma differentiable_compose:
-  "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
+  "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
+    (\<lambda>x. f (g x)) differentiable (at x within s)"
   by (blast intro: differentiable_in_compose differentiable_subset)
 
 lemma differentiable_sum [simp, derivative_intros]:
@@ -528,57 +566,59 @@
   unfolding differentiable_def by (blast intro: has_derivative_diff)
 
 lemma differentiable_mult [simp, derivative_intros]:
-  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
-  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x * g x) differentiable (at x within s)"
+  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
+  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
+    (\<lambda>x. f x * g x) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_mult)
 
 lemma differentiable_inverse [simp, derivative_intros]:
-  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
-  shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable (at x within s)"
+  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
+  shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
+    (\<lambda>x. inverse (f x)) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_inverse)
 
 lemma differentiable_divide [simp, derivative_intros]:
-  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
-  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
+  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
+  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
+    g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
   unfolding divide_inverse by simp
 
 lemma differentiable_power [simp, derivative_intros]:
-  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
+  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
   shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_power)
 
 lemma differentiable_scaleR [simp, derivative_intros]:
-  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
+  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
+    (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
   unfolding differentiable_def by (blast intro: has_derivative_scaleR)
 
 lemma has_derivative_imp_has_field_derivative:
   "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
-  unfolding has_field_derivative_def 
+  unfolding has_field_derivative_def
   by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
 
-lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
+lemma has_field_derivative_imp_has_derivative:
+  "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
   by (simp add: has_field_derivative_def)
 
-lemma DERIV_subset: 
-  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
-   \<Longrightarrow> (f has_field_derivative f') (at x within t)"
+lemma DERIV_subset:
+  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
+    (f has_field_derivative f') (at x within t)"
   by (simp add: has_field_derivative_def has_derivative_within_subset)
 
 lemma has_field_derivative_at_within:
-    "(f has_field_derivative f') (at x) \<Longrightarrow> (f has_field_derivative f') (at x within s)"
+  "(f has_field_derivative f') (at x) \<Longrightarrow> (f has_field_derivative f') (at x within s)"
   using DERIV_subset by blast
 
 abbreviation (input)
   DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
-  ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
-where
-  "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
+    ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
+  where "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
 
-abbreviation 
-  has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
-  (infix "(has'_real'_derivative)" 50)
-where
-  "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
+abbreviation has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
+    (infix "(has'_real'_derivative)" 50)
+  where "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
 
 lemma real_differentiable_def:
   "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
@@ -593,31 +633,36 @@
 qed (auto simp: differentiable_def has_field_derivative_def)
 
 lemma real_differentiableE [elim?]:
-  assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"
+  assumes f: "f differentiable (at x within s)"
+  obtains df where "(f has_real_derivative df) (at x within s)"
   using assms by (auto simp: real_differentiable_def)
 
-lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
+lemma differentiableD:
+  "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
   by (auto elim: real_differentiableE)
 
-lemma differentiableI: "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
+lemma differentiableI:
+  "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
   by (force simp add: real_differentiable_def)
 
 lemma has_field_derivative_iff:
   "(f has_field_derivative D) (at x within S) \<longleftrightarrow>
     ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
   apply (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right
-    LIM_zero_iff[symmetric, of _ D])
+      LIM_zero_iff[symmetric, of _ D])
   apply (subst (2) tendsto_norm_zero_iff[symmetric])
   apply (rule filterlim_cong)
-  apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
+    apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
   done
 
 lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
   unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..
 
-lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
+lemma mult_commute_abs: "(\<lambda>x. x * c) = op * c"
+  for c :: "'a::ab_semigroup_mult"
   by (simp add: fun_eq_iff mult.commute)
 
+
 subsection \<open>Vector derivative\<close>
 
 lemma has_field_derivative_iff_has_vector_derivative:
@@ -625,7 +670,8 @@
   unfolding has_vector_derivative_def has_field_derivative_def real_scaleR_def mult_commute_abs ..
 
 lemma has_field_derivative_subset:
-  "(f has_field_derivative y) (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_field_derivative y) (at x within t)"
+  "(f has_field_derivative y) (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
+    (f has_field_derivative y) (at x within t)"
   unfolding has_field_derivative_def by (rule has_derivative_subset)
 
 lemma has_vector_derivative_const[simp, derivative_intros]: "((\<lambda>x. c) has_vector_derivative 0) net"
@@ -654,16 +700,18 @@
   by (auto simp: has_vector_derivative_def scaleR_diff_right)
 
 lemma has_vector_derivative_add_const:
-     "((\<lambda>t. g t + z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
-apply (intro iffI)
-apply (drule has_vector_derivative_diff [where g = "\<lambda>t. z", OF _ has_vector_derivative_const], simp)
-apply (drule has_vector_derivative_add [OF _ has_vector_derivative_const], simp)
-done
+  "((\<lambda>t. g t + z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
+  apply (intro iffI)
+   apply (drule has_vector_derivative_diff [where g = "\<lambda>t. z", OF _ has_vector_derivative_const])
+   apply simp
+  apply (drule has_vector_derivative_add [OF _ has_vector_derivative_const])
+  apply simp
+  done
 
 lemma has_vector_derivative_diff_const:
-     "((\<lambda>t. g t - z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
-using has_vector_derivative_add_const [where z = "-z"]
-by simp
+  "((\<lambda>t. g t - z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
+  using has_vector_derivative_add_const [where z = "-z"]
+  by simp
 
 lemma (in bounded_linear) has_vector_derivative:
   assumes "(g has_vector_derivative g') F"
@@ -686,24 +734,26 @@
 
 lemma has_vector_derivative_mult[derivative_intros]:
   "(f has_vector_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
-    ((\<lambda>x. f x * g x) has_vector_derivative (f x * g' + f' * g x :: 'a :: real_normed_algebra)) (at x within s)"
+    ((\<lambda>x. f x * g x) has_vector_derivative (f x * g' + f' * g x)) (at x within s)"
+  for f g :: "real \<Rightarrow> 'a::real_normed_algebra"
   by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_mult])
 
 lemma has_vector_derivative_of_real[derivative_intros]:
   "(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_vector_derivative (of_real D)) F"
   by (rule bounded_linear.has_vector_derivative[OF bounded_linear_of_real])
-     (simp add: has_field_derivative_iff_has_vector_derivative)
+    (simp add: has_field_derivative_iff_has_vector_derivative)
 
-lemma has_vector_derivative_continuous: "(f has_vector_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
+lemma has_vector_derivative_continuous:
+  "(f has_vector_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
   by (auto intro: has_derivative_continuous simp: has_vector_derivative_def)
 
 lemma has_vector_derivative_mult_right[derivative_intros]:
-  fixes a :: "'a :: real_normed_algebra"
+  fixes a :: "'a::real_normed_algebra"
   shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. a * f x) has_vector_derivative (a * x)) F"
   by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_right])
 
 lemma has_vector_derivative_mult_left[derivative_intros]:
-  fixes a :: "'a :: real_normed_algebra"
+  fixes a :: "'a::real_normed_algebra"
   shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. f x * a) has_vector_derivative (x * a)) F"
   by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_left])
 
@@ -725,14 +775,14 @@
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
 
 lemma field_differentiable_add[derivative_intros]:
-  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> 
+  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow>
     ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
      (auto simp: has_field_derivative_def field_simps mult_commute_abs)
 
 corollary DERIV_add:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
-  ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
+    ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
   by (rule field_differentiable_add)
 
 lemma field_differentiable_minus[derivative_intros]:
@@ -740,16 +790,20 @@
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
      (auto simp: has_field_derivative_def field_simps mult_commute_abs)
 
-corollary DERIV_minus: "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
+corollary DERIV_minus:
+  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
+    ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
   by (rule field_differentiable_minus)
 
 lemma field_differentiable_diff[derivative_intros]:
-  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
+  "(f has_field_derivative f') F \<Longrightarrow>
+    (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
   by (simp only: diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
 
 corollary DERIV_diff:
-  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
-  ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
+  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
+    (g has_field_derivative E) (at x within s) \<Longrightarrow>
+    ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
   by (rule field_differentiable_diff)
 
 lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
@@ -761,51 +815,54 @@
 lemma DERIV_continuous_on:
   "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative (D x)) (at x within s)) \<Longrightarrow> continuous_on s f"
   unfolding continuous_on_eq_continuous_within
-  by (intro continuous_at_imp_continuous_on ballI DERIV_continuous) 
+  by (intro continuous_at_imp_continuous_on ballI DERIV_continuous)
 
 lemma DERIV_mult':
   "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
-  ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
+    ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
      (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
 
 lemma DERIV_mult[derivative_intros]:
   "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
-  ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
+    ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
      (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
 
 text \<open>Derivative of linear multiplication\<close>
 
 lemma DERIV_cmult:
-  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
-  by (drule DERIV_mult' [OF DERIV_const], simp)
+  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
+    ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
+  by (drule DERIV_mult' [OF DERIV_const]) simp
 
 lemma DERIV_cmult_right:
-  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
-  using DERIV_cmult by (force simp add: ac_simps)
+  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
+    ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
+  using DERIV_cmult by (auto simp add: ac_simps)
 
 lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
-  by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
+  using DERIV_ident [THEN DERIV_cmult, where c = c and x = x] by simp
 
 lemma DERIV_cdivide:
-  "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
+  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
+    ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
   using DERIV_cmult_right[of f D x s "1 / c"] by simp
 
-lemma DERIV_unique:
-  "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
-  unfolding DERIV_def by (rule LIM_unique) 
+lemma DERIV_unique: "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
+  unfolding DERIV_def by (rule LIM_unique)
 
 lemma DERIV_setsum[derivative_intros]:
-  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow> 
+  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow>
     ((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
-  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])
+  by (rule has_derivative_imp_has_field_derivative [OF has_derivative_setsum])
      (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
 
 lemma DERIV_inverse'[derivative_intros]:
   assumes "(f has_field_derivative D) (at x within s)"
     and "f x \<noteq> 0"
-  shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
+  shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x)))
+    (at x within s)"
 proof -
   have "(f has_derivative (\<lambda>x. x * D)) = (f has_derivative op * D)"
     by (rule arg_cong [of "\<lambda>x. x * D"]) (simp add: fun_eq_iff)
@@ -825,40 +882,40 @@
 
 lemma DERIV_inverse_fun:
   "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
-  ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
+    ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
   by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)
 
 text \<open>Derivative of quotient\<close>
 
 lemma DERIV_divide[derivative_intros]:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
-  (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
-  ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
+    (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
+    ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
      (auto dest: has_field_derivative_imp_has_derivative simp: field_simps)
 
 lemma DERIV_quotient:
   "(f has_field_derivative d) (at x within s) \<Longrightarrow>
-  (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 
-  ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
+    (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
+    ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
   by (drule (2) DERIV_divide) (simp add: mult.commute)
 
 lemma DERIV_power_Suc:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
-  ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
+    ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
      (auto simp: has_field_derivative_def)
 
 lemma DERIV_power[derivative_intros]:
   "(f has_field_derivative D) (at x within s) \<Longrightarrow>
-  ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
+    ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
   by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
      (auto simp: has_field_derivative_def)
 
 lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
   using DERIV_power [OF DERIV_ident] by simp
 
-lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 
+lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow>
   ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
   using has_derivative_compose[of f "op * D" x s g "op * E"]
   by (simp only: has_field_derivative_def mult_commute_abs ac_simps)
@@ -870,42 +927,42 @@
 text \<open>Standard version\<close>
 
 lemma DERIV_chain:
-  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 
-  (f o g has_field_derivative Da * Db) (at x within s)"
+  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
+    (f \<circ> g has_field_derivative Da * Db) (at x within s)"
   by (drule (1) DERIV_chain', simp add: o_def mult.commute)
 
-lemma DERIV_image_chain: 
-  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
-  (f o g has_field_derivative Da * Db) (at x within s)"
+lemma DERIV_image_chain:
+  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow>
+    (g has_field_derivative Db) (at x within s) \<Longrightarrow>
+    (f \<circ> g has_field_derivative Da * Db) (at x within s)"
   using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
   by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
 
 (*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
 lemma DERIV_chain_s:
   assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
-      and "DERIV f x :> f'" 
-      and "f x \<in> s"
-    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
+    and "DERIV f x :> f'"
+    and "f x \<in> s"
+  shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
   by (metis (full_types) DERIV_chain' mult.commute assms)
 
 lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
   assumes "(\<And>x. DERIV g x :> g'(x))"
-      and "DERIV f x :> f'" 
-    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
+    and "DERIV f x :> f'"
+  shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
   by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
 
-text\<open>Alternative definition for differentiability\<close>
+text \<open>Alternative definition for differentiability\<close>
 
 lemma DERIV_LIM_iff:
-  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
-     "((%h. (f(a + h) - f(a)) / h) \<midarrow>0\<rightarrow> D) =
-      ((%x. (f(x)-f(a)) / (x-a)) \<midarrow>a\<rightarrow> D)"
-apply (rule iffI)
-apply (drule_tac k="- a" in LIM_offset)
-apply simp
-apply (drule_tac k="a" in LIM_offset)
-apply (simp add: add.commute)
-done
+  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a"
+  shows "((\<lambda>h. (f (a + h) - f a) / h) \<midarrow>0\<rightarrow> D) = ((\<lambda>x. (f x - f a) / (x - a)) \<midarrow>a\<rightarrow> D)"
+  apply (rule iffI)
+   apply (drule_tac k="- a" in LIM_offset)
+   apply simp
+  apply (drule_tac k="a" in LIM_offset)
+  apply (simp add: add.commute)
+  done
 
 lemmas DERIV_iff2 = has_field_derivative_iff
 
@@ -913,16 +970,18 @@
   assumes "x = y"
     and *: "eventually (\<lambda>x. x \<in> s \<longrightarrow> f x = g x) (nhds x)"
     and "u = v" "s = t" "x \<in> s"
-    shows "(f has_field_derivative u) (at x within s) = (g has_field_derivative v) (at y within t)"
+  shows "(f has_field_derivative u) (at x within s) = (g has_field_derivative v) (at y within t)"
   unfolding DERIV_iff2
 proof (rule filterlim_cong)
-  from assms have "f y = g y" by (auto simp: eventually_nhds)
+  from assms have "f y = g y"
+    by (auto simp: eventually_nhds)
   with * show "\<forall>\<^sub>F xa in at x within s. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)"
     unfolding eventually_at_filter
     by eventually_elim (auto simp: assms \<open>f y = g y\<close>)
 qed (simp_all add: assms)
 
-lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
+lemma DERIV_cong_ev:
+  "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
     DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
   by (rule has_field_derivative_cong_ev) simp_all
 
@@ -930,18 +989,19 @@
   "(f has_field_derivative y) (at (x + z)) = ((\<lambda>x. f (x + z)) has_field_derivative y) (at x)"
   by (simp add: DERIV_def field_simps)
 
-lemma DERIV_mirror:
-  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
+lemma DERIV_mirror: "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x)) x :> - y)"
+  for f :: "real \<Rightarrow> real" and x y :: real
   by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
-                tendsto_minus_cancel_left field_simps conj_commute)
+      tendsto_minus_cancel_left field_simps conj_commute)
 
 lemma floor_has_real_derivative:
-  fixes f::"real \<Rightarrow> 'a::{floor_ceiling,order_topology}"
+  fixes f :: "real \<Rightarrow> 'a::{floor_ceiling,order_topology}"
   assumes "isCont f x"
-  assumes "f x \<notin> \<int>"
+    and "f x \<notin> \<int>"
   shows "((\<lambda>x. floor (f x)) has_real_derivative 0) (at x)"
 proof (subst DERIV_cong_ev[OF refl _ refl])
-  show "((\<lambda>_. floor (f x)) has_real_derivative 0) (at x)" by simp
+  show "((\<lambda>_. floor (f x)) has_real_derivative 0) (at x)"
+    by simp
   have "\<forall>\<^sub>F y in at x. \<lfloor>f y\<rfloor> = \<lfloor>f x\<rfloor>"
     by (rule eventually_floor_eq[OF assms[unfolded continuous_at]])
   then show "\<forall>\<^sub>F y in nhds x. real_of_int \<lfloor>f y\<rfloor> = real_of_int \<lfloor>f x\<rfloor>"
@@ -954,382 +1014,387 @@
 
 lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
   "(DERIV f x :> l) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
-      (is "?lhs = ?rhs")
+  (is "?lhs = ?rhs")
 proof
-  assume der: "DERIV f x :> l"
-  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
+  assume ?lhs
+  show "\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l"
   proof (intro exI conjI)
-    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
-    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
-    show "isCont ?g x" using der
-      by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
-    show "?g x = l" by simp
+    let ?g = "(\<lambda>z. if z = x then l else (f z - f x) / (z-x))"
+    show "\<forall>z. f z - f x = ?g z * (z - x)"
+      by simp
+    show "isCont ?g x"
+      using \<open>?lhs\<close> by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
+    show "?g x = l"
+      by simp
   qed
 next
-  assume "?rhs"
-  then obtain g where
-    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
-  thus "(DERIV f x :> l)"
-     by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
+  assume ?rhs
+  then obtain g where "(\<forall>z. f z - f x = g z * (z - x))" and "isCont g x" and "g x = l"
+    by blast
+  then show ?lhs
+    by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
 qed
 
 
 subsection \<open>Local extrema\<close>
 
-text\<open>If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right\<close>
+text \<open>If @{term "0 < f' x"} then @{term x} is Locally Strictly Increasing At The Right.\<close>
 
 lemma has_real_derivative_pos_inc_right:
-  fixes f :: "real => real"
+  fixes f :: "real \<Rightarrow> real"
   assumes der: "(f has_real_derivative l) (at x within S)"
-      and l:   "0 < l"
+    and l: "0 < l"
   shows "\<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x + h)"
   using assms
 proof -
   from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
-  obtain s where s:   "0 < s"
-    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> abs ((f xa - f x) / (xa - x) - l) < l"
+  obtain s where s: "0 < s"
+    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < l"
     by (auto simp: dist_real_def)
   then show ?thesis
   proof (intro exI conjI strip)
-    show "0<s" using s .
-    fix h::real
+    show "0 < s" by (rule s)
+  next
+    fix h :: real
     assume "0 < h" "h < s" "x + h \<in> S"
     with all [of "x + h"] show "f x < f (x+h)"
     proof (simp add: abs_if dist_real_def pos_less_divide_eq split: if_split_asm)
-      assume "\<not> (f (x+h) - f x) / h < l" and h: "0 < h"
-      with l
-      have "0 < (f (x+h) - f x) / h" by arith
-      thus "f x < f (x+h)"
+      assume "\<not> (f (x + h) - f x) / h < l" and h: "0 < h"
+      with l have "0 < (f (x + h) - f x) / h"
+        by arith
+      then show "f x < f (x + h)"
         by (simp add: pos_less_divide_eq h)
     qed
   qed
 qed
 
 lemma DERIV_pos_inc_right:
-  fixes f :: "real => real"
+  fixes f :: "real \<Rightarrow> real"
   assumes der: "DERIV f x :> l"
-      and l:   "0 < l"
-  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
+    and l: "0 < l"
+  shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x + h)"
   using has_real_derivative_pos_inc_right[OF assms]
   by auto
 
 lemma has_real_derivative_neg_dec_left:
-  fixes f :: "real => real"
+  fixes f :: "real \<Rightarrow> real"
   assumes der: "(f has_real_derivative l) (at x within S)"
-      and "l < 0"
+    and "l < 0"
   shows "\<exists>d > 0. \<forall>h > 0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x - h)"
 proof -
-  from \<open>l < 0\<close> have l: "- l > 0" by simp
+  from \<open>l < 0\<close> have l: "- l > 0"
+    by simp
   from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
-  obtain s where s:   "0 < s"
-    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> abs ((f xa - f x) / (xa - x) - l) < - l"
+  obtain s where s: "0 < s"
+    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < - l"
     by (auto simp: dist_real_def)
-  thus ?thesis
+  then show ?thesis
   proof (intro exI conjI strip)
-    show "0<s" using s .
-    fix h::real
+    show "0 < s" by (rule s)
+  next
+    fix h :: real
     assume "0 < h" "h < s" "x - h \<in> S"
     with all [of "x - h"] show "f x < f (x-h)"
     proof (simp add: abs_if pos_less_divide_eq dist_real_def split add: if_split_asm)
-      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
-      with l
-      have "0 < (f (x-h) - f x) / h" by arith
-      thus "f x < f (x-h)"
+      assume "- ((f (x-h) - f x) / h) < l" and h: "0 < h"
+      with l have "0 < (f (x-h) - f x) / h"
+        by arith
+      then show "f x < f (x - h)"
         by (simp add: pos_less_divide_eq h)
     qed
   qed
 qed
 
 lemma DERIV_neg_dec_left:
-  fixes f :: "real => real"
+  fixes f :: "real \<Rightarrow> real"
   assumes der: "DERIV f x :> l"
-      and l:   "l < 0"
-  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
+    and l: "l < 0"
+  shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x - h)"
   using has_real_derivative_neg_dec_left[OF assms]
   by auto
 
 lemma has_real_derivative_pos_inc_left:
-  fixes f :: "real => real"
-  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x - h) < f x"
-  by (rule has_real_derivative_neg_dec_left [of "%x. - f x" "-l" x S, simplified])
+  fixes f :: "real \<Rightarrow> real"
+  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> 0 < l \<Longrightarrow>
+    \<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x - h) < f x"
+  by (rule has_real_derivative_neg_dec_left [of "\<lambda>x. - f x" "-l" x S, simplified])
       (auto simp add: DERIV_minus)
 
 lemma DERIV_pos_inc_left:
-  fixes f :: "real => real"
-  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
+  fixes f :: "real \<Rightarrow> real"
+  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f (x - h) < f x"
   using has_real_derivative_pos_inc_left
   by blast
 
 lemma has_real_derivative_neg_dec_right:
-  fixes f :: "real => real"
-  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f(x) > f(x + h)"
-  by (rule has_real_derivative_pos_inc_right [of "%x. - f x" "-l" x S, simplified])
+  fixes f :: "real \<Rightarrow> real"
+  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> l < 0 \<Longrightarrow>
+    \<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x > f (x + h)"
+  by (rule has_real_derivative_pos_inc_right [of "\<lambda>x. - f x" "-l" x S, simplified])
       (auto simp add: DERIV_minus)
 
 lemma DERIV_neg_dec_right:
-  fixes f :: "real => real"
-  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
+  fixes f :: "real \<Rightarrow> real"
+  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x > f (x + h)"
   using has_real_derivative_neg_dec_right by blast
 
 lemma DERIV_local_max:
-  fixes f :: "real => real"
+  fixes f :: "real \<Rightarrow> real"
   assumes der: "DERIV f x :> l"
-      and d:   "0 < d"
-      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
+    and d: "0 < d"
+    and le: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
   shows "l = 0"
 proof (cases rule: linorder_cases [of l 0])
-  case equal thus ?thesis .
+  case equal
+  then show ?thesis .
 next
   case less
   from DERIV_neg_dec_left [OF der less]
-  obtain d' where d': "0 < d'"
-             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
-  from real_lbound_gt_zero [OF d d']
-  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
-  with lt le [THEN spec [where x="x-e"]]
-  show ?thesis by (auto simp add: abs_if)
+  obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x - h)"
+    by blast
+  obtain e where "0 < e \<and> e < d \<and> e < d'"
+    using real_lbound_gt_zero [OF d d']  ..
+  with lt le [THEN spec [where x="x - e"]] show ?thesis
+    by (auto simp add: abs_if)
 next
   case greater
   from DERIV_pos_inc_right [OF der greater]
-  obtain d' where d': "0 < d'"
-             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
-  from real_lbound_gt_zero [OF d d']
-  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
-  with lt le [THEN spec [where x="x+e"]]
-  show ?thesis by (auto simp add: abs_if)
+  obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)"
+    by blast
+  obtain e where "0 < e \<and> e < d \<and> e < d'"
+    using real_lbound_gt_zero [OF d d'] ..
+  with lt le [THEN spec [where x="x + e"]] show ?thesis
+    by (auto simp add: abs_if)
 qed
 
-
-text\<open>Similar theorem for a local minimum\<close>
+text \<open>Similar theorem for a local minimum\<close>
 lemma DERIV_local_min:
-  fixes f :: "real => real"
-  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
-by (drule DERIV_minus [THEN DERIV_local_max], auto)
+  fixes f :: "real \<Rightarrow> real"
+  shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x \<le> f y \<Longrightarrow> l = 0"
+  by (drule DERIV_minus [THEN DERIV_local_max]) auto
 
 
 text\<open>In particular, if a function is locally flat\<close>
 lemma DERIV_local_const:
-  fixes f :: "real => real"
-  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
-by (auto dest!: DERIV_local_max)
+  fixes f :: "real \<Rightarrow> real"
+  shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x = f y \<Longrightarrow> l = 0"
+  by (auto dest!: DERIV_local_max)
 
 
 subsection \<open>Rolle's Theorem\<close>
 
-text\<open>Lemma about introducing open ball in open interval\<close>
-lemma lemma_interval_lt:
-     "[| a < x;  x < b |]
-      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
+text \<open>Lemma about introducing open ball in open interval\<close>
+lemma lemma_interval_lt: "a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a < y \<and> y < b)"
+  for a b x :: real
+  apply (simp add: abs_less_iff)
+  apply (insert linorder_linear [of "x - a" "b - x"])
+  apply safe
+   apply (rule_tac x = "x - a" in exI)
+   apply (rule_tac [2] x = "b - x" in exI)
+   apply auto
+  done
 
-apply (simp add: abs_less_iff)
-apply (insert linorder_linear [of "x-a" "b-x"], safe)
-apply (rule_tac x = "x-a" in exI)
-apply (rule_tac [2] x = "b-x" in exI, auto)
-done
+lemma lemma_interval: "a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b)"
+  for a b x :: real
+  apply (drule lemma_interval_lt)
+   apply auto
+  apply force
+  done
 
-lemma lemma_interval: "[| a < x;  x < b |] ==>
-        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
-apply (drule lemma_interval_lt, auto)
-apply force
-done
-
-text\<open>Rolle's Theorem.
+text \<open>Rolle's Theorem.
    If @{term f} is defined and continuous on the closed interval
    \<open>[a,b]\<close> and differentiable on the open interval \<open>(a,b)\<close>,
-   and @{term "f(a) = f(b)"},
-   then there exists \<open>x0 \<in> (a,b)\<close> such that @{term "f'(x0) = 0"}\<close>
+   and @{term "f a = f b"},
+   then there exists \<open>x0 \<in> (a,b)\<close> such that @{term "f' x0 = 0"}\<close>
 theorem Rolle:
+  fixes a b :: real
   assumes lt: "a < b"
-      and eq: "f(a) = f(b)"
-      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
-      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
-  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
+    and eq: "f a = f b"
+    and con: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
+    and dif [rule_format]: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
+  shows "\<exists>z. a < z \<and> z < b \<and> DERIV f z :> 0"
 proof -
-  have le: "a \<le> b" using lt by simp
+  have le: "a \<le> b"
+    using lt by simp
   from isCont_eq_Ub [OF le con]
-  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
-             and alex: "a \<le> x" and xleb: "x \<le> b"
+  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" and "a \<le> x" "x \<le> b"
     by blast
   from isCont_eq_Lb [OF le con]
-  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
-              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
+  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" and "a \<le> x'" "x' \<le> b"
     by blast
-  show ?thesis
+  consider "a < x" "x < b" | "x = a \<or> x = b"
+    using \<open>a \<le> x\<close> \<open>x \<le> b\<close> by arith
+  then show ?thesis
   proof cases
-    assume axb: "a < x & x < b"
-        \<comment>\<open>@{term f} attains its maximum within the interval\<close>
-    hence ax: "a<x" and xb: "x<b" by arith + 
-    from lemma_interval [OF ax xb]
-    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
-      by blast
-    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
-      by blast
-    from differentiableD [OF dif [OF axb]]
-    obtain l where der: "DERIV f x :> l" ..
-    have "l=0" by (rule DERIV_local_max [OF der d bound'])
-        \<comment>\<open>the derivative at a local maximum is zero\<close>
-    thus ?thesis using ax xb der by auto
+    case 1
+    \<comment>\<open>@{term f} attains its maximum within the interval\<close>
+    obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+      using lemma_interval [OF 1] by blast
+    then have bound': "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
+      using x_max by blast
+    obtain l where der: "DERIV f x :> l"
+      using differentiableD [OF dif [OF conjI [OF 1]]] ..
+    \<comment>\<open>the derivative at a local maximum is zero\<close>
+    have "l = 0"
+      by (rule DERIV_local_max [OF der d bound'])
+    with 1 der show ?thesis by auto
   next
-    assume notaxb: "~ (a < x & x < b)"
-    hence xeqab: "x=a | x=b" using alex xleb by arith
-    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
-    show ?thesis
+    case 2
+    then have fx: "f b = f x" by (auto simp add: eq)
+    consider "a < x'" "x' < b" | "x' = a \<or> x' = b"
+      using \<open>a \<le> x'\<close> \<open>x' \<le> b\<close> by arith
+    then show ?thesis
     proof cases
-      assume ax'b: "a < x' & x' < b"
-        \<comment>\<open>@{term f} attains its minimum within the interval\<close>
-      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
-      from lemma_interval [OF ax' x'b]
+      case 1
+        \<comment> \<open>@{term f} attains its minimum within the interval\<close>
+      from lemma_interval [OF 1]
       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
-  by blast
-      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
-  by blast
-      from differentiableD [OF dif [OF ax'b]]
+        by blast
+      then have bound': "\<forall>y. \<bar>x' - y\<bar> < d \<longrightarrow> f x' \<le> f y"
+        using x'_min by blast
+      from differentiableD [OF dif [OF conjI [OF 1]]]
       obtain l where der: "DERIV f x' :> l" ..
-      have "l=0" by (rule DERIV_local_min [OF der d bound'])
-        \<comment>\<open>the derivative at a local minimum is zero\<close>
-      thus ?thesis using ax' x'b der by auto
+      have "l = 0" by (rule DERIV_local_min [OF der d bound'])
+        \<comment> \<open>the derivative at a local minimum is zero\<close>
+      then show ?thesis using 1 der by auto
     next
-      assume notax'b: "~ (a < x' & x' < b)"
-        \<comment>\<open>@{term f} is constant througout the interval\<close>
-      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
-      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
-      from dense [OF lt]
-      obtain r where ar: "a < r" and rb: "r < b" by blast
-      from lemma_interval [OF ar rb]
-      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
-  by blast
-      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
-      proof (clarify)
-        fix z::real
-        assume az: "a \<le> z" and zb: "z \<le> b"
-        show "f z = f b"
-        proof (rule order_antisym)
-          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
-          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
-        qed
+      case 2
+        \<comment> \<open>@{term f} is constant throughout the interval\<close>
+      then have fx': "f b = f x'" by (auto simp: eq)
+      from dense [OF lt] obtain r where r: "a < r" "r < b" by blast
+      obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
+        using lemma_interval [OF r] by blast
+      have eq_fb: "f z = f b" if "a \<le> z" and "z \<le> b" for z
+      proof (rule order_antisym)
+        show "f z \<le> f b" by (simp add: fx x_max that)
+        show "f b \<le> f z" by (simp add: fx' x'_min that)
       qed
-      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
+      have bound': "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> f r = f y"
       proof (intro strip)
-        fix y::real
-        assume lt: "\<bar>r-y\<bar> < d"
-        hence "f y = f b" by (simp add: eq_fb bound)
-        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
+        fix y :: real
+        assume lt: "\<bar>r - y\<bar> < d"
+        then have "f y = f b" by (simp add: eq_fb bound)
+        then show "f r = f y" by (simp add: eq_fb r order_less_imp_le)
       qed
-      from differentiableD [OF dif [OF conjI [OF ar rb]]]
-      obtain l where der: "DERIV f r :> l" ..
-      have "l=0" by (rule DERIV_local_const [OF der d bound'])
-        \<comment>\<open>the derivative of a constant function is zero\<close>
-      thus ?thesis using ar rb der by auto
+      obtain l where der: "DERIV f r :> l"
+        using differentiableD [OF dif [OF conjI [OF r]]] ..
+      have "l = 0"
+        by (rule DERIV_local_const [OF der d bound'])
+        \<comment> \<open>the derivative of a constant function is zero\<close>
+      with r der show ?thesis by auto
     qed
   qed
 qed
 
 
-subsection\<open>Mean Value Theorem\<close>
+subsection \<open>Mean Value Theorem\<close>
 
-lemma lemma_MVT:
-     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
+lemma lemma_MVT: "f a - (f b - f a) / (b - a) * a = f b - (f b - f a) / (b - a) * b"
+  for a b :: real
   by (cases "a = b") (simp_all add: field_simps)
 
 theorem MVT:
-  assumes lt:  "a < b"
-      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
-      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
-  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
-                   (f(b) - f(a) = (b-a) * l)"
+  fixes a b :: real
+  assumes lt: "a < b"
+    and con: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
+    and dif [rule_format]: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
+  shows "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
 proof -
-  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
-  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
+  let ?F = "\<lambda>x. f x - ((f b - f a) / (b - a)) * x"
+  have cont_f: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
     using con by (fast intro: continuous_intros)
-  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
-  proof (clarify)
-    fix x::real
-    assume ax: "a < x" and xb: "x < b"
-    from differentiableD [OF dif [OF conjI [OF ax xb]]]
-    obtain l where der: "DERIV f x :> l" ..
+  have dif_f: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
+  proof clarify
+    fix x :: real
+    assume x: "a < x" "x < b"
+    obtain l where der: "DERIV f x :> l"
+      using differentiableD [OF dif [OF conjI [OF x]]] ..
     show "?F differentiable (at x)"
-      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
+      by (rule differentiableI [where D = "l - (f b - f a) / (b - a)"],
           blast intro: DERIV_diff DERIV_cmult_Id der)
   qed
-  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
-  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
+  from Rolle [where f = ?F, OF lt lemma_MVT cont_f dif_f]
+  obtain z where z: "a < z" "z < b" and der: "DERIV ?F z :> 0"
     by blast
-  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
+  have "DERIV (\<lambda>x. ((f b - f a) / (b - a)) * x) z :> (f b - f a) / (b - a)"
     by (rule DERIV_cmult_Id)
-  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
-                   :> 0 + (f b - f a) / (b - a)"
+  then have der_f: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z :> 0 + (f b - f a) / (b - a)"
     by (rule DERIV_add [OF der])
   show ?thesis
   proof (intro exI conjI)
-    show "a < z" using az .
-    show "z < b" using zb .
-    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
-    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
+    show "a < z" and "z < b" using z .
+    show "f b - f a = (b - a) * ((f b - f a) / (b - a))" by simp
+    show "DERIV f z :> ((f b - f a) / (b - a))" using der_f by simp
   qed
 qed
 
 lemma MVT2:
-     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
-      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
-apply (drule MVT)
-apply (blast intro: DERIV_isCont)
-apply (force dest: order_less_imp_le simp add: real_differentiable_def)
-apply (blast dest: DERIV_unique order_less_imp_le)
-done
+  "a < b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> DERIV f x :> f' x \<Longrightarrow>
+    \<exists>z::real. a < z \<and> z < b \<and> (f b - f a = (b - a) * f' z)"
+  apply (drule MVT)
+    apply (blast intro: DERIV_isCont)
+   apply (force dest: order_less_imp_le simp add: real_differentiable_def)
+  apply (blast dest: DERIV_unique order_less_imp_le)
+  done
 
 
-text\<open>A function is constant if its derivative is 0 over an interval.\<close>
+text \<open>A function is constant if its derivative is 0 over an interval.\<close>
 
 lemma DERIV_isconst_end:
-  fixes f :: "real => real"
-  shows "[| a < b;
-         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
-         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
-        ==> f b = f a"
-apply (drule MVT, assumption)
-apply (blast intro: differentiableI)
-apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
-done
+  fixes f :: "real \<Rightarrow> real"
+  shows "a < b \<Longrightarrow>
+    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
+    \<forall>x. a < x \<and> x < b \<longrightarrow> DERIV f x :> 0 \<Longrightarrow> f b = f a"
+  apply (drule (1) MVT)
+   apply (blast intro: differentiableI)
+  apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
+  done
 
 lemma DERIV_isconst1:
-  fixes f :: "real => real"
-  shows "[| a < b;
-         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
-         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
-        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
-apply safe
-apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
-apply (drule_tac b = x in DERIV_isconst_end, auto)
-done
+  fixes f :: "real \<Rightarrow> real"
+  shows "a < b \<Longrightarrow>
+    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
+    \<forall>x. a < x \<and> x < b \<longrightarrow> DERIV f x :> 0 \<Longrightarrow>
+    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x = f a"
+  apply safe
+  apply (drule_tac x = a in order_le_imp_less_or_eq)
+  apply safe
+  apply (drule_tac b = x in DERIV_isconst_end)
+    apply auto
+  done
 
 lemma DERIV_isconst2:
-  fixes f :: "real => real"
-  shows "[| a < b;
-         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
-         \<forall>x. a < x & x < b --> DERIV f x :> 0;
-         a \<le> x; x \<le> b |]
-        ==> f x = f a"
-apply (blast dest: DERIV_isconst1)
-done
+  fixes f :: "real \<Rightarrow> real"
+  shows "a < b \<Longrightarrow>
+    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
+    \<forall>x. a < x \<and> x < b \<longrightarrow> DERIV f x :> 0 \<Longrightarrow>
+    a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> f x = f a"
+  by (blast dest: DERIV_isconst1)
 
-lemma DERIV_isconst3: fixes a b x y :: real
-  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
-  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
+lemma DERIV_isconst3:
+  fixes a b x y :: real
+  assumes "a < b"
+    and "x \<in> {a <..< b}"
+    and "y \<in> {a <..< b}"
+    and derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
   shows "f x = f y"
 proof (cases "x = y")
   case False
   let ?a = "min x y"
   let ?b = "max x y"
-  
+
   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
   proof (rule allI, rule impI)
-    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
-    hence "a < z" and "z < b" using \<open>x \<in> {a <..< b}\<close> and \<open>y \<in> {a <..< b}\<close> by auto
-    hence "z \<in> {a<..<b}" by auto
-    thus "DERIV f z :> 0" by (rule derivable)
+    fix z :: real
+    assume "?a \<le> z \<and> z \<le> ?b"
+    then have "a < z" and "z < b"
+      using \<open>x \<in> {a <..< b}\<close> and \<open>y \<in> {a <..< b}\<close> by auto
+    then have "z \<in> {a<..<b}" by auto
+    then show "DERIV f z :> 0" by (rule derivable)
   qed
-  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
-    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
+  then have isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
+    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0"
+    using DERIV_isCont by auto
 
   have "?a < ?b" using \<open>x \<noteq> y\<close> by auto
   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
@@ -1337,184 +1402,196 @@
 qed auto
 
 lemma DERIV_isconst_all:
-  fixes f :: "real => real"
-  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
-apply (rule linorder_cases [of x y])
-apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
-done
+  fixes f :: "real \<Rightarrow> real"
+  shows "\<forall>x. DERIV f x :> 0 \<Longrightarrow> f x = f y"
+  apply (rule linorder_cases [of x y])
+    apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
+  done
 
 lemma DERIV_const_ratio_const:
-  fixes f :: "real => real"
-  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
-apply (rule linorder_cases [of a b], auto)
-apply (drule_tac [!] f = f in MVT)
-apply (auto dest: DERIV_isCont DERIV_unique simp add: real_differentiable_def)
-apply (auto dest: DERIV_unique simp add: ring_distribs)
-done
+  fixes f :: "real \<Rightarrow> real"
+  shows "a \<noteq> b \<Longrightarrow> \<forall>x. DERIV f x :> k \<Longrightarrow> f b - f a = (b - a) * k"
+  apply (rule linorder_cases [of a b])
+    apply auto
+   apply (drule_tac [!] f = f in MVT)
+       apply (auto dest: DERIV_isCont DERIV_unique simp: real_differentiable_def)
+  apply (auto dest: DERIV_unique simp: ring_distribs)
+  done
 
 lemma DERIV_const_ratio_const2:
-  fixes f :: "real => real"
-  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
-apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
-apply (auto dest!: DERIV_const_ratio_const simp add: mult.assoc)
-done
+  fixes f :: "real \<Rightarrow> real"
+  shows "a \<noteq> b \<Longrightarrow> \<forall>x. DERIV f x :> k \<Longrightarrow> (f b - f a) / (b - a) = k"
+  apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
+   apply (auto dest!: DERIV_const_ratio_const simp add: mult.assoc)
+  done
 
-lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
-by (simp)
+lemma real_average_minus_first [simp]: "(a + b) / 2 - a = (b - a) / 2"
+  for a b :: real
+  by simp
 
-lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
-by (simp)
+lemma real_average_minus_second [simp]: "(b + a) / 2 - a = (b - a) / 2"
+  for a b :: real
+  by simp
 
-text\<open>Gallileo's "trick": average velocity = av. of end velocities\<close>
+text \<open>Gallileo's "trick": average velocity = av. of end velocities.\<close>
 
 lemma DERIV_const_average:
-  fixes v :: "real => real"
-  assumes neq: "a \<noteq> (b::real)"
-      and der: "\<forall>x. DERIV v x :> k"
-  shows "v ((a + b)/2) = (v a + v b)/2"
+  fixes v :: "real \<Rightarrow> real"
+    and a b :: real
+  assumes neq: "a \<noteq> b"
+    and der: "\<forall>x. DERIV v x :> k"
+  shows "v ((a + b) / 2) = (v a + v b) / 2"
 proof (cases rule: linorder_cases [of a b])
-  case equal with neq show ?thesis by simp
+  case equal
+  with neq show ?thesis by simp
 next
   case less
   have "(v b - v a) / (b - a) = k"
     by (rule DERIV_const_ratio_const2 [OF neq der])
-  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
+  then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k"
+    by simp
   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
-    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
-  ultimately show ?thesis using neq by force
+    by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq)
+  ultimately show ?thesis
+    using neq by force
 next
   case greater
   have "(v b - v a) / (b - a) = k"
     by (rule DERIV_const_ratio_const2 [OF neq der])
-  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
+  then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k"
+    by simp
   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
-    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
-  ultimately show ?thesis using neq by (force simp add: add.commute)
+    by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq)
+  ultimately show ?thesis
+    using neq by (force simp add: add.commute)
 qed
 
-(* A function with positive derivative is increasing. 
-   A simple proof using the MVT, by Jeremy Avigad. And variants.
-*)
+text \<open>
+  A function with positive derivative is increasing.
+  A simple proof using the MVT, by Jeremy Avigad. And variants.
+\<close>
 lemma DERIV_pos_imp_increasing_open:
-  fixes a::real and b::real and f::"real => real"
-  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
-      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
+  fixes a b :: real
+    and f :: "real \<Rightarrow> real"
+  assumes "a < b"
+    and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
+    and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
   shows "f a < f b"
 proof (rule ccontr)
-  assume f: "~ f a < f b"
-  have "EX l z. a < z & z < b & DERIV f z :> l
-      & f b - f a = (b - a) * l"
-    apply (rule MVT)
-      using assms Deriv.differentiableI
-      apply force+
-    done
-  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
-      and "f b - f a = (b - a) * l"
+  assume f: "\<not> ?thesis"
+  have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
+    by (rule MVT) (use assms Deriv.differentiableI in \<open>force+\<close>)
+  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" and "f b - f a = (b - a) * l"
     by auto
-  with assms f have "~(l > 0)"
+  with assms f have "\<not> l > 0"
     by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
   with assms z show False
     by (metis DERIV_unique)
 qed
 
 lemma DERIV_pos_imp_increasing:
-  fixes a::real and b::real and f::"real => real"
-  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
+  fixes a b :: real
+    and f :: "real \<Rightarrow> real"
+  assumes "a < b"
+    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
   shows "f a < f b"
-by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)
+  by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)
 
 lemma DERIV_nonneg_imp_nondecreasing:
-  fixes a::real and b::real and f::"real => real"
-  assumes "a \<le> b" and
-    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
+  fixes a b :: real
+    and f :: "real \<Rightarrow> real"
+  assumes "a \<le> b"
+    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y \<ge> 0)"
   shows "f a \<le> f b"
 proof (rule ccontr, cases "a = b")
-  assume "~ f a \<le> f b" and "a = b"
+  assume "\<not> ?thesis" and "a = b"
   then show False by auto
 next
-  assume A: "~ f a \<le> f b"
-  assume B: "a ~= b"
-  with assms have "EX l z. a < z & z < b & DERIV f z :> l
-      & f b - f a = (b - a) * l"
+  assume *: "\<not> ?thesis"
+  assume "a \<noteq> b"
+  with assms have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
     apply -
     apply (rule MVT)
       apply auto
-      apply (metis DERIV_isCont)
-     apply (metis differentiableI less_le)
+     apply (metis DERIV_isCont)
+    apply (metis differentiableI less_le)
     done
-  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
-      and C: "f b - f a = (b - a) * l"
+  then obtain l z where lz: "a < z" "z < b" "DERIV f z :> l" and **: "f b - f a = (b - a) * l"
     by auto
-  with A have "a < b" "f b < f a" by auto
-  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
-    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
-  with assms z show False
+  with * have "a < b" "f b < f a" by auto
+  with ** have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
+    (metis * add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
+  with assms lz show False
     by (metis DERIV_unique order_less_imp_le)
 qed
 
 lemma DERIV_neg_imp_decreasing_open:
-  fixes a::real and b::real and f::"real => real"
-  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
-      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
+  fixes a b :: real
+    and f :: "real \<Rightarrow> real"
+  assumes "a < b"
+    and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y < 0)"
+    and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
   shows "f a > f b"
 proof -
-  have "(%x. -f x) a < (%x. -f x) b"
-    apply (rule DERIV_pos_imp_increasing_open [of a b "%x. -f x"])
+  have "(\<lambda>x. -f x) a < (\<lambda>x. -f x) b"
+    apply (rule DERIV_pos_imp_increasing_open [of a b "\<lambda>x. -f x"])
     using assms
-    apply auto
+      apply auto
     apply (metis field_differentiable_minus neg_0_less_iff_less)
     done
-  thus ?thesis
+  then show ?thesis
     by simp
 qed
 
 lemma DERIV_neg_imp_decreasing:
-  fixes a::real and b::real and f::"real => real"
-  assumes "a < b" and
-    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
+  fixes a b :: real
+    and f :: "real \<Rightarrow> real"
+  assumes "a < b"
+    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y < 0)"
   shows "f a > f b"
-by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)
+  by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)
 
 lemma DERIV_nonpos_imp_nonincreasing:
-  fixes a::real and b::real and f::"real => real"
-  assumes "a \<le> b" and
-    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
+  fixes a b :: real
+    and f :: "real \<Rightarrow> real"
+  assumes "a \<le> b"
+    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y \<le> 0)"
   shows "f a \<ge> f b"
 proof -
-  have "(%x. -f x) a \<le> (%x. -f x) b"
-    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
+  have "(\<lambda>x. -f x) a \<le> (\<lambda>x. -f x) b"
+    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "\<lambda>x. -f x"])
     using assms
-    apply auto
+     apply auto
     apply (metis DERIV_minus neg_0_le_iff_le)
     done
-  thus ?thesis
+  then show ?thesis
     by simp
 qed
 
 lemma DERIV_pos_imp_increasing_at_bot:
-  fixes f :: "real => real"
-  assumes "\<And>x. x \<le> b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
-      and lim: "(f \<longlongrightarrow> flim) at_bot"
+  fixes f :: "real \<Rightarrow> real"
+  assumes "\<And>x. x \<le> b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
+    and lim: "(f \<longlongrightarrow> flim) at_bot"
   shows "flim < f b"
 proof -
   have "flim \<le> f (b - 1)"
     apply (rule tendsto_ge_const [OF _ lim])
-    apply (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder)
+     apply (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder)
     apply (rule_tac x="b - 2" in exI)
     apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
     done
-  also have "... < f b"
+  also have "\<dots> < f b"
     by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
   finally show ?thesis .
 qed
 
 lemma DERIV_neg_imp_decreasing_at_top:
-  fixes f :: "real => real"
-  assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
-      and lim: "(f \<longlongrightarrow> flim) at_top"
+  fixes f :: "real \<Rightarrow> real"
+  assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y < 0)"
+    and lim: "(f \<longlongrightarrow> flim) at_top"
   shows "flim < f b"
   apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
-  apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
+   apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
   apply (metis filterlim_at_top_mirror lim)
   done
 
@@ -1523,47 +1600,45 @@
 lemma DERIV_inverse_function:
   fixes f g :: "real \<Rightarrow> real"
   assumes der: "DERIV f (g x) :> D"
-  assumes neq: "D \<noteq> 0"
-  assumes a: "a < x" and b: "x < b"
-  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
-  assumes cont: "isCont g x"
+    and neq: "D \<noteq> 0"
+    and x: "a < x" "x < b"
+    and inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
+    and cont: "isCont g x"
   shows "DERIV g x :> inverse D"
 unfolding DERIV_iff2
 proof (rule LIM_equal2)
   show "0 < min (x - a) (b - x)"
-    using a b by arith 
+    using x by arith
 next
   fix y
   assume "norm (y - x) < min (x - a) (b - x)"
-  hence "a < y" and "y < b" 
+  then have "a < y" and "y < b"
     by (simp_all add: abs_less_iff)
-  thus "(g y - g x) / (y - x) =
-        inverse ((f (g y) - x) / (g y - g x))"
+  then show "(g y - g x) / (y - x) = inverse ((f (g y) - x) / (g y - g x))"
     by (simp add: inj)
 next
   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) \<midarrow>g x\<rightarrow> D"
     by (rule der [unfolded DERIV_iff2])
-  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) \<midarrow>g x\<rightarrow> D"
-    using inj a b by simp
+  then have 1: "(\<lambda>z. (f z - x) / (z - g x)) \<midarrow>g x\<rightarrow> D"
+    using inj x by simp
   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
   proof (rule exI, safe)
     show "0 < min (x - a) (b - x)"
-      using a b by simp
+      using x by simp
   next
     fix y
     assume "norm (y - x) < min (x - a) (b - x)"
-    hence y: "a < y" "y < b"
+    then have y: "a < y" "y < b"
       by (simp_all add: abs_less_iff)
     assume "g y = g x"
-    hence "f (g y) = f (g x)" by simp
-    hence "y = x" using inj y a b by simp
+    then have "f (g y) = f (g x)" by simp
+    then have "y = x" using inj y x by simp
     also assume "y \<noteq> x"
     finally show False by simp
   qed
   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) \<midarrow>x\<rightarrow> D"
     using cont 1 2 by (rule isCont_LIM_compose2)
-  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
-        \<midarrow>x\<rightarrow> inverse D"
+  then show "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x))) \<midarrow>x\<rightarrow> inverse D"
     using neq by (rule tendsto_inverse)
 qed
 
@@ -1577,65 +1652,67 @@
     and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
     and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
   shows "\<exists>g'c f'c c.
-    DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
+    DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
 proof -
-  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
-  from assms have "a < b" by simp
-  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
-    using fc gc by simp
-  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
-    using fd gd by simp
-  ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
-  then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
-  then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
+  let ?h = "\<lambda>x. (f b - f a) * g x - (g b - g a) * f x"
+  have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l"
+  proof (rule MVT)
+    from assms show "a < b" by simp
+    show "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
+      using fc gc by simp
+    show "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
+      using fd gd by simp
+  qed
+  then obtain l where l: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
+  then obtain c where c: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
 
-  from cdef have cint: "a < c \<and> c < b" by auto
+  from c have cint: "a < c \<and> c < b" by auto
   with gd have "g differentiable (at c)" by simp
-  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
-  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
+  then have "\<exists>D. DERIV g c :> D" by (rule differentiableD)
+  then obtain g'c where g'c: "DERIV g c :> g'c" ..
 
-  from cdef have "a < c \<and> c < b" by auto
+  from c have "a < c \<and> c < b" by auto
   with fd have "f differentiable (at c)" by simp
-  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
-  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
+  then have "\<exists>D. DERIV f c :> D" by (rule differentiableD)
+  then obtain f'c where f'c: "DERIV f c :> f'c" ..
 
-  from cdef have "DERIV ?h c :> l" by auto
+  from c have "DERIV ?h c :> l" by auto
   moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
-    using g'cdef f'cdef by (auto intro!: derivative_eq_intros)
+    using g'c f'c by (auto intro!: derivative_eq_intros)
   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
 
-  {
-    from cdef have "?h b - ?h a = (b - a) * l" by auto
+  have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))"
+  proof -
+    from c have "?h b - ?h a = (b - a) * l" by auto
     also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
-    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
-  }
-  moreover
-  {
+    finally show ?thesis by simp
+  qed
+  moreover have "?h b - ?h a = 0"
+  proof -
     have "?h b - ?h a =
-         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
-          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
+      ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
+      ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
       by (simp add: algebra_simps)
-    hence "?h b - ?h a = 0" by auto
-  }
+    then show ?thesis  by auto
+  qed
   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
-  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
-  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
-
-  with g'cdef f'cdef cint show ?thesis by auto
+  then have "g'c * (f b - f a) = f'c * (g b - g a)" by simp
+  then have "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
+  with g'c f'c cint show ?thesis by auto
 qed
 
 lemma GMVT':
   fixes f g :: "real \<Rightarrow> real"
   assumes "a < b"
-  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
-  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
-  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
-  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
+    and isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
+    and isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
+    and DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
+    and DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
   shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
 proof -
   have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
-    a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
+      a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
     using assms by (intro GMVT) (force simp: real_differentiable_def)+
   then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
     using DERIV_f DERIV_g by (force dest: DERIV_unique)
@@ -1648,25 +1725,26 @@
 
 lemma isCont_If_ge:
   fixes a :: "'a :: linorder_topology"
-  shows "continuous (at_left a) g \<Longrightarrow> (f \<longlongrightarrow> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
+  shows "continuous (at_left a) g \<Longrightarrow> (f \<longlongrightarrow> g a) (at_right a) \<Longrightarrow>
+    isCont (\<lambda>x. if x \<le> a then g x else f x) a"
   unfolding isCont_def continuous_within
   apply (intro filterlim_split_at)
-  apply (subst filterlim_cong[OF refl refl, where g=g])
-  apply (simp_all add: eventually_at_filter less_le)
+   apply (subst filterlim_cong[OF refl refl, where g=g])
+    apply (simp_all add: eventually_at_filter less_le)
   apply (subst filterlim_cong[OF refl refl, where g=f])
-  apply (simp_all add: eventually_at_filter less_le)
+   apply (simp_all add: eventually_at_filter less_le)
   done
 
 lemma lhopital_right_0:
   fixes f0 g0 :: "real \<Rightarrow> real"
   assumes f_0: "(f0 \<longlongrightarrow> 0) (at_right 0)"
-  assumes g_0: "(g0 \<longlongrightarrow> 0) (at_right 0)"
-  assumes ev:
-    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
-    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
-    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
-    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
-  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
+    and g_0: "(g0 \<longlongrightarrow> 0) (at_right 0)"
+    and ev:
+      "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
+      "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
+      "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
+      "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
+    and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
   shows "((\<lambda> x. f0 x / g0 x) \<longlongrightarrow> x) (at_right 0)"
 proof -
   define f where [abs_def]: "f x = (if x \<le> 0 then 0 else f0 x)" for x
@@ -1688,15 +1766,15 @@
   have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
     using g0_neq_0 by (simp add: g_def)
 
-  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
-      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
-         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
-  note f = this
+  have f: "DERIV f x :> (f' x)" if x: "0 < x" "x < a" for x
+    using that
+    by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
+      (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])
 
-  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
-      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
-         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
-  note g = this
+  have g: "DERIV g x :> (g' x)" if x: "0 < x" "x < a" for x
+    using that
+    by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
+         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])
 
   have "isCont f 0"
     unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
@@ -1705,8 +1783,9 @@
     unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
 
   have "\<exists>\<zeta>. \<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
-  proof (rule bchoice, rule)
-    fix x assume "x \<in> {0 <..< a}"
+  proof (rule bchoice, rule ballI)
+    fix x
+    assume "x \<in> {0 <..< a}"
     then have x[arith]: "0 < x" "x < a" by auto
     with g'_neq_0 g_neq_0 \<open>g 0 = 0\<close> have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
       by auto
@@ -1747,53 +1826,57 @@
 qed
 
 lemma lhopital_right:
-  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at_right x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_right x) \<Longrightarrow>
+  "(f \<longlongrightarrow> 0) (at_right x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
     ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_right x) \<Longrightarrow>
   ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_right x)"
+  for x :: real
   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
   by (rule lhopital_right_0)
 
 lemma lhopital_left:
-  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at_left x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_left x) \<Longrightarrow>
+  "(f \<longlongrightarrow> 0) (at_left x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
     ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
   ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
+  for x :: real
   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
   by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
 
 lemma lhopital:
-  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at x) \<Longrightarrow>
+  "(f \<longlongrightarrow> 0) (at x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at x) \<Longrightarrow>
     eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
     ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at x) \<Longrightarrow>
   ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at x)"
+  for x :: real
   unfolding eventually_at_split filterlim_at_split
   by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
 
 lemma lhopital_right_0_at_top:
   fixes f g :: "real \<Rightarrow> real"
   assumes g_0: "LIM x at_right 0. g x :> at_top"
-  assumes ev:
-    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
-    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
-    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
-  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
+    and ev:
+      "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
+      "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
+      "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
+    and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
   shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) (at_right 0)"
   unfolding tendsto_iff
 proof safe
-  fix e :: real assume "0 < e"
-
+  fix e :: real
+  assume "0 < e"
   with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
-  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
+  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)"
+    by simp
   from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
   obtain a where [arith]: "0 < a"
     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
@@ -1801,10 +1884,8 @@
     and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
     and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
     unfolding eventually_at_le by (auto simp: dist_real_def)
-    
 
-  from Df have
-    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
+  from Df have "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
     unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
 
   moreover
@@ -1824,7 +1905,8 @@
     by (auto elim!: eventually_mono simp: dist_real_def)
 
   moreover
-  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) \<longlongrightarrow> norm ((f a - x * g a) * 0)) (at_right 0)"
+  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) \<longlongrightarrow> norm ((f a - x * g a) * 0))
+      (at_right 0)"
     by (intro tendsto_intros)
   then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) \<longlongrightarrow> 0) (at_right 0)"
     by (simp add: inverse_eq_divide)
@@ -1870,12 +1952,13 @@
   by (rule lhopital_right_0_at_top)
 
 lemma lhopital_left_at_top:
-  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
+  "LIM x at_left x. g x :> at_top \<Longrightarrow>
     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
     ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
     ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
+  for x :: real
   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
   by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
 
@@ -1892,10 +1975,10 @@
 lemma lhospital_at_top_at_top:
   fixes f g :: "real \<Rightarrow> real"
   assumes g_0: "LIM x at_top. g x :> at_top"
-  assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
-  assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
-  assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
-  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) at_top"
+    and g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
+    and Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
+    and Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
+    and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) at_top"
   shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) at_top"
   unfolding filterlim_at_top_to_right
 proof (rule lhopital_right_0_at_top)
@@ -1903,37 +1986,32 @@
   let ?G = "\<lambda>x. g (inverse x)"
   let ?R = "at_right (0::real)"
   let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
-
   show "LIM x ?R. ?G x :> at_top"
     using g_0 unfolding filterlim_at_top_to_right .
-
   show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
     unfolding eventually_at_right_to_top
-    using Dg eventually_ge_at_top[where c="1::real"]
+    using Dg eventually_ge_at_top[where c=1]
     apply eventually_elim
     apply (rule DERIV_cong)
-    apply (rule DERIV_chain'[where f=inverse])
-    apply (auto intro!:  DERIV_inverse)
+     apply (rule DERIV_chain'[where f=inverse])
+      apply (auto intro!:  DERIV_inverse)
     done
-
   show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
     unfolding eventually_at_right_to_top
-    using Df eventually_ge_at_top[where c="1::real"]
+    using Df eventually_ge_at_top[where c=1]
     apply eventually_elim
     apply (rule DERIV_cong)
-    apply (rule DERIV_chain'[where f=inverse])
-    apply (auto intro!:  DERIV_inverse)
+     apply (rule DERIV_chain'[where f=inverse])
+      apply (auto intro!:  DERIV_inverse)
     done
-
   show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
     unfolding eventually_at_right_to_top
-    using g' eventually_ge_at_top[where c="1::real"]
+    using g' eventually_ge_at_top[where c=1]
     by eventually_elim auto
-    
   show "((\<lambda>x. ?D f' x / ?D g' x) \<longlongrightarrow> x) ?R"
     unfolding filterlim_at_right_to_top
     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
-    using eventually_ge_at_top[where c="1::real"]
+    using eventually_ge_at_top[where c=1]
     by eventually_elim simp
 qed
 
--- a/src/HOL/NthRoot.thy	Wed Jul 27 10:24:50 2016 +0200
+++ b/src/HOL/NthRoot.thy	Thu Jul 28 20:39:46 2016 +0200
@@ -110,10 +110,14 @@
 qed
 
 lemma split_root: "P (root n x) \<longleftrightarrow> (n = 0 \<longrightarrow> P 0) \<and> (0 < n \<longrightarrow> (\<forall>y. sgn y * \<bar>y\<bar>^n = x \<longrightarrow> P y))"
-  apply (cases "n = 0")
-  apply simp_all
-  apply (metis root_sgn_power sgn_power_root)
-  done
+proof (cases "n = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
+  then show ?thesis
+    by simp (metis root_sgn_power sgn_power_root)
+qed
 
 lemma real_root_zero [simp]: "root n 0 = 0"
   by (simp split: split_root add: sgn_zero_iff)
@@ -176,16 +180,10 @@
   by (auto simp add: order_le_less real_root_less_mono)
 
 lemma real_root_less_iff [simp]: "0 < n \<Longrightarrow> root n x < root n y \<longleftrightarrow> x < y"
-  apply (cases "x < y")
-  apply (simp add: real_root_less_mono)
-  apply (simp add: linorder_not_less real_root_le_mono)
-  done
+  by (cases "x < y") (simp_all add: real_root_less_mono linorder_not_less real_root_le_mono)
 
 lemma real_root_le_iff [simp]: "0 < n \<Longrightarrow> root n x \<le> root n y \<longleftrightarrow> x \<le> y"
-  apply (cases "x \<le> y")
-  apply (simp add: real_root_le_mono)
-  apply (simp add: linorder_not_le real_root_less_mono)
-  done
+  by (cases "x \<le> y") (simp_all add: real_root_le_mono linorder_not_le real_root_less_mono)
 
 lemma real_root_eq_iff [simp]: "0 < n \<Longrightarrow> root n x = root n y \<longleftrightarrow> x = y"
   by (simp add: order_eq_iff)
@@ -248,17 +246,23 @@
 
 text \<open>Monotonicity in first argument.\<close>
 
-lemma real_root_strict_decreasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 1 < x \<Longrightarrow> root N x < root n x"
-  apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N")
-  apply simp
-  apply (simp add: real_root_commute power_strict_increasing del: real_root_pow_pos2)
-  done
+lemma real_root_strict_decreasing:
+  assumes "0 < n" "n < N" "1 < x"
+  shows "root N x < root n x"
+proof -
+  from assms have "root n (root N x) ^ n < root N (root n x) ^ N"
+    by (simp add: real_root_commute power_strict_increasing del: real_root_pow_pos2)
+  with assms show ?thesis by simp
+qed
 
-lemma real_root_strict_increasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 0 < x \<Longrightarrow> x < 1 \<Longrightarrow> root n x < root N x"
-  apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n")
-  apply simp
-  apply (simp add: real_root_commute power_strict_decreasing del: real_root_pow_pos2)
-  done
+lemma real_root_strict_increasing:
+  assumes "0 < n" "n < N" "0 < x" "x < 1"
+  shows "root n x < root N x"
+proof -
+  from assms have "root N (root n x) ^ N < root n (root N x) ^ n"
+    by (simp add: real_root_commute power_strict_decreasing del: real_root_pow_pos2)
+  with assms show ?thesis by simp
+qed
 
 lemma real_root_decreasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 1 \<le> x \<Longrightarrow> root N x \<le> root n x"
   by (auto simp add: order_le_less real_root_strict_decreasing)
@@ -365,16 +369,17 @@
 qed
 
 lemma DERIV_real_root_generic:
-  assumes "0 < n" and "x \<noteq> 0"
-    and "\<lbrakk> even n ; 0 < x \<rbrakk> \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
-    and "\<lbrakk> even n ; x < 0 \<rbrakk> \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
+  assumes "0 < n"
+    and "x \<noteq> 0"
+    and "even n \<Longrightarrow> 0 < x \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
+    and "even n \<Longrightarrow> x < 0 \<Longrightarrow> D = - inverse (real n * root n x ^ (n - Suc 0))"
     and "odd n \<Longrightarrow> D = inverse (real n * root n x ^ (n - Suc 0))"
   shows "DERIV (root n) x :> D"
   using assms
-  by (cases "even n", cases "0 < x",
-      auto intro: DERIV_real_root[THEN DERIV_cong]
-              DERIV_odd_real_root[THEN DERIV_cong]
-              DERIV_even_real_root[THEN DERIV_cong])
+  by (cases "even n", cases "0 < x")
+    (auto intro: DERIV_real_root[THEN DERIV_cong]
+      DERIV_odd_real_root[THEN DERIV_cong]
+      DERIV_even_real_root[THEN DERIV_cong])
 
 
 subsection \<open>Square Root\<close>
@@ -390,17 +395,17 @@
 
 lemma real_sqrt_abs [simp]: "sqrt (x\<^sup>2) = \<bar>x\<bar>"
   apply (rule real_sqrt_unique)
-  apply (rule power2_abs)
+   apply (rule power2_abs)
   apply (rule abs_ge_zero)
   done
 
 lemma real_sqrt_pow2 [simp]: "0 \<le> x \<Longrightarrow> (sqrt x)\<^sup>2 = x"
   unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])
 
-lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<^sup>2 = x) = (0 \<le> x)"
+lemma real_sqrt_pow2_iff [simp]: "(sqrt x)\<^sup>2 = x \<longleftrightarrow> 0 \<le> x"
   apply (rule iffI)
-  apply (erule subst)
-  apply (rule zero_le_power2)
+   apply (erule subst)
+   apply (rule zero_le_power2)
   apply (erule real_sqrt_pow2)
   done
 
@@ -443,13 +448,13 @@
 lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> sqrt y"
   unfolding sqrt_def by (rule real_root_le_mono [OF pos2])
 
-lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
+lemma real_sqrt_less_iff [simp]: "sqrt x < sqrt y \<longleftrightarrow> x < y"
   unfolding sqrt_def by (rule real_root_less_iff [OF pos2])
 
-lemma real_sqrt_le_iff [simp]: "(sqrt x \<le> sqrt y) = (x \<le> y)"
+lemma real_sqrt_le_iff [simp]: "sqrt x \<le> sqrt y \<longleftrightarrow> x \<le> y"
   unfolding sqrt_def by (rule real_root_le_iff [OF pos2])
 
-lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
+lemma real_sqrt_eq_iff [simp]: "sqrt x = sqrt y \<longleftrightarrow> x = y"
   unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])
 
 lemma real_less_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y\<^sup>2 \<Longrightarrow> sqrt x < y"
@@ -525,9 +530,10 @@
   DERIV_real_sqrt_generic[THEN DERIV_chain2, derivative_intros]
   DERIV_real_root_generic[THEN DERIV_chain2, derivative_intros]
 
-lemma not_real_square_gt_zero [simp]: "\<not> 0 < x * x \<longleftrightarrow> x = 0" for x :: real
+lemma not_real_square_gt_zero [simp]: "\<not> 0 < x * x \<longleftrightarrow> x = 0"
+  for x :: real
   apply auto
-  apply (cut_tac x = x and y = 0 in linorder_less_linear)
+  using linorder_less_linear [where x = x and y = 0]
   apply (simp add: zero_less_mult_iff)
   done
 
@@ -566,22 +572,17 @@
 qed
 
 lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
-  apply (cases "x = 0")
-  apply simp_all
-  using sqrt_divide_self_eq[of x]
-  apply (simp add: field_simps)
-  done
+  by (cases "x = 0") (simp_all add: sqrt_divide_self_eq [of x] field_simps)
 
-lemma real_divide_square_eq [simp]: "(r * a) / (r * r) = a / r" for a r :: real
-  apply (simp add: divide_inverse)
-  apply (case_tac "r = 0")
-  apply (auto simp add: ac_simps)
-  done
+lemma real_divide_square_eq [simp]: "(r * a) / (r * r) = a / r"
+  for a r :: real
+  by (cases "r = 0") (simp_all add: divide_inverse ac_simps)
 
 lemma lemma_real_divide_sqrt_less: "0 < u \<Longrightarrow> u / sqrt 2 < u"
   by (simp add: divide_less_eq)
 
-lemma four_x_squared: "4 * x\<^sup>2 = (2 * x)\<^sup>2" for x :: real
+lemma four_x_squared: "4 * x\<^sup>2 = (2 * x)\<^sup>2"
+  for x :: real
   by (simp add: power2_eq_square)
 
 lemma sqrt_at_top: "LIM x at_top. sqrt x :: real :> at_top"
@@ -591,7 +592,8 @@
 
 subsection \<open>Square Root of Sum of Squares\<close>
 
-lemma sum_squares_bound: "2 * x * y \<le> x\<^sup>2 + y\<^sup>2" for x y :: "'a::linordered_field"
+lemma sum_squares_bound: "2 * x * y \<le> x\<^sup>2 + y\<^sup>2"
+  for x y :: "'a::linordered_field"
 proof -
   have "(x - y)\<^sup>2 = x * x - 2 * x * y + y * y"
     by algebra
@@ -612,13 +614,15 @@
   done
 
 lemma arith_geo_mean_sqrt:
-  fixes x::real assumes "x\<ge>0" "y\<ge>0" shows "sqrt(x*y) \<le> (x + y)/2"
+  fixes x :: real
+  assumes "x \<ge> 0" "y \<ge> 0"
+  shows "sqrt (x * y) \<le> (x + y)/2"
   apply (rule arith_geo_mean)
   using assms
   apply (auto simp: zero_le_mult_iff)
   done
 
-lemma real_sqrt_sum_squares_mult_ge_zero [simp]: "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2)*(xa\<^sup>2 + ya\<^sup>2))"
+lemma real_sqrt_sum_squares_mult_ge_zero [simp]: "0 \<le> sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2))"
   by (metis real_sqrt_ge_0_iff split_mult_pos_le sum_power2_ge_zero)
 
 lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
@@ -649,29 +653,29 @@
 lemma real_sqrt_sum_squares_triangle_ineq:
   "sqrt ((a + c)\<^sup>2 + (b + d)\<^sup>2) \<le> sqrt (a\<^sup>2 + b\<^sup>2) + sqrt (c\<^sup>2 + d\<^sup>2)"
   apply (rule power2_le_imp_le)
-  apply simp
-  apply (simp add: power2_sum)
-  apply (simp only: mult.assoc distrib_left [symmetric])
-  apply (rule mult_left_mono)
-  apply (rule power2_le_imp_le)
-  apply (simp add: power2_sum power_mult_distrib)
-  apply (simp add: ring_distribs)
-  apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)")
-  apply simp
-  apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
-  apply (rule zero_le_power2)
-  apply (simp add: power2_diff power_mult_distrib)
-  apply simp
-  apply simp
+   apply simp
+   apply (simp add: power2_sum)
+   apply (simp only: mult.assoc distrib_left [symmetric])
+   apply (rule mult_left_mono)
+    apply (rule power2_le_imp_le)
+     apply (simp add: power2_sum power_mult_distrib)
+     apply (simp add: ring_distribs)
+     apply (subgoal_tac "0 \<le> b\<^sup>2 * c\<^sup>2 + a\<^sup>2 * d\<^sup>2 - 2 * (a * c) * (b * d)")
+      apply simp
+     apply (rule_tac b="(a * d - b * c)\<^sup>2" in ord_le_eq_trans)
+      apply (rule zero_le_power2)
+     apply (simp add: power2_diff power_mult_distrib)
+    apply simp
+   apply simp
   apply (simp add: add_increasing)
   done
 
 lemma real_sqrt_sum_squares_less: "\<bar>x\<bar> < u / sqrt 2 \<Longrightarrow> \<bar>y\<bar> < u / sqrt 2 \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
   apply (rule power2_less_imp_less)
-  apply simp
-  apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
-  apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
-  apply (simp add: power_divide)
+   apply simp
+   apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
+   apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
+   apply (simp add: power_divide)
   apply (drule order_le_less_trans [OF abs_ge_zero])
   apply (simp add: zero_less_divide_iff)
   done
@@ -685,14 +689,14 @@
 lemma lemma_sqrt_hcomplex_capprox:
   "0 < u \<Longrightarrow> x < u/2 \<Longrightarrow> y < u/2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> sqrt (x\<^sup>2 + y\<^sup>2) < u"
   apply (rule real_sqrt_sum_squares_less)
-  apply (auto simp add: abs_if field_simps)
-  apply (rule le_less_trans [where y = "x*2"])
+   apply (auto simp add: abs_if field_simps)
+   apply (rule le_less_trans [where y = "x*2"])
   using less_eq_real_def sqrt2_less_2
-  apply force
-  apply assumption
+    apply force
+   apply assumption
   apply (rule le_less_trans [where y = "y*2"])
   using less_eq_real_def sqrt2_less_2 mult_le_cancel_left
-  apply auto
+   apply auto
   done
 
 lemma LIMSEQ_root: "(\<lambda>n. root n n) \<longlonglongrightarrow> 1"
--- a/src/HOL/Transcendental.thy	Wed Jul 27 10:24:50 2016 +0200
+++ b/src/HOL/Transcendental.thy	Thu Jul 28 20:39:46 2016 +0200
@@ -4,7 +4,7 @@
     Author:     Jeremy Avigad
 *)
 
-section\<open>Power Series, Transcendental Functions etc.\<close>
+section \<open>Power Series, Transcendental Functions etc.\<close>
 
 theory Transcendental
 imports Binomial Series Deriv NthRoot
@@ -79,8 +79,7 @@
 proof -
   have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
     by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
-  then show ?thesis
-    by (simp add: One_nat_def)
+  then show ?thesis by simp
 qed
 
 lemma powser_sums_zero: "(\<lambda>n. a n * 0^n) sums a 0"
@@ -105,18 +104,17 @@
   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
   from 1 have "(\<lambda>n. f n * x^n) \<longlonglongrightarrow> 0"
     by (rule summable_LIMSEQ_zero)
-  hence "convergent (\<lambda>n. f n * x^n)"
+  then have "convergent (\<lambda>n. f n * x^n)"
     by (rule convergentI)
-  hence "Cauchy (\<lambda>n. f n * x^n)"
+  then have "Cauchy (\<lambda>n. f n * x^n)"
     by (rule convergent_Cauchy)
-  hence "Bseq (\<lambda>n. f n * x^n)"
+  then have "Bseq (\<lambda>n. f n * x^n)"
     by (rule Cauchy_Bseq)
   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K"
-    by (simp add: Bseq_def, safe)
-  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
-                   K * norm (z ^ n) * inverse (norm (x^n))"
+    by (auto simp add: Bseq_def)
+  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
   proof (intro exI allI impI)
-    fix n::nat
+    fix n :: nat
     assume "0 \<le> n"
     have "norm (norm (f n * z ^ n)) * norm (x^n) =
           norm (f n * x^n) * norm (z ^ n)"
@@ -127,8 +125,7 @@
       by (simp add: x_neq_0)
     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
       by (simp only: mult.assoc)
-    finally show "norm (norm (f n * z ^ n)) \<le>
-                  K * norm (z ^ n) * inverse (norm (x^n))"
+    finally show "norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
       by (simp add: mult_le_cancel_right x_neq_0)
   qed
   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
@@ -136,14 +133,14 @@
     from 2 have "norm (norm (z * inverse x)) < 1"
       using x_neq_0
       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
-    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
+    then have "summable (\<lambda>n. norm (z * inverse x) ^ n)"
       by (rule summable_geometric)
-    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
+    then have "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
       by (rule summable_mult)
-    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
+    then show "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
       using x_neq_0
       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
-                    power_inverse norm_power mult.assoc)
+          power_inverse norm_power mult.assoc)
   qed
   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
     by (rule summable_comparison_test)
@@ -162,45 +159,38 @@
     shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0"
 proof -
   have "norm x / (1 - norm x) \<ge> 0"
-    using assms
-    by (auto simp: divide_simps)
+    using assms by (auto simp: divide_simps)
   moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
-    using ex_le_of_int
-    by (meson ex_less_of_int)
+    using ex_le_of_int by (meson ex_less_of_int)
   ultimately have N0: "N>0"
     by auto
   then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
-    using N assms
-    by (auto simp: field_simps)
-  { fix n::nat
-    assume "N \<le> int n"
-    then have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)"
+    using N assms by (auto simp: field_simps)
+  have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) \<le>
+      real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N \<le> int n" for n :: nat
+  proof -
+    from that have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)"
       by (simp add: algebra_simps)
-    then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n))
-               \<le> (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
+    then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n)) \<le>
+        (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
       using N0 mult_mono by fastforce
-    then have "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n)))
-         \<le> real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))"
+    then show ?thesis
       by (simp add: algebra_simps)
-  } note ** = this
+  qed
   show ?thesis using *
-    apply (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
-    apply (simp add: N0 norm_mult field_simps **
-                del: of_nat_Suc of_int_add)
-    done
+    by (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
+      (simp add: N0 norm_mult field_simps ** del: of_nat_Suc of_int_add)
 qed
 
 corollary lim_n_over_pown:
   fixes x :: "'a::{real_normed_field,banach}"
   shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially"
-using powser_times_n_limit_0 [of "inverse x"]
-by (simp add: norm_divide divide_simps)
+  using powser_times_n_limit_0 [of "inverse x"]
+  by (simp add: norm_divide divide_simps)
 
 lemma sum_split_even_odd:
   fixes f :: "nat \<Rightarrow> real"
-  shows
-    "(\<Sum>i<2 * n. if even i then f i else g i) =
-     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
+  shows "(\<Sum>i<2 * n. if even i then f i else g i) = (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
 proof (induct n)
   case 0
   then show ?case by simp
@@ -223,33 +213,35 @@
   fix r :: real
   assume "0 < r"
   from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this]
-  obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast
+  obtain no where no_eq: "\<And>n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)"
+    by blast
 
   let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
-  {
-    fix m
-    assume "m \<ge> 2 * no"
-    hence "m div 2 \<ge> no" by auto
+  have "(norm (?SUM m - x) < r)" if "m \<ge> 2 * no" for m
+  proof -
+    from that have "m div 2 \<ge> no" by auto
     have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}"
       using sum_split_even_odd by auto
-    hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
+    then have "(norm (?SUM (2 * (m div 2)) - x) < r)"
       using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto
     moreover
     have "?SUM (2 * (m div 2)) = ?SUM m"
     proof (cases "even m")
       case True
-      then show ?thesis by (auto simp add: even_two_times_div_two)
+      then show ?thesis
+        by (auto simp add: even_two_times_div_two)
     next
       case False
       then have eq: "Suc (2 * (m div 2)) = m" by simp
-      hence "even (2 * (m div 2))" using \<open>odd m\<close> by auto
+      then have "even (2 * (m div 2))" using \<open>odd m\<close> by auto
       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
       also have "\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto
       finally show ?thesis by auto
     qed
-    ultimately have "(norm (?SUM m - x) < r)" by auto
-  }
-  thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
+    ultimately show ?thesis by auto
+  qed
+  then show "\<exists>no. \<forall> m \<ge> no. norm (?SUM m - x) < r"
+    by blast
 qed
 
 lemma sums_if:
@@ -258,70 +250,70 @@
   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
 proof -
   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
-  {
-    fix B T E
-    have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
-      by (cases B) auto
-  } note if_sum = this
+  have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
+    for B T E
+    by (cases B) auto
   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
     using sums_if'[OF \<open>g sums x\<close>] .
-  {
-    have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
-
-    have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] .
-    from this[unfolded sums_def, THEN LIMSEQ_Suc]
-    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
-      by (simp add: lessThan_Suc_eq_insert_0 setsum_atLeast1_atMost_eq image_Suc_lessThan
+  have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)"
+    by auto
+  have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] .
+  from this[unfolded sums_def, THEN LIMSEQ_Suc]
+  have "(\<lambda>n. if even n then f (n div 2) else 0) sums y"
+    by (simp add: lessThan_Suc_eq_insert_0 setsum_atLeast1_atMost_eq image_Suc_lessThan
         if_eq sums_def cong del: if_cong)
-  }
-  from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
+  from sums_add[OF g_sums this] show ?thesis
+    by (simp only: if_sum)
 qed
 
 subsection \<open>Alternating series test / Leibniz formula\<close>
-text\<open>FIXME: generalise these results from the reals via type classes?\<close>
+(* FIXME: generalise these results from the reals via type classes? *)
 
 lemma sums_alternating_upper_lower:
   fixes a :: "nat \<Rightarrow> real"
-  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a \<longlonglongrightarrow> 0"
+  assumes mono: "\<And>n. a (Suc n) \<le> a n"
+    and a_pos: "\<And>n. 0 \<le> a n"
+    and "a \<longlonglongrightarrow> 0"
   shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> l) \<and>
              ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)"
   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
 proof (rule nested_sequence_unique)
-  have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
+  have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" by auto
 
   show "\<forall>n. ?f n \<le> ?f (Suc n)"
   proof
-    fix n
-    show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
+    show "?f n \<le> ?f (Suc n)" for n
+      using mono[of "2*n"] by auto
   qed
   show "\<forall>n. ?g (Suc n) \<le> ?g n"
   proof
-    fix n
-    show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
-      unfolding One_nat_def by auto
+    show "?g (Suc n) \<le> ?g n" for n
+      using mono[of "Suc (2*n)"] by auto
   qed
   show "\<forall>n. ?f n \<le> ?g n"
   proof
-    fix n
-    show "?f n \<le> ?g n" using fg_diff a_pos
-      unfolding One_nat_def by auto
+    show "?f n \<le> ?g n" for n
+      using fg_diff a_pos by auto
   qed
-  show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0" unfolding fg_diff
+  show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0"
+    unfolding fg_diff
   proof (rule LIMSEQ_I)
     fix r :: real
     assume "0 < r"
     with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
       by auto
-    hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
-    thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
+    then have "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
+      by auto
+    then show "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
+      by auto
   qed
 qed
 
 lemma summable_Leibniz':
   fixes a :: "nat \<Rightarrow> real"
   assumes a_zero: "a \<longlonglongrightarrow> 0"
-    and a_pos: "\<And> n. 0 \<le> a n"
-    and a_monotone: "\<And> n. a (Suc n) \<le> a n"
+    and a_pos: "\<And>n. 0 \<le> a n"
+    and a_monotone: "\<And>n. a (Suc n) \<le> a n"
   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
     and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
     and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
@@ -345,44 +337,44 @@
     fix r :: real
     assume "0 < r"
     with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
-    obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
-
+    obtain f_no where f: "\<And>n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r"
+      by auto
     from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
-    obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
-
-    {
-      fix n :: nat
-      assume "n \<ge> (max (2 * f_no) (2 * g_no))"
-      hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
-      have "norm (?Sa n - l) < r"
+    obtain g_no where g: "\<And>n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r"
+      by auto
+    have "norm (?Sa n - l) < r" if "n \<ge> (max (2 * f_no) (2 * g_no))" for n
+    proof -
+      from that have "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
+      show ?thesis
       proof (cases "even n")
         case True
-        then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two)
+        then have n_eq: "2 * (n div 2) = n"
+          by (simp add: even_two_times_div_two)
         with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no"
           by auto
         from f[OF this] show ?thesis
           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
       next
         case False
-        hence "even (n - 1)" by simp
+        then have "even (n - 1)" by simp
         then have n_eq: "2 * ((n - 1) div 2) = n - 1"
           by (simp add: even_two_times_div_two)
-        hence range_eq: "n - 1 + 1 = n"
+        then have range_eq: "n - 1 + 1 = n"
           using odd_pos[OF False] by auto
-
         from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no"
           by auto
         from g[OF this] show ?thesis
-          unfolding n_eq range_eq .
+          by (simp only: n_eq range_eq)
       qed
-    }
-    thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
+    qed
+    then show "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
   qed
-  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
-    unfolding sums_def .
-  thus "summable ?S" using summable_def by auto
-
-  have "l = suminf ?S" using sums_unique[OF sums_l] .
+  then have sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
+    by (simp only: sums_def)
+  then show "summable ?S"
+    by (auto simp: summable_def)
+
+  have "l = suminf ?S" by (rule sums_unique[OF sums_l])
 
   fix n
   show "suminf ?S \<le> ?g n"
@@ -397,7 +389,8 @@
 
 theorem summable_Leibniz:
   fixes a :: "nat \<Rightarrow> real"
-  assumes a_zero: "a \<longlonglongrightarrow> 0" and "monoseq a"
+  assumes a_zero: "a \<longlonglongrightarrow> 0"
+    and "monoseq a"
   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
     and "0 < a 0 \<longrightarrow>
       (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos")
@@ -407,30 +400,25 @@
     and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?g")
 proof -
   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
-  proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
+  proof (cases "(\<forall>n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
     case True
-    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
+    then have ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m"
+      and ge0: "\<And>n. 0 \<le> a n"
       by auto
-    {
-      fix n
-      have "a (Suc n) \<le> a n"
-        using ord[where n="Suc n" and m=n] by auto
-    } note mono = this
+    have mono: "a (Suc n) \<le> a n" for n
+      using ord[where n="Suc n" and m=n] by auto
     note leibniz = summable_Leibniz'[OF \<open>a \<longlonglongrightarrow> 0\<close> ge0]
     from leibniz[OF mono]
     show ?thesis using \<open>0 \<le> a 0\<close> by auto
   next
-    let ?a = "\<lambda> n. - a n"
+    let ?a = "\<lambda>n. - a n"
     case False
     with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>]
     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
-    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
+    then have ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
       by auto
-    {
-      fix n
-      have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
-        by auto
-    } note monotone = this
+    have monotone: "?a (Suc n) \<le> ?a n" for n
+      using ord[where n="Suc n" and m=n] by auto
     note leibniz =
       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
         OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone]
@@ -440,9 +428,9 @@
       unfolding summable_def by auto
     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
       by auto
-    hence ?summable unfolding summable_def by auto
+    then have ?summable by (auto simp: summable_def)
     moreover
-    have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
+    have "\<bar>- a - - b\<bar> = \<bar>a - b\<bar>" for a b :: real
       unfolding minus_diff_minus by auto
 
     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
@@ -463,60 +451,60 @@
     by safe
 qed
 
+
 subsection \<open>Term-by-Term Differentiability of Power Series\<close>
 
 definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
   where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
 
-text\<open>Lemma about distributing negation over it\<close>
+text \<open>Lemma about distributing negation over it.\<close>
 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
   by (simp add: diffs_def)
 
 lemma diffs_equiv:
-  fixes x :: "'a::{real_normed_vector, ring_1}"
+  fixes x :: "'a::{real_normed_vector,ring_1}"
   shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
-      (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
+    (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
   unfolding diffs_def
   by (simp add: summable_sums sums_Suc_imp)
 
 lemma lemma_termdiff1:
-  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
-  "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
-   (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
+  fixes z :: "'a :: {monoid_mult,comm_ring}"
+  shows "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
+    (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
   by (auto simp add: algebra_simps power_add [symmetric])
 
-lemma sumr_diff_mult_const2:
-  "setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)"
+lemma sumr_diff_mult_const2: "setsum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)"
+  for r :: "'a::ring_1"
   by (simp add: setsum_subtractf)
 
 lemma lemma_realpow_rev_sumr:
-   "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) =
-    (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
+  "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) = (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
   by (subst nat_diff_setsum_reindex[symmetric]) simp
 
 lemma lemma_termdiff2:
-  fixes h :: "'a :: {field}"
+  fixes h :: "'a::field"
   assumes h: "h \<noteq> 0"
-  shows
-    "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
-     h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
-          (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
-  apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
+  shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
+    h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
+    (is "?lhs = ?rhs")
+  apply (subgoal_tac "h * ?lhs = h * ?rhs")
+   apply (simp add: h)
   apply (simp add: right_diff_distrib diff_divide_distrib h)
   apply (simp add: mult.assoc [symmetric])
-  apply (cases "n", simp)
-  apply (simp add: diff_power_eq_setsum h
-                   right_diff_distrib [symmetric] mult.assoc
-              del: power_Suc setsum_lessThan_Suc of_nat_Suc)
+  apply (cases n)
+  apply simp
+  apply (simp add: diff_power_eq_setsum h right_diff_distrib [symmetric] mult.assoc
+      del: power_Suc setsum_lessThan_Suc of_nat_Suc)
   apply (subst lemma_realpow_rev_sumr)
   apply (subst sumr_diff_mult_const2)
   apply simp
   apply (simp only: lemma_termdiff1 setsum_right_distrib)
   apply (rule setsum.cong [OF refl])
   apply (simp add: less_iff_Suc_add)
-  apply (clarify)
+  apply clarify
   apply (simp add: setsum_right_distrib diff_power_eq_setsum ac_simps
-              del: setsum_lessThan_Suc power_Suc)
+      del: setsum_lessThan_Suc power_Suc)
   apply (subst mult.assoc [symmetric], subst power_add [symmetric])
   apply (simp add: ac_simps)
   done
@@ -527,21 +515,21 @@
     and K: "0 \<le> K"
   shows "setsum f {..<n-k} \<le> of_nat n * K"
   apply (rule order_trans [OF setsum_mono])
-  apply (rule f, simp)
+   apply (rule f)
+   apply simp
   apply (simp add: mult_right_mono K)
   done
 
 lemma lemma_termdiff3:
-  fixes h z :: "'a::{real_normed_field}"
+  fixes h z :: "'a::real_normed_field"
   assumes 1: "h \<noteq> 0"
     and 2: "norm z \<le> K"
     and 3: "norm (z + h) \<le> K"
-  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
-          \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
+  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \<le>
+    of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
 proof -
   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
-        norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
-          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
+    norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
   proof (rule mult_right_mono [OF _ norm_ge_zero])
@@ -552,15 +540,16 @@
       apply (simp only: norm_mult norm_power power_add)
       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
       done
-    show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
-          \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
+    show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) \<le>
+        of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
       apply (intro
-         order_trans [OF norm_setsum]
-         real_setsum_nat_ivl_bounded2
-         mult_nonneg_nonneg
-         of_nat_0_le_iff
-         zero_le_power K)
-      apply (rule le_Kn, simp)
+          order_trans [OF norm_setsum]
+          real_setsum_nat_ivl_bounded2
+          mult_nonneg_nonneg
+          of_nat_0_le_iff
+          zero_le_power K)
+      apply (rule le_Kn)
+      apply simp
       done
   qed
   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
@@ -570,8 +559,9 @@
 
 lemma lemma_termdiff4:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
-  assumes k: "0 < (k::real)"
-    and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
+    and k :: real
+  assumes k: "0 < k"
+    and le: "\<And>h. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (f h) \<le> K * norm h"
   shows "f \<midarrow>0\<rightarrow> 0"
 proof (rule tendsto_norm_zero_cancel)
   show "(\<lambda>h. norm (f h)) \<midarrow>0\<rightarrow> 0"
@@ -591,24 +581,25 @@
 
 lemma lemma_termdiff5:
   fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
-  assumes k: "0 < (k::real)"
-  assumes f: "summable f"
-  assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
+    and k :: real
+  assumes k: "0 < k"
+    and f: "summable f"
+    and le: "\<And>h n. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (g h n) \<le> f n * norm h"
   shows "(\<lambda>h. suminf (g h)) \<midarrow>0\<rightarrow> 0"
 proof (rule lemma_termdiff4 [OF k])
-  fix h::'a
+  fix h :: 'a
   assume "h \<noteq> 0" and "norm h < k"
-  hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
+  then have 1: "\<forall>n. norm (g h n) \<le> f n * norm h"
     by (simp add: le)
-  hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
+  then have "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
     by simp
-  moreover from f have B: "summable (\<lambda>n. f n * norm h)"
+  moreover from f have 2: "summable (\<lambda>n. f n * norm h)"
     by (rule summable_mult2)
-  ultimately have C: "summable (\<lambda>n. norm (g h n))"
+  ultimately have 3: "summable (\<lambda>n. norm (g h n))"
     by (rule summable_comparison_test)
-  hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
+  then have "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
     by (rule summable_norm)
-  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
+  also from 1 3 2 have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
     by (rule suminf_le)
   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
     by (rule suminf_mult2 [symmetric])
@@ -616,62 +607,63 @@
 qed
 
 
-text\<open>FIXME: Long proofs\<close>
+(* FIXME: Long proofs *)
 
 lemma termdiffs_aux:
   fixes x :: "'a::{real_normed_field,banach}"
   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
     and 2: "norm x < norm K"
-  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h
-             - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
+  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
 proof -
-  from dense [OF 2]
-  obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
+  from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K"
+    by fast
   from norm_ge_zero r1 have r: "0 < r"
     by (rule order_le_less_trans)
-  hence r_neq_0: "r \<noteq> 0" by simp
+  then have r_neq_0: "r \<noteq> 0" by simp
   show ?thesis
   proof (rule lemma_termdiff5)
-    show "0 < r - norm x" using r1 by simp
+    show "0 < r - norm x"
+      using r1 by simp
     from r r2 have "norm (of_real r::'a) < norm K"
       by simp
     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
       by (rule powser_insidea)
-    hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
-      using r
-      by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
-    hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
+    then have "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
+      using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
+    then have "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
       by (rule diffs_equiv [THEN sums_summable])
     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
       apply (rule ext)
       apply (simp add: diffs_def)
-      apply (case_tac n, simp_all add: r_neq_0)
+      apply (case_tac n)
+       apply (simp_all add: r_neq_0)
       done
     finally have "summable
       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
       by (rule diffs_equiv [THEN sums_summable])
     also have
-      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
-           r ^ (n - Suc 0)) =
+      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) =
        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
       apply (rule ext)
-      apply (case_tac "n", simp)
+      apply (case_tac n)
+       apply simp
       apply (rename_tac nat)
-      apply (case_tac "nat", simp)
+      apply (case_tac nat)
+       apply simp
       apply (simp add: r_neq_0)
       done
-    finally
-    show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
+    finally show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
   next
-    fix h::'a and n::nat
+    fix h :: 'a
+    fix n :: nat
     assume h: "h \<noteq> 0"
     assume "norm h < r - norm x"
-    hence "norm x + norm h < r" by simp
+    then have "norm x + norm h < r" by simp
     with norm_triangle_ineq have xh: "norm (x + h) < r"
       by (rule order_le_less_trans)
-    show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))
-          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
+    show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<le>
+      norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
       apply (simp only: norm_mult mult.assoc)
       apply (rule mult_left_mono [OF _ norm_ge_zero])
       apply (simp add: mult.assoc [symmetric])
@@ -683,21 +675,22 @@
 lemma termdiffs:
   fixes K x :: "'a::{real_normed_field,banach}"
   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
-      and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
-      and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
-      and 4: "norm x < norm K"
+    and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
+    and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
+    and 4: "norm x < norm K"
   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)"
   unfolding DERIV_def
 proof (rule LIM_zero_cancel)
   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
             - suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0"
   proof (rule LIM_equal2)
-    show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
+    show "0 < norm K - norm x"
+      using 4 by (simp add: less_diff_eq)
   next
     fix h :: 'a
     assume "norm (h - 0) < norm K - norm x"
-    hence "norm x + norm h < norm K" by simp
-    hence 5: "norm (x + h) < norm K"
+    then have "norm x + norm h < norm K" by simp
+    then have 5: "norm (x + h) < norm K"
       by (rule norm_triangle_ineq [THEN order_le_less_trans])
     have "summable (\<lambda>n. c n * x^n)"
       and "summable (\<lambda>n. c n * (x + h) ^ n)"
@@ -720,16 +713,17 @@
 lemma termdiff_converges:
   fixes x :: "'a::{real_normed_field,banach}"
   assumes K: "norm x < K"
-      and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
-    shows "summable (\<lambda>n. diffs c n * x ^ n)"
+    and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
+  shows "summable (\<lambda>n. diffs c n * x ^ n)"
 proof (cases "x = 0")
-  case True then show ?thesis
-  using powser_sums_zero sums_summable by auto
+  case True
+  then show ?thesis
+    using powser_sums_zero sums_summable by auto
 next
   case False
-  then have "K>0"
+  then have "K > 0"
     using K less_trans zero_less_norm_iff by blast
-  then obtain r::real where r: "norm x < norm r" "norm r < K" "r>0"
+  then obtain r :: real where r: "norm x < norm r" "norm r < K" "r > 0"
     using K False
     by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
   have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> 0"
@@ -741,10 +735,11 @@
     done
   have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
     apply (rule summable_comparison_test' [of "\<lambda>n. norm(c n * (of_real r) ^ n)" N])
-    apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
-    using N r norm_of_real [of "r+K", where 'a = 'a]
-    apply (auto simp add: norm_divide norm_mult norm_power field_simps)
-    using less_eq_real_def by fastforce
+     apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
+    using N r norm_of_real [of "r + K", where 'a = 'a]
+      apply (auto simp add: norm_divide norm_mult norm_power field_simps)
+    apply (fastforce simp: less_eq_real_def)
+    done
   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
     using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1]
     by simp
@@ -758,23 +753,23 @@
 lemma termdiff_converges_all:
   fixes x :: "'a::{real_normed_field,banach}"
   assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
-    shows "summable (\<lambda>n. diffs c n * x^n)"
+  shows "summable (\<lambda>n. diffs c n * x^n)"
   apply (rule termdiff_converges [where K = "1 + norm x"])
   using assms
-  apply auto
+   apply auto
   done
 
 lemma termdiffs_strong:
   fixes K x :: "'a::{real_normed_field,banach}"
   assumes sm: "summable (\<lambda>n. c n * K ^ n)"
-      and K: "norm x < norm K"
+    and K: "norm x < norm K"
   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)"
 proof -
   have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
     using K
     apply (auto simp: norm_divide field_simps)
     apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"])
-    apply (auto simp: mult_2_right norm_triangle_mono)
+     apply (auto simp: mult_2_right norm_triangle_mono)
     done
   then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
     by simp
@@ -786,16 +781,16 @@
     by (blast intro: sm termdiff_converges powser_inside)
   ultimately show ?thesis
     apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
-    apply (auto simp: field_simps)
+      apply (auto simp: field_simps)
     using K
     apply (simp_all add: of_real_add [symmetric] del: of_real_add)
     done
 qed
 
 lemma termdiffs_strong_converges_everywhere:
-    fixes K x :: "'a::{real_normed_field,banach}"
+  fixes K x :: "'a::{real_normed_field,banach}"
   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
-  shows   "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)"
+  shows "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)"
   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
   by (force simp del: of_real_add)
 
@@ -803,7 +798,7 @@
   fixes K x :: "'a::{real_normed_field,banach}"
   assumes "summable (\<lambda>n. c n * K ^ n)"
   assumes "norm x < norm K"
-  shows   "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
+  shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
   using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)
 
 lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]
@@ -811,15 +806,15 @@
 lemma isCont_powser_converges_everywhere:
   fixes K x :: "'a::{real_normed_field,banach}"
   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
-  shows   "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
+  shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
   by (force intro!: DERIV_isCont simp del: of_real_add)
 
 lemma powser_limit_0:
   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
   assumes s: "0 < s"
-      and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
-    shows "(f \<longlongrightarrow> a 0) (at 0)"
+    and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
+  shows "(f \<longlongrightarrow> a 0) (at 0)"
 proof -
   have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
     apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
@@ -834,7 +829,7 @@
   then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
     by (blast intro: DERIV_continuous)
   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) \<longlongrightarrow> a 0) (at 0)"
-    by (simp add: continuous_within powser_zero)
+    by (simp add: continuous_within)
   then show ?thesis
     apply (rule Lim_transform)
     apply (auto simp add: LIM_eq)
@@ -847,17 +842,17 @@
 lemma powser_limit_0_strong:
   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
   assumes s: "0 < s"
-      and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
-    shows "(f \<longlongrightarrow> a 0) (at 0)"
+    and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
+  shows "(f \<longlongrightarrow> a 0) (at 0)"
 proof -
   have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)"
     apply (rule powser_limit_0 [OF s])
-    apply (case_tac "x=0")
-    apply (auto simp add: powser_sums_zero sm)
+    apply (case_tac "x = 0")
+     apply (auto simp add: powser_sums_zero sm)
     done
   show ?thesis
     apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"])
-    apply (simp_all add: *)
+     apply (simp_all add: *)
     done
 qed
 
@@ -867,15 +862,16 @@
 lemma DERIV_series':
   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
-    and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
+    and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)"
+    and x0_in_I: "x0 \<in> {a <..< b}"
     and "summable (f' x0)"
     and "summable L"
-    and L_def: "\<And>n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
+    and L_def: "\<And>n x y. x \<in> {a <..< b} \<Longrightarrow> y \<in> {a <..< b} \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
   unfolding DERIV_def
 proof (rule LIM_I)
   fix r :: real
-  assume "0 < r" hence "0 < r/3" by auto
+  assume "0 < r" then have "0 < r/3" by auto
 
   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto
@@ -884,8 +880,9 @@
     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
 
   let ?N = "Suc (max N_L N_f')"
-  have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
-    L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
+  have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3")
+    and L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3"
+    using N_L[of "?N"] and N_f' [of "?N"] by auto
 
   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
 
@@ -895,7 +892,8 @@
   let ?s = "\<lambda>n. SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
   define S' where "S' = Min (?s ` {..< ?N })"
 
-  have "0 < S'" unfolding S'_def
+  have "0 < S'"
+    unfolding S'_def
   proof (rule iffD2[OF Min_gr_iff])
     show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
     proof
@@ -906,20 +904,21 @@
       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
       obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)"
         by auto
-      have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound simp del: of_nat_Suc)
-      thus "0 < x" unfolding \<open>x = ?s n\<close> .
+      have "0 < ?s n"
+        by (rule someI2[where a=s]) (auto simp add: s_bound simp del: of_nat_Suc)
+      then show "0 < x" by (simp only: \<open>x = ?s n\<close>)
     qed
   qed auto
 
   define S where "S = min (min (x0 - a) (b - x0)) S'"
-  hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
+  then have "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
     and "S \<le> S'" using x0_in_I and \<open>0 < S'\<close>
     by auto
 
-  {
-    fix x
-    assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
-    hence x_in_I: "x0 + x \<in> { a <..< b }"
+  have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
+    if "x \<noteq> 0" and "\<bar>x\<bar> < S" for x
+  proof -
+    from that have x_in_I: "x0 + x \<in> {a <..< b}"
       using S_a S_b by auto
 
     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
@@ -930,42 +929,49 @@
     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
     note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]]
 
-    { fix n
-      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
+    have 1: "\<bar>(\<bar>?diff (n + ?N) x\<bar>)\<bar> \<le> L (n + ?N)" for n
+    proof -
+      have "\<bar>?diff (n + ?N) x\<bar> \<le> L (n + ?N) * \<bar>(x0 + x) - x0\<bar> / \<bar>x\<bar>"
         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
-        unfolding abs_divide .
-      hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
-        using \<open>x \<noteq> 0\<close> by auto }
-    note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
-    then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
-      by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
-    then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
+        by (simp only: abs_divide)
+      with \<open>x \<noteq> 0\<close> show ?thesis by auto
+    qed
+    note 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
+    from 1 have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
+      by (metis (lifting) abs_idempotent
+          order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
+    then have "\<bar>\<Sum>i. ?diff (i + ?N) x\<bar> \<le> r / 3" (is "?L_part \<le> r/3")
       using L_estimate by auto
 
-    have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" ..
+    have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n\<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n\<bar>)" ..
     also have "\<dots> < (\<Sum>n<?N. ?r)"
     proof (rule setsum_strict_mono)
       fix n
       assume "n \<in> {..< ?N}"
       have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> .
       also have "S \<le> S'" using \<open>S \<le> S'\<close> .
-      also have "S' \<le> ?s n" unfolding S'_def
+      also have "S' \<le> ?s n"
+        unfolding S'_def
       proof (rule Min_le_iff[THEN iffD2])
         have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
           using \<open>n \<in> {..< ?N}\<close> by auto
-        thus "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" by blast
+        then show "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n"
+          by blast
       qed auto
       finally have "\<bar>x\<bar> < ?s n" .
 
-      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
+      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>,
+          unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
       with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
         by blast
     qed auto
     also have "\<dots> = of_nat (card {..<?N}) * ?r"
       by (rule setsum_constant)
-    also have "\<dots> = real ?N * ?r" by simp
-    also have "\<dots> = r/3" by (auto simp del: of_nat_Suc)
+    also have "\<dots> = real ?N * ?r"
+      by simp
+    also have "\<dots> = r/3"
+      by (auto simp del: of_nat_Suc)
     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
 
     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
@@ -973,148 +979,148 @@
         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
       unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
       using suminf_divide[OF diff_smbl, symmetric] by auto
-    also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
+    also have "\<dots> \<le> ?diff_part + \<bar>(\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N))\<bar>"
       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
       unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
       apply (subst (5) add.commute)
-      by (rule abs_triangle_ineq)
+      apply (rule abs_triangle_ineq)
+      done
     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
       using abs_triangle_ineq4 by auto
     also have "\<dots> < r /3 + r/3 + r/3"
       using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close>
       by (rule add_strict_mono [OF add_less_le_mono])
-    finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
+    finally show ?thesis
       by auto
-  }
-  thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
+  qed
+  then show "\<exists>s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
-    using \<open>0 < S\<close> unfolding real_norm_def diff_0_right by blast
+    using \<open>0 < S\<close> by auto
 qed
 
 lemma DERIV_power_series':
   fixes f :: "nat \<Rightarrow> real"
-  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
-    and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
-  shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
-  (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
+  assumes converges: "\<And>x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda>n. f n * real (Suc n) * x^n)"
+    and x0_in_I: "x0 \<in> {-R <..< R}"
+    and "0 < R"
+  shows "DERIV (\<lambda>x. (\<Sum>n. f n * x^(Suc n))) x0 :> (\<Sum>n. f n * real (Suc n) * x0^n)"
+    (is "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)")
 proof -
-  {
-    fix R'
-    assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
-    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
+  have for_subinterval: "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)"
+    if "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" for R'
+  proof -
+    from that have "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
       by auto
-    have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
+    show ?thesis
     proof (rule DERIV_series')
       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
       proof -
         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
-        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
+        then have in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
           using \<open>R' < R\<close> by auto
         have "norm R' < norm ((R' + R) / 2)"
           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
           by auto
       qed
-      {
-        fix n x y
-        assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
-        show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
-        proof -
-          have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
-            (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
-            unfolding right_diff_distrib[symmetric] diff_power_eq_setsum abs_mult
-            by auto
-          also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
-          proof (rule mult_left_mono)
-            have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
-              by (rule setsum_abs)
-            also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
-            proof (rule setsum_mono)
-              fix p
-              assume "p \<in> {..<Suc n}"
-              hence "p \<le> n" by auto
-              {
-                fix n
-                fix x :: real
-                assume "x \<in> {-R'<..<R'}"
-                hence "\<bar>x\<bar> \<le> R'"  by auto
-                hence "\<bar>x^n\<bar> \<le> R'^n"
-                  unfolding power_abs by (rule power_mono, auto)
-              }
-              from mult_mono[OF this[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]] \<open>0 < R'\<close>
-              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
-                unfolding abs_mult by auto
-              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
-                unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
+    next
+      fix n x y
+      assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
+      show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
+      proof -
+        have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
+          (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
+          unfolding right_diff_distrib[symmetric] diff_power_eq_setsum abs_mult
+          by auto
+        also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
+        proof (rule mult_left_mono)
+          have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
+            by (rule setsum_abs)
+          also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
+          proof (rule setsum_mono)
+            fix p
+            assume "p \<in> {..<Suc n}"
+            then have "p \<le> n" by auto
+            have "\<bar>x^n\<bar> \<le> R'^n" if  "x \<in> {-R'<..<R'}" for n and x :: real
+            proof -
+              from that have "\<bar>x\<bar> \<le> R'" by auto
+              then show ?thesis
+                unfolding power_abs by (rule power_mono) auto
             qed
-            also have "\<dots> = real (Suc n) * R' ^ n"
-              unfolding setsum_constant card_atLeastLessThan by auto
-            finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
-              unfolding  abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
-              by linarith
-            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
-              unfolding abs_mult[symmetric] by auto
+            from mult_mono[OF this[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]]
+              and \<open>0 < R'\<close>
+            have "\<bar>x^p * y^(n - p)\<bar> \<le> R'^p * R'^(n - p)"
+              unfolding abs_mult by auto
+            then show "\<bar>x^p * y^(n - p)\<bar> \<le> R'^n"
+              unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
           qed
-          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
-            unfolding abs_mult mult.assoc[symmetric] by algebra
-          finally show ?thesis .
+          also have "\<dots> = real (Suc n) * R' ^ n"
+            unfolding setsum_constant card_atLeastLessThan by auto
+          finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
+            unfolding abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
+            by linarith
+          show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
+            unfolding abs_mult[symmetric] by auto
         qed
-      }
-      {
+        also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
+          unfolding abs_mult mult.assoc[symmetric] by algebra
+        finally show ?thesis .
+      qed
+    next
+      show "DERIV (\<lambda>x. ?f x n) x0 :> ?f' x0 n" for n
+        by (auto intro!: derivative_eq_intros simp del: power_Suc)
+    next
+      fix x
+      assume "x \<in> {-R' <..< R'}"
+      then have "R' \<in> {-R <..< R}" and "norm x < norm R'"
+        using assms \<open>R' < R\<close> by auto
+      have "summable (\<lambda>n. f n * x^n)"
+      proof (rule summable_comparison_test, intro exI allI impI)
         fix n
-        show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
-          by (auto intro!: derivative_eq_intros simp del: power_Suc)
-      }
-      {
-        fix x
-        assume "x \<in> {-R' <..< R'}"
-        hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
-          using assms \<open>R' < R\<close> by auto
-        have "summable (\<lambda> n. f n * x^n)"
-        proof (rule summable_comparison_test, intro exI allI impI)
-          fix n
-          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
-            by (rule mult_left_mono) auto
-          show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
-            unfolding real_norm_def abs_mult
-            using le mult_right_mono by fastforce
-        qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
-        from this[THEN summable_mult2[where c=x], simplified mult.assoc, simplified mult.commute]
-        show "summable (?f x)" by auto
-      }
+        have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
+          by (rule mult_left_mono) auto
+        show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
+          unfolding real_norm_def abs_mult
+          using le mult_right_mono by fastforce
+      qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
+      from this[THEN summable_mult2[where c=x], simplified mult.assoc, simplified mult.commute]
+      show "summable (?f x)" by auto
+    next
       show "summable (?f' x0)"
         using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] .
       show "x0 \<in> {-R' <..< R'}"
         using \<open>x0 \<in> {-R' <..< R'}\<close> .
     qed
-  } note for_subinterval = this
+  qed
   let ?R = "(R + \<bar>x0\<bar>) / 2"
-  have "\<bar>x0\<bar> < ?R" using assms by (auto simp: field_simps)
-  hence "- ?R < x0"
+  have "\<bar>x0\<bar> < ?R"
+    using assms by (auto simp: field_simps)
+  then have "- ?R < x0"
   proof (cases "x0 < 0")
     case True
-    hence "- x0 < ?R" using \<open>\<bar>x0\<bar> < ?R\<close> by auto
-    thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
+    then have "- x0 < ?R"
+      using \<open>\<bar>x0\<bar> < ?R\<close> by auto
+    then show ?thesis
+      unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
   next
     case False
     have "- ?R < 0" using assms by auto
     also have "\<dots> \<le> x0" using False by auto
     finally show ?thesis .
   qed
-  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
+  then have "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
     using assms by (auto simp: field_simps)
-  from for_subinterval[OF this]
-  show ?thesis .
+  from for_subinterval[OF this] show ?thesis .
 qed
 
 
-lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z::'a::real_normed_field. pochhammer z n) z"
-  by (induction n) (auto intro!: continuous_intros simp: pochhammer_rec')
-
-lemma continuous_on_pochhammer [continuous_intros]:
-  fixes A :: "'a :: real_normed_field set"
-  shows "continuous_on A (\<lambda>z. pochhammer z n)"
+lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z. pochhammer z n) z"
+  for z :: "'a::real_normed_field"
+  by (induct n) (auto simp: pochhammer_rec')
+
+lemma continuous_on_pochhammer [continuous_intros]: "continuous_on A (\<lambda>z. pochhammer z n)"
+  for A :: "'a::real_normed_field set"
   by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer)
 
 
@@ -1144,29 +1150,26 @@
       using r0 n by (simp add: mult_right_mono)
     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
       using norm_ge_zero by (rule mult_right_mono)
-    hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
+    then have "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
       by (rule order_trans [OF norm_mult_ineq])
-    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
+    then have "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
       by (simp add: pos_divide_le_eq ac_simps)
-    thus "norm (S (Suc n)) \<le> r * norm (S n)"
+    then show "norm (S (Suc n)) \<le> r * norm (S n)"
       by (simp add: S_Suc inverse_eq_divide)
   qed
 qed
 
-lemma summable_norm_exp:
-  fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
+lemma summable_norm_exp: "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
+  for x :: "'a::{real_normed_algebra_1,banach}"
 proof (rule summable_norm_comparison_test [OF exI, rule_format])
   show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
     by (rule summable_exp_generic)
-  fix n
-  show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n"
+  show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n" for n
     by (simp add: norm_power_ineq)
 qed
 
-lemma summable_exp:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "summable (\<lambda>n. inverse (fact n) * x^n)"
+lemma summable_exp: "summable (\<lambda>n. inverse (fact n) * x^n)"
+  for x :: "'a::{real_normed_field,banach}"
   using summable_exp_generic [where x=x]
   by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
 
@@ -1176,22 +1179,22 @@
 lemma exp_fdiffs:
   "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a::{real_normed_field,banach}))"
   by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
-           del: mult_Suc of_nat_Suc)
+      del: mult_Suc of_nat_Suc)
 
 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
   by (simp add: diffs_def)
 
-lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
+lemma DERIV_exp [simp]: "DERIV exp x :> exp x"
   unfolding exp_def scaleR_conv_of_real
   apply (rule DERIV_cong)
-  apply (rule termdiffs [where K="of_real (1 + norm x)"])
-  apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
-  apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
+   apply (rule termdiffs [where K="of_real (1 + norm x)"])
+      apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
+     apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
   apply (simp del: of_real_add)
   done
 
 declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
-        DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
+  and DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
 
 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
 proof -
@@ -1204,39 +1207,34 @@
   finally show ?thesis .
 qed
 
-lemma isCont_exp:
-  fixes x::"'a::{real_normed_field,banach}"
-  shows "isCont exp x"
+lemma isCont_exp: "isCont exp x"
+  for x :: "'a::{real_normed_field,banach}"
   by (rule DERIV_exp [THEN DERIV_isCont])
 
-lemma isCont_exp' [simp]:
-  fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
-  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
+lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
+  for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
   by (rule isCont_o2 [OF _ isCont_exp])
 
-lemma tendsto_exp [tendsto_intros]:
-  fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
-  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
+lemma tendsto_exp [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
+  for f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
   by (rule isCont_tendsto_compose [OF isCont_exp])
 
-lemma continuous_exp [continuous_intros]:
-  fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
-  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
+lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
+  for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
   unfolding continuous_def by (rule tendsto_exp)
 
-lemma continuous_on_exp [continuous_intros]:
-  fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
-  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
+lemma continuous_on_exp [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
+  for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
   unfolding continuous_on_def by (auto intro: tendsto_exp)
 
 
 subsubsection \<open>Properties of the Exponential Function\<close>
 
 lemma exp_zero [simp]: "exp 0 = 1"
-  unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
+  unfolding exp_def by (simp add: scaleR_conv_of_real)
 
 lemma exp_series_add_commuting:
-  fixes x y :: "'a::{real_normed_algebra_1, banach}"
+  fixes x y :: "'a::{real_normed_algebra_1,banach}"
   defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n"
   assumes comm: "x * y = y * x"
   shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
@@ -1248,55 +1246,49 @@
   case (Suc n)
   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
     unfolding S_def by (simp del: mult_Suc)
-  hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
+  then have times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
     by simp
   have S_comm: "\<And>n. S x n * y = y * S x n"
     by (simp add: power_commuting_commutes comm S_def)
 
   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
     by (simp only: times_S)
-  also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))"
+  also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n - i))"
     by (simp only: Suc)
-  also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i))
-                + y * (\<Sum>i\<le>n. S x i * S y (n-i))"
+  also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n - i)) + y * (\<Sum>i\<le>n. S x i * S y (n - i))"
     by (rule distrib_right)
-  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
-                + (\<Sum>i\<le>n. S x i * y * S y (n-i))"
+  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * y * S y (n - i))"
     by (simp add: setsum_right_distrib ac_simps S_comm)
-  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
-                + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
+  also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * (y * S y (n - i)))"
     by (simp add: ac_simps)
-  also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
-                + (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
+  also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) +
+      (\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
     by (simp add: times_S Suc_diff_le)
-  also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
-             (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
+  also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) =
+      (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))"
     by (subst setsum_atMost_Suc_shift) simp
-  also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
-             (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
+  also have "(\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) =
+      (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
     by simp
-  also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
-             (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
-             (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
+  also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i))) +
+        (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) =
+      (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n - i)))"
     by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric]
-                   of_nat_add [symmetric]) simp
-  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
+        of_nat_add [symmetric]) simp
+  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
     by (simp only: scaleR_right.setsum)
-  finally show
-    "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
+  finally show "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
     by (simp del: setsum_cl_ivl_Suc)
 qed
 
 lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
-  unfolding exp_def
-  by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting)
+  by (simp only: exp_def Cauchy_product summable_norm_exp exp_series_add_commuting)
 
 lemma exp_times_arg_commute: "exp A * A = A * exp A"
   by (simp add: exp_def suminf_mult[symmetric] summable_exp_generic power_commutes suminf_mult2)
 
-lemma exp_add:
-  fixes x y::"'a::{real_normed_field,banach}"
-  shows "exp (x + y) = exp x * exp y"
+lemma exp_add: "exp (x + y) = exp x * exp y"
+  for x y :: "'a::{real_normed_field,banach}"
   by (rule exp_add_commuting) (simp add: ac_simps)
 
 lemma exp_double: "exp(2 * z) = exp z ^ 2"
@@ -1307,7 +1299,7 @@
 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
   unfolding exp_def
   apply (subst suminf_of_real)
-  apply (rule summable_exp_generic)
+   apply (rule summable_exp_generic)
   apply (simp add: scaleR_conv_of_real)
   done
 
@@ -1316,47 +1308,47 @@
 
 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
 proof
-  have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric])
+  have "exp x * exp (- x) = 1"
+    by (simp add: exp_add_commuting[symmetric])
   also assume "exp x = 0"
-  finally show "False" by simp
+  finally show False by simp
 qed
 
-lemma exp_minus_inverse:
-  shows "exp x * exp (- x) = 1"
+lemma exp_minus_inverse: "exp x * exp (- x) = 1"
   by (simp add: exp_add_commuting[symmetric])
 
-lemma exp_minus:
-  fixes x :: "'a::{real_normed_field, banach}"
-  shows "exp (- x) = inverse (exp x)"
+lemma exp_minus: "exp (- x) = inverse (exp x)"
+  for x :: "'a::{real_normed_field,banach}"
   by (intro inverse_unique [symmetric] exp_minus_inverse)
 
-lemma exp_diff:
-  fixes x :: "'a::{real_normed_field, banach}"
-  shows "exp (x - y) = exp x / exp y"
+lemma exp_diff: "exp (x - y) = exp x / exp y"
+  for x :: "'a::{real_normed_field,banach}"
   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
 
-lemma exp_of_nat_mult:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "exp(of_nat n * x) = exp(x) ^ n"
-    by (induct n) (auto simp add: distrib_left exp_add mult.commute)
-
-corollary exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
+lemma exp_of_nat_mult: "exp (of_nat n * x) = exp x ^ n"
+  for x :: "'a::{real_normed_field,banach}"
+  by (induct n) (auto simp add: distrib_left exp_add mult.commute)
+
+corollary exp_real_of_nat_mult: "exp (real n * x) = exp x ^ n"
   by (simp add: exp_of_nat_mult)
 
-lemma exp_setsum: "finite I \<Longrightarrow> exp(setsum f I) = setprod (\<lambda>x. exp(f x)) I"
-  by (induction I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
+lemma exp_setsum: "finite I \<Longrightarrow> exp (setsum f I) = setprod (\<lambda>x. exp (f x)) I"
+  by (induct I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
 
 lemma exp_divide_power_eq:
-  fixes x:: "'a::{real_normed_field,banach}"
-  assumes "n>0" shows "exp (x / of_nat n) ^ n = exp x"
-using assms
+  fixes x :: "'a::{real_normed_field,banach}"
+  assumes "n > 0"
+  shows "exp (x / of_nat n) ^ n = exp x"
+  using assms
 proof (induction n arbitrary: x)
-  case 0 then show ?case by simp
+  case 0
+  then show ?case by simp
 next
   case (Suc n)
   show ?case
-  proof (cases "n=0")
-    case True then show ?thesis by simp
+  proof (cases "n = 0")
+    case True
+    then show ?thesis by simp
   next
     case False
     then have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)"
@@ -1376,40 +1368,49 @@
 
 text \<open>Comparisons of @{term "exp x"} with zero.\<close>
 
-text\<open>Proof: because every exponential can be seen as a square.\<close>
-lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
+text \<open>Proof: because every exponential can be seen as a square.\<close>
+lemma exp_ge_zero [simp]: "0 \<le> exp x"
+  for x :: real
 proof -
-  have "0 \<le> exp (x/2) * exp (x/2)" by simp
-  thus ?thesis by (simp add: exp_add [symmetric])
+  have "0 \<le> exp (x/2) * exp (x/2)"
+    by simp
+  then show ?thesis
+    by (simp add: exp_add [symmetric])
 qed
 
-lemma exp_gt_zero [simp]: "0 < exp (x::real)"
+lemma exp_gt_zero [simp]: "0 < exp x"
+  for x :: real
   by (simp add: order_less_le)
 
-lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
+lemma not_exp_less_zero [simp]: "\<not> exp x < 0"
+  for x :: real
   by (simp add: not_less)
 
-lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
+lemma not_exp_le_zero [simp]: "\<not> exp x \<le> 0"
+  for x :: real
   by (simp add: not_le)
 
-lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
+lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
+  for x :: real
   by simp
 
 text \<open>Strict monotonicity of exponential.\<close>
 
 lemma exp_ge_add_one_self_aux:
-  assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
-using order_le_imp_less_or_eq [OF assms]
+  fixes x :: real
+  assumes "0 \<le> x"
+  shows "1 + x \<le> exp x"
+  using order_le_imp_less_or_eq [OF assms]
 proof
   assume "0 < x"
-  have "1+x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
+  have "1 + x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
     by (auto simp add: numeral_2_eq_2)
-  also have "... \<le> (\<Sum>n. inverse (fact n) * x^n)"
+  also have "\<dots> \<le> (\<Sum>n. inverse (fact n) * x^n)"
     apply (rule setsum_le_suminf [OF summable_exp])
     using \<open>0 < x\<close>
     apply (auto  simp add:  zero_le_mult_iff)
     done
-  finally show "1+x \<le> exp x"
+  finally show "1 + x \<le> exp x"
     by (simp add: exp_def)
 next
   assume "0 = x"
@@ -1417,10 +1418,11 @@
     by auto
 qed
 
-lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
+lemma exp_gt_one: "0 < x \<Longrightarrow> 1 < exp x"
+  for x :: real
 proof -
   assume x: "0 < x"
-  hence "1 < 1 + x" by simp
+  then have "1 < 1 + x" by simp
   also from x have "1 + x \<le> exp x"
     by (simp add: exp_ge_add_one_self_aux)
   finally show ?thesis .
@@ -1432,59 +1434,75 @@
   shows "exp x < exp y"
 proof -
   from \<open>x < y\<close> have "0 < y - x" by simp
-  hence "1 < exp (y - x)" by (rule exp_gt_one)
-  hence "1 < exp y / exp x" by (simp only: exp_diff)
-  thus "exp x < exp y" by simp
+  then have "1 < exp (y - x)" by (rule exp_gt_one)
+  then have "1 < exp y / exp x" by (simp only: exp_diff)
+  then show "exp x < exp y" by simp
 qed
 
-lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
+lemma exp_less_cancel: "exp x < exp y \<Longrightarrow> x < y"
+  for x y :: real
   unfolding linorder_not_le [symmetric]
   by (auto simp add: order_le_less exp_less_mono)
 
-lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
+lemma exp_less_cancel_iff [iff]: "exp x < exp y \<longleftrightarrow> x < y"
+  for x y :: real
   by (auto intro: exp_less_mono exp_less_cancel)
 
-lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
+lemma exp_le_cancel_iff [iff]: "exp x \<le> exp y \<longleftrightarrow> x \<le> y"
+  for x y :: real
   by (auto simp add: linorder_not_less [symmetric])
 
-lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
+lemma exp_inj_iff [iff]: "exp x = exp y \<longleftrightarrow> x = y"
+  for x y :: real
   by (simp add: order_eq_iff)
 
 text \<open>Comparisons of @{term "exp x"} with one.\<close>
 
-lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
-  using exp_less_cancel_iff [where x=0 and y=x] by simp
-
-lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
-  using exp_less_cancel_iff [where x=x and y=0] by simp
-
-lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
-  using exp_le_cancel_iff [where x=0 and y=x] by simp
-
-lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
-  using exp_le_cancel_iff [where x=x and y=0] by simp
-
-lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
-  using exp_inj_iff [where x=x and y=0] by simp
-
-lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
+lemma one_less_exp_iff [simp]: "1 < exp x \<longleftrightarrow> 0 < x"
+  for x :: real
+  using exp_less_cancel_iff [where x = 0 and y = x] by simp
+
+lemma exp_less_one_iff [simp]: "exp x < 1 \<longleftrightarrow> x < 0"
+  for x :: real
+  using exp_less_cancel_iff [where x = x and y = 0] by simp
+
+lemma one_le_exp_iff [simp]: "1 \<le> exp x \<longleftrightarrow> 0 \<le> x"
+  for x :: real
+  using exp_le_cancel_iff [where x = 0 and y = x] by simp
+
+lemma exp_le_one_iff [simp]: "exp x \<le> 1 \<longleftrightarrow> x \<le> 0"
+  for x :: real
+  using exp_le_cancel_iff [where x = x and y = 0] by simp
+
+lemma exp_eq_one_iff [simp]: "exp x = 1 \<longleftrightarrow> x = 0"
+  for x :: real
+  using exp_inj_iff [where x = x and y = 0] by simp
+
+lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x \<le> y - 1 \<and> exp x = y"
+  for y :: real
 proof (rule IVT)
   assume "1 \<le> y"
-  hence "0 \<le> y - 1" by simp
-  hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
-  thus "y \<le> exp (y - 1)" by simp
+  then have "0 \<le> y - 1" by simp
+  then have "1 + (y - 1) \<le> exp (y - 1)"
+    by (rule exp_ge_add_one_self_aux)
+  then show "y \<le> exp (y - 1)" by simp
 qed (simp_all add: le_diff_eq)
 
-lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
+lemma exp_total: "0 < y \<Longrightarrow> \<exists>x. exp x = y"
+  for y :: real
 proof (rule linorder_le_cases [of 1 y])
   assume "1 \<le> y"
-  thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
+  then show "\<exists>x. exp x = y"
+    by (fast dest: lemma_exp_total)
 next
   assume "0 < y" and "y \<le> 1"
-  hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
-  then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
-  hence "exp (- x) = y" by (simp add: exp_minus)
-  thus "\<exists>x. exp x = y" ..
+  then have "1 \<le> inverse y"
+    by (simp add: one_le_inverse_iff)
+  then obtain x where "exp x = inverse y"
+    by (fast dest: lemma_exp_total)
+  then have "exp (- x) = y"
+    by (simp add: exp_minus)
+  then show "\<exists>x. exp x = y" ..
 qed
 
 
@@ -1494,9 +1512,9 @@
   fixes ln :: "'a \<Rightarrow> 'a"
   assumes ln_one [simp]: "ln 1 = 0"
 
-definition powr :: "['a,'a] => 'a::ln"     (infixr "powr" 80)
+definition powr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a::ln"  (infixr "powr" 80)
   \<comment> \<open>exponentation via ln and exp\<close>
-  where  [code del]: "x powr a \<equiv> if x = 0 then 0 else exp(a * ln x)"
+  where  [code del]: "x powr a \<equiv> if x = 0 then 0 else exp (a * ln x)"
 
 lemma powr_0 [simp]: "0 powr z = 0"
   by (simp add: powr_def)
@@ -1509,118 +1527,116 @@
   where "ln_real x = (THE u. exp u = x)"
 
 instance
-by intro_classes (simp add: ln_real_def)
+  by intro_classes (simp add: ln_real_def)
 
 end
 
 lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
   by (simp add: powr_def)
 
-lemma ln_exp [simp]:
-  fixes x::real shows "ln (exp x) = x"
+lemma ln_exp [simp]: "ln (exp x) = x"
+  for x :: real
   by (simp add: ln_real_def)
 
-lemma exp_ln [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> exp (ln x) = x"
+lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
+  for x :: real
   by (auto dest: exp_total)
 
-lemma exp_ln_iff [simp]:
-  fixes x::real shows "exp (ln x) = x \<longleftrightarrow> 0 < x"
+lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
+  for x :: real
   by (metis exp_gt_zero exp_ln)
 
-lemma ln_unique:
-  fixes x::real shows "exp y = x \<Longrightarrow> ln x = y"
-  by (erule subst, rule ln_exp)
-
-lemma ln_mult:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
+lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
+  for x :: real
+  by (erule subst) (rule ln_exp)
+
+lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
+  for x :: real
   by (rule ln_unique) (simp add: exp_add)
 
-lemma ln_setprod:
-  fixes f:: "'a => real"
-  shows
-    "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i > 0\<rbrakk> \<Longrightarrow> ln(setprod f I) = setsum (\<lambda>x. ln(f x)) I"
-  by (induction I rule: finite_induct) (auto simp: ln_mult setprod_pos)
-
-lemma ln_inverse:
-  fixes x::real shows "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
+lemma ln_setprod: "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i > 0) \<Longrightarrow> ln (setprod f I) = setsum (\<lambda>x. ln(f x)) I"
+  for f :: "'a \<Rightarrow> real"
+  by (induct I rule: finite_induct) (auto simp: ln_mult setprod_pos)
+
+lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
+  for x :: real
   by (rule ln_unique) (simp add: exp_minus)
 
-lemma ln_div:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
+lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
+  for x :: real
   by (rule ln_unique) (simp add: exp_diff)
 
 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
 
-lemma ln_less_cancel_iff [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
+lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
+  for x :: real
   by (subst exp_less_cancel_iff [symmetric]) simp
 
-lemma ln_le_cancel_iff [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
+lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
+  for x :: real
   by (simp add: linorder_not_less [symmetric])
 
-lemma ln_inj_iff [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
+lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
+  for x :: real
   by (simp add: order_eq_iff)
 
-lemma ln_add_one_self_le_self [simp]:
-  fixes x::real shows "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
-  apply (rule exp_le_cancel_iff [THEN iffD1])
-  apply (simp add: exp_ge_add_one_self_aux)
-  done
-
-lemma ln_less_self [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> ln x < x"
-  by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
-
-lemma ln_ge_zero [simp]:
-  fixes x::real shows "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
+lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
+  for x :: real
+  by (rule exp_le_cancel_iff [THEN iffD1]) (simp add: exp_ge_add_one_self_aux)
+
+lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
+  for x :: real
+  by (rule order_less_le_trans [where y = "ln (1 + x)"]) simp_all
+
+lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
+  for x :: real
   using ln_le_cancel_iff [of 1 x] by simp
 
-lemma ln_ge_zero_imp_ge_one:
-  fixes x::real shows "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
+lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
+  for x :: real
   using ln_le_cancel_iff [of 1 x] by simp
 
-lemma ln_ge_zero_iff [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
+lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
+  for x :: real
   using ln_le_cancel_iff [of 1 x] by simp
 
-lemma ln_less_zero_iff [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
+lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
+  for x :: real
   using ln_less_cancel_iff [of x 1] by simp
 
-lemma ln_gt_zero:
-  fixes x::real shows "1 < x \<Longrightarrow> 0 < ln x"
+lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
+  for x :: real
   using ln_less_cancel_iff [of 1 x] by simp
 
-lemma ln_gt_zero_imp_gt_one:
-  fixes x::real shows "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
+lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
+  for x :: real
   using ln_less_cancel_iff [of 1 x] by simp
 
-lemma ln_gt_zero_iff [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
+lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
+  for x :: real
   using ln_less_cancel_iff [of 1 x] by simp
 
-lemma ln_eq_zero_iff [simp]:
-  fixes x::real shows "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
+lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
+  for x :: real
   using ln_inj_iff [of x 1] by simp
 
-lemma ln_less_zero:
-  fixes x::real shows "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
+lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
+  for x :: real
   by simp
 
-lemma ln_neg_is_const:
-  fixes x::real shows "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
-  by (auto simp add: ln_real_def intro!: arg_cong[where f=The])
+lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
+  for x :: real
+  by (auto simp: ln_real_def intro!: arg_cong[where f = The])
 
 lemma isCont_ln:
-  fixes x::real assumes "x \<noteq> 0" shows "isCont ln x"
+  fixes x :: real
+  assumes "x \<noteq> 0"
+  shows "isCont ln x"
 proof (cases "0 < x")
   case True
   then have "isCont ln (exp (ln x))"
-    by (intro isCont_inv_fun[where d="\<bar>x\<bar>" and f=exp]) auto
+    by (intro isCont_inv_fun[where d = "\<bar>x\<bar>" and f = exp]) auto
   with True show ?thesis
     by simp
 next
@@ -1629,12 +1645,11 @@
     unfolding isCont_def
     by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
        (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
-                intro!: exI[of _ "\<bar>x\<bar>"])
+         intro!: exI[of _ "\<bar>x\<bar>"])
 qed
 
-lemma tendsto_ln [tendsto_intros]:
-  fixes a::real shows
-  "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
+lemma tendsto_ln [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
+  for a :: real
   by (rule isCont_tendsto_compose [OF isCont_ln])
 
 lemma continuous_ln:
@@ -1653,32 +1668,30 @@
   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))"
   unfolding continuous_on_def by (auto intro: tendsto_ln)
 
-lemma DERIV_ln:
-  fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
-  apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
-  apply (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
-  done
-
-lemma DERIV_ln_divide:
-  fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
-  by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
+lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
+  for x :: real
+  by (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
+    (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
+
+lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
+  for x :: real
+  by (rule DERIV_ln[THEN DERIV_cong]) (simp_all add: divide_inverse)
 
 declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
-        DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
-
+  and DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
 
 lemma ln_series:
   assumes "0 < x" and "x < 2"
   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
-  (is "ln x = suminf (?f (x - 1))")
+    (is "ln x = suminf (?f (x - 1))")
 proof -
   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
 
   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
-  proof (rule DERIV_isconst3[where x=x])
+  proof (rule DERIV_isconst3 [where x = x])
     fix x :: real
     assume "x \<in> {0 <..< 2}"
-    hence "0 < x" and "x < 2" by auto
+    then have "0 < x" and "x < 2" by auto
     have "norm (1 - x) < 1"
       using \<open>0 < x\<close> and \<open>x < 2\<close> by auto
     have "1 / x = 1 / (1 - (1 - x))" by auto
@@ -1696,87 +1709,88 @@
     proof (rule DERIV_power_series')
       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
         using \<open>0 < x\<close> \<open>x < 2\<close> by auto
+    next
       fix x :: real
       assume "x \<in> {- 1<..<1}"
-      hence "norm (-x) < 1" by auto
+      then have "norm (-x) < 1" by auto
       show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)"
         unfolding One_nat_def
         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF \<open>norm (-x) < 1\<close>])
     qed
-    hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
+    then have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
       unfolding One_nat_def by auto
-    hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
+    then have "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
       unfolding DERIV_def repos .
-    ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
+    ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> suminf (?f' x) - suminf (?f' x)"
       by (rule DERIV_diff)
-    thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
+    then show "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
   qed (auto simp add: assms)
-  thus ?thesis by auto
+  then show ?thesis by auto
 qed
 
 lemma exp_first_terms:
   fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "exp x = (\<Sum>n<k. inverse(fact n) *\<^sub>R (x ^ n)) + (\<Sum>n. inverse(fact (n+k)) *\<^sub>R (x ^ (n+k)))"
+  shows "exp x = (\<Sum>n<k. inverse(fact n) *\<^sub>R (x ^ n)) + (\<Sum>n. inverse(fact (n + k)) *\<^sub>R (x ^ (n + k)))"
 proof -
   have "exp x = suminf (\<lambda>n. inverse(fact n) *\<^sub>R (x^n))"
     by (simp add: exp_def)
-  also from summable_exp_generic have "... = (\<Sum> n. inverse(fact(n+k)) *\<^sub>R (x ^ (n+k))) +
+  also from summable_exp_generic have "\<dots> = (\<Sum> n. inverse(fact(n+k)) *\<^sub>R (x ^ (n + k))) +
     (\<Sum> n::nat<k. inverse(fact n) *\<^sub>R (x^n))" (is "_ = _ + ?a")
     by (rule suminf_split_initial_segment)
   finally show ?thesis by simp
 qed
 
-lemma exp_first_term:
-  fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "exp x = 1 + (\<Sum> n. inverse(fact (Suc n)) *\<^sub>R (x ^ (Suc n)))"
+lemma exp_first_term: "exp x = 1 + (\<Sum>n. inverse (fact (Suc n)) *\<^sub>R (x ^ Suc n))"
+  for x :: "'a::{real_normed_algebra_1,banach}"
   using exp_first_terms[of x 1] by simp
 
-lemma exp_first_two_terms:
-  fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) *\<^sub>R (x ^ (n+2)))"
-  using exp_first_terms[of x 2]
-  by (simp add: eval_nat_numeral)
-
-lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
+lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum>n. inverse (fact (n + 2)) *\<^sub>R (x ^ (n + 2)))"
+  for x :: "'a::{real_normed_algebra_1,banach}"
+  using exp_first_terms[of x 2] by (simp add: eval_nat_numeral)
+
+lemma exp_bound:
+  fixes x :: real
+  assumes a: "0 \<le> x"
+    and b: "x \<le> 1"
+  shows "exp x \<le> 1 + x + x\<^sup>2"
 proof -
-  assume a: "0 <= x"
-  assume b: "x <= 1"
-  {
-    fix n :: nat
+  have aux1: "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" for n :: nat
+  proof -
     have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
       by (induct n) simp_all
-    hence "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
+    then have "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
       by (simp only: of_nat_le_iff)
-    hence "((2::real) * 2 ^ n) \<le> fact (n + 2)"
-      unfolding of_nat_fact
-      by (simp add: of_nat_mult of_nat_power)
-    hence "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
+    then have "((2::real) * 2 ^ n) \<le> fact (n + 2)"
+      unfolding of_nat_fact by simp
+    then have "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
       by (rule le_imp_inverse_le) simp
-    hence "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
+    then have "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
       by (simp add: power_inverse [symmetric])
-    hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
-      by (rule mult_mono)
-        (rule mult_mono, simp_all add: power_le_one a b)
-    hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
-      unfolding power_add by (simp add: ac_simps del: fact_Suc) }
-  note aux1 = this
+    then have "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
+      by (rule mult_mono) (rule mult_mono, simp_all add: power_le_one a b)
+    then show ?thesis
+      unfolding power_add by (simp add: ac_simps del: fact_Suc)
+  qed
   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
-    by (intro sums_mult geometric_sums, simp)
-  hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
+    by (intro sums_mult geometric_sums) simp
+  then have aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
     by simp
-  have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
+  have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> x\<^sup>2"
   proof -
-    have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
-        suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
+    have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
       apply (rule suminf_le)
-      apply (rule allI, rule aux1)
-      apply (rule summable_exp [THEN summable_ignore_initial_segment])
-      by (rule sums_summable, rule aux2)
-    also have "... = x\<^sup>2"
-      by (rule sums_unique [THEN sym], rule aux2)
+        apply (rule allI)
+        apply (rule aux1)
+       apply (rule summable_exp [THEN summable_ignore_initial_segment])
+      apply (rule sums_summable)
+      apply (rule aux2)
+      done
+    also have "\<dots> = x\<^sup>2"
+      by (rule sums_unique [THEN sym]) (rule aux2)
     finally show ?thesis .
   qed
-  thus ?thesis unfolding exp_first_two_terms by auto
+  then show ?thesis
+    unfolding exp_first_two_terms by auto
 qed
 
 corollary exp_half_le2: "exp(1/2) \<le> (2::real)"
@@ -1787,172 +1801,177 @@
   using exp_bound [of 1]
   by (simp add: field_simps)
 
-lemma exp_bound_half: "norm(z) \<le> 1/2 \<Longrightarrow> norm(exp z) \<le> 2"
+lemma exp_bound_half: "norm z \<le> 1/2 \<Longrightarrow> norm (exp z) \<le> 2"
   by (blast intro: order_trans intro!: exp_half_le2 norm_exp)
 
 lemma exp_bound_lemma:
-  assumes "norm(z) \<le> 1/2" shows "norm(exp z) \<le> 1 + 2 * norm(z)"
+  assumes "norm z \<le> 1/2"
+  shows "norm (exp z) \<le> 1 + 2 * norm z"
 proof -
-  have n: "(norm z)\<^sup>2 \<le> norm z * 1"
+  have *: "(norm z)\<^sup>2 \<le> norm z * 1"
     unfolding power2_eq_square
     apply (rule mult_left_mono)
     using assms
-    apply auto
+     apply auto
     done
   show ?thesis
     apply (rule order_trans [OF norm_exp])
     apply (rule order_trans [OF exp_bound])
-    using assms n
-    apply auto
+    using assms *
+      apply auto
     done
 qed
 
-lemma real_exp_bound_lemma:
-  fixes x :: real
-  shows "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp(x) \<le> 1 + 2 * x"
-using exp_bound_lemma [of x]
-by simp
+lemma real_exp_bound_lemma: "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp x \<le> 1 + 2 * x"
+  for x :: real
+  using exp_bound_lemma [of x] by simp
 
 lemma ln_one_minus_pos_upper_bound:
-  fixes x::real shows "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
+  fixes x :: real
+  assumes a: "0 \<le> x" and b: "x < 1"
+  shows "ln (1 - x) \<le> - x"
 proof -
-  assume a: "0 <= (x::real)" and b: "x < 1"
-  have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
+  have "(1 - x) * (1 + x + x\<^sup>2) = 1 - x^3"
     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
-  also have "... <= 1"
+  also have "\<dots> \<le> 1"
     by (auto simp add: a)
-  finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
+  finally have "(1 - x) * (1 + x + x\<^sup>2) \<le> 1" .
   moreover have c: "0 < 1 + x + x\<^sup>2"
     by (simp add: add_pos_nonneg a)
-  ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
+  ultimately have "1 - x \<le> 1 / (1 + x + x\<^sup>2)"
     by (elim mult_imp_le_div_pos)
-  also have "... <= 1 / exp x"
+  also have "\<dots> \<le> 1 / exp x"
     by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
-              real_sqrt_pow2_iff real_sqrt_power)
-  also have "... = exp (-x)"
+        real_sqrt_pow2_iff real_sqrt_power)
+  also have "\<dots> = exp (- x)"
     by (auto simp add: exp_minus divide_inverse)
-  finally have "1 - x <= exp (- x)" .
+  finally have "1 - x \<le> exp (- x)" .
   also have "1 - x = exp (ln (1 - x))"
     by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
-  finally have "exp (ln (1 - x)) <= exp (- x)" .
-  thus ?thesis by (auto simp only: exp_le_cancel_iff)
+  finally have "exp (ln (1 - x)) \<le> exp (- x)" .
+  then show ?thesis
+    by (auto simp only: exp_le_cancel_iff)
 qed
 
-lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
-  apply (case_tac "0 <= x")
-  apply (erule exp_ge_add_one_self_aux)
-  apply (case_tac "x <= -1")
-  apply (subgoal_tac "1 + x <= 0")
-  apply (erule order_trans)
-  apply simp
-  apply simp
-  apply (subgoal_tac "1 + x = exp(ln (1 + x))")
-  apply (erule ssubst)
-  apply (subst exp_le_cancel_iff)
-  apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
-  apply simp
-  apply (rule ln_one_minus_pos_upper_bound)
-  apply auto
-done
+lemma exp_ge_add_one_self [simp]: "1 + x \<le> exp x"
+  for x :: real
+  apply (cases "0 \<le> x")
+   apply (erule exp_ge_add_one_self_aux)
+  apply (cases "x \<le> -1")
+   apply (subgoal_tac "1 + x \<le> 0")
+    apply (erule order_trans)
+    apply simp
+   apply simp
+  apply (subgoal_tac "1 + x = exp (ln (1 + x))")
+   apply (erule ssubst)
+   apply (subst exp_le_cancel_iff)
+   apply (subgoal_tac "ln (1 - (- x)) \<le> - (- x)")
+    apply simp
+   apply (rule ln_one_minus_pos_upper_bound)
+    apply auto
+  done
 
 lemma ln_one_plus_pos_lower_bound:
-  fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
+  fixes x :: real
+  assumes a: "0 \<le> x" and b: "x \<le> 1"
+  shows "x - x\<^sup>2 \<le> ln (1 + x)"
 proof -
-  assume a: "0 <= x" and b: "x <= 1"
   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
     by (rule exp_diff)
-  also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
+  also have "\<dots> \<le> (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
     by (metis a b divide_right_mono exp_bound exp_ge_zero)
-  also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
+  also have "\<dots> \<le> (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
     by (simp add: a divide_left_mono add_pos_nonneg)
-  also from a have "... <= 1 + x"
+  also from a have "\<dots> \<le> 1 + x"
     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
-  finally have "exp (x - x\<^sup>2) <= 1 + x" .
-  also have "... = exp (ln (1 + x))"
+  finally have "exp (x - x\<^sup>2) \<le> 1 + x" .
+  also have "\<dots> = exp (ln (1 + x))"
   proof -
     from a have "0 < 1 + x" by auto
-    thus ?thesis
+    then show ?thesis
       by (auto simp only: exp_ln_iff [THEN sym])
   qed
-  finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
-  thus ?thesis
+  finally have "exp (x - x\<^sup>2) \<le> exp (ln (1 + x))" .
+  then show ?thesis
     by (metis exp_le_cancel_iff)
 qed
 
 lemma ln_one_minus_pos_lower_bound:
-  fixes x::real
-  shows "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
+  fixes x :: real
+  assumes a: "0 \<le> x" and b: "x \<le> 1 / 2"
+  shows "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
 proof -
-  assume a: "0 <= x" and b: "x <= (1 / 2)"
   from b have c: "x < 1" by auto
   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
     apply (subst ln_inverse [symmetric])
-    apply (simp add: field_simps)
+     apply (simp add: field_simps)
     apply (rule arg_cong [where f=ln])
     apply (simp add: field_simps)
     done
-  also have "- (x / (1 - x)) <= ..."
+  also have "- (x / (1 - x)) \<le> \<dots>"
   proof -
-    have "ln (1 + x / (1 - x)) <= x / (1 - x)"
+    have "ln (1 + x / (1 - x)) \<le> x / (1 - x)"
       using a c by (intro ln_add_one_self_le_self) auto
-    thus ?thesis
+    then show ?thesis
       by auto
   qed
-  also have "- (x / (1 - x)) = -x / (1 - x)"
+  also have "- (x / (1 - x)) = - x / (1 - x)"
     by auto
-  finally have d: "- x / (1 - x) <= ln (1 - x)" .
+  finally have d: "- x / (1 - x) \<le> ln (1 - x)" .
   have "0 < 1 - x" using a b by simp
-  hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
-    using mult_right_le_one_le[of "x*x" "2*x"] a b
+  then have e: "- x - 2 * x\<^sup>2 \<le> - x / (1 - x)"
+    using mult_right_le_one_le[of "x * x" "2 * x"] a b
     by (simp add: field_simps power2_eq_square)
-  from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
+  from e d show "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
     by (rule order_trans)
 qed
 
 lemma ln_add_one_self_le_self2:
-  fixes x::real shows "-1 < x \<Longrightarrow> ln(1 + x) <= x"
-  apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
+  fixes x :: real
+  shows "-1 < x \<Longrightarrow> ln (1 + x) \<le> x"
+  apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)")
+   apply simp
   apply (subst ln_le_cancel_iff)
-  apply auto
+    apply auto
   done
 
 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
-  fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> <= x\<^sup>2"
+  fixes x :: real
+  assumes x: "0 \<le> x" and x1: "x \<le> 1"
+  shows "\<bar>ln (1 + x) - x\<bar> \<le> x\<^sup>2"
 proof -
-  assume x: "0 <= x"
-  assume x1: "x <= 1"
-  from x have "ln (1 + x) <= x"
+  from x have "ln (1 + x) \<le> x"
     by (rule ln_add_one_self_le_self)
-  then have "ln (1 + x) - x <= 0"
+  then have "ln (1 + x) - x \<le> 0"
     by simp
   then have "\<bar>ln(1 + x) - x\<bar> = - (ln(1 + x) - x)"
     by (rule abs_of_nonpos)
-  also have "... = x - ln (1 + x)"
+  also have "\<dots> = x - ln (1 + x)"
     by simp
-  also have "... <= x\<^sup>2"
+  also have "\<dots> \<le> x\<^sup>2"
   proof -
-    from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
+    from x x1 have "x - x\<^sup>2 \<le> ln (1 + x)"
       by (intro ln_one_plus_pos_lower_bound)
-    thus ?thesis
+    then show ?thesis
       by simp
   qed
   finally show ?thesis .
 qed
 
 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
-  fixes x::real shows "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> <= 2 * x\<^sup>2"
+  fixes x :: real
+  assumes a: "-(1 / 2) \<le> x" and b: "x \<le> 0"
+  shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
 proof -
-  assume a: "-(1 / 2) <= x"
-  assume b: "x <= 0"
-  have "\<bar>ln (1 + x) - x\<bar> = x - ln(1 - (-x))"
+  have "\<bar>ln (1 + x) - x\<bar> = x - ln (1 - (- x))"
     apply (subst abs_of_nonpos)
-    apply simp
-    apply (rule ln_add_one_self_le_self2)
+     apply simp
+     apply (rule ln_add_one_self_le_self2)
     using a apply auto
     done
-  also have "... <= 2 * x\<^sup>2"
-    apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
-    apply (simp add: algebra_simps)
+  also have "\<dots> \<le> 2 * x\<^sup>2"
+    apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 \<le> ln (1 - (- x))")
+     apply (simp add: algebra_simps)
     apply (rule ln_one_minus_pos_lower_bound)
     using a b apply auto
     done
@@ -1960,63 +1979,68 @@
 qed
 
 lemma abs_ln_one_plus_x_minus_x_bound:
-  fixes x::real shows "\<bar>x\<bar> <= 1 / 2 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> <= 2 * x\<^sup>2"
-  apply (case_tac "0 <= x")
-  apply (rule order_trans)
-  apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
-  apply auto
+  fixes x :: real
+  shows "\<bar>x\<bar> \<le> 1 / 2 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
+  apply (cases "0 \<le> x")
+   apply (rule order_trans)
+    apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
+     apply auto
   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
-  apply auto
+   apply auto
   done
 
 lemma ln_x_over_x_mono:
-  fixes x::real shows "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
+  fixes x :: real
+  assumes x: "exp 1 \<le> x" "x \<le> y"
+  shows "ln y / y \<le> ln x / x"
 proof -
-  assume x: "exp 1 <= x" "x <= y"
+  note x
   moreover have "0 < exp (1::real)" by simp
   ultimately have a: "0 < x" and b: "0 < y"
     by (fast intro: less_le_trans order_trans)+
   have "x * ln y - x * ln x = x * (ln y - ln x)"
     by (simp add: algebra_simps)
-  also have "... = x * ln(y / x)"
+  also have "\<dots> = x * ln (y / x)"
     by (simp only: ln_div a b)
   also have "y / x = (x + (y - x)) / x"
     by simp
-  also have "... = 1 + (y - x) / x"
+  also have "\<dots> = 1 + (y - x) / x"
     using x a by (simp add: field_simps)
-  also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
+  also have "x * ln (1 + (y - x) / x) \<le> x * ((y - x) / x)"
     using x a
     by (intro mult_left_mono ln_add_one_self_le_self) simp_all
-  also have "... = y - x" using a by simp
-  also have "... = (y - x) * ln (exp 1)" by simp
-  also have "... <= (y - x) * ln x"
+  also have "\<dots> = y - x"
+    using a by simp
+  also have "\<dots> = (y - x) * ln (exp 1)" by simp
+  also have "\<dots> \<le> (y - x) * ln x"
     apply (rule mult_left_mono)
-    apply (subst ln_le_cancel_iff)
-    apply fact
-    apply (rule a)
-    apply (rule x)
+     apply (subst ln_le_cancel_iff)
+       apply fact
+      apply (rule a)
+     apply (rule x)
     using x apply simp
     done
-  also have "... = y * ln x - x * ln x"
+  also have "\<dots> = y * ln x - x * ln x"
     by (rule left_diff_distrib)
-  finally have "x * ln y <= y * ln x"
+  finally have "x * ln y \<le> y * ln x"
     by arith
-  then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
-  also have "... = y * (ln x / x)" by simp
-  finally show ?thesis using b by (simp add: field_simps)
+  then have "ln y \<le> (y * ln x) / x"
+    using a by (simp add: field_simps)
+  also have "\<dots> = y * (ln x / x)" by simp
+  finally show ?thesis
+    using b by (simp add: field_simps)
 qed
 
-lemma ln_le_minus_one:
-  fixes x::real shows "0 < x \<Longrightarrow> ln x \<le> x - 1"
+lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
+  for x :: real
   using exp_ge_add_one_self[of "ln x"] by simp
 
-corollary ln_diff_le:
-  fixes x::real
-  shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
+corollary ln_diff_le: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
+  for x :: real
   by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one)
 
 lemma ln_eq_minus_one:
-  fixes x::real
+  fixes x :: real
   assumes "0 < x" "ln x = x - 1"
   shows "x = 1"
 proof -
@@ -2060,19 +2084,18 @@
   qed
 qed
 
-lemma ln_x_over_x_tendsto_0:
-  "((\<lambda>x::real. ln x / x) \<longlongrightarrow> 0) at_top"
+lemma ln_x_over_x_tendsto_0: "((\<lambda>x::real. ln x / x) \<longlongrightarrow> 0) at_top"
 proof (rule lhospital_at_top_at_top[where f' = inverse and g' = "\<lambda>_. 1"])
   from eventually_gt_at_top[of "0::real"]
-    show "\<forall>\<^sub>F x in at_top. (ln has_real_derivative inverse x) (at x)"
-     by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
-qed (insert tendsto_inverse_0,
-     auto simp: filterlim_ident dest!: tendsto_mono[OF at_top_le_at_infinity])
+  show "\<forall>\<^sub>F x in at_top. (ln has_real_derivative inverse x) (at x)"
+    by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
+qed (use tendsto_inverse_0 in
+      \<open>auto simp: filterlim_ident dest!: tendsto_mono[OF at_top_le_at_infinity]\<close>)
 
 lemma exp_ge_one_plus_x_over_n_power_n:
   assumes "x \<ge> - real n" "n > 0"
-  shows   "(1 + x / of_nat n) ^ n \<le> exp x"
-proof (cases "x = -of_nat n")
+  shows "(1 + x / of_nat n) ^ n \<le> exp x"
+proof (cases "x = - of_nat n")
   case False
   from assms False have "(1 + x / of_nat n) ^ n = exp (of_nat n * ln (1 + x / of_nat n))"
     by (subst exp_of_nat_mult, subst exp_ln) (simp_all add: field_simps)
@@ -2081,35 +2104,37 @@
   with assms have "exp (of_nat n * ln (1 + x / of_nat n)) \<le> exp x"
     by (simp add: field_simps)
   finally show ?thesis .
-qed (simp_all add: zero_power)
+next
+  case True
+  then show ?thesis by (simp add: zero_power)
+qed
 
 lemma exp_ge_one_minus_x_over_n_power_n:
   assumes "x \<le> real n" "n > 0"
-  shows   "(1 - x / of_nat n) ^ n \<le> exp (-x)"
+  shows "(1 - x / of_nat n) ^ n \<le> exp (-x)"
   using exp_ge_one_plus_x_over_n_power_n[of n "-x"] assms by simp
 
 lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot"
   unfolding tendsto_Zfun_iff
 proof (rule ZfunI, simp add: eventually_at_bot_dense)
-  fix r :: real assume "0 < r"
-  {
-    fix x
-    assume "x < ln r"
-    then have "exp x < exp (ln r)"
+  fix r :: real
+  assume "0 < r"
+  have "exp x < r" if "x < ln r" for x
+  proof -
+    from that have "exp x < exp (ln r)"
       by simp
-    with \<open>0 < r\<close> have "exp x < r"
+    with \<open>0 < r\<close> show ?thesis
       by simp
-  }
+  qed
   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
 qed
 
 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
-     (auto intro: eventually_gt_at_top)
-
-lemma lim_exp_minus_1:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)"
+    (auto intro: eventually_gt_at_top)
+
+lemma lim_exp_minus_1: "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)"
+  for x :: "'a::{real_normed_field,banach}"
 proof -
   have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
     by (intro derivative_eq_intros | simp)+
@@ -2138,7 +2163,7 @@
   show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top"
     by (simp add: inverse_eq_divide[symmetric])
        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
-              at_top_le_at_infinity order_refl)
+         at_top_le_at_infinity order_refl)
 next
   case (Suc k)
   show ?case
@@ -2155,14 +2180,13 @@
   qed (rule exp_at_top)
 qed
 
-
-definition log :: "[real,real] => real"
+definition log :: "real \<Rightarrow> real \<Rightarrow> real"
   \<comment> \<open>logarithm of @{term x} to base @{term a}\<close>
   where "log a x = ln x / ln a"
 
-
 lemma tendsto_log [tendsto_intros]:
-  "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
+  "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow>
+    ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
   unfolding log_def by (intro tendsto_intros) auto
 
 lemma continuous_log:
@@ -2197,77 +2221,77 @@
 lemma powr_one_eq_one [simp]: "1 powr a = 1"
   by (simp add: powr_def)
 
-lemma powr_zero_eq_one [simp]: "x powr 0 = (if x=0 then 0 else 1)"
+lemma powr_zero_eq_one [simp]: "x powr 0 = (if x = 0 then 0 else 1)"
   by (simp add: powr_def)
 
-lemma powr_one_gt_zero_iff [simp]:
-  fixes x::real shows "(x powr 1 = x) = (0 \<le> x)"
+lemma powr_one_gt_zero_iff [simp]: "x powr 1 = x \<longleftrightarrow> 0 \<le> x"
+  for x :: real
   by (auto simp: powr_def)
 declare powr_one_gt_zero_iff [THEN iffD2, simp]
 
-lemma powr_mult:
-  fixes x::real shows "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
+lemma powr_mult: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
+  for a x y :: real
   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
 
-lemma powr_ge_pzero [simp]:
-  fixes x::real shows "0 <= x powr y"
+lemma powr_ge_pzero [simp]: "0 \<le> x powr y"
+  for x y :: real
   by (simp add: powr_def)
 
-lemma powr_divide:
-  fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
+lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
+  for a b x :: real
   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
   done
 
-lemma powr_divide2:
-  fixes x::real shows "x powr a / x powr b = x powr (a - b)"
+lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
+  for a b x :: real
   apply (simp add: powr_def)
   apply (subst exp_diff [THEN sym])
   apply (simp add: left_diff_distrib)
   done
 
-lemma powr_add:
-  fixes x::real shows "x powr (a + b) = (x powr a) * (x powr b)"
+lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
+  for a b x :: real
   by (simp add: powr_def exp_add [symmetric] distrib_right)
 
-lemma powr_mult_base:
-  fixes x::real shows "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
+lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
+  for x :: real
   by (auto simp: powr_add)
 
-lemma powr_powr:
-  fixes x::real shows "(x powr a) powr b = x powr (a * b)"
+lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
+  for a b x :: real
   by (simp add: powr_def)
 
-lemma powr_powr_swap:
-  fixes x::real shows "(x powr a) powr b = (x powr b) powr a"
+lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
+  for a b x :: real
   by (simp add: powr_powr mult.commute)
 
-lemma powr_minus:
-  fixes x::real shows "x powr (-a) = inverse (x powr a)"
+lemma powr_minus: "x powr (- a) = inverse (x powr a)"
+  for x a :: real
   by (simp add: powr_def exp_minus [symmetric])
 
-lemma powr_minus_divide:
-  fixes x::real shows "x powr (-a) = 1/(x powr a)"
+lemma powr_minus_divide: "x powr (- a) = 1/(x powr a)"
+  for x a :: real
   by (simp add: divide_inverse powr_minus)
 
-lemma divide_powr_uminus:
-  fixes a::real shows "a / b powr c = a * b powr (- c)"
+lemma divide_powr_uminus: "a / b powr c = a * b powr (- c)"
+  for a b c :: real
   by (simp add: powr_minus_divide)
 
-lemma powr_less_mono:
-  fixes x::real shows "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
+lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
+  for a b x :: real
   by (simp add: powr_def)
 
-lemma powr_less_cancel:
-  fixes x::real shows "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
+lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
+  for a b x :: real
   by (simp add: powr_def)
 
-lemma powr_less_cancel_iff [simp]:
-  fixes x::real shows "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
+lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a < x powr b \<longleftrightarrow> a < b"
+  for a b x :: real
   by (blast intro: powr_less_cancel powr_less_mono)
 
-lemma powr_le_cancel_iff [simp]:
-  fixes x::real shows "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
+lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a \<le> x powr b \<longleftrightarrow> a \<le> b"
+  for a b x :: real
   by (simp add: linorder_not_less [symmetric])
 
 lemma log_ln: "ln x = log (exp(1)) x"
@@ -2285,7 +2309,7 @@
 qed
 
 lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
-       DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
+  and DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
 
 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
   by (simp add: powr_def log_def)
@@ -2313,18 +2337,19 @@
 lemma log_one [simp]: "log a 1 = 0"
   by (simp add: log_def)
 
-lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
+lemma log_eq_one [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a a = 1"
   by (simp add: log_def)
 
 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
-  apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
+  apply (rule add_left_cancel [THEN iffD1, where a1 = "log a x"])
   apply (simp add: log_mult [symmetric])
   done
 
 lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y"
   by (simp add: log_mult divide_inverse log_inverse)
 
-lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> (x::real) \<noteq> 0"
+lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> x \<noteq> 0"
+  for a x :: real
   by (simp add: powr_def)
 
 lemma log_add_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x + y = log b (x * b powr y)"
@@ -2333,11 +2358,11 @@
   and minus_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y - log b x = log b (b powr y / x)"
   by (simp_all add: log_mult log_divide)
 
-lemma log_less_cancel_iff [simp]:
-  "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
+lemma log_less_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
   apply safe
-  apply (rule_tac [2] powr_less_cancel)
-  apply (drule_tac a = "log a x" in powr_less_mono, auto)
+   apply (rule_tac [2] powr_less_cancel)
+    apply (drule_tac a = "log a x" in powr_less_mono)
+     apply auto
   done
 
 lemma log_inj:
@@ -2351,18 +2376,19 @@
     assume "x = y"
     then show ?thesis by simp
   next
-    assume "x < y" hence "log b x < log b y"
+    assume "x < y"
+    then have "log b x < log b y"
       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
     then show ?thesis using * by simp
   next
-    assume "y < x" hence "log b y < log b x"
+    assume "y < x"
+    then have "log b y < log b x"
       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
     then show ?thesis using * by simp
   qed
 qed
 
-lemma log_le_cancel_iff [simp]:
-  "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
+lemma log_le_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x \<le> log a y \<longleftrightarrow> x \<le> y"
   by (simp add: linorder_not_less [symmetric])
 
 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
@@ -2390,12 +2416,13 @@
   using log_le_cancel_iff[of a x a] by simp
 
 lemma le_log_iff:
+  fixes b x y :: real
   assumes "1 < b" "x > 0"
-  shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> (x::real)"
+  shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> x"
   using assms
   apply auto
-  apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
-               powr_log_cancel zero_less_one)
+   apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
+      powr_log_cancel zero_less_one)
   apply (metis not_less order.trans order_refl powr_le_cancel_iff powr_log_cancel zero_le_one)
   done
 
@@ -2417,27 +2444,28 @@
   and less_powr_iff = log_less_iff[symmetric]
   and le_powr_iff = log_le_iff[symmetric]
 
-lemma floor_log_eq_powr_iff:
-  "x > 0 \<Longrightarrow> b > 1 \<Longrightarrow> \<lfloor>log b x\<rfloor> = k \<longleftrightarrow> b powr k \<le> x \<and> x < b powr (k + 1)"
+lemma floor_log_eq_powr_iff: "x > 0 \<Longrightarrow> b > 1 \<Longrightarrow> \<lfloor>log b x\<rfloor> = k \<longleftrightarrow> b powr k \<le> x \<and> x < b powr (k + 1)"
   by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff)
 
-lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
-  by (induct n) (simp_all add: ac_simps powr_add of_nat_Suc)
+lemma powr_realpow: "0 < x \<Longrightarrow> x powr (real n) = x^n"
+  by (induct n) (simp_all add: ac_simps powr_add)
 
 lemma powr_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
   by (metis of_nat_numeral powr_realpow)
 
 lemma powr_real_of_int:
-  "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (-n)))"
+  "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (- n)))"
   using powr_realpow[of x "nat n"] powr_realpow[of x "nat (-n)"]
   by (auto simp: field_simps powr_minus)
 
 lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x"
-by(simp add: powr_numeral)
-
-lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
-  apply (case_tac "x = 0", simp, simp)
-  apply (rule powr_realpow [THEN sym], simp)
+  by (simp add: powr_numeral)
+
+lemma powr_realpow2: "0 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x^n = (if (x = 0) then 0 else x powr (real n))"
+  apply (cases "x = 0")
+   apply simp_all
+  apply (rule powr_realpow [THEN sym])
+  apply simp
   done
 
 lemma powr_int:
@@ -2445,49 +2473,51 @@
   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
 proof (cases "i < 0")
   case True
-  have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
-  show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close> by (simp add: r field_simps powr_realpow[symmetric])
+  have r: "x powr i = 1 / x powr (- i)"
+    by (simp add: powr_minus field_simps)
+  show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close>
+    by (simp add: r field_simps powr_realpow[symmetric])
 next
   case False
-  then show ?thesis by (simp add: assms powr_realpow[symmetric])
+  then show ?thesis
+    by (simp add: assms powr_realpow[symmetric])
 qed
 
 lemma compute_powr[code]:
-  fixes i::real
+  fixes i :: real
   shows "b powr i =
     (if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i)
-    else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>)
-    else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))"
+     else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>)
+     else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))"
   by (auto simp: powr_int)
 
-lemma powr_one:
-  fixes x::real shows "0 \<le> x \<Longrightarrow> x powr 1 = x"
-  using powr_realpow [of x 1]
-  by simp
-
-lemma powr_neg_one:
-  fixes x::real shows "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
+lemma powr_one: "0 \<le> x \<Longrightarrow> x powr 1 = x"
+  for x :: real
+  using powr_realpow [of x 1] by simp
+
+lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
+  for x :: real
   using powr_int [of x "- 1"] by simp
 
-lemma powr_neg_numeral:
-  fixes x::real shows "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
+lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
+  for x :: real
   using powr_int [of x "- numeral n"] by simp
 
 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
 
-lemma ln_powr:
-  fixes x::real shows "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
+lemma ln_powr: "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
+  for x :: real
   by (simp add: powr_def)
 
-lemma ln_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> ln (root n b) =  ln b / n"
-by(simp add: root_powr_inverse ln_powr)
+lemma ln_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> ln (root n b) =  ln b / n"
+  by (simp add: root_powr_inverse ln_powr)
 
 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
   by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult.commute)
 
-lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) =  log b a / n"
-by(simp add: log_def ln_root)
+lemma log_root: "n > 0 \<Longrightarrow> a > 0 \<Longrightarrow> log b (root n a) =  log b a / n"
+  by (simp add: log_def ln_root)
 
 lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x"
   by (simp add: log_def ln_powr)
@@ -2495,16 +2525,21 @@
 lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x"
   by (simp add: log_powr powr_realpow [symmetric])
 
-lemma le_log_of_power: assumes "1 < b" "b ^ n \<le> m" shows "n \<le> log b m"
+lemma le_log_of_power:
+  assumes "1 < b" "b ^ n \<le> m"
+  shows "n \<le> log b m"
 proof -
    from assms have "0 < m"
      by (metis less_trans zero_less_power less_le_trans zero_less_one)
-   have "n = log b (b ^ n)" using assms(1) by (simp add: log_nat_power)
-   also have "\<dots> \<le> log b m" using assms \<open>0 < m\<close> by simp
+   have "n = log b (b ^ n)"
+     using assms(1) by (simp add: log_nat_power)
+   also have "\<dots> \<le> log b m"
+     using assms \<open>0 < m\<close> by simp
    finally show ?thesis .
 qed
 
-lemma le_log2_of_power: "2 ^ n \<le> (m::nat) \<Longrightarrow> n \<le> log 2 m"
+lemma le_log2_of_power: "2 ^ n \<le> m \<Longrightarrow> n \<le> log 2 m"
+  for m n :: nat
   using le_log_of_power[of 2] by simp
 
 lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
@@ -2516,122 +2551,140 @@
 lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b"
   by (simp add: log_def ln_powr)
 
-lemma log_base_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> log (root n b) x = n * (log b x)"
-by(simp add: log_def ln_root)
-
-lemma ln_bound:
-  fixes x::real shows "1 <= x ==> ln x <= x"
-  apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
+lemma log_base_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> log (root n b) x = n * (log b x)"
+  by (simp add: log_def ln_root)
+
+lemma ln_bound: "1 \<le> x \<Longrightarrow> ln x \<le> x"
+  for x :: real
+  apply (subgoal_tac "ln (1 + (x - 1)) \<le> x - 1")
+   apply simp
+  apply (rule ln_add_one_self_le_self)
   apply simp
-  apply (rule ln_add_one_self_le_self, simp)
-  done
-
-lemma powr_mono:
-  fixes x::real shows "a <= b ==> 1 <= x ==> x powr a <= x powr b"
-  apply (cases "x = 1", simp)
-  apply (cases "a = b", simp)
-  apply (rule order_less_imp_le)
-  apply (rule powr_less_mono, auto)
   done
 
-lemma ge_one_powr_ge_zero:
-  fixes x::real shows "1 <= x ==> 0 <= a ==> 1 <= x powr a"
-using powr_mono by fastforce
-
-lemma powr_less_mono2:
-  fixes x::real shows "0 < a ==> 0 \<le> x ==> x < y ==> x powr a < y powr a"
+lemma powr_mono: "a \<le> b \<Longrightarrow> 1 \<le> x \<Longrightarrow> x powr a \<le> x powr b"
+  for x :: real
+  apply (cases "x = 1")
+   apply simp
+  apply (cases "a = b")
+   apply simp
+  apply (rule order_less_imp_le)
+  apply (rule powr_less_mono)
+   apply auto
+  done
+
+lemma ge_one_powr_ge_zero: "1 \<le> x \<Longrightarrow> 0 \<le> a \<Longrightarrow> 1 \<le> x powr a"
+  for x :: real
+  using powr_mono by fastforce
+
+lemma powr_less_mono2: "0 < a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> x powr a < y powr a"
+  for x :: real
   by (simp add: powr_def)
 
-lemma powr_less_mono2_neg:
-  fixes x::real shows "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
+lemma powr_less_mono2_neg: "a < 0 \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> y powr a < x powr a"
+  for x :: real
   by (simp add: powr_def)
 
-lemma powr_mono2:
-  fixes x::real shows "0 <= a ==> 0 \<le> x ==> x <= y ==> x powr a <= y powr a"
-  apply (case_tac "a = 0", simp)
-  apply (case_tac "x = y", simp)
+lemma powr_mono2: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> x powr a \<le> y powr a"
+  for x :: real
+  apply (case_tac "a = 0")
+   apply simp
+  apply (case_tac "x = y")
+   apply simp
   apply (metis dual_order.strict_iff_order powr_less_mono2)
   done
 
 lemma powr_mono2':
-  assumes "a \<le> 0" "x > 0" "x \<le> (y::real)"
-  shows   "x powr a \<ge> y powr a"
+  fixes a x y :: real
+  assumes "a \<le> 0" "x > 0" "x \<le> y"
+  shows "x powr a \<ge> y powr a"
 proof -
-  from assms have "x powr -a \<le> y powr -a" by (intro powr_mono2) simp_all
-  with assms show ?thesis by (auto simp add: powr_minus field_simps)
+  from assms have "x powr - a \<le> y powr - a"
+    by (intro powr_mono2) simp_all
+  with assms show ?thesis
+    by (auto simp add: powr_minus field_simps)
 qed
 
-lemma powr_inj:
-  fixes x::real shows "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
+lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
+  for x :: real
   unfolding powr_def exp_inj_iff by simp
 
 lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x"
   by (simp add: powr_def root_powr_inverse sqrt_def)
 
-lemma ln_powr_bound:
-  fixes x::real shows "1 \<le> x \<Longrightarrow> 0 < a \<Longrightarrow> ln x \<le> (x powr a) / a"
+lemma ln_powr_bound: "1 \<le> x \<Longrightarrow> 0 < a \<Longrightarrow> ln x \<le> (x powr a) / a"
+  for x :: real
   by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute
-            mult_imp_le_div_pos not_less powr_gt_zero)
+      mult_imp_le_div_pos not_less powr_gt_zero)
 
 lemma ln_powr_bound2:
-  fixes x::real
+  fixes x :: real
   assumes "1 < x" and "0 < a"
-  shows "(ln x) powr a <= (a powr a) * x"
+  shows "(ln x) powr a \<le> (a powr a) * x"
 proof -
-  from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
+  from assms have "ln x \<le> (x powr (1 / a)) / (1 / a)"
     by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
-  also have "... = a * (x powr (1 / a))"
+  also have "\<dots> = a * (x powr (1 / a))"
     by simp
-  finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
+  finally have "(ln x) powr a \<le> (a * (x powr (1 / a))) powr a"
     by (metis assms less_imp_le ln_gt_zero powr_mono2)
-  also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
+  also have "\<dots> = (a powr a) * ((x powr (1 / a)) powr a)"
     using assms powr_mult by auto
   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
     by (rule powr_powr)
-  also have "... = x" using assms
+  also have "\<dots> = x" using assms
     by auto
   finally show ?thesis .
 qed
 
 lemma tendsto_powr:
-  fixes a::real
-  assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F" and a: "a \<noteq> 0"
+  fixes a b :: real
+  assumes f: "(f \<longlongrightarrow> a) F"
+    and g: "(g \<longlongrightarrow> b) F"
+    and a: "a \<noteq> 0"
   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
   unfolding powr_def
 proof (rule filterlim_If)
   from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
     by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
-qed (insert f g a, auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
+  from f g a show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a)))
+      (inf F (principal {x. f x \<noteq> 0}))"
+    by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
+qed
 
 lemma tendsto_powr'[tendsto_intros]:
-  fixes a::real
-  assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F" 
-      and a: "a \<noteq> 0 \<or> (b > 0 \<and> eventually (\<lambda>x. f x \<ge> 0) F)"
+  fixes a :: real
+  assumes f: "(f \<longlongrightarrow> a) F"
+    and g: "(g \<longlongrightarrow> b) F"
+    and a: "a \<noteq> 0 \<or> (b > 0 \<and> eventually (\<lambda>x. f x \<ge> 0) F)"
   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
 proof -
-  from a consider "a \<noteq> 0" | "a = 0" "b > 0" "eventually (\<lambda>x. f x \<ge> 0) F" by auto
-  thus ?thesis
+  from a consider "a \<noteq> 0" | "a = 0" "b > 0" "eventually (\<lambda>x. f x \<ge> 0) F"
+    by auto
+  then show ?thesis
   proof cases
-    assume "a \<noteq> 0"
-    from f g this show ?thesis by (rule tendsto_powr)
+    case 1
+    with f g show ?thesis by (rule tendsto_powr)
   next
-    assume a: "a = 0" and b: "b > 0" and f_nonneg: "eventually (\<lambda>x. f x \<ge> 0) F"
-    hence "((\<lambda>x. if f x = 0 then 0 else exp (g x * ln (f x))) \<longlongrightarrow> 0) F"
+    case 2
+    have "((\<lambda>x. if f x = 0 then 0 else exp (g x * ln (f x))) \<longlongrightarrow> 0) F"
     proof (intro filterlim_If)
       have "filterlim f (principal {0<..}) (inf F (principal {z. f z \<noteq> 0}))"
-        using f_nonneg by (auto simp add: filterlim_iff eventually_inf_principal 
-                             eventually_principal elim: eventually_mono)
+        using \<open>eventually (\<lambda>x. f x \<ge> 0) F\<close>
+        by (auto simp add: filterlim_iff eventually_inf_principal
+            eventually_principal elim: eventually_mono)
       moreover have "filterlim f (nhds a) (inf F (principal {z. f z \<noteq> 0}))"
         by (rule tendsto_mono[OF _ f]) simp_all
       ultimately have f: "filterlim f (at_right 0) (inf F (principal {x. f x \<noteq> 0}))"
-        by (simp add: at_within_def filterlim_inf a)
+        by (simp add: at_within_def filterlim_inf \<open>a = 0\<close>)
       have g: "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))"
         by (rule tendsto_mono[OF _ g]) simp_all
       show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))"
         by (rule filterlim_compose[OF exp_at_bot] filterlim_tendsto_pos_mult_at_bot
-                 filterlim_compose[OF ln_at_0] f g b)+
+                 filterlim_compose[OF ln_at_0] f g \<open>b > 0\<close>)+
     qed simp_all
-    with a show ?thesis by (simp add: powr_def)
+    with \<open>a = 0\<close> show ?thesis
+      by (simp add: powr_def)
   qed
 qed
 
@@ -2643,32 +2696,40 @@
   using assms unfolding continuous_def by (rule tendsto_powr)
 
 lemma continuous_at_within_powr[continuous_intros]:
+  fixes f g :: "_ \<Rightarrow> real"
   assumes "continuous (at a within s) f"
     and "continuous (at a within s) g"
     and "f a \<noteq> 0"
-  shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x :: real))"
+  shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
   using assms unfolding continuous_within by (rule tendsto_powr)
 
 lemma isCont_powr[continuous_intros, simp]:
-  assumes "isCont f a" "isCont g a" "f a \<noteq> (0::real)"
+  fixes f g :: "_ \<Rightarrow> real"
+  assumes "isCont f a" "isCont g a" "f a \<noteq> 0"
   shows "isCont (\<lambda>x. (f x) powr g x) a"
   using assms unfolding continuous_at by (rule tendsto_powr)
 
 lemma continuous_on_powr[continuous_intros]:
-  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> (0::real)"
+  fixes f g :: "_ \<Rightarrow> real"
+  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0"
   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
-  
+
 lemma tendsto_powr2:
-  fixes a::real
-  assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F" and f_nonneg: "\<forall>\<^sub>F x in F. 0 \<le> f x" and b: "0 < b"
+  fixes a :: real
+  assumes f: "(f \<longlongrightarrow> a) F"
+    and g: "(g \<longlongrightarrow> b) F"
+    and "\<forall>\<^sub>F x in F. 0 \<le> f x"
+    and b: "0 < b"
   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
   using tendsto_powr'[of f a F g b] assms by auto
 
 lemma DERIV_powr:
-  fixes r::real
-  assumes g: "DERIV g x :> m" and pos: "g x > 0" and f: "DERIV f x :> r"
-  shows  "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
+  fixes r :: real
+  assumes g: "DERIV g x :> m"
+    and pos: "g x > 0"
+    and f: "DERIV f x :> r"
+  shows "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
 proof -
   have "DERIV (\<lambda>x. exp (f x * ln (g x))) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
     using pos
@@ -2683,9 +2744,10 @@
 qed
 
 lemma DERIV_fun_powr:
-  fixes r::real
-  assumes g: "DERIV g x :> m" and pos: "g x > 0"
-  shows  "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m"
+  fixes r :: real
+  assumes g: "DERIV g x :> m"
+    and pos: "g x > 0"
+  shows "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m"
   using DERIV_powr[OF g pos DERIV_const, of r] pos
   by (simp add: powr_divide2[symmetric] field_simps)
 
@@ -2693,8 +2755,9 @@
   assumes "z > 0"
   shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)"
 proof (subst DERIV_cong_ev[OF refl _ refl])
-  from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)" by (intro t1_space_nhds) auto
-  thus "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)"
+  from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)"
+    by (intro t1_space_nhds) auto
+  then show "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)"
     unfolding powr_def by eventually_elim simp
   from assms show "((\<lambda>z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)"
     by (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff)
@@ -2708,12 +2771,14 @@
   using tendsto_powr2[OF assms] by simp
 
 lemma continuous_on_powr':
-  assumes "continuous_on s f" "continuous_on s g" and
-    "\<forall>x\<in>s. f x \<ge> (0::real) \<and> (f x = 0 \<longrightarrow> g x > 0)"
+  fixes f g :: "_ \<Rightarrow> real"
+  assumes "continuous_on s f" "continuous_on s g"
+    and "\<forall>x\<in>s. f x \<ge> 0 \<and> (f x = 0 \<longrightarrow> g x > 0)"
   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
   unfolding continuous_on_def
 proof
-  fix x assume x: "x \<in> s"
+  fix x
+  assume x: "x \<in> s"
   from assms x show "((\<lambda>x. f x powr g x) \<longlongrightarrow> f x powr g x) (at x within s)"
   proof (cases "f x = 0")
     case True
@@ -2735,18 +2800,20 @@
 proof -
   have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X")
     by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top]
-                     filterlim_tendsto_neg_mult_at_bot assms)
+        filterlim_tendsto_neg_mult_at_bot assms)
   also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
     using f filterlim_at_top_dense[of f F]
     by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono)
   finally show ?thesis .
 qed
 
-lemma tendsto_exp_limit_at_right:
-  fixes x :: real
-  shows "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
-proof cases
-  assume "x \<noteq> 0"
+lemma tendsto_exp_limit_at_right: "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
+  for x :: real
+proof (cases "x = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
   have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
     by (auto intro!: derivative_eq_intros)
   then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)"
@@ -2757,32 +2824,32 @@
   proof (rule filterlim_mono_eventually)
     show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
       unfolding eventually_at_right[OF zero_less_one]
-      using \<open>x \<noteq> 0\<close>
-      apply  (intro exI[of _ "1 / \<bar>x\<bar>"])
+      using False
+      apply (intro exI[of _ "1 / \<bar>x\<bar>"])
       apply (auto simp: field_simps powr_def abs_if)
-      by (metis add_less_same_cancel1 mult_less_0_iff not_less_iff_gr_or_eq zero_less_one)
+      apply (metis add_less_same_cancel1 mult_less_0_iff not_less_iff_gr_or_eq zero_less_one)
+      done
   qed (simp_all add: at_eq_sup_left_right)
-qed simp
-
-lemma tendsto_exp_limit_at_top:
-  fixes x :: real
-  shows "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
+qed
+
+lemma tendsto_exp_limit_at_top: "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
+  for x :: real
   apply (subst filterlim_at_top_to_right)
   apply (simp add: inverse_eq_divide)
   apply (rule tendsto_exp_limit_at_right)
   done
 
-lemma tendsto_exp_limit_sequentially:
-  fixes x :: real
-  shows "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x"
+lemma tendsto_exp_limit_sequentially: "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x"
+  for x :: real
 proof (rule filterlim_mono_eventually)
   from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" ..
-  hence "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
+  then have "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
     apply (intro eventually_sequentiallyI [of n])
-    apply (case_tac "x \<ge> 0")
-    apply (rule add_pos_nonneg, auto intro: divide_nonneg_nonneg)
-    apply (subgoal_tac "x / real xa > -1")
-    apply (auto simp add: field_simps)
+    apply (cases "x \<ge> 0")
+     apply (rule add_pos_nonneg)
+      apply (auto intro: divide_nonneg_nonneg)
+    apply (subgoal_tac "x / real xa > - 1")
+     apply (auto simp add: field_simps)
     done
   then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
     by (rule eventually_mono) (erule powr_realpow)
@@ -2790,13 +2857,14 @@
     by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
 qed auto
 
+
 subsection \<open>Sine and Cosine\<close>
 
-definition sin_coeff :: "nat \<Rightarrow> real" where
-  "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
-
-definition cos_coeff :: "nat \<Rightarrow> real" where
-  "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
+definition sin_coeff :: "nat \<Rightarrow> real"
+  where "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
+
+definition cos_coeff :: "nat \<Rightarrow> real"
+  where "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
 
 definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)"
@@ -2818,41 +2886,37 @@
   unfolding cos_coeff_def sin_coeff_def
   by (simp del: mult_Suc) (auto elim: oddE)
 
-lemma summable_norm_sin:
-  fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
+lemma summable_norm_sin: "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
+  for x :: "'a::{real_normed_algebra_1,banach}"
   unfolding sin_coeff_def
   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
   done
 
-lemma summable_norm_cos:
-  fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
+lemma summable_norm_cos: "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
+  for x :: "'a::{real_normed_algebra_1,banach}"
   unfolding cos_coeff_def
   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
   done
 
-lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin(x)"
-unfolding sin_def
+lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin x"
+  unfolding sin_def
   by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums)
 
-lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos(x)"
-unfolding cos_def
+lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos x"
+  unfolding cos_def
   by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums)
 
-lemma sin_of_real:
-  fixes x::real
-  shows "sin (of_real x) = of_real (sin x)"
+lemma sin_of_real: "sin (of_real x) = of_real (sin x)"
+  for x :: real
 proof -
   have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R  x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R  (of_real x)^n)"
   proof
-    fix n
-    show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n"
+    show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n" for n
       by (simp add: scaleR_conv_of_real)
   qed
-  also have "... sums (sin (of_real x))"
+  also have "\<dots> sums (sin (of_real x))"
     by (rule sin_converges)
   finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" .
   then show ?thesis
@@ -2863,17 +2927,15 @@
 corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
   by (metis Reals_cases Reals_of_real sin_of_real)
 
-lemma cos_of_real:
-  fixes x::real
-  shows "cos (of_real x) = of_real (cos x)"
+lemma cos_of_real: "cos (of_real x) = of_real (cos x)"
+  for x :: real
 proof -
   have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R  x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R  (of_real x)^n)"
   proof
-    fix n
-    show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n"
+    show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n" for n
       by (simp add: scaleR_conv_of_real)
   qed
-  also have "... sums (cos (of_real x))"
+  also have "\<dots> sums (cos (of_real x))"
     by (rule cos_converges)
   finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" .
   then show ?thesis
@@ -2890,30 +2952,28 @@
 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
   by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc)
 
-text\<open>Now at last we can get the derivatives of exp, sin and cos\<close>
-
-lemma DERIV_sin [simp]:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "DERIV sin x :> cos(x)"
+text \<open>Now at last we can get the derivatives of exp, sin and cos.\<close>
+
+lemma DERIV_sin [simp]: "DERIV sin x :> cos x"
+  for x :: "'a::{real_normed_field,banach}"
   unfolding sin_def cos_def scaleR_conv_of_real
   apply (rule DERIV_cong)
-  apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
-  apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
+   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
+      apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
               summable_minus_iff scaleR_conv_of_real [symmetric]
               summable_norm_sin [THEN summable_norm_cancel]
               summable_norm_cos [THEN summable_norm_cancel])
   done
 
 declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
-        DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
-
-lemma DERIV_cos [simp]:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "DERIV cos x :> -sin(x)"
+  and DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
+
+lemma DERIV_cos [simp]: "DERIV cos x :> - sin x"
+  for x :: "'a::{real_normed_field,banach}"
   unfolding sin_def cos_def scaleR_conv_of_real
   apply (rule DERIV_cong)
-  apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
-  apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
+   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
+      apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
               diffs_sin_coeff diffs_cos_coeff
               summable_minus_iff scaleR_conv_of_real [symmetric]
               summable_norm_sin [THEN summable_norm_cancel]
@@ -2921,151 +2981,147 @@
   done
 
 declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
-        DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
-
-lemma isCont_sin:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "isCont sin x"
+  and DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
+
+lemma isCont_sin: "isCont sin x"
+  for x :: "'a::{real_normed_field,banach}"
   by (rule DERIV_sin [THEN DERIV_isCont])
 
-lemma isCont_cos:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "isCont cos x"
+lemma isCont_cos: "isCont cos x"
+  for x :: "'a::{real_normed_field,banach}"
   by (rule DERIV_cos [THEN DERIV_isCont])
 
-lemma isCont_sin' [simp]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
+lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   by (rule isCont_o2 [OF _ isCont_sin])
 
-(*FIXME A CONTEXT FOR F WOULD BE BETTER*)
-
-lemma isCont_cos' [simp]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
+(* FIXME a context for f would be better *)
+
+lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   by (rule isCont_o2 [OF _ isCont_cos])
 
-lemma tendsto_sin [tendsto_intros]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
+lemma tendsto_sin [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   by (rule isCont_tendsto_compose [OF isCont_sin])
 
-lemma tendsto_cos [tendsto_intros]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
+lemma tendsto_cos [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   by (rule isCont_tendsto_compose [OF isCont_cos])
 
-lemma continuous_sin [continuous_intros]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
+lemma continuous_sin [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   unfolding continuous_def by (rule tendsto_sin)
 
-lemma continuous_on_sin [continuous_intros]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
+lemma continuous_on_sin [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   unfolding continuous_on_def by (auto intro: tendsto_sin)
 
-lemma continuous_within_sin:
-  fixes z :: "'a::{real_normed_field,banach}"
-  shows "continuous (at z within s) sin"
+lemma continuous_within_sin: "continuous (at z within s) sin"
+  for z :: "'a::{real_normed_field,banach}"
   by (simp add: continuous_within tendsto_sin)
 
-lemma continuous_cos [continuous_intros]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
+lemma continuous_cos [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   unfolding continuous_def by (rule tendsto_cos)
 
-lemma continuous_on_cos [continuous_intros]:
-  fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
+lemma continuous_on_cos [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
+  for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
   unfolding continuous_on_def by (auto intro: tendsto_cos)
 
-lemma continuous_within_cos:
-  fixes z :: "'a::{real_normed_field,banach}"
-  shows "continuous (at z within s) cos"
+lemma continuous_within_cos: "continuous (at z within s) cos"
+  for z :: "'a::{real_normed_field,banach}"
   by (simp add: continuous_within tendsto_cos)
 
+
 subsection \<open>Properties of Sine and Cosine\<close>
 
 lemma sin_zero [simp]: "sin 0 = 0"
-  unfolding sin_def sin_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
+  by (simp add: sin_def sin_coeff_def scaleR_conv_of_real)
 
 lemma cos_zero [simp]: "cos 0 = 1"
-  unfolding cos_def cos_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
-
-lemma DERIV_fun_sin:
-     "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
+  by (simp add: cos_def cos_coeff_def scaleR_conv_of_real)
+
+lemma DERIV_fun_sin: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin (g x)) x :> cos (g x) * m"
   by (auto intro!: derivative_intros)
 
-lemma DERIV_fun_cos:
-     "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
+lemma DERIV_fun_cos: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> - sin (g x) * m"
   by (auto intro!: derivative_eq_intros)
 
+
 subsection \<open>Deriving the Addition Formulas\<close>
 
-text\<open>The the product of two cosine series\<close>
+text \<open>The product of two cosine series.\<close>
 lemma cos_x_cos_y:
   fixes x :: "'a::{real_normed_field,banach}"
-  shows "(\<lambda>p. \<Sum>n\<le>p.
-          if even p \<and> even n
-          then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
-         sums (cos x * cos y)"
+  shows
+    "(\<lambda>p. \<Sum>n\<le>p.
+        if even p \<and> even n
+        then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
+      sums (cos x * cos y)"
 proof -
-  { fix n p::nat
-    assume "n\<le>p"
-    then have *: "even n \<Longrightarrow> even p \<Longrightarrow> (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
+  have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p - n)) =
+    (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p - n)
+     else 0)"
+    if "n \<le> p" for n p :: nat
+  proof -
+    from that have *: "even n \<Longrightarrow> even p \<Longrightarrow>
+        (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
       by (metis div_add power_add le_add_diff_inverse odd_add)
-    have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
-          (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
-    using \<open>n\<le>p\<close>
-      by (auto simp: * algebra_simps cos_coeff_def binomial_fact)
-  }
+    with that show ?thesis
+      by (auto simp: algebra_simps cos_coeff_def binomial_fact)
+  qed
   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
              (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
     by simp
-  also have "... = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
+  also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
     by (simp add: algebra_simps)
-  also have "... sums (cos x * cos y)"
+  also have "\<dots> sums (cos x * cos y)"
     using summable_norm_cos
     by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums)
   finally show ?thesis .
 qed
 
-text\<open>The product of two sine series\<close>
+text \<open>The product of two sine series.\<close>
 lemma sin_x_sin_y:
   fixes x :: "'a::{real_normed_field,banach}"
-  shows "(\<lambda>p. \<Sum>n\<le>p.
-          if even p \<and> odd n
-               then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
-         sums (sin x * sin y)"
+  shows
+    "(\<lambda>p. \<Sum>n\<le>p.
+        if even p \<and> odd n
+        then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
+        else 0)
+      sums (sin x * sin y)"
 proof -
-  { fix n p::nat
-    assume "n\<le>p"
-    { assume np: "odd n" "even p"
-        with \<open>n\<le>p\<close> have "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
+  have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
+    (if even p \<and> odd n
+     then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
+     else 0)"
+    if "n \<le> p" for n p :: nat
+  proof -
+    have "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
+      if np: "odd n" "even p"
+    proof -
+      from \<open>n \<le> p\<close> np have *: "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
         by arith+
-      moreover have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
+      have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
         by simp
-      ultimately have *: "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
-        using np \<open>n\<le>p\<close>
+      with \<open>n \<le> p\<close> np * show ?thesis
         apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add)
-        apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc)
+        apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus
+            mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc)
         done
-    } then
-    have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
-          (if even p \<and> odd n
-          then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
-    using \<open>n\<le>p\<close>
-      by (auto simp:  algebra_simps sin_coeff_def binomial_fact)
-  }
+    qed
+    then show ?thesis
+      using \<open>n\<le>p\<close> by (auto simp: algebra_simps sin_coeff_def binomial_fact)
+  qed
   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
              (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
     by simp
-  also have "... = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
+  also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
     by (simp add: algebra_simps)
-  also have "... sums (sin x * sin y)"
+  also have "\<dots> sums (sin x * sin y)"
     using summable_norm_sin
     by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums)
   finally show ?thesis .
@@ -3074,36 +3130,38 @@
 lemma sums_cos_x_plus_y:
   fixes x :: "'a::{real_normed_field,banach}"
   shows
-  "(\<lambda>p. \<Sum>n\<le>p. if even p
-               then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
-               else 0)
-        sums cos (x + y)"
+    "(\<lambda>p. \<Sum>n\<le>p.
+        if even p
+        then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
+        else 0)
+      sums cos (x + y)"
 proof -
-  { fix p::nat
-    have "(\<Sum>n\<le>p. if even p
-                  then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
-                  else 0) =
-          (if even p
-                  then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
-                  else 0)"
+  have
+    "(\<Sum>n\<le>p.
+      if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
+      else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)"
+    for p :: nat
+  proof -
+    have
+      "(\<Sum>n\<le>p. if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
+       (if even p then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
       by simp
-    also have "... = (if even p
-                  then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
-                  else 0)"
+    also have "\<dots> =
+       (if even p
+        then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
+        else 0)"
       by (auto simp: setsum_right_distrib field_simps scaleR_conv_of_real nonzero_of_real_divide)
-    also have "... = cos_coeff p *\<^sub>R ((x + y) ^ p)"
+    also have "\<dots> = cos_coeff p *\<^sub>R ((x + y) ^ p)"
       by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real atLeast0AtMost)
-    finally have "(\<Sum>n\<le>p. if even p
-                  then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
-                  else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" .
-  }
-  then have "(\<lambda>p. \<Sum>n\<le>p.
-               if even p
-               then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
-               else 0)
-        = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
-        by simp
-   also have "... sums cos (x + y)"
+    finally show ?thesis .
+  qed
+  then have
+    "(\<lambda>p. \<Sum>n\<le>p.
+        if even p
+        then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
+        else 0) = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
+    by simp
+   also have "\<dots> sums cos (x + y)"
     by (rule cos_converges)
    finally show ?thesis .
 qed
@@ -3112,26 +3170,24 @@
   fixes x :: "'a::{real_normed_field,banach}"
   shows "cos (x + y) = cos x * cos y - sin x * sin y"
 proof -
-  { fix n p::nat
-    assume "n\<le>p"
-    then have "(if even p \<and> even n
-               then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
-          (if even p \<and> odd n
-               then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
-          = (if even p
-               then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
-      by simp
-  }
-  then have "(\<lambda>p. \<Sum>n\<le>p. (if even p
-               then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
-        sums (cos x * cos y - sin x * sin y)"
+  have
+    "(if even p \<and> even n
+      then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
+     (if even p \<and> odd n
+      then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
+     (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
+    if "n \<le> p" for n p :: nat
+    by simp
+  then have
+    "(\<lambda>p. \<Sum>n\<le>p. (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
+      sums (cos x * cos y - sin x * sin y)"
     using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]]
     by (simp add: setsum_subtractf [symmetric])
   then show ?thesis
     by (blast intro: sums_cos_x_plus_y sums_unique2)
 qed
 
-lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin(x)"
+lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin x"
 proof -
   have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)"
     by (auto simp: sin_coeff_def elim!: oddE)
@@ -3139,13 +3195,13 @@
     by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums])
 qed
 
-lemma sin_minus [simp]:
-  fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "sin (-x) = -sin(x)"
-using sin_minus_converges [of x]
-by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] suminf_minus sums_iff equation_minus_iff)
-
-lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos(x)"
+lemma sin_minus [simp]: "sin (- x) = - sin x"
+  for x :: "'a::{real_normed_algebra_1,banach}"
+  using sin_minus_converges [of x]
+  by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel]
+      suminf_minus sums_iff equation_minus_iff)
+
+lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos x"
 proof -
   have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)"
     by (auto simp: Transcendental.cos_coeff_def elim!: evenE)
@@ -3153,110 +3209,91 @@
     by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums])
 qed
 
-lemma cos_minus [simp]:
-  fixes x :: "'a::{real_normed_algebra_1,banach}"
-  shows "cos (-x) = cos(x)"
-using cos_minus_converges [of x]
-by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
-              suminf_minus sums_iff equation_minus_iff)
-
-lemma sin_cos_squared_add [simp]:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
-using cos_add [of x "-x"]
-by (simp add: power2_eq_square algebra_simps)
-
-lemma sin_cos_squared_add2 [simp]:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
+lemma cos_minus [simp]: "cos (-x) = cos x"
+  for x :: "'a::{real_normed_algebra_1,banach}"
+  using cos_minus_converges [of x]
+  by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
+      suminf_minus sums_iff equation_minus_iff)
+
+lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
+  for x :: "'a::{real_normed_field,banach}"
+  using cos_add [of x "-x"]
+  by (simp add: power2_eq_square algebra_simps)
+
+lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
+  for x :: "'a::{real_normed_field,banach}"
   by (subst add.commute, rule sin_cos_squared_add)
 
-lemma sin_cos_squared_add3 [simp]:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "cos x * cos x + sin x * sin x = 1"
+lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
+  for x :: "'a::{real_normed_field,banach}"
   using sin_cos_squared_add2 [unfolded power2_eq_square] .
 
-lemma sin_squared_eq:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
+lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
+  for x :: "'a::{real_normed_field,banach}"
   unfolding eq_diff_eq by (rule sin_cos_squared_add)
 
-lemma cos_squared_eq:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
+lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
+  for x :: "'a::{real_normed_field,banach}"
   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
 
-lemma abs_sin_le_one [simp]:
-  fixes x :: real
-  shows "\<bar>sin x\<bar> \<le> 1"
-  by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
-
-lemma sin_ge_minus_one [simp]:
-  fixes x :: real
-  shows "-1 \<le> sin x"
-  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
-
-lemma sin_le_one [simp]:
-  fixes x :: real
-  shows "sin x \<le> 1"
-  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
-
-lemma abs_cos_le_one [simp]:
-  fixes x :: real
-  shows "\<bar>cos x\<bar> \<le> 1"
-  by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
-
-lemma cos_ge_minus_one [simp]:
-  fixes x :: real
-  shows "-1 \<le> cos x"
-  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
-
-lemma cos_le_one [simp]:
-  fixes x :: real
-  shows "cos x \<le> 1"
-  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
-
-lemma cos_diff:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "cos (x - y) = cos x * cos y + sin x * sin y"
+lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
+  for x :: real
+  by (rule power2_le_imp_le) (simp_all add: sin_squared_eq)
+
+lemma sin_ge_minus_one [simp]: "- 1 \<le> sin x"
+  for x :: real
+  using abs_sin_le_one [of x] by (simp add: abs_le_iff)
+
+lemma sin_le_one [simp]: "sin x \<le> 1"
+  for x :: real
+  using abs_sin_le_one [of x] by (simp add: abs_le_iff)
+
+lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
+  for x :: real
+  by (rule power2_le_imp_le) (simp_all add: cos_squared_eq)
+
+lemma cos_ge_minus_one [simp]: "- 1 \<le> cos x"
+  for x :: real
+  using abs_cos_le_one [of x] by (simp add: abs_le_iff)
+
+lemma cos_le_one [simp]: "cos x \<le> 1"
+  for x :: real
+  using abs_cos_le_one [of x] by (simp add: abs_le_iff)
+
+lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
+  for x :: "'a::{real_normed_field,banach}"
   using cos_add [of x "- y"] by simp
 
-lemma cos_double:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
-  using cos_add [where x=x and y=x]
-  by (simp add: power2_eq_square)
-
-lemma sin_cos_le1:
-  fixes x::real shows "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1"
-  using cos_diff [of x y]
-  by (metis abs_cos_le_one add.commute)
-
-lemma DERIV_fun_pow: "DERIV g x :> m ==>
-      DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
+lemma cos_double: "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
+  for x :: "'a::{real_normed_field,banach}"
+  using cos_add [where x=x and y=x] by (simp add: power2_eq_square)
+
+lemma sin_cos_le1: "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1"
+  for x :: real
+  using cos_diff [of x y] by (metis abs_cos_le_one add.commute)
+
+lemma DERIV_fun_pow: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
   by (auto intro!: derivative_eq_intros simp:)
 
-lemma DERIV_fun_exp:
-     "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
+lemma DERIV_fun_exp: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. exp (g x)) x :> exp (g x) * m"
   by (auto intro!: derivative_intros)
 
+
 subsection \<open>The Constant Pi\<close>
 
 definition pi :: real
-  where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
-
-text\<open>Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
+  where "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
+
+text \<open>Show that there's a least positive @{term x} with @{term "cos x = 0"};
    hence define pi.\<close>
 
-lemma sin_paired:
-  fixes x :: real
-  shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
+lemma sin_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
+  for x :: real
 proof -
   have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
-    apply (rule sums_group)
-    using sin_converges [of x, unfolded scaleR_conv_of_real]
-    by auto
-  thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: ac_simps)
+    by (rule sums_group) (use sin_converges [of x, unfolded scaleR_conv_of_real] in auto)
+  then show ?thesis
+    by (simp add: sin_coeff_def ac_simps)
 qed
 
 lemma sin_gt_zero_02:
@@ -3272,52 +3309,51 @@
     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
     have "x * x < ?k2 * ?k3"
       using assms by (intro mult_strict_mono', simp_all)
-    hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
+    then have "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
       by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
-    thus "0 < ?f n"
+    then show "0 < ?f n"
       by (simp add: divide_simps mult_ac del: mult_Suc)
 qed
   have sums: "?f sums sin x"
-    by (rule sin_paired [THEN sums_group], simp)
+    by (rule sin_paired [THEN sums_group]) simp
   show "0 < sin x"
     unfolding sums_unique [OF sums]
     using sums_summable [OF sums] pos
     by (rule suminf_pos)
 qed
 
-lemma cos_double_less_one:
-  fixes x :: real
-  shows "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
+lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
+  for x :: real
   using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double)
 
-lemma cos_paired:
-  fixes x :: real
-  shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
+lemma cos_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
+  for x :: real
 proof -
   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
-    apply (rule sums_group)
-    using cos_converges [of x, unfolded scaleR_conv_of_real]
-    by auto
-  thus ?thesis unfolding cos_coeff_def by (simp add: ac_simps)
+    by (rule sums_group) (use cos_converges [of x, unfolded scaleR_conv_of_real] in auto)
+  then show ?thesis
+    by (simp add: cos_coeff_def ac_simps)
 qed
 
 lemmas realpow_num_eq_if = power_eq_if
 
 lemma sumr_pos_lt_pair:
   fixes f :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>summable f;
-        \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
-      \<Longrightarrow> setsum f {..<k} < suminf f"
-apply (simp only: One_nat_def)
-apply (subst suminf_split_initial_segment [where k=k], assumption, simp)
-apply (drule_tac k=k in summable_ignore_initial_segment)
-apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums], simp)
-apply simp
-apply (metis (no_types, lifting) add.commute suminf_pos summable_def sums_unique)
-done
-
-lemma cos_two_less_zero [simp]:
-  "cos 2 < (0::real)"
+  shows "summable f \<Longrightarrow>
+    (\<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc (Suc 0) * d) + 1))) \<Longrightarrow>
+    setsum f {..<k} < suminf f"
+  apply (simp only: One_nat_def)
+  apply (subst suminf_split_initial_segment [where k=k])
+   apply assumption
+  apply simp
+  apply (drule_tac k=k in summable_ignore_initial_segment)
+  apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums])
+   apply simp
+  apply simp
+  apply (metis (no_types, lifting) add.commute suminf_pos summable_def sums_unique)
+  done
+
+lemma cos_two_less_zero [simp]: "cos 2 < (0::real)"
 proof -
   note fact_Suc [simp del]
   from sums_minus [OF cos_paired]
@@ -3327,13 +3363,14 @@
     by (rule sums_summable)
   have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
     by (simp add: fact_num_eq_if realpow_num_eq_if)
-  moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n))))
-                 < (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
+  moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n)))) <
+    (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
   proof -
-    { fix d
+    {
+      fix d
       let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))"
       have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
-        unfolding of_nat_mult   by (rule mult_strict_mono) (simp_all add: fact_less_mono)
+        unfolding of_nat_mult by (rule mult_strict_mono) (simp_all add: fact_less_mono)
       then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
         by (simp only: fact_Suc [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
       then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
@@ -3353,40 +3390,40 @@
 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
 
-lemma cos_is_zero: "EX! x::real. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
+lemma cos_is_zero: "\<exists>!x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
 proof (rule ex_ex1I)
-  show "\<exists>x::real. 0 \<le> x & x \<le> 2 & cos x = 0"
-    by (rule IVT2, simp_all)
+  show "\<exists>x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
+    by (rule IVT2) simp_all
 next
-  fix x::real and y::real
+  fix x y :: real
   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
   have [simp]: "\<forall>x::real. cos differentiable (at x)"
     unfolding real_differentiable_def by (auto intro: DERIV_cos)
-  from x y show "x = y"
-    apply (cut_tac less_linear [of x y], auto)
-    apply (drule_tac f = cos in Rolle)
-    apply (drule_tac [5] f = cos in Rolle)
-    apply (auto dest!: DERIV_cos [THEN DERIV_unique])
-    apply (metis order_less_le_trans less_le sin_gt_zero_02)
+  from x y less_linear [of x y] show "x = y"
+    apply auto
+     apply (drule_tac f = cos in Rolle)
+        apply (drule_tac [5] f = cos in Rolle)
+           apply (auto dest!: DERIV_cos [THEN DERIV_unique])
+     apply (metis order_less_le_trans less_le sin_gt_zero_02)
     apply (metis order_less_le_trans less_le sin_gt_zero_02)
     done
 qed
 
-lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
+lemma pi_half: "pi/2 = (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
   by (simp add: pi_def)
 
 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
   by (simp add: pi_half cos_is_zero [THEN theI'])
 
-lemma cos_of_real_pi_half [simp]:
-  fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
-  shows "cos ((of_real pi / 2) :: 'a) = 0"
-by (metis cos_pi_half cos_of_real eq_numeral_simps(4) nonzero_of_real_divide of_real_0 of_real_numeral)
+lemma cos_of_real_pi_half [simp]: "cos ((of_real pi / 2) :: 'a) = 0"
+  if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
+  by (metis cos_pi_half cos_of_real eq_numeral_simps(4)
+      nonzero_of_real_divide of_real_0 of_real_numeral)
 
 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
   apply (rule order_le_neq_trans)
-  apply (simp add: pi_half cos_is_zero [THEN theI'])
+   apply (simp add: pi_half cos_is_zero [THEN theI'])
   apply (metis cos_pi_half cos_zero zero_neq_one)
   done
 
@@ -3395,7 +3432,7 @@
 
 lemma pi_half_less_two [simp]: "pi / 2 < 2"
   apply (rule order_le_neq_trans)
-  apply (simp add: pi_half cos_is_zero [THEN theI'])
+   apply (simp add: pi_half cos_is_zero [THEN theI'])
   apply (metis cos_pi_half cos_two_neq_zero)
   done
 
@@ -3425,45 +3462,36 @@
   using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]
   by (simp add: power2_eq_1_iff)
 
-lemma sin_of_real_pi_half [simp]:
-  fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
-  shows "sin ((of_real pi / 2) :: 'a) = 1"
+lemma sin_of_real_pi_half [simp]: "sin ((of_real pi / 2) :: 'a) = 1"
+  if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
   using sin_pi_half
-by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
-
-lemma sin_cos_eq:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "sin x = cos (of_real pi / 2 - x)"
+  by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
+
+lemma sin_cos_eq: "sin x = cos (of_real pi / 2 - x)"
+  for x :: "'a::{real_normed_field,banach}"
   by (simp add: cos_diff)
 
-lemma minus_sin_cos_eq:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "-sin x = cos (x + of_real pi / 2)"
+lemma minus_sin_cos_eq: "- sin x = cos (x + of_real pi / 2)"
+  for x :: "'a::{real_normed_field,banach}"
   by (simp add: cos_add nonzero_of_real_divide)
 
-lemma cos_sin_eq:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "cos x = sin (of_real pi / 2 - x)"
-  using sin_cos_eq [of "of_real pi / 2 - x"]
-  by simp
-
-lemma sin_add:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "sin (x + y) = sin x * cos y + cos x * sin y"
+lemma cos_sin_eq: "cos x = sin (of_real pi / 2 - x)"
+  for x :: "'a::{real_normed_field,banach}"
+  using sin_cos_eq [of "of_real pi / 2 - x"] by simp
+
+lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
+  for x :: "'a::{real_normed_field,banach}"
   using cos_add [of "of_real pi / 2 - x" "-y"]
   by (simp add: cos_sin_eq) (simp add: sin_cos_eq)
 
-lemma sin_diff:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "sin (x - y) = sin x * cos y - cos x * sin y"
+lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
+  for x :: "'a::{real_normed_field,banach}"
   using sin_add [of x "- y"] by simp
 
-lemma sin_double:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "sin(2 * x) = 2 * sin x * cos x"
+lemma sin_double: "sin(2 * x) = 2 * sin x * cos x"
+  for x :: "'a::{real_normed_field,banach}"
   using sin_add [where x=x and y=x] by simp
 
-
 lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1"
   using cos_add [where x = "pi/2" and y = "pi/2"]
   by (simp add: cos_of_real)
@@ -3490,10 +3518,10 @@
 lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x"
   by (simp add: cos_add)
 
-lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
+lemma sin_periodic [simp]: "sin (x + 2 * pi) = sin x"
   by (simp add: sin_add sin_double cos_double)
 
-lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
+lemma cos_periodic [simp]: "cos (x + 2 * pi) = cos x"
   by (simp add: cos_add sin_double cos_double)
 
 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
@@ -3502,204 +3530,197 @@
 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
   by (metis cos_npi mult.commute)
 
-lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
-  by (induct n) (auto simp: of_nat_Suc distrib_right)
-
-lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
+lemma sin_npi [simp]: "sin (real n * pi) = 0"
+  for n :: nat
+  by (induct n) (auto simp: distrib_right)
+
+lemma sin_npi2 [simp]: "sin (pi * real n) = 0"
+  for n :: nat
   by (simp add: mult.commute [of pi])
 
-lemma cos_two_pi [simp]: "cos (2*pi) = 1"
+lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
   by (simp add: cos_double)
 
-lemma sin_two_pi [simp]: "sin (2*pi) = 0"
+lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
   by (simp add: sin_double)
 
-
-lemma sin_times_sin:
-  fixes w :: "'a::{real_normed_field,banach}"
-  shows "sin(w) * sin(z) = (cos(w - z) - cos(w + z)) / 2"
+lemma sin_times_sin: "sin w * sin z = (cos (w - z) - cos (w + z)) / 2"
+  for w :: "'a::{real_normed_field,banach}"
   by (simp add: cos_diff cos_add)
 
-lemma sin_times_cos:
-  fixes w :: "'a::{real_normed_field,banach}"
-  shows "sin(w) * cos(z) = (sin(w + z) + sin(w - z)) / 2"
+lemma sin_times_cos: "sin w * cos z = (sin (w + z) + sin (w - z)) / 2"
+  for w :: "'a::{real_normed_field,banach}"
   by (simp add: sin_diff sin_add)
 
-lemma cos_times_sin:
-  fixes w :: "'a::{real_normed_field,banach}"
-  shows "cos(w) * sin(z) = (sin(w + z) - sin(w - z)) / 2"
+lemma cos_times_sin: "cos w * sin z = (sin (w + z) - sin (w - z)) / 2"
+  for w :: "'a::{real_normed_field,banach}"
   by (simp add: sin_diff sin_add)
 
-lemma cos_times_cos:
-  fixes w :: "'a::{real_normed_field,banach}"
-  shows "cos(w) * cos(z) = (cos(w - z) + cos(w + z)) / 2"
+lemma cos_times_cos: "cos w * cos z = (cos (w - z) + cos (w + z)) / 2"
+  for w :: "'a::{real_normed_field,banach}"
   by (simp add: cos_diff cos_add)
 
-lemma sin_plus_sin:  (*FIXME field should not be necessary*)
-  fixes w :: "'a::{real_normed_field,banach,field}"
-  shows "sin(w) + sin(z) = 2 * sin((w + z) / 2) * cos((w - z) / 2)"
+lemma sin_plus_sin: "sin w + sin z = 2 * sin ((w + z) / 2) * cos ((w - z) / 2)"
+  for w :: "'a::{real_normed_field,banach,field}"  (* FIXME field should not be necessary *)
   apply (simp add: mult.assoc sin_times_cos)
   apply (simp add: field_simps)
   done
 
-lemma sin_diff_sin:
-  fixes w :: "'a::{real_normed_field,banach,field}"
-  shows "sin(w) - sin(z) = 2 * sin((w - z) / 2) * cos((w + z) / 2)"
+lemma sin_diff_sin: "sin w - sin z = 2 * sin ((w - z) / 2) * cos ((w + z) / 2)"
+  for w :: "'a::{real_normed_field,banach,field}"
   apply (simp add: mult.assoc sin_times_cos)
   apply (simp add: field_simps)
   done
 
-lemma cos_plus_cos:
-  fixes w :: "'a::{real_normed_field,banach,field}"
-  shows "cos(w) + cos(z) = 2 * cos((w + z) / 2) * cos((w - z) / 2)"
+lemma cos_plus_cos: "cos w + cos z = 2 * cos ((w + z) / 2) * cos ((w - z) / 2)"
+  for w :: "'a::{real_normed_field,banach,field}"
   apply (simp add: mult.assoc cos_times_cos)
   apply (simp add: field_simps)
   done
 
-lemma cos_diff_cos:
-  fixes w :: "'a::{real_normed_field,banach,field}"
-  shows "cos(w) - cos(z) = 2 * sin((w + z) / 2) * sin((z - w) / 2)"
+lemma cos_diff_cos: "cos w - cos z = 2 * sin ((w + z) / 2) * sin ((z - w) / 2)"
+  for w :: "'a::{real_normed_field,banach,field}"
   apply (simp add: mult.assoc sin_times_sin)
   apply (simp add: field_simps)
   done
 
-lemma cos_double_cos:
-  fixes z :: "'a::{real_normed_field,banach}"
-  shows "cos(2 * z) = 2 * cos z ^ 2 - 1"
-by (simp add: cos_double sin_squared_eq)
-
-lemma cos_double_sin:
-  fixes z :: "'a::{real_normed_field,banach}"
-  shows "cos(2 * z) = 1 - 2 * sin z ^ 2"
-by (simp add: cos_double sin_squared_eq)
+lemma cos_double_cos: "cos (2 * z) = 2 * cos z ^ 2 - 1"
+  for z :: "'a::{real_normed_field,banach}"
+  by (simp add: cos_double sin_squared_eq)
+
+lemma cos_double_sin: "cos (2 * z) = 1 - 2 * sin z ^ 2"
+  for z :: "'a::{real_normed_field,banach}"
+  by (simp add: cos_double sin_squared_eq)
 
 lemma sin_pi_minus [simp]: "sin (pi - x) = sin x"
   by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff)
 
-lemma cos_pi_minus [simp]: "cos (pi - x) = -(cos x)"
+lemma cos_pi_minus [simp]: "cos (pi - x) = - (cos x)"
   by (metis cos_minus cos_periodic_pi uminus_add_conv_diff)
 
 lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)"
   by (simp add: sin_diff)
 
-lemma cos_minus_pi [simp]: "cos (x - pi) = -(cos x)"
+lemma cos_minus_pi [simp]: "cos (x - pi) = - (cos x)"
   by (simp add: cos_diff)
 
-lemma sin_2pi_minus [simp]: "sin (2*pi - x) = -(sin x)"
+lemma sin_2pi_minus [simp]: "sin (2 * pi - x) = - (sin x)"
   by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
 
-lemma cos_2pi_minus [simp]: "cos (2*pi - x) = cos x"
+lemma cos_2pi_minus [simp]: "cos (2 * pi - x) = cos x"
   by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
-           diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
-
-lemma sin_gt_zero2: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < sin x"
+      diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
+
+lemma sin_gt_zero2: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < sin x"
   by (metis sin_gt_zero_02 order_less_trans pi_half_less_two)
 
 lemma sin_less_zero:
   assumes "- pi/2 < x" and "x < 0"
   shows "sin x < 0"
 proof -
-  have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
-  thus ?thesis by simp
+  have "0 < sin (- x)"
+    using assms by (simp only: sin_gt_zero2)
+  then show ?thesis by simp
 qed
 
 lemma pi_less_4: "pi < 4"
   using pi_half_less_two by auto
 
-lemma cos_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
+lemma cos_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
   by (simp add: cos_sin_eq sin_gt_zero2)
 
-lemma cos_gt_zero_pi: "\<lbrakk>-(pi/2) < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
+lemma cos_gt_zero_pi: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
   using cos_gt_zero [of x] cos_gt_zero [of "-x"]
   by (cases rule: linorder_cases [of x 0]) auto
 
-lemma cos_ge_zero: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> 0 \<le> cos x"
-  apply (auto simp: order_le_less cos_gt_zero_pi)
-  by (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
-
-lemma sin_gt_zero: "\<lbrakk>0 < x; x < pi \<rbrakk> \<Longrightarrow> 0 < sin x"
+lemma cos_ge_zero: "-(pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> 0 \<le> cos x"
+  by (auto simp: order_le_less cos_gt_zero_pi)
+    (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
+
+lemma sin_gt_zero: "0 < x \<Longrightarrow> x < pi \<Longrightarrow> 0 < sin x"
   by (simp add: sin_cos_eq cos_gt_zero_pi)
 
-lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x < 0"
-  using sin_gt_zero [of "x-pi"]
+lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x < 0"
+  using sin_gt_zero [of "x - pi"]
   by (simp add: sin_diff)
 
 lemma pi_ge_two: "2 \<le> pi"
 proof (rule ccontr)
-  assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
-  have "\<exists>y > pi. y < 2 \<and> y < 2*pi"
-  proof (cases "2 < 2*pi")
-    case True with dense[OF \<open>pi < 2\<close>] show ?thesis by auto
+  assume "\<not> ?thesis"
+  then have "pi < 2" by auto
+  have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
+  proof (cases "2 < 2 * pi")
+    case True
+    with dense[OF \<open>pi < 2\<close>] show ?thesis by auto
   next
-    case False have "pi < 2*pi" by auto
+    case False
+    have "pi < 2 * pi" by auto
     from dense[OF this] and False show ?thesis by auto
   qed
-  then obtain y where "pi < y" and "y < 2" and "y < 2*pi" by blast
-  hence "0 < sin y" using sin_gt_zero_02 by auto
-  moreover
-  have "sin y < 0" using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2*pi\<close> sin_periodic_pi[of "y - pi"] by auto
+  then obtain y where "pi < y" and "y < 2" and "y < 2 * pi"
+    by blast
+  then have "0 < sin y"
+    using sin_gt_zero_02 by auto
+  moreover have "sin y < 0"
+    using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2 * pi\<close> sin_periodic_pi[of "y - pi"]
+    by auto
   ultimately show False by auto
 qed
 
-lemma sin_ge_zero: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> 0 \<le> sin x"
+lemma sin_ge_zero: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> sin x"
   by (auto simp: order_le_less sin_gt_zero)
 
-lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x \<le> 0"
-  using sin_ge_zero [of "x-pi"]
-  by (simp add: sin_diff)
+lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x \<le> 0"
+  using sin_ge_zero [of "x - pi"] by (simp add: sin_diff)
 
 lemma sin_pi_divide_n_ge_0 [simp]:
-  assumes "n \<noteq> 0" shows "0 \<le> sin (pi / real n)"
-apply (rule sin_ge_zero)
-using assms
-apply (simp_all add: divide_simps)
-done
+  assumes "n \<noteq> 0"
+  shows "0 \<le> sin (pi / real n)"
+  by (rule sin_ge_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
 
 lemma sin_pi_divide_n_gt_0:
-  assumes "2 \<le> n" shows "0 < sin (pi / real n)"
-apply (rule sin_gt_zero)
-using assms
-apply (simp_all add: divide_simps)
-done
-
-text \<open>FIXME: This proof is almost identical to lemma \<open>cos_is_zero\<close>.
-  It should be possible to factor out some of the common parts.\<close>
-
-lemma cos_total: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
+  assumes "2 \<le> n"
+  shows "0 < sin (pi / real n)"
+  by (rule sin_gt_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
+
+(* FIXME: This proof is almost identical to lemma \<open>cos_is_zero\<close>.
+   It should be possible to factor out some of the common parts. *)
+lemma cos_total:
+  assumes y: "- 1 \<le> y" "y \<le> 1"
+  shows "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
 proof (rule ex_ex1I)
-  assume y: "-1 \<le> y" "y \<le> 1"
-  show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
-    by (rule IVT2, simp_all add: y)
+  show "\<exists>x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
+    by (rule IVT2) (simp_all add: y)
 next
   fix a b
   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
   have [simp]: "\<forall>x::real. cos differentiable (at x)"
     unfolding real_differentiable_def by (auto intro: DERIV_cos)
-  from a b show "a = b"
-    apply (cut_tac less_linear [of a b], auto)
-    apply (drule_tac f = cos in Rolle)
-    apply (drule_tac [5] f = cos in Rolle)
-    apply (auto dest!: DERIV_cos [THEN DERIV_unique])
-    apply (metis order_less_le_trans less_le sin_gt_zero)
+  from a b less_linear [of a b] show "a = b"
+    apply auto
+     apply (drule_tac f = cos in Rolle)
+        apply (drule_tac [5] f = cos in Rolle)
+           apply (auto dest!: DERIV_cos [THEN DERIV_unique])
+     apply (metis order_less_le_trans less_le sin_gt_zero)
     apply (metis order_less_le_trans less_le sin_gt_zero)
     done
 qed
 
 lemma sin_total:
   assumes y: "-1 \<le> y" "y \<le> 1"
-    shows "\<exists>! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
+  shows "\<exists>!x. - (pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y"
 proof -
   from cos_total [OF y]
   obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y"
-           and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
+    and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
     by blast
   show ?thesis
     apply (simp add: sin_cos_eq)
     apply (rule ex1I [where a="pi/2 - x"])
-    apply (cut_tac [2] x'="pi/2 - xa" in uniq)
+     apply (cut_tac [2] x'="pi/2 - xa" in uniq)
     using x
-    apply auto
+        apply auto
     done
 qed
 
@@ -3713,7 +3734,7 @@
   obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi"
     apply (rule that [of "nat \<lfloor>x/pi\<rfloor>"])
     using assms
-    apply (simp_all add: xle)
+     apply (simp_all add: xle)
     apply (metis floor_less_iff less_irrefl mult_imp_div_pos_less not_le pi_gt_zero)
     done
   then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0"
@@ -3721,7 +3742,7 @@
   then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0"
     by (auto simp: intro!: cos_total)
   then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0"
-                  and uniq: "\<And>\<phi>. \<lbrakk>0 \<le> \<phi>; \<phi> \<le> pi; cos \<phi> = 0\<rbrakk> \<Longrightarrow> \<phi> = \<theta>"
+    and uniq: "\<And>\<phi>. 0 \<le> \<phi> \<Longrightarrow> \<phi> \<le> pi \<Longrightarrow> cos \<phi> = 0 \<Longrightarrow> \<phi> = \<theta>"
     by blast
   then have "x - real n * pi = \<theta>"
     using x by blast
@@ -3731,8 +3752,7 @@
     by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps)
 qed
 
-lemma sin_zero_lemma:
-     "\<lbrakk>0 \<le> x; sin x = 0\<rbrakk> \<Longrightarrow> \<exists>n::nat. even n & x = real n * (pi/2)"
+lemma sin_zero_lemma: "0 \<le> x \<Longrightarrow> sin x = 0 \<Longrightarrow> \<exists>n::nat. even n \<and> x = real n * (pi/2)"
   using cos_zero_lemma [of "x + pi/2"]
   apply (clarsimp simp add: cos_add)
   apply (rule_tac x = "n - 1" in exI)
@@ -3740,79 +3760,78 @@
   done
 
 lemma cos_zero_iff:
-     "(cos x = 0) \<longleftrightarrow>
-      ((\<exists>n. odd n & (x = real n * (pi/2))) \<or> (\<exists>n. odd n & (x = -(real n * (pi/2)))))"
-      (is "?lhs = ?rhs")
+  "cos x = 0 \<longleftrightarrow> ((\<exists>n. odd n \<and> x = real n * (pi/2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi/2))))"
+  (is "?lhs = ?rhs")
 proof -
-  { fix n :: nat
-    assume "odd n"
-    then obtain m where "n = 2 * m + 1" ..
-    then have "cos (real n * pi / 2) = 0"
+  have *: "cos (real n * pi / 2) = 0" if "odd n" for n :: nat
+  proof -
+    from that obtain m where "n = 2 * m + 1" ..
+    then show ?thesis
       by (simp add: field_simps) (simp add: cos_add add_divide_distrib)
-  } note * = this
+  qed
   show ?thesis
   proof
-    assume "cos x = 0" then show ?rhs
-      using cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force
-  next
-    assume ?rhs then show "cos x = 0"
-    by (auto dest: * simp del: eq_divide_eq_numeral1)
+    show ?rhs if ?lhs
+      using that cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force
+    show ?lhs if ?rhs
+      using that by (auto dest: * simp del: eq_divide_eq_numeral1)
   qed
 qed
 
 lemma sin_zero_iff:
-     "(sin x = 0) \<longleftrightarrow>
-      ((\<exists>n. even n & (x = real n * (pi/2))) \<or> (\<exists>n. even n & (x = -(real n * (pi/2)))))"
-      (is "?lhs = ?rhs")
+  "sin x = 0 \<longleftrightarrow> ((\<exists>n. even n \<and> x = real n * (pi/2)) \<or> (\<exists>n. even n \<and> x = - (real n * (pi/2))))"
+  (is "?lhs = ?rhs")
 proof
-  assume "sin x = 0" then show ?rhs
-    using sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force
-next
-  assume ?rhs then show "sin x = 0"
-    by (auto elim: evenE)
+  show ?rhs if ?lhs
+    using that sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force
+  show ?lhs if ?rhs
+    using that by (auto elim: evenE)
 qed
 
-lemma cos_zero_iff_int:
-     "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))"
+lemma cos_zero_iff_int: "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))"
 proof safe
   assume "cos x = 0"
-  then show "\<exists>n. odd n & x = of_int n * (pi/2)"
-    apply (simp add: cos_zero_iff, safe)
-    apply (metis even_int_iff of_int_of_nat_eq)
-    apply (rule_tac x="- (int n)" in exI, simp)
+  then show "\<exists>n. odd n \<and> x = of_int n * (pi/2)"
+    apply (simp add: cos_zero_iff)
+    apply safe
+     apply (metis even_int_iff of_int_of_nat_eq)
+    apply (rule_tac x="- (int n)" in exI)
+    apply simp
     done
 next
-  fix n::int
+  fix n :: int
   assume "odd n"
   then show "cos (of_int n * (pi / 2)) = 0"
     apply (simp add: cos_zero_iff)
-    apply (case_tac n rule: int_cases2, simp_all)
+    apply (cases n rule: int_cases2)
+     apply simp_all
     done
 qed
 
-lemma sin_zero_iff_int:
-     "sin x = 0 \<longleftrightarrow> (\<exists>n. even n & (x = of_int n * (pi/2)))"
+lemma sin_zero_iff_int: "sin x = 0 \<longleftrightarrow> (\<exists>n. even n \<and> x = of_int n * (pi/2))"
 proof safe
   assume "sin x = 0"
   then show "\<exists>n. even n \<and> x = of_int n * (pi / 2)"
-    apply (simp add: sin_zero_iff, safe)
-    apply (metis even_int_iff of_int_of_nat_eq)
-    apply (rule_tac x="- (int n)" in exI, simp)
+    apply (simp add: sin_zero_iff)
+    apply safe
+     apply (metis even_int_iff of_int_of_nat_eq)
+    apply (rule_tac x="- (int n)" in exI)
+    apply simp
     done
 next
-  fix n::int
+  fix n :: int
   assume "even n"
   then show "sin (of_int n * (pi / 2)) = 0"
     apply (simp add: sin_zero_iff)
-    apply (case_tac n rule: int_cases2, simp_all)
+    apply (cases n rule: int_cases2)
+     apply simp_all
     done
 qed
 
-lemma sin_zero_iff_int2:
-  "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)"
+lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)"
   apply (simp only: sin_zero_iff_int)
   apply (safe elim!: evenE)
-  apply (simp_all add: field_simps)
+   apply (simp_all add: field_simps)
   using dvd_triv_left apply fastforce
   done
 
@@ -3824,12 +3843,14 @@
   from MVT2[OF \<open>y < x\<close> DERIV_cos[THEN impI, THEN allI]]
   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
     by auto
-  hence "0 < z" and "z < pi" using assms by auto
-  hence "0 < sin z" using sin_gt_zero by auto
-  hence "cos x - cos y < 0"
+  then have "0 < z" and "z < pi"
+    using assms by auto
+  then have "0 < sin z"
+    using sin_gt_zero by auto
+  then have "cos x - cos y < 0"
     unfolding cos_diff minus_mult_commute[symmetric]
     using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2)
-  thus ?thesis by auto
+  then show ?thesis by auto
 qed
 
 lemma cos_monotone_0_pi_le:
@@ -3841,22 +3862,22 @@
     using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<close>] by auto
 next
   case False
-  hence "y = x" using \<open>y \<le> x\<close> by auto
-  thus ?thesis by auto
+  then have "y = x" using \<open>y \<le> x\<close> by auto
+  then show ?thesis by auto
 qed
 
 lemma cos_monotone_minus_pi_0:
-  assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
+  assumes "- pi \<le> y" and "y < x" and "x \<le> 0"
   shows "cos y < cos x"
 proof -
-  have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
+  have "0 \<le> - x" and "- x < - y" and "- y \<le> pi"
     using assms by auto
   from cos_monotone_0_pi[OF this] show ?thesis
     unfolding cos_minus .
 qed
 
 lemma cos_monotone_minus_pi_0':
-  assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
+  assumes "- pi \<le> y" and "y \<le> x" and "x \<le> 0"
   shows "cos y \<le> cos x"
 proof (cases "y < x")
   case True
@@ -3864,18 +3885,18 @@
     by auto
 next
   case False
-  hence "y = x" using \<open>y \<le> x\<close> by auto
-  thus ?thesis by auto
+  then have "y = x" using \<open>y \<le> x\<close> by auto
+  then show ?thesis by auto
 qed
 
 lemma sin_monotone_2pi:
   assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2"
   shows "sin y < sin x"
-    apply (simp add: sin_cos_eq)
-    apply (rule cos_monotone_0_pi)
-    using assms
+  apply (simp add: sin_cos_eq)
+  apply (rule cos_monotone_0_pi)
+  using assms
     apply auto
-    done
+  done
 
 lemma sin_monotone_2pi_le:
   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
@@ -3883,7 +3904,9 @@
   by (metis assms le_less sin_monotone_2pi)
 
 lemma sin_x_le_x:
-  fixes x::real assumes x: "x \<ge> 0" shows "sin x \<le> x"
+  fixes x :: real
+  assumes x: "x \<ge> 0"
+  shows "sin x \<le> x"
 proof -
   let ?f = "\<lambda>x. x - sin x"
   from x have "?f x \<ge> ?f 0"
@@ -3891,11 +3914,13 @@
     apply (intro allI impI exI[of _ "1 - cos x" for x])
     apply (auto intro!: derivative_eq_intros simp: field_simps)
     done
-  thus "sin x \<le> x" by simp
+  then show "sin x \<le> x" by simp
 qed
 
 lemma sin_x_ge_neg_x:
-  fixes x::real assumes x: "x \<ge> 0" shows "sin x \<ge> - x"
+  fixes x :: real
+  assumes x: "x \<ge> 0"
+  shows "sin x \<ge> - x"
 proof -
   let ?f = "\<lambda>x. x + sin x"
   from x have "?f x \<ge> ?f 0"
@@ -3903,11 +3928,11 @@
     apply (intro allI impI exI[of _ "1 + cos x" for x])
     apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff)
     done
-  thus "sin x \<ge> -x" by simp
+  then show "sin x \<ge> -x" by simp
 qed
 
-lemma abs_sin_x_le_abs_x:
-  fixes x::real shows "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
+lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
+  for x :: real
   using sin_x_ge_neg_x [of x] sin_x_le_x [of x] sin_x_ge_neg_x [of "-x"] sin_x_le_x [of "-x"]
   by (auto simp: abs_real_def)
 
@@ -3918,35 +3943,42 @@
 proof -
   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
     by (auto simp: algebra_simps sin_add)
-  thus ?thesis
+  then show ?thesis
     by (simp add: distrib_right add_divide_distrib add.commute mult.commute [of pi])
 qed
 
-lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
+lemma cos_2npi [simp]: "cos (2 * real n * pi) = 1"
+  for n :: nat
   by (cases "even n") (simp_all add: cos_double mult.assoc)
 
 lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0"
-  apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
-  apply (subst cos_add, simp)
+  apply (subgoal_tac "cos (pi + pi/2) = 0")
+   apply simp
+  apply (subst cos_add)
+  apply simp
   done
 
-lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
+lemma sin_2npi [simp]: "sin (2 * real n * pi) = 0"
+  for n :: nat
   by (auto simp: mult.assoc sin_double)
 
 lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1"
-  apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
-  apply (subst sin_add, simp)
+  apply (subgoal_tac "sin (pi + pi/2) = - 1")
+   apply simp
+  apply (subst sin_add)
+  apply simp
   done
 
 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
-by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
+  by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
 
 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
   by (auto intro!: derivative_eq_intros)
 
 lemma sin_zero_norm_cos_one:
   fixes x :: "'a::{real_normed_field,banach}"
-  assumes "sin x = 0" shows "norm (cos x) = 1"
+  assumes "sin x = 0"
+  shows "norm (cos x) = 1"
   using sin_cos_squared_add [of x, unfolded assms]
   by (simp add: square_norm_one)
 
@@ -3955,23 +3987,23 @@
 
 lemma cos_one_sin_zero:
   fixes x :: "'a::{real_normed_field,banach}"
-  assumes "cos x = 1" shows "sin x = 0"
+  assumes "cos x = 1"
+  shows "sin x = 0"
   using sin_cos_squared_add [of x, unfolded assms]
   by simp
 
-lemma sin_times_pi_eq_0: "sin(x * pi) = 0 \<longleftrightarrow> x \<in> \<int>"
+lemma sin_times_pi_eq_0: "sin (x * pi) = 0 \<longleftrightarrow> x \<in> \<int>"
   by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int)
 
-lemma cos_one_2pi:
-    "cos(x) = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2*pi) | (\<exists>n::nat. x = -(n * 2*pi))"
-    (is "?lhs = ?rhs")
+lemma cos_one_2pi: "cos x = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2 * pi) | (\<exists>n::nat. x = - (n * 2 * pi))"
+  (is "?lhs = ?rhs")
 proof
-  assume "cos(x) = 1"
+  assume ?lhs
   then have "sin x = 0"
     by (simp add: cos_one_sin_zero)
   then show ?rhs
   proof (simp only: sin_zero_iff, elim exE disjE conjE)
-    fix n::nat
+    fix n :: nat
     assume n: "even n" "x = real n * (pi/2)"
     then obtain m where m: "n = 2 * m"
       using dvdE by blast
@@ -3981,7 +4013,7 @@
       using m me n
       by (auto simp: field_simps elim!: evenE)
   next
-    fix n::nat
+    fix n :: nat
     assume n: "even n" "x = - (real n * (pi/2))"
     then obtain m where m: "n = 2 * m"
       using dvdE by blast
@@ -3992,33 +4024,33 @@
       by (auto simp: field_simps elim!: evenE)
   qed
 next
-  assume "?rhs"
+  assume ?rhs
   then show "cos x = 1"
     by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
 qed
 
-lemma cos_one_2pi_int: "cos(x) = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2*pi)"
-  apply auto  \<comment>\<open>FIXME simproc bug\<close>
-  apply (auto simp: cos_one_2pi)
-  apply (metis of_int_of_nat_eq)
-  apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
-  by (metis mult_minus_right of_int_of_nat )
-
-lemma sin_cos_sqrt: "0 \<le> sin(x) \<Longrightarrow> (sin(x) = sqrt(1 - (cos(x) ^ 2)))"
+lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)"
+  apply auto  (* FIXME simproc bug? *)
+   apply (auto simp: cos_one_2pi)
+    apply (metis of_int_of_nat_eq)
+   apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
+  apply (metis mult_minus_right of_int_of_nat)
+  done
+
+lemma sin_cos_sqrt: "0 \<le> sin x \<Longrightarrow> sin x = sqrt (1 - (cos(x) ^ 2))"
   using sin_squared_eq real_sqrt_unique by fastforce
 
-lemma sin_eq_0_pi: "-pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin(x) = 0 \<Longrightarrow> x = 0"
+lemma sin_eq_0_pi: "- pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin x = 0 \<Longrightarrow> x = 0"
   by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
 
-lemma cos_treble_cos:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "cos(3 * x) = 4 * cos(x) ^ 3 - 3 * cos x"
+lemma cos_treble_cos: "cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x"
+  for x :: "'a::{real_normed_field,banach}"
 proof -
   have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))"
     by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])
   have "cos(3 * x) = cos(2*x + x)"
     by simp
-  also have "... = 4 * cos(x) ^ 3 - 3 * cos x"
+  also have "\<dots> = 4 * cos x ^ 3 - 3 * cos x"
     apply (simp only: cos_add cos_double sin_double)
     apply (simp add: * field_simps power2_eq_square power3_eq_cube)
     done
@@ -4027,7 +4059,8 @@
 
 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
 proof -
-  let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
+  let ?c = "cos (pi / 4)"
+  let ?s = "sin (pi / 4)"
   have nonneg: "0 \<le> ?c"
     by (simp add: cos_ge_zero)
   have "0 = cos (pi / 4 + pi / 4)"
@@ -4038,15 +4071,16 @@
     by (simp add: sin_squared_eq)
   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
     by (simp add: power_divide)
-  thus ?thesis
+  then show ?thesis
     using nonneg by (rule power2_eq_imp_eq) simp
 qed
 
 lemma cos_30: "cos (pi / 6) = sqrt 3/2"
 proof -
-  let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
+  let ?c = "cos (pi / 6)"
+  let ?s = "sin (pi / 6)"
   have pos_c: "0 < ?c"
-    by (rule cos_gt_zero, simp, simp)
+    by (rule cos_gt_zero) simp_all
   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
     by simp
   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
@@ -4055,7 +4089,7 @@
     by (simp add: algebra_simps power2_eq_square)
   finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2"
     using pos_c by (simp add: sin_squared_eq power_divide)
-  thus ?thesis
+  then show ?thesis
     using pos_c [THEN order_less_imp_le]
     by (rule power2_eq_imp_eq) simp
 qed
@@ -4068,31 +4102,35 @@
 
 lemma cos_60: "cos (pi / 3) = 1 / 2"
   apply (rule power2_eq_imp_eq)
-  apply (simp add: cos_squared_eq sin_60 power_divide)
-  apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
+    apply (simp add: cos_squared_eq sin_60 power_divide)
+   apply (rule cos_ge_zero)
+    apply (rule order_trans [where y=0])
+     apply simp_all
   done
 
 lemma sin_30: "sin (pi / 6) = 1 / 2"
   by (simp add: sin_cos_eq cos_60)
 
-lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2*pi * n) = 1"
+lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2 * pi * n) = 1"
   by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute)
 
-lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2*pi * n) = 0"
+lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0"
   by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
 
-lemma cos_int_2npi [simp]: "cos (2 * of_int (n::int) * pi) = 1"
+lemma cos_int_2npi [simp]: "cos (2 * of_int n * pi) = 1"
+  for n :: int
   by (simp add: cos_one_2pi_int)
 
-lemma sin_int_2npi [simp]: "sin (2 * of_int (n::int) * pi) = 0"
+lemma sin_int_2npi [simp]: "sin (2 * of_int n * pi) = 0"
+  for n :: int
   by (metis Ints_of_int mult.assoc mult.commute sin_integer_2pi)
 
-lemma sincos_principal_value: "\<exists>y. (-pi < y \<and> y \<le> pi) \<and> (sin(y) = sin(x) \<and> cos(y) = cos(x))"
-  apply (rule exI [where x="pi - (2*pi) * frac((pi - x) / (2*pi))"])
+lemma sincos_principal_value: "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> (sin y = sin x \<and> cos y = cos x)"
+  apply (rule exI [where x="pi - (2 * pi) * frac ((pi - x) / (2 * pi))"])
   apply (auto simp: field_simps frac_lt_1)
-  apply (simp_all add: frac_def divide_simps)
-  apply (simp_all add: add_divide_distrib diff_divide_distrib)
-  apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
+   apply (simp_all add: frac_def divide_simps)
+   apply (simp_all add: add_divide_distrib diff_divide_distrib)
+   apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
   done
 
 
@@ -4101,13 +4139,11 @@
 definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
   where "tan = (\<lambda>x. sin x / cos x)"
 
-lemma tan_of_real:
-  "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})"
+lemma tan_of_real: "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})"
   by (simp add: tan_def sin_of_real cos_of_real)
 
-lemma tan_in_Reals [simp]:
-  fixes z :: "'a::{real_normed_field,banach}"
-  shows "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
+lemma tan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
+  for z :: "'a::{real_normed_field,banach}"
   by (simp add: tan_def)
 
 lemma tan_zero [simp]: "tan 0 = 0"
@@ -4116,52 +4152,46 @@
 lemma tan_pi [simp]: "tan pi = 0"
   by (simp add: tan_def)
 
-lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
+lemma tan_npi [simp]: "tan (real n * pi) = 0"
+  for n :: nat
   by (simp add: tan_def)
 
-lemma tan_minus [simp]: "tan (-x) = - tan x"
-  by (simp add: tan_def)
-
-lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
+lemma tan_minus [simp]: "tan (- x) = - tan x"
   by (simp add: tan_def)
 
-lemma lemma_tan_add1:
-  "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
+lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x"
+  by (simp add: tan_def)
+
+lemma lemma_tan_add1: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
   by (simp add: tan_def cos_add field_simps)
 
-lemma add_tan_eq:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
+lemma add_tan_eq: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
+  for x :: "'a::{real_normed_field,banach}"
   by (simp add: tan_def sin_add field_simps)
 
 lemma tan_add:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows
-     "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0\<rbrakk>
-      \<Longrightarrow> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
-      by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def)
-
-lemma tan_double:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows
-     "\<lbrakk>cos x \<noteq> 0; cos (2 * x) \<noteq> 0\<rbrakk>
-      \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
+  "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x + y) \<noteq> 0 \<Longrightarrow> tan (x + y) = (tan x + tan y)/(1 - tan x * tan y)"
+  for x :: "'a::{real_normed_field,banach}"
+  by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def)
+
+lemma tan_double: "cos x \<noteq> 0 \<Longrightarrow> cos (2 * x) \<noteq> 0 \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
+  for x :: "'a::{real_normed_field,banach}"
   using tan_add [of x x] by (simp add: power2_eq_square)
 
-lemma tan_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < tan x"
+lemma tan_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < tan x"
   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
 
 lemma tan_less_zero:
-  assumes lb: "- pi/2 < x" and "x < 0"
+  assumes "- pi/2 < x" and "x < 0"
   shows "tan x < 0"
 proof -
-  have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
-  thus ?thesis by simp
+  have "0 < tan (- x)"
+    using assms by (simp only: tan_gt_zero)
+  then show ?thesis by simp
 qed
 
-lemma tan_half:
-  fixes x :: "'a::{real_normed_field,banach,field}"
-  shows  "tan x = sin (2 * x) / (cos (2 * x) + 1)"
+lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
+  for x :: "'a::{real_normed_field,banach,field}"
   unfolding tan_def sin_double cos_double sin_squared_eq
   by (simp add: power2_eq_square)
 
@@ -4174,25 +4204,23 @@
 lemma tan_60: "tan (pi / 3) = sqrt 3"
   unfolding tan_def by (simp add: sin_60 cos_60)
 
-lemma DERIV_tan [simp]:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
+lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
+  for x :: "'a::{real_normed_field,banach}"
   unfolding tan_def
   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
 
-lemma isCont_tan:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
+lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
+  for x :: "'a::{real_normed_field,banach}"
   by (rule DERIV_tan [THEN DERIV_isCont])
 
 lemma isCont_tan' [simp,continuous_intros]:
   fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
-  shows "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
+  shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
   by (rule isCont_o2 [OF _ isCont_tan])
 
 lemma tendsto_tan [tendsto_intros]:
   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "\<lbrakk>(f \<longlongrightarrow> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
+  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cos a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
   by (rule isCont_tendsto_compose [OF isCont_tan])
 
 lemma continuous_tan:
@@ -4207,61 +4235,69 @@
 
 lemma continuous_within_tan [continuous_intros]:
   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows
-  "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
+  shows "continuous (at x within s) f \<Longrightarrow>
+    cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
   unfolding continuous_within by (rule tendsto_tan)
 
 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) \<midarrow>pi/2\<rightarrow> 0"
   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
 
-lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
-  apply (cut_tac LIM_cos_div_sin)
+lemma lemma_tan_total: "0 < y \<Longrightarrow> \<exists>x. 0 < x \<and> x < pi/2 \<and> y < tan x"
+  apply (insert LIM_cos_div_sin)
   apply (simp only: LIM_eq)
-  apply (drule_tac x = "inverse y" in spec, safe, force)
-  apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
+  apply (drule_tac x = "inverse y" in spec)
+  apply safe
+   apply force
+  apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero])
+  apply safe
   apply (rule_tac x = "(pi/2) - e" in exI)
   apply (simp (no_asm_simp))
   apply (drule_tac x = "(pi/2) - e" in spec)
   apply (auto simp add: tan_def sin_diff cos_diff)
   apply (rule inverse_less_iff_less [THEN iffD1])
-  apply (auto simp add: divide_inverse)
-  apply (rule mult_pos_pos)
-  apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
-  apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute)
+    apply (auto simp add: divide_inverse)
+   apply (rule mult_pos_pos)
+    apply (subgoal_tac [3] "0 < sin e \<and> 0 < cos e")
+     apply (auto intro: cos_gt_zero sin_gt_zero2 simp: mult.commute)
   done
 
-lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
-  apply (frule order_le_imp_less_or_eq, safe)
+lemma tan_total_pos: "0 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x < pi/2 \<and> tan x = y"
+  apply (frule order_le_imp_less_or_eq)
+  apply safe
    prefer 2 apply force
-  apply (drule lemma_tan_total, safe)
+  apply (drule lemma_tan_total)
+  apply safe
   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
   apply (drule_tac y = xa in order_le_imp_less_or_eq)
   apply (auto dest: cos_gt_zero)
   done
 
-lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
-  apply (cut_tac linorder_linear [of 0 y], safe)
-  apply (drule tan_total_pos)
-  apply (cut_tac [2] y="-y" in tan_total_pos, safe)
-  apply (rule_tac [3] x = "-x" in exI)
-  apply (auto del: exI intro!: exI)
+lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
+  apply (insert linorder_linear [of 0 y])
+  apply safe
+   apply (drule tan_total_pos)
+   apply (cut_tac [2] y="-y" in tan_total_pos)
+    apply safe
+    apply (rule_tac [3] x = "-x" in exI)
+    apply (auto del: exI intro!: exI)
   done
 
-lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
-  apply (cut_tac y = y in lemma_tan_total1, auto)
+lemma tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
+  apply (insert lemma_tan_total1 [where y = y])
+  apply auto
   apply hypsubst_thin
-  apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
-  apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
-  apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
-  apply (rule_tac [4] Rolle)
-  apply (rule_tac [2] Rolle)
-  apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
-              simp add: real_differentiable_def)
-  txt\<open>Now, simulate TRYALL\<close>
-  apply (rule_tac [!] DERIV_tan asm_rl)
-  apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
-              simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
+  apply (cut_tac x = xa and y = y in linorder_less_linear)
+  apply auto
+   apply (subgoal_tac [2] "\<exists>z. y < z \<and> z < xa \<and> DERIV tan z :> 0")
+    apply (subgoal_tac "\<exists>z. xa < z \<and> z < y \<and> DERIV tan z :> 0")
+     apply (rule_tac [4] Rolle)
+        apply (rule_tac [2] Rolle)
+           apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
+            simp add: real_differentiable_def)
+       apply (rule_tac [!] DERIV_tan asm_rl)
+       apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
+        simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
   done
 
 lemma tan_monotone:
@@ -4272,20 +4308,26 @@
   proof (rule allI, rule impI)
     fix x' :: real
     assume "y \<le> x' \<and> x' \<le> x"
-    hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
+    then have "-(pi/2) < x'" and "x' < pi/2"
+      using assms by auto
     from cos_gt_zero_pi[OF this]
     have "cos x' \<noteq> 0" by auto
-    thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
+    then show "DERIV tan x' :> inverse ((cos x')\<^sup>2)"
+      by (rule DERIV_tan)
   qed
   from MVT2[OF \<open>y < x\<close> this]
   obtain z where "y < z" and "z < x"
     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
-  hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
-  hence "0 < cos z" using cos_gt_zero_pi by auto
-  hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
+  then have "- (pi / 2) < z" and "z < pi / 2"
+    using assms by auto
+  then have "0 < cos z"
+    using cos_gt_zero_pi by auto
+  then have inv_pos: "0 < inverse ((cos z)\<^sup>2)"
+    by auto
   have "0 < x - y" using \<open>y < x\<close> by auto
-  with inv_pos have "0 < tan x - tan y" unfolding tan_diff by auto
-  thus ?thesis by auto
+  with inv_pos have "0 < tan x - tan y"
+    unfolding tan_diff by auto
+  then show ?thesis by auto
 qed
 
 lemma tan_monotone':
@@ -4293,24 +4335,29 @@
     and "y < pi / 2"
     and "- (pi / 2) < x"
     and "x < pi / 2"
-  shows "(y < x) = (tan y < tan x)"
+  shows "y < x \<longleftrightarrow> tan y < tan x"
 proof
   assume "y < x"
-  thus "tan y < tan x"
+  then show "tan y < tan x"
     using tan_monotone and \<open>- (pi / 2) < y\<close> and \<open>x < pi / 2\<close> by auto
 next
   assume "tan y < tan x"
   show "y < x"
   proof (rule ccontr)
-    assume "\<not> y < x" hence "x \<le> y" by auto
-    hence "tan x \<le> tan y"
+    assume "\<not> ?thesis"
+    then have "x \<le> y" by auto
+    then have "tan x \<le> tan y"
     proof (cases "x = y")
-      case True thus ?thesis by auto
+      case True
+      then show ?thesis by auto
     next
-      case False hence "x < y" using \<open>x \<le> y\<close> by auto
-      from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi / 2\<close>] show ?thesis by auto
+      case False
+      then have "x < y" using \<open>x \<le> y\<close> by auto
+      from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi / 2\<close>] show ?thesis
+        by auto
     qed
-    thus False using \<open>tan y < tan x\<close> by auto
+    then show False
+      using \<open>tan y < tan x\<close> by auto
   qed
 qed
 
@@ -4320,9 +4367,8 @@
 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
   by (simp add: tan_def)
 
-lemma tan_periodic_nat[simp]:
-  fixes n :: nat
-  shows "tan (x + real n * pi) = tan x"
+lemma tan_periodic_nat[simp]: "tan (x + real n * pi) = tan x"
+  for n :: nat
 proof (induct n arbitrary: x)
   case 0
   then show ?case by simp
@@ -4330,18 +4376,19 @@
   case (Suc n)
   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
     unfolding Suc_eq_plus1 of_nat_add  distrib_right by auto
-  show ?case unfolding split_pi_off using Suc by auto
+  show ?case
+    unfolding split_pi_off using Suc by auto
 qed
 
-lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + of_int i * pi) = tan x"
+lemma tan_periodic_int[simp]: "tan (x + of_int i * pi) = tan x"
 proof (cases "0 \<le> i")
   case True
-  hence i_nat: "of_int i = of_int (nat i)" by auto
+  then have i_nat: "of_int i = of_int (nat i)" by auto
   show ?thesis unfolding i_nat
     by (metis of_int_of_nat_eq tan_periodic_nat)
 next
   case False
-  hence i_nat: "of_int i = - of_int (nat (-i))" by auto
+  then have i_nat: "of_int i = - of_int (nat (- i))" by auto
   have "tan x = tan (x + of_int i * pi - of_int i * pi)"
     by auto
   also have "\<dots> = tan (x + of_int i * pi)"
@@ -4357,34 +4404,30 @@
   unfolding tan_def by (simp add: sin_45 cos_45)
 
 lemma tan_diff:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows
-     "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x - y) \<noteq> 0\<rbrakk>
-      \<Longrightarrow> tan(x - y) = (tan(x) - tan(y))/(1 + tan(x) * tan(y))"
-  using tan_add [of x "-y"]
-  by simp
-
-
-lemma tan_pos_pi2_le: "0 \<le> x ==> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
+  "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x - y) \<noteq> 0 \<Longrightarrow> tan (x - y) = (tan x - tan y)/(1 + tan x * tan y)"
+  for x :: "'a::{real_normed_field,banach}"
+  using tan_add [of x "-y"] by simp
+
+lemma tan_pos_pi2_le: "0 \<le> x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
   using less_eq_real_def tan_gt_zero by auto
 
-lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos(x) = 1 / sqrt(1 + tan(x) ^ 2)"
+lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos x = 1 / sqrt (1 + tan x ^ 2)"
   using cos_gt_zero_pi [of x]
   by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
 
-lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin(x) = tan(x) / sqrt(1 + tan(x) ^ 2)"
+lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin x = tan x / sqrt (1 + tan x ^ 2)"
   using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
   by (force simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
 
-lemma tan_mono_le: "-(pi/2) < x ==> x \<le> y ==> y < pi/2 \<Longrightarrow> tan(x) \<le> tan(y)"
+lemma tan_mono_le: "-(pi/2) < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y"
   using less_eq_real_def tan_monotone by auto
 
-lemma tan_mono_lt_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
-         \<Longrightarrow> (tan(x) < tan(y) \<longleftrightarrow> x < y)"
+lemma tan_mono_lt_eq:
+  "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x < tan y \<longleftrightarrow> x < y"
   using tan_monotone' by blast
 
-lemma tan_mono_le_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
-         \<Longrightarrow> (tan(x) \<le> tan(y) \<longleftrightarrow> x \<le> y)"
+lemma tan_mono_le_eq:
+  "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y \<longleftrightarrow> x \<le> y"
   by (meson tan_mono_le not_le tan_monotone)
 
 lemma tan_bound_pi2: "\<bar>x\<bar> < pi/4 \<Longrightarrow> \<bar>tan x\<bar> < 1"
@@ -4394,18 +4437,17 @@
 lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
   by (simp add: tan_def sin_diff cos_diff)
 
+
 subsection \<open>Cotangent\<close>
 
 definition cot :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
   where "cot = (\<lambda>x. cos x / sin x)"
 
-lemma cot_of_real:
-  "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})"
+lemma cot_of_real: "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})"
   by (simp add: cot_def sin_of_real cos_of_real)
 
-lemma cot_in_Reals [simp]:
-  fixes z :: "'a::{real_normed_field,banach}"
-  shows "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>"
+lemma cot_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>"
+  for z :: "'a::{real_normed_field,banach}"
   by (simp add: cot_def)
 
 lemma cot_zero [simp]: "cot 0 = 0"
@@ -4414,13 +4456,14 @@
 lemma cot_pi [simp]: "cot pi = 0"
   by (simp add: cot_def)
 
-lemma cot_npi [simp]: "cot (real (n::nat) * pi) = 0"
+lemma cot_npi [simp]: "cot (real n * pi) = 0"
+  for n :: nat
   by (simp add: cot_def)
 
-lemma cot_minus [simp]: "cot (-x) = - cot x"
+lemma cot_minus [simp]: "cot (- x) = - cot x"
   by (simp add: cot_def)
 
-lemma cot_periodic [simp]: "cot (x + 2*pi) = cot x"
+lemma cot_periodic [simp]: "cot (x + 2 * pi) = cot x"
   by (simp add: cot_def)
 
 lemma cot_altdef: "cot x = inverse (tan x)"
@@ -4429,44 +4472,42 @@
 lemma tan_altdef: "tan x = inverse (cot x)"
   by (simp add: cot_def tan_def)
 
-lemma tan_cot': "tan(pi/2 - x) = cot x"
+lemma tan_cot': "tan (pi/2 - x) = cot x"
   by (simp add: tan_cot cot_altdef)
 
-lemma cot_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cot x"
+lemma cot_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cot x"
   by (simp add: cot_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
 
 lemma cot_less_zero:
   assumes lb: "- pi/2 < x" and "x < 0"
   shows "cot x < 0"
 proof -
-  have "0 < cot (- x)" using assms by (simp only: cot_gt_zero)
-  thus ?thesis by simp
+  have "0 < cot (- x)"
+    using assms by (simp only: cot_gt_zero)
+  then show ?thesis by simp
 qed
 
-lemma DERIV_cot [simp]:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)"
+lemma DERIV_cot [simp]: "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)"
+  for x :: "'a::{real_normed_field,banach}"
   unfolding cot_def using cos_squared_eq[of x]
-  by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
-
-lemma isCont_cot:
-  fixes x :: "'a::{real_normed_field,banach}"
-  shows "sin x \<noteq> 0 \<Longrightarrow> isCont cot x"
+  by (auto intro!: derivative_eq_intros) (simp add: divide_inverse power2_eq_square)
+
+lemma isCont_cot: "sin x \<noteq> 0 \<Longrightarrow> isCont cot x"
+  for x :: "'a::{real_normed_field,banach}"
   by (rule DERIV_cot [THEN DERIV_isCont])
 
 lemma isCont_cot' [simp,continuous_intros]:
-  fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
-  shows "\<lbrakk>isCont f a; sin (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a"
+  "isCont f a \<Longrightarrow> sin (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a"
+  for a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
   by (rule isCont_o2 [OF _ isCont_cot])
 
-lemma tendsto_cot [tendsto_intros]:
-  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "\<lbrakk>(f \<longlongrightarrow> a) F; sin a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
+lemma tendsto_cot [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> sin a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
+  for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
   by (rule isCont_tendsto_compose [OF isCont_cot])
 
 lemma continuous_cot:
-  fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))"
+  "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))"
+  for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
   unfolding continuous_def by (rule tendsto_cot)
 
 lemma continuous_on_cot [continuous_intros]:
@@ -4476,167 +4517,164 @@
 
 lemma continuous_within_cot [continuous_intros]:
   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
-  shows
-  "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))"
+  shows "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))"
   unfolding continuous_within by (rule tendsto_cot)
 
 
 subsection \<open>Inverse Trigonometric Functions\<close>
 
-definition arcsin :: "real => real"
-  where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
-
-definition arccos :: "real => real"
-  where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
-
-definition arctan :: "real => real"
-  where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
-
-lemma arcsin:
-  "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
-    -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
+definition arcsin :: "real \<Rightarrow> real"
+  where "arcsin y = (THE x. -(pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y)"
+
+definition arccos :: "real \<Rightarrow> real"
+  where "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)"
+
+definition arctan :: "real \<Rightarrow> real"
+  where "arctan y = (THE x. -(pi/2) < x \<and> x < pi/2 \<and> tan x = y)"
+
+lemma arcsin: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2 \<and> sin (arcsin y) = y"
   unfolding arcsin_def by (rule theI' [OF sin_total])
 
-lemma arcsin_pi:
-  "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
-  apply (drule (1) arcsin)
-  apply (force intro: order_trans)
-  done
-
-lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
+lemma arcsin_pi: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi \<and> sin (arcsin y) = y"
+  by (drule (1) arcsin) (force intro: order_trans)
+
+lemma sin_arcsin [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin (arcsin y) = y"
+  by (blast dest: arcsin)
+
+lemma arcsin_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2"
   by (blast dest: arcsin)
 
-lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
-  by (blast dest: arcsin)
-
-lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
+lemma arcsin_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y"
   by (blast dest: arcsin)
 
-lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
+lemma arcsin_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
   by (blast dest: arcsin)
 
-lemma arcsin_lt_bounded:
-     "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> -(pi/2) < arcsin y & arcsin y < pi/2"
+lemma arcsin_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> - (pi/2) < arcsin y \<and> arcsin y < pi/2"
   apply (frule order_less_imp_le)
   apply (frule_tac y = y in order_less_imp_le)
   apply (frule arcsin_bounded)
-  apply (safe, simp)
-  apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
-  apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
-  apply (drule_tac [!] f = sin in arg_cong, auto)
+   apply safe
+    apply simp
+   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
+   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq)
+   apply safe
+   apply (drule_tac [!] f = sin in arg_cong)
+   apply auto
   done
 
-lemma arcsin_sin: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> arcsin(sin x) = x"
+lemma arcsin_sin: "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> arcsin (sin x) = x"
   apply (unfold arcsin_def)
   apply (rule the1_equality)
-  apply (rule sin_total, auto)
+   apply (rule sin_total)
+    apply auto
   done
 
 lemma arcsin_0 [simp]: "arcsin 0 = 0"
-  using arcsin_sin [of 0]
-  by simp
+  using arcsin_sin [of 0] by simp
 
 lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
-  using arcsin_sin [of "pi/2"]
-  by simp
-
-lemma arcsin_minus_1 [simp]: "arcsin (-1) = - (pi/2)"
-  using arcsin_sin [of "-pi/2"]
-  by simp
-
-lemma arcsin_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin(-x) = -arcsin x"
+  using arcsin_sin [of "pi/2"] by simp
+
+lemma arcsin_minus_1 [simp]: "arcsin (- 1) = - (pi/2)"
+  using arcsin_sin [of "- pi/2"] by simp
+
+lemma arcsin_minus: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin (- x) = - arcsin x"
   by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
 
-lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> (arcsin x = arcsin y \<longleftrightarrow> x = y)"
+lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x = arcsin y \<longleftrightarrow> x = y"
   by (metis abs_le_iff arcsin minus_le_iff)
 
-lemma cos_arcsin_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos(arcsin x) \<noteq> 0"
+lemma cos_arcsin_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos (arcsin x) \<noteq> 0"
   using arcsin_lt_bounded cos_gt_zero_pi by force
 
-lemma arccos:
-     "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk>
-      \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
+lemma arccos: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y"
   unfolding arccos_def by (rule theI' [OF cos_total])
 
-lemma cos_arccos [simp]: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> cos(arccos y) = y"
+lemma cos_arccos [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> cos (arccos y) = y"
   by (blast dest: arccos)
 
-lemma arccos_bounded: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi"
+lemma arccos_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi"
   by (blast dest: arccos)
 
-lemma arccos_lbound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y"
+lemma arccos_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y"
   by (blast dest: arccos)
 
-lemma arccos_ubound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi"
+lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi"
   by (blast dest: arccos)
 
-lemma arccos_lt_bounded:
-     "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> 0 < arccos y & arccos y < pi"
+lemma arccos_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> 0 < arccos y \<and> arccos y < pi"
   apply (frule order_less_imp_le)
   apply (frule_tac y = y in order_less_imp_le)
-  apply (frule arccos_bounded, auto)
-  apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
-  apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
-  apply (drule_tac [!] f = cos in arg_cong, auto)
+  apply (frule arccos_bounded)
+   apply auto
+   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
+   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq)
+   apply auto
+   apply (drule_tac [!] f = cos in arg_cong)
+   apply auto
   done
 
-lemma arccos_cos: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> arccos(cos x) = x"
-  apply (simp add: arccos_def)
-  apply (auto intro!: the1_equality cos_total)
-  done
-
-lemma arccos_cos2: "\<lbrakk>x \<le> 0; -pi \<le> x\<rbrakk> \<Longrightarrow> arccos(cos x) = -x"
-  apply (simp add: arccos_def)
-  apply (auto intro!: the1_equality cos_total)
-  done
-
-lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
+lemma arccos_cos: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> arccos (cos x) = x"
+  by (auto simp: arccos_def intro!: the1_equality cos_total)
+
+lemma arccos_cos2: "x \<le> 0 \<Longrightarrow> - pi \<le> x \<Longrightarrow> arccos (cos x) = -x"
+  by (auto simp: arccos_def intro!: the1_equality cos_total)
+
+lemma cos_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
   apply (subgoal_tac "x\<^sup>2 \<le> 1")
-  apply (rule power2_eq_imp_eq)
-  apply (simp add: cos_squared_eq)
-  apply (rule cos_ge_zero)
-  apply (erule (1) arcsin_lbound)
-  apply (erule (1) arcsin_ubound)
+   apply (rule power2_eq_imp_eq)
+     apply (simp add: cos_squared_eq)
+    apply (rule cos_ge_zero)
+     apply (erule (1) arcsin_lbound)
+    apply (erule (1) arcsin_ubound)
+   apply simp
+  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2")
+   apply simp
+  apply (rule power_mono)
+   apply simp
   apply simp
-  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
-  apply (rule power_mono, simp, simp)
   done
 
-lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
+lemma sin_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
   apply (subgoal_tac "x\<^sup>2 \<le> 1")
-  apply (rule power2_eq_imp_eq)
-  apply (simp add: sin_squared_eq)
-  apply (rule sin_ge_zero)
-  apply (erule (1) arccos_lbound)
-  apply (erule (1) arccos_ubound)
+   apply (rule power2_eq_imp_eq)
+     apply (simp add: sin_squared_eq)
+    apply (rule sin_ge_zero)
+     apply (erule (1) arccos_lbound)
+    apply (erule (1) arccos_ubound)
+   apply simp
+  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2")
+   apply simp
+  apply (rule power_mono)
+   apply simp
   apply simp
-  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
-  apply (rule power_mono, simp, simp)
   done
 
 lemma arccos_0 [simp]: "arccos 0 = pi/2"
-by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One)
+  by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero
+      pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One)
 
 lemma arccos_1 [simp]: "arccos 1 = 0"
   using arccos_cos by force
 
-lemma arccos_minus_1 [simp]: "arccos(-1) = pi"
+lemma arccos_minus_1 [simp]: "arccos (- 1) = pi"
   by (metis arccos_cos cos_pi order_refl pi_ge_zero)
 
-lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos(-x) = pi - arccos x"
+lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos (- x) = pi - arccos x"
   by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
-    minus_diff_eq uminus_add_conv_diff)
-
-lemma sin_arccos_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> ~(sin(arccos x) = 0)"
+      minus_diff_eq uminus_add_conv_diff)
+
+lemma sin_arccos_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> \<not> sin (arccos x) = 0"
   using arccos_lt_bounded sin_gt_zero by force
 
-lemma arctan: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
+lemma arctan: "- (pi/2) < arctan y \<and> arctan y < pi/2 \<and> tan (arctan y) = y"
   unfolding arctan_def by (rule theI' [OF tan_total])
 
 lemma tan_arctan: "tan (arctan y) = y"
   by (simp add: arctan)
 
-lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
+lemma arctan_bounded: "- (pi/2) < arctan y \<and> arctan y < pi/2"
   by (auto simp only: arctan)
 
 lemma arctan_lbound: "- (pi/2) < arctan y"
@@ -4660,13 +4698,13 @@
 
 lemma arctan_minus: "arctan (- x) = - arctan x"
   apply (rule arctan_unique)
-  apply (simp only: neg_less_iff_less arctan_ubound)
-  apply (metis minus_less_iff arctan_lbound, simp add: arctan)
+    apply (simp only: neg_less_iff_less arctan_ubound)
+   apply (metis minus_less_iff arctan_lbound)
+  apply (simp add: arctan)
   done
 
 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
-  by (intro less_imp_neq [symmetric] cos_gt_zero_pi
-    arctan_lbound arctan_ubound)
+  by (intro less_imp_neq [symmetric] cos_gt_zero_pi arctan_lbound arctan_ubound)
 
 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
 proof (rule power2_eq_imp_eq)
@@ -4676,7 +4714,7 @@
     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
     unfolding tan_def by (simp add: distrib_left power_divide)
-  thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
+  then show "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
     using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq)
 qed
 
@@ -4685,12 +4723,11 @@
   using tan_arctan [of x] unfolding tan_def cos_arctan
   by (simp add: eq_divide_eq)
 
-lemma tan_sec:
-  fixes x :: "'a::{real_normed_field,banach,field}"
-  shows "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
+lemma tan_sec: "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
+  for x :: "'a::{real_normed_field,banach,field}"
   apply (rule power_inverse [THEN subst])
   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
-  apply (auto simp add: tan_def field_simps)
+   apply (auto simp add: tan_def field_simps)
   done
 
 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
@@ -4766,11 +4803,14 @@
   by (auto simp: continuous_on_eq_continuous_at subset_eq)
 
 lemma isCont_arctan: "isCont arctan x"
-  apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
-  apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
-  apply (subgoal_tac "isCont arctan (tan (arctan x))", simp add: arctan)
+  apply (rule arctan_lbound [of x, THEN dense, THEN exE])
+  apply clarify
+  apply (rule arctan_ubound [of x, THEN dense, THEN exE])
+  apply clarify
+  apply (subgoal_tac "isCont arctan (tan (arctan x))")
+   apply (simp add: arctan)
   apply (erule (1) isCont_inverse_function2 [where f=tan])
-  apply (metis arctan_tan order_le_less_trans order_less_le_trans)
+   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
   done
 
@@ -4780,38 +4820,50 @@
 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
   unfolding continuous_def by (rule tendsto_arctan)
 
-lemma continuous_on_arctan [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
+lemma continuous_on_arctan [continuous_intros]:
+  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
   unfolding continuous_on_def by (auto intro: tendsto_arctan)
 
-lemma DERIV_arcsin:
-  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
+lemma DERIV_arcsin: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1])
-  apply (rule DERIV_cong [OF DERIV_sin])
-  apply (simp add: cos_arcsin)
-  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
-  apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
-  apply simp
+       apply (rule DERIV_cong [OF DERIV_sin])
+       apply (simp add: cos_arcsin)
+      apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2")
+       apply simp
+      apply (rule power_strict_mono)
+        apply simp
+       apply simp
+      apply simp
+     apply assumption
+    apply assumption
+   apply simp
   apply (erule (1) isCont_arcsin)
   done
 
-lemma DERIV_arccos:
-  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
+lemma DERIV_arccos: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1])
-  apply (rule DERIV_cong [OF DERIV_cos])
-  apply (simp add: sin_arccos)
-  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
-  apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
-  apply simp
+       apply (rule DERIV_cong [OF DERIV_cos])
+       apply (simp add: sin_arccos)
+      apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2")
+       apply simp
+      apply (rule power_strict_mono)
+        apply simp
+       apply simp
+      apply simp
+     apply assumption
+    apply assumption
+   apply simp
   apply (erule (1) isCont_arccos)
   done
 
 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
-  apply (rule DERIV_cong [OF DERIV_tan])
-  apply (rule cos_arctan_not_zero)
-  apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
-  apply (simp add: arctan power_inverse [symmetric] tan_sec [symmetric])
-  apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
+       apply (rule DERIV_cong [OF DERIV_tan])
+        apply (rule cos_arctan_not_zero)
+       apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
+   apply (simp add: arctan power_inverse [symmetric] tan_sec [symmetric])
+  apply (subgoal_tac "0 < 1 + x\<^sup>2")
+   apply simp
   apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
   done
 
@@ -4840,7 +4892,6 @@
   define y where "y = pi/2 - min (pi/2) e"
   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
     using \<open>0 < e\<close> by auto
-
   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
     fix x
@@ -4860,7 +4911,7 @@
   by (intro tendsto_minus tendsto_arctan_at_top)
 
 
-subsection\<open>Prove Totality of the Trigonometric Functions\<close>
+subsection \<open>Prove Totality of the Trigonometric Functions\<close>
 
 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
   by (simp add: abs_le_iff)
@@ -4868,29 +4919,26 @@
 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
   by (simp add: sin_arccos abs_le_iff)
 
-lemma sin_mono_less_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
-         \<Longrightarrow> (sin(x) < sin(y) \<longleftrightarrow> x < y)"
-by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
-
-lemma sin_mono_le_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
-         \<Longrightarrow> (sin(x) \<le> sin(y) \<longleftrightarrow> x \<le> y)"
-by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
+lemma sin_mono_less_eq:
+  "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x < sin y \<longleftrightarrow> x < y"
+  by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
+
+lemma sin_mono_le_eq:
+  "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x \<le> sin y \<longleftrightarrow> x \<le> y"
+  by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
 
 lemma sin_inj_pi:
-    "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2;-(pi/2) \<le> y; y \<le> pi/2; sin(x) = sin(y)\<rbrakk> \<Longrightarrow> x = y"
-by (metis arcsin_sin)
-
-lemma cos_mono_less_eq:
-    "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi \<Longrightarrow> (cos(x) < cos(y) \<longleftrightarrow> y < x)"
-by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
-
-lemma cos_mono_le_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi
-         \<Longrightarrow> (cos(x) \<le> cos(y) \<longleftrightarrow> y \<le> x)"
+  "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x = sin y \<Longrightarrow> x = y"
+  by (metis arcsin_sin)
+
+lemma cos_mono_less_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x < cos y \<longleftrightarrow> y < x"
+  by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
+
+lemma cos_mono_le_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x \<le> cos y \<longleftrightarrow> y \<le> x"
   by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear)
 
-lemma cos_inj_pi: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi ==> cos(x) = cos(y)
-         \<Longrightarrow> x = y"
-by (metis arccos_cos)
+lemma cos_inj_pi: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x = cos y \<Longrightarrow> x = y"
+  by (metis arccos_cos)
 
 lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2"
   by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le
@@ -4898,109 +4946,97 @@
 
 lemma sincos_total_pi_half:
   assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
-    shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
+  shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
 proof -
   have x1: "x \<le> 1"
-    using assms
-    by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
-  with assms have ax: "0 \<le> arccos x" "cos (arccos x) = x"
+    using assms by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
+  with assms have *: "0 \<le> arccos x" "cos (arccos x) = x"
     by (auto simp: arccos)
   from assms have "y = sqrt (1 - x\<^sup>2)"
     by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)
-  with x1 ax assms arccos_le_pi2 [of x] show ?thesis
+  with x1 * assms arccos_le_pi2 [of x] show ?thesis
     by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
 qed
 
 lemma sincos_total_pi:
-  assumes "0 \<le> y" and "x\<^sup>2 + y\<^sup>2 = 1"
-    shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
+  assumes "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
+  shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
 proof (cases rule: le_cases [of 0 x])
-  case le from sincos_total_pi_half [OF le]
-  show ?thesis
+  case le
+  from sincos_total_pi_half [OF le] show ?thesis
     by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms)
 next
   case ge
   then have "0 \<le> -x"
     by simp
-  then obtain t where "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
+  then obtain t where t: "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
     using sincos_total_pi_half assms
-    apply auto
-    by (metis \<open>0 \<le> - x\<close> power2_minus)
-  then show ?thesis
-    by (rule_tac x="pi-t" in exI, auto)
+    by auto (metis \<open>0 \<le> - x\<close> power2_minus)
+  show ?thesis
+    by (rule exI [where x = "pi -t"]) (use t in auto)
 qed
 
 lemma sincos_total_2pi_le:
   assumes "x\<^sup>2 + y\<^sup>2 = 1"
-    shows "\<exists>t. 0 \<le> t \<and> t \<le> 2*pi \<and> x = cos t \<and> y = sin t"
+  shows "\<exists>t. 0 \<le> t \<and> t \<le> 2 * pi \<and> x = cos t \<and> y = sin t"
 proof (cases rule: le_cases [of 0 y])
-  case le from sincos_total_pi [OF le]
-  show ?thesis
+  case le
+  from sincos_total_pi [OF le] show ?thesis
     by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
 next
   case ge
   then have "0 \<le> -y"
     by simp
-  then obtain t where "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
+  then obtain t where t: "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
     using sincos_total_pi assms
-    apply auto
-    by (metis \<open>0 \<le> - y\<close> power2_minus)
-  then show ?thesis
-    by (rule_tac x="2*pi-t" in exI, auto)
+    by auto (metis \<open>0 \<le> - y\<close> power2_minus)
+  show ?thesis
+    by (rule exI [where x = "2 * pi - t"]) (use t in auto)
 qed
 
 lemma sincos_total_2pi:
   assumes "x\<^sup>2 + y\<^sup>2 = 1"
-    obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
+  obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
 proof -
   from sincos_total_2pi_le [OF assms]
   obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t"
     by blast
   show ?thesis
-    apply (cases "t = 2*pi")
-    using t that
-    apply force+
-    done
+    by (cases "t = 2 * pi") (use t that in \<open>force+\<close>)
 qed
 
 lemma arcsin_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
-  apply (rule trans [OF sin_mono_less_eq [symmetric]])
-  using arcsin_ubound arcsin_lbound
-  apply auto
-  done
+  by (rule trans [OF sin_mono_less_eq [symmetric]]) (use arcsin_ubound arcsin_lbound in auto)
 
 lemma arcsin_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y"
   using arcsin_less_mono not_le by blast
 
-lemma arcsin_less_arcsin: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
+lemma arcsin_less_arcsin: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
   using arcsin_less_mono by auto
 
-lemma arcsin_le_arcsin: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
+lemma arcsin_le_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
   using arcsin_le_mono by auto
 
-lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> (arccos x < arccos y \<longleftrightarrow> y < x)"
-  apply (rule trans [OF cos_mono_less_eq [symmetric]])
-  using arccos_ubound arccos_lbound
-  apply auto
-  done
+lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x < arccos y \<longleftrightarrow> y < x"
+  by (rule trans [OF cos_mono_less_eq [symmetric]]) (use arccos_ubound arccos_lbound in auto)
 
 lemma arccos_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
-  using arccos_less_mono [of y x]
-  by (simp add: not_le [symmetric])
-
-lemma arccos_less_arccos: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
+  using arccos_less_mono [of y x] by (simp add: not_le [symmetric])
+
+lemma arccos_less_arccos: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
   using arccos_less_mono by auto
 
-lemma arccos_le_arccos: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
+lemma arccos_le_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
   using arccos_le_mono by auto
 
-lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 & \<bar>y\<bar> \<le> 1 \<Longrightarrow> (arccos x = arccos y \<longleftrightarrow> x = y)"
+lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 \<and> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x = arccos y \<longleftrightarrow> x = y"
   using cos_arccos_abs by fastforce
 
-subsection \<open>Machins formula\<close>
+
+subsection \<open>Machin's formula\<close>
 
 lemma arctan_one: "arctan 1 = pi / 4"
-  by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
+  by (rule arctan_unique) (simp_all add: tan_45 m2pi_less_pi)
 
 lemma tan_total_pi4:
   assumes "\<bar>x\<bar> < 1"
@@ -5008,70 +5044,78 @@
 proof
   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
     unfolding arctan_one [symmetric] arctan_minus [symmetric]
-    unfolding arctan_less_iff using assms  by (auto simp add: arctan)
-
+    unfolding arctan_less_iff
+    using assms by (auto simp add: arctan)
 qed
 
 lemma arctan_add:
-  assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
+  assumes "\<bar>x\<bar> \<le> 1" "\<bar>y\<bar> < 1"
   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
 proof (rule arctan_unique [symmetric])
-  have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
+  have "- (pi / 4) \<le> arctan x" "- (pi / 4) < arctan y"
     unfolding arctan_one [symmetric] arctan_minus [symmetric]
-    unfolding arctan_le_iff arctan_less_iff using assms by auto
-  from add_le_less_mono [OF this]
-  show 1: "- (pi / 2) < arctan x + arctan y" by simp
-  have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
+    unfolding arctan_le_iff arctan_less_iff
+    using assms by auto
+  from add_le_less_mono [OF this] show 1: "- (pi / 2) < arctan x + arctan y"
+    by simp
+  have "arctan x \<le> pi / 4" "arctan y < pi / 4"
     unfolding arctan_one [symmetric]
-    unfolding arctan_le_iff arctan_less_iff using assms by auto
-  from add_le_less_mono [OF this]
-  show 2: "arctan x + arctan y < pi / 2" by simp
+    unfolding arctan_le_iff arctan_less_iff
+    using assms by auto
+  from add_le_less_mono [OF this] show 2: "arctan x + arctan y < pi / 2"
+    by simp
   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
     using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add)
 qed
 
-lemma arctan_double:
-  assumes "\<bar>x\<bar> < 1"
-  shows "2 * arctan x = arctan ((2*x) / (1 - x\<^sup>2))"
-  by (metis assms arctan_add linear mult_2 not_less power2_eq_square)
-
-theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
+lemma arctan_double: "\<bar>x\<bar> < 1 \<Longrightarrow> 2 * arctan x = arctan ((2 * x) / (1 - x\<^sup>2))"
+  by (metis arctan_add linear mult_2 not_less power2_eq_square)
+
+theorem machin: "pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)"
 proof -
-  have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
-  from arctan_add[OF less_imp_le[OF this] this]
-  have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
+  have "\<bar>1 / 5\<bar> < (1 :: real)"
+    by auto
+  from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (1 / 5) = arctan (5 / 12)"
+    by auto
   moreover
-  have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
-  from arctan_add[OF less_imp_le[OF this] this]
-  have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
+  have "\<bar>5 / 12\<bar> < (1 :: real)"
+    by auto
+  from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (5 / 12) = arctan (120 / 119)"
+    by auto
   moreover
-  have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
-  from arctan_add[OF this]
-  have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
-  ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
-  thus ?thesis unfolding arctan_one by algebra
+  have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)"
+    by auto
+  from arctan_add[OF this] have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)"
+    by auto
+  ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)"
+    by auto
+  then show ?thesis
+    unfolding arctan_one by algebra
 qed
 
-lemma machin_Euler: "5 * arctan(1/7) + 2 * arctan(3/79) = pi/4"
+lemma machin_Euler: "5 * arctan (1 / 7) + 2 * arctan (3 / 79) = pi / 4"
 proof -
-  have 17: "\<bar>1/7\<bar> < (1 :: real)" by auto
-  with arctan_double have "2 * arctan (1/7) = arctan (7/24)"
+  have 17: "\<bar>1 / 7\<bar> < (1 :: real)" by auto
+  with arctan_double have "2 * arctan (1 / 7) = arctan (7 / 24)"
     by simp (simp add: field_simps)
-  moreover have "\<bar>7/24\<bar> < (1 :: real)" by auto
-  with arctan_double have "2 * arctan (7/24) = arctan (336/527)"  by simp (simp add: field_simps)
-  moreover have "\<bar>336/527\<bar> < (1 :: real)" by auto
+  moreover
+  have "\<bar>7 / 24\<bar> < (1 :: real)" by auto
+  with arctan_double have "2 * arctan (7 / 24) = arctan (336 / 527)"
+    by simp (simp add: field_simps)
+  moreover
+  have "\<bar>336 / 527\<bar> < (1 :: real)" by auto
   from arctan_add[OF less_imp_le[OF 17] this]
-  have "arctan(1/7) + arctan (336/527) = arctan (2879/3353)"  by auto
-  ultimately have I: "5 * arctan(1/7) = arctan (2879/3353)"  by auto
-  have 379: "\<bar>3/79\<bar> < (1 :: real)" by auto
-  with arctan_double have II: "2 * arctan (3/79) = arctan (237/3116)"  by simp (simp add: field_simps)
-  have *: "\<bar>2879/3353\<bar> < (1 :: real)" by auto
-  have "\<bar>237/3116\<bar> < (1 :: real)" by auto
-  from arctan_add[OF less_imp_le[OF *] this]
-  have "arctan (2879/3353) + arctan (237/3116) = pi/4"
+  have "arctan(1/7) + arctan (336 / 527) = arctan (2879 / 3353)"
+    by auto
+  ultimately have I: "5 * arctan (1 / 7) = arctan (2879 / 3353)" by auto
+  have 379: "\<bar>3 / 79\<bar> < (1 :: real)" by auto
+  with arctan_double have II: "2 * arctan (3 / 79) = arctan (237 / 3116)"
+    by simp (simp add: field_simps)
+  have *: "\<bar>2879 / 3353\<bar> < (1 :: real)" by auto
+  have "\<bar>237 / 3116\<bar> < (1 :: real)" by auto
+  from arctan_add[OF less_imp_le[OF *] this] have "arctan (2879/3353) + arctan (237/3116) = pi/4"
     by (simp add: arctan_one)
-  then show ?thesis using I II
-    by auto
+  with I II show ?thesis by auto
 qed
 
 (*But could also prove MACHIN_GAUSS:
@@ -5083,45 +5127,46 @@
 lemma monoseq_arctan_series:
   fixes x :: real
   assumes "\<bar>x\<bar> \<le> 1"
-  shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
+  shows "monoseq (\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1))"
+    (is "monoseq ?a")
 proof (cases "x = 0")
   case True
-  thus ?thesis unfolding monoseq_def One_nat_def by auto
+  then show ?thesis by (auto simp: monoseq_def)
 next
   case False
-  have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
+  have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
+    using assms by auto
   show "monoseq ?a"
   proof -
-    {
-      fix n
-      fix x :: real
-      assume "0 \<le> x" and "x \<le> 1"
-      have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
+    have mono: "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
-      proof (rule mult_mono)
-        show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
-          by (rule frac_le) simp_all
-        show "0 \<le> 1 / real (Suc (n * 2))"
-          by auto
-        show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
-          by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>)
-        show "0 \<le> x ^ Suc (Suc n * 2)"
-          by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>)
-      qed
-    } note mono = this
-
+      if "0 \<le> x" and "x \<le> 1" for n and x :: real
+    proof (rule mult_mono)
+      show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
+        by (rule frac_le) simp_all
+      show "0 \<le> 1 / real (Suc (n * 2))"
+        by auto
+      show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
+        by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>)
+      show "0 \<le> x ^ Suc (Suc n * 2)"
+        by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>)
+    qed
     show ?thesis
     proof (cases "0 \<le> x")
-      case True from mono[OF this \<open>x \<le> 1\<close>, THEN allI]
-      show ?thesis unfolding Suc_eq_plus1[symmetric]
-        by (rule mono_SucI2)
+      case True
+      from mono[OF this \<open>x \<le> 1\<close>, THEN allI]
+      show ?thesis
+        unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
     next
       case False
-      hence "0 \<le> -x" and "-x \<le> 1" using \<open>-1 \<le> x\<close> by auto
+      then have "0 \<le> - x" and "- x \<le> 1"
+        using \<open>-1 \<le> x\<close> by auto
       from mono[OF this]
-      have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
-        1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using \<open>0 \<le> -x\<close> by auto
-      thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
+      have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
+          1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" for n
+        using \<open>0 \<le> -x\<close> by auto
+      then show ?thesis
+        unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
     qed
   qed
 qed
@@ -5129,29 +5174,33 @@
 lemma zeroseq_arctan_series:
   fixes x :: real
   assumes "\<bar>x\<bar> \<le> 1"
-  shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) \<longlonglongrightarrow> 0" (is "?a \<longlonglongrightarrow> 0")
+  shows "(\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1)) \<longlonglongrightarrow> 0"
+    (is "?a \<longlonglongrightarrow> 0")
 proof (cases "x = 0")
   case True
-  thus ?thesis
-    unfolding One_nat_def by auto
+  then show ?thesis by simp
 next
   case False
-  have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
+  have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
+    using assms by auto
   show "?a \<longlonglongrightarrow> 0"
   proof (cases "\<bar>x\<bar> < 1")
     case True
-    hence "norm x < 1" by auto
+    then have "norm x < 1" by auto
     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]]
     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0"
       unfolding inverse_eq_divide Suc_eq_plus1 by simp
-    then show ?thesis using pos2 by (rule LIMSEQ_linear)
+    then show ?thesis
+      using pos2 by (rule LIMSEQ_linear)
   next
     case False
-    hence "x = -1 \<or> x = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
-    hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
+    then have "x = -1 \<or> x = 1"
+      using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
+    then have n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
       unfolding One_nat_def by auto
     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
-    show ?thesis unfolding n_eq Suc_eq_plus1 by auto
+    show ?thesis
+      unfolding n_eq Suc_eq_plus1 by auto
   qed
 qed
 
@@ -5159,100 +5208,109 @@
   fixes n :: nat
   assumes "\<bar>x\<bar> \<le> 1"
   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
-  (is "summable (?c x)")
-  by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
+    (is "summable (?c x)")
+  by (rule summable_Leibniz(1),
+      rule zeroseq_arctan_series[OF assms],
+      rule monoseq_arctan_series[OF assms])
 
 lemma DERIV_arctan_series:
-  assumes "\<bar> x \<bar> < 1"
-  shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))"
-  (is "DERIV ?arctan _ :> ?Int")
+  assumes "\<bar>x\<bar> < 1"
+  shows "DERIV (\<lambda>x'. \<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x' ^ (k * 2 + 1))) x :>
+      (\<Sum>k. (-1)^k * x^(k * 2))"
+    (is "DERIV ?arctan _ :> ?Int")
 proof -
   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
 
-  have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
+  have n_even: "even n \<Longrightarrow> 2 * (n div 2) = n" for n :: nat
     by presburger
-  then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
-    (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
+  then have if_eq: "?f n * real (Suc n) * x'^n =
+      (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
+    for n x'
     by auto
 
-  {
-    fix x :: real
-    assume "\<bar>x\<bar> < 1"
-    hence "x\<^sup>2 < 1" by (simp add: abs_square_less_1)
+  have summable_Integral: "summable (\<lambda> n. (- 1) ^ n * x^(2 * n))" if "\<bar>x\<bar> < 1" for x :: real
+  proof -
+    from that have "x\<^sup>2 < 1"
+      by (simp add: abs_square_less_1)
     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
-      by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>])
-    hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult .
-  } note summable_Integral = this
-
-  {
-    fix f :: "nat \<Rightarrow> real"
-    have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
+      by (rule summable_Leibniz(1))
+        (auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>])
+    then show ?thesis
+      by (simp only: power_mult)
+  qed
+
+  have sums_even: "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)"
+    for f :: "nat \<Rightarrow> real"
+  proof -
+    have "f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x" for x :: real
     proof
-      fix x :: real
       assume "f sums x"
-      from sums_if[OF sums_zero this]
-      show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
+      from sums_if[OF sums_zero this] show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
         by auto
     next
-      fix x :: real
       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
       from LIMSEQ_linear[OF this[simplified sums_def] pos2, simplified sum_split_even_odd[simplified mult.commute]]
-      show "f sums x" unfolding sums_def by auto
+      show "f sums x"
+        unfolding sums_def by auto
     qed
-    hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
-  } note sums_even = this
+    then show ?thesis ..
+  qed
 
   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
-    unfolding if_eq mult.commute[of _ 2] suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
+    unfolding if_eq mult.commute[of _ 2]
+      suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
     by auto
 
-  {
-    fix x :: real
+  have arctan_eq: "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x" for x
+  proof -
     have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
       (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
       using n_even by auto
-    have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
-    have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
-      unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
+    have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)"
       by auto
-  } note arctan_eq = this
-
-  have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
+    then show ?thesis
+      unfolding if_eq' idx_eq suminf_def
+        sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
+      by auto
+  qed
+
+  have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum>n. ?f n * real (Suc n) * x^n)"
   proof (rule DERIV_power_series')
-    show "x \<in> {- 1 <..< 1}" using \<open>\<bar> x \<bar> < 1\<close> by auto
-    {
-      fix x' :: real
-      assume x'_bounds: "x' \<in> {- 1 <..< 1}"
-      then have "\<bar>x'\<bar> < 1" by auto
-      then
-        have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
+    show "x \<in> {- 1 <..< 1}"
+      using \<open>\<bar> x \<bar> < 1\<close> by auto
+    show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)"
+      if x'_bounds: "x' \<in> {- 1 <..< 1}" for x' :: real
+    proof -
+      from that have "\<bar>x'\<bar> < 1" by auto
+      then have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
         by (rule summable_Integral)
-      let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
-      show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
-        apply (rule sums_summable [where l="0 + ?S"])
+      show ?thesis
+        unfolding if_eq
+        apply (rule sums_summable [where l="0 + (\<Sum>n. (-1)^n * x'^(2 * n))"])
         apply (rule sums_if)
-        apply (rule sums_zero)
+         apply (rule sums_zero)
         apply (rule summable_sums)
         apply (rule *)
         done
-    }
+    qed
   qed auto
-  thus ?thesis unfolding Int_eq arctan_eq .
+  then show ?thesis
+    by (simp only: Int_eq arctan_eq)
 qed
 
 lemma arctan_series:
-  assumes "\<bar> x \<bar> \<le> 1"
-  shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
-  (is "_ = suminf (\<lambda> n. ?c x n)")
+  assumes "\<bar>x\<bar> \<le> 1"
+  shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
+    (is "_ = suminf (\<lambda> n. ?c x n)")
 proof -
   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
 
-  {
-    fix r x :: real
-    assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
-    have "\<bar>x\<bar> < 1" using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto
-    from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
-  } note DERIV_arctan_suminf = this
+  have DERIV_arctan_suminf: "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))"
+    if "0 < r" and "r < 1" and "\<bar>x\<bar> < r" for r x :: real
+  proof (rule DERIV_arctan_series)
+    from that show "\<bar>x\<bar> < 1"
+      using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto
+  qed
 
   {
     fix x :: real
@@ -5260,141 +5318,133 @@
     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
   } note arctan_series_borders = this
 
-  {
-    fix x :: real
-    assume "\<bar>x\<bar> < 1"
-    have "arctan x = (\<Sum>k. ?c x k)"
+  have when_less_one: "arctan x = (\<Sum>k. ?c x k)" if "\<bar>x\<bar> < 1" for x :: real
+  proof -
+    obtain r where "\<bar>x\<bar> < r" and "r < 1"
+      using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast
+    then have "0 < r" and "- r < x" and "x < r" by auto
+
+    have suminf_eq_arctan_bounded: "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
+      if "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" for x a b
     proof -
-      obtain r where "\<bar>x\<bar> < r" and "r < 1"
-        using dense[OF \<open>\<bar>