# HG changeset patch
# User wenzelm
# Date 1468574800 -7200
# Node ID 7f0e36eb73b4e956007e2c199353cc306eb55ee0
# Parent  d51a0a772094bf71ca220dec4648d08966055a21# Parent  b0f8845e349895e843530ce915e93a7c0911fefb
merged

diff -r d51a0a772094 -r 7f0e36eb73b4 src/HOL/Rat.thy
--- a/src/HOL/Rat.thy	Thu Jul 14 16:49:22 2016 +0100
+++ b/src/HOL/Rat.thy	Fri Jul 15 11:26:40 2016 +0200
@@ -1,11 +1,11 @@
-(*  Title:  HOL/Rat.thy
-    Author: Markus Wenzel, TU Muenchen
+(*  Title:      HOL/Rat.thy
+    Author:     Markus Wenzel, TU Muenchen
 *)
 
 section \<open>Rational numbers\<close>
 
 theory Rat
-imports GCD Archimedean_Field
+  imports GCD Archimedean_Field
 begin
 
 subsection \<open>Rational numbers as quotient\<close>
@@ -27,17 +27,17 @@
 lemma transp_ratrel: "transp ratrel"
 proof (rule transpI, unfold split_paired_all)
   fix a b a' b' a'' b'' :: int
-  assume A: "ratrel (a, b) (a', b')"
-  assume B: "ratrel (a', b') (a'', b'')"
+  assume *: "ratrel (a, b) (a', b')"
+  assume **: "ratrel (a', b') (a'', b'')"
   have "b' * (a * b'') = b'' * (a * b')" by simp
-  also from A have "a * b' = a' * b" by auto
+  also from * have "a * b' = a' * b" by auto
   also have "b'' * (a' * b) = b * (a' * b'')" by simp
-  also from B have "a' * b'' = a'' * b'" by auto
+  also from ** have "a' * b'' = a'' * b'" by auto
   also have "b * (a'' * b') = b' * (a'' * b)" by simp
   finally have "b' * (a * b'') = b' * (a'' * b)" .
-  moreover from B have "b' \<noteq> 0" by auto
+  moreover from ** have "b' \<noteq> 0" by auto
   ultimately have "a * b'' = a'' * b" by simp
-  with A B show "ratrel (a, b) (a'', b'')" by auto
+  with * ** show "ratrel (a, b) (a'', b'')" by auto
 qed
 
 lemma part_equivp_ratrel: "part_equivp ratrel"
@@ -120,7 +120,7 @@
 
 lift_definition plus_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
   is "\<lambda>x y. (fst x * snd y + fst y * snd x, snd x * snd y)"
-  by (clarsimp, simp add: distrib_right, simp add: ac_simps)
+  by (auto simp: distrib_right) (simp add: ac_simps)
 
 lemma add_rat [simp]:
   assumes "b \<noteq> 0" and "d \<noteq> 0"
@@ -139,9 +139,8 @@
 definition diff_rat_def: "q - r = q + - r" for q r :: rat
 
 lemma diff_rat [simp]:
-  assumes "b \<noteq> 0" and "d \<noteq> 0"
-  shows "Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
-  using assms by (simp add: diff_rat_def)
+  "b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b - Fract c d = Fract (a * d - c * b) (b * d)"
+  by (simp add: diff_rat_def)
 
 lift_definition times_rat :: "rat \<Rightarrow> rat \<Rightarrow> rat"
   is "\<lambda>x y. (fst x * fst y, snd x * snd y)"
@@ -150,10 +149,8 @@
 lemma mult_rat [simp]: "Fract a b * Fract c d = Fract (a * c) (b * d)"
   by transfer simp
 
-lemma mult_rat_cancel:
-  assumes "c \<noteq> 0"
-  shows "Fract (c * a) (c * b) = Fract a b"
-  using assms by transfer simp
+lemma mult_rat_cancel: "c \<noteq> 0 \<Longrightarrow> Fract (c * a) (c * b) = Fract a b"
+  by transfer simp
 
 lift_definition inverse_rat :: "rat \<Rightarrow> rat"
   is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)"
@@ -220,7 +217,7 @@
   "Fract (- 1) 1 = - 1"
   "Fract k 0 = 0"
   using Fract_of_int_eq [of "numeral w"]
-  using Fract_of_int_eq [of "- numeral w"]
+    and Fract_of_int_eq [of "- numeral w"]
   by (simp_all add: Zero_rat_def One_rat_def eq_rat)
 
 lemma rat_number_expand:
@@ -255,7 +252,8 @@
 lemma Fract_coprime: "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
 proof (cases "b = 0")
   case True
-  then show ?thesis by (simp add: eq_rat)
+  then show ?thesis
+    by (simp add: eq_rat)
 next
   case False
   moreover have "b div gcd a b * gcd a b = b"
@@ -282,28 +280,27 @@
   have *: "p * s = q * r"
     if "p * gcd r s = sgn (q * s) * r * gcd p q" and "q * gcd r s = sgn (q * s) * s * gcd p q"
   proof -
-    from that
-    have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) = (q * gcd r s) * (sgn (q * s) * r * gcd p q)"
+    from that have "(p * gcd r s) * (sgn (q * s) * s * gcd p q) =
+        (q * gcd r s) * (sgn (q * s) * r * gcd p q)"
       by simp
     with assms show ?thesis
       by (auto simp add: ac_simps sgn_times sgn_0_0)
   qed
   from assms show ?thesis
-    by (auto simp add: normalize_def Let_def dvd_div_div_eq_mult mult.commute sgn_times
+    by (auto simp: normalize_def Let_def dvd_div_div_eq_mult mult.commute sgn_times
         split: if_splits intro: *)
 qed
 
 lemma normalize_eq: "normalize (a, b) = (p, q) \<Longrightarrow> Fract p q = Fract a b"
-  by (auto simp add: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
+  by (auto simp: normalize_def Let_def Fract_coprime dvd_div_neg rat_number_collapse
       split: if_split_asm)
 
 lemma normalize_denom_pos: "normalize r = (p, q) \<Longrightarrow> q > 0"
-  by (auto simp add: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
+  by (auto simp: normalize_def Let_def dvd_div_neg pos_imp_zdiv_neg_iff nonneg1_imp_zdiv_pos_iff
       split: if_split_asm)
 
 lemma normalize_coprime: "normalize r = (p, q) \<Longrightarrow> coprime p q"
-  by (auto simp add: normalize_def Let_def dvd_div_neg div_gcd_coprime
-      split: if_split_asm)
+  by (auto simp: normalize_def Let_def dvd_div_neg div_gcd_coprime split: if_split_asm)
 
 lemma normalize_stable [simp]: "q > 0 \<Longrightarrow> coprime p q \<Longrightarrow> normalize (p, q) = (p, q)"
   by (simp add: normalize_def)
@@ -325,7 +322,8 @@
 lemma quotient_of_unique: "\<exists>!p. r = Fract (fst p) (snd p) \<and> snd p > 0 \<and> coprime (fst p) (snd p)"
 proof (cases r)
   case (Fract a b)
-  then have "r = Fract (fst (a, b)) (snd (a, b)) \<and> snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))"
+  then have "r = Fract (fst (a, b)) (snd (a, b)) \<and>
+      snd (a, b) > 0 \<and> coprime (fst (a, b)) (snd (a, b))"
     by auto
   then show ?thesis
   proof (rule ex1I)
@@ -453,7 +451,8 @@
 lemma positive_add: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
   apply transfer
   apply (simp add: zero_less_mult_iff)
-  apply (elim disjE, simp_all add: add_pos_pos add_neg_neg mult_pos_neg mult_neg_pos mult_neg_neg)
+  apply (elim disjE)
+     apply (simp_all add: add_pos_pos add_neg_neg mult_pos_neg mult_neg_pos mult_neg_neg)
   done
 
 lemma positive_mult: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
@@ -484,8 +483,8 @@
   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
     unfolding less_eq_rat_def less_rat_def
     apply auto
-    apply (drule (1) positive_add)
-    apply (simp_all add: positive_zero)
+     apply (drule (1) positive_add)
+     apply (simp_all add: positive_zero)
     done
   show "a \<le> a"
     unfolding less_eq_rat_def by simp
@@ -533,15 +532,12 @@
   by transfer simp
 
 lemma less_rat [simp]:
-  assumes "b \<noteq> 0" and "d \<noteq> 0"
-  shows "Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
-  using assms unfolding less_rat_def
-  by (simp add: positive_rat algebra_simps)
+  "b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)"
+  by (simp add: less_rat_def positive_rat algebra_simps)
 
 lemma le_rat [simp]:
-  assumes "b \<noteq> 0" and "d \<noteq> 0"
-  shows "Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
-  using assms unfolding le_less by (simp add: eq_rat)
+  "b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)"
+  by (simp add: le_less eq_rat)
 
 lemma abs_rat [simp, code]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>"
   by (auto simp add: abs_rat_def zabs_def Zero_rat_def not_less le_less eq_rat zero_less_mult_iff)
@@ -565,7 +561,8 @@
     then show "P (Fract a b)"
       by (simp add: order_less_imp_not_eq [OF b])
   qed
-  from Fract show "P q" by (auto simp add: linorder_neq_iff step step')
+  from Fract show "P q"
+    by (auto simp add: linorder_neq_iff step step')
 qed
 
 lemma zero_less_Fract_iff: "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a"
@@ -662,9 +659,7 @@
 
 lift_definition of_rat :: "rat \<Rightarrow> 'a"
   is "\<lambda>x. of_int (fst x) / of_int (snd x)"
-  apply (clarsimp simp add: nonzero_divide_eq_eq nonzero_eq_divide_eq)
-  apply (simp only: of_int_mult [symmetric])
-  done
+  by (auto simp: nonzero_divide_eq_eq nonzero_eq_divide_eq) (simp only: of_int_mult [symmetric])
 
 end
 
@@ -779,7 +774,7 @@
     by (induct a) (simp add: of_rat_rat Fract_of_int_eq [symmetric])
 qed
 
-text \<open>Collapse nested embeddings\<close>
+text \<open>Collapse nested embeddings.\<close>
 lemma of_rat_of_nat_eq [simp]: "of_rat (of_nat n) = of_nat n"
   by (induct n) (simp_all add: of_rat_add)
 
@@ -854,31 +849,36 @@
   apply (rule of_rat_mult [symmetric])
   done
 
-lemma nonzero_Rats_inverse: "a \<in> \<rat> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> inverse a \<in> \<rat>" for a :: "'a::field_char_0"
+lemma nonzero_Rats_inverse: "a \<in> \<rat> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> inverse a \<in> \<rat>"
+  for a :: "'a::field_char_0"
   apply (auto simp add: Rats_def)
   apply (rule range_eqI)
   apply (erule nonzero_of_rat_inverse [symmetric])
   done
 
-lemma Rats_inverse [simp]: "a \<in> \<rat> \<Longrightarrow> inverse a \<in> \<rat>" for a :: "'a::{field_char_0,field}"
+lemma Rats_inverse [simp]: "a \<in> \<rat> \<Longrightarrow> inverse a \<in> \<rat>"
+  for a :: "'a::{field_char_0,field}"
   apply (auto simp add: Rats_def)
   apply (rule range_eqI)
   apply (rule of_rat_inverse [symmetric])
   done
 
-lemma nonzero_Rats_divide: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a / b \<in> \<rat>" for a b :: "'a::field_char_0"
+lemma nonzero_Rats_divide: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a / b \<in> \<rat>"
+  for a b :: "'a::field_char_0"
   apply (auto simp add: Rats_def)
   apply (rule range_eqI)
   apply (erule nonzero_of_rat_divide [symmetric])
   done
 
-lemma Rats_divide [simp]: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> a / b \<in> \<rat>" for a b :: "'a::{field_char_0, field}"
+lemma Rats_divide [simp]: "a \<in> \<rat> \<Longrightarrow> b \<in> \<rat> \<Longrightarrow> a / b \<in> \<rat>"
+  for a b :: "'a::{field_char_0, field}"
   apply (auto simp add: Rats_def)
   apply (rule range_eqI)
   apply (rule of_rat_divide [symmetric])
   done
 
-lemma Rats_power [simp]: "a \<in> \<rat> \<Longrightarrow> a ^ n \<in> \<rat>" for a :: "'a::field_char_0"
+lemma Rats_power [simp]: "a \<in> \<rat> \<Longrightarrow> a ^ n \<in> \<rat>"
+  for a :: "'a::field_char_0"
   apply (auto simp add: Rats_def)
   apply (rule range_eqI)
   apply (rule of_rat_power [symmetric])
@@ -888,7 +888,8 @@
   assumes "q \<in> \<rat>"
   obtains (of_rat) r where "q = of_rat r"
 proof -
-  from \<open>q \<in> \<rat>\<close> have "q \<in> range of_rat" unfolding Rats_def .
+  from \<open>q \<in> \<rat>\<close> have "q \<in> range of_rat"
+    by (simp only: Rats_def)
   then obtain r where "q = of_rat r" ..
   then show thesis ..
 qed
@@ -1028,7 +1029,8 @@
 text \<open>Quickcheck\<close>
 
 definition (in term_syntax)
-  valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
+  valterm_fract :: "int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
+    int \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow>
     rat \<times> (unit \<Rightarrow> Code_Evaluation.term)"
   where [code_unfold]: "valterm_fract k l = Code_Evaluation.valtermify Fract {\<cdot>} k {\<cdot>} l"
 
diff -r d51a0a772094 -r 7f0e36eb73b4 src/HOL/Real.thy
--- a/src/HOL/Real.thy	Thu Jul 14 16:49:22 2016 +0100
+++ b/src/HOL/Real.thy	Fri Jul 15 11:26:40 2016 +0200
@@ -23,19 +23,24 @@
 
 subsection \<open>Preliminary lemmas\<close>
 
-lemma inj_add_left [simp]: "inj (op + x)" for x :: "'a::cancel_semigroup_add"
+lemma inj_add_left [simp]: "inj (op + x)"
+  for x :: "'a::cancel_semigroup_add"
   by (meson add_left_imp_eq injI)
 
-lemma inj_mult_left [simp]: "inj (op * x) \<longleftrightarrow> x \<noteq> 0" for x :: "'a::idom"
+lemma inj_mult_left [simp]: "inj (op * x) \<longleftrightarrow> x \<noteq> 0"
+  for x :: "'a::idom"
   by (metis injI mult_cancel_left the_inv_f_f zero_neq_one)
 
-lemma add_diff_add: "(a + c) - (b + d) = (a - b) + (c - d)" for a b c d :: "'a::ab_group_add"
+lemma add_diff_add: "(a + c) - (b + d) = (a - b) + (c - d)"
+  for a b c d :: "'a::ab_group_add"
   by simp
 
-lemma minus_diff_minus: "- a - - b = - (a - b)" for a b :: "'a::ab_group_add"
+lemma minus_diff_minus: "- a - - b = - (a - b)"
+  for a b :: "'a::ab_group_add"
   by simp
 
-lemma mult_diff_mult: "(x * y - a * b) = x * (y - b) + (x - a) * b" for x y a b :: "'a::ring"
+lemma mult_diff_mult: "(x * y - a * b) = x * (y - b) + (x - a) * b"
+  for x y a b :: "'a::ring"
   by (simp add: algebra_simps)
 
 lemma inverse_diff_inverse:
@@ -68,7 +73,7 @@
 lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
   unfolding vanishes_def
   apply (cases "c = 0")
-  apply auto
+   apply auto
   apply (rule exI [where x = "\<bar>c\<bar>"])
   apply auto
   done
@@ -93,9 +98,12 @@
   proof clarsimp
     fix n
     assume n: "i \<le> n" "j \<le> n"
-    have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
-    also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
-    finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
+    have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>"
+      by (rule abs_triangle_ineq)
+    also have "\<dots> < s + t"
+      by (simp add: add_strict_mono i j n)
+    finally show "\<bar>X n + Y n\<bar> < r"
+      by (simp only: r)
   qed
   then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
 qed
@@ -242,7 +250,8 @@
       unfolding abs_mult ..
     also have "\<dots> < a * t + s * b"
       by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
-    finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" by (simp only: r)
+    finally show "\<bar>X m * Y m - X n * Y n\<bar> < r"
+      by (simp only: r)
   qed
   then show "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
 qed
@@ -273,7 +282,7 @@
 
 lemma cauchy_not_vanishes:
   assumes X: "cauchy X"
-  assumes nz: "\<not> vanishes X"
+    and nz: "\<not> vanishes X"
   shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
   using cauchy_not_vanishes_cases [OF assms]
   apply clarify
@@ -285,7 +294,7 @@
 
 lemma cauchy_inverse [simp]:
   assumes X: "cauchy X"
-  assumes nz: "\<not> vanishes X"
+    and nz: "\<not> vanishes X"
   shows "cauchy (\<lambda>n. inverse (X n))"
 proof (rule cauchyI)
   fix r :: rat
@@ -328,8 +337,10 @@
     using cauchy_not_vanishes [OF Y] by blast
   obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
   proof
-    show "0 < a * r * b" using a r b by simp
-    show "inverse a * (a * r * b) * inverse b = r" using a r b by simp
+    show "0 < a * r * b"
+      using a r b by simp
+    show "inverse a * (a * r * b) * inverse b = r"
+      using a r b by simp
   qed
   obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
     using vanishesD [OF XY s] ..
@@ -435,16 +446,19 @@
   fix X Y
   assume "realrel X Y"
   then have X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
-    unfolding realrel_def by simp_all
+    by (simp_all add: realrel_def)
   have "vanishes X \<longleftrightarrow> vanishes Y"
   proof
     assume "vanishes X"
-    from vanishes_diff [OF this XY] show "vanishes Y" by simp
+    from vanishes_diff [OF this XY] show "vanishes Y"
+      by simp
   next
     assume "vanishes Y"
-    from vanishes_add [OF this XY] show "vanishes X" by simp
+    from vanishes_add [OF this XY] show "vanishes X"
+      by simp
   qed
-  then show "?thesis X Y" by (simp add: vanishes_diff_inverse X Y XY realrel_def)
+  then show "?thesis X Y"
+    by (simp add: vanishes_diff_inverse X Y XY realrel_def)
 qed
 
 definition "x - y = x + - y" for x y :: real
@@ -495,9 +509,12 @@
     apply transfer
     apply (simp add: realrel_def)
     apply (rule vanishesI)
-    apply (frule (1) cauchy_not_vanishes, clarify)
-    apply (rule_tac x=k in exI, clarify)
-    apply (drule_tac x=n in spec, simp)
+    apply (frule (1) cauchy_not_vanishes)
+    apply clarify
+    apply (rule_tac x=k in exI)
+    apply clarify
+    apply (drule_tac x=n in spec)
+    apply simp
     done
   show "a div b = a * inverse b"
     by (rule divide_real_def)
@@ -567,14 +584,15 @@
   apply (rule_tac x = "max i j" in exI)
   apply clarsimp
   apply (rule mult_strict_mono)
-  apply auto
+     apply auto
   done
 
 lemma positive_minus: "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
   apply transfer
   apply (simp add: realrel_def)
-  apply (drule (1) cauchy_not_vanishes_cases, safe)
-  apply blast+
+  apply (drule (1) cauchy_not_vanishes_cases)
+  apply safe
+   apply blast+
   done
 
 instantiation real :: linordered_field
@@ -596,8 +614,8 @@
   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
     unfolding less_eq_real_def less_real_def
     apply auto
-    apply (drule (1) positive_add)
-    apply (simp_all add: positive_zero)
+     apply (drule (1) positive_add)
+     apply (simp_all add: positive_zero)
     done
   show "a \<le> a"
     unfolding less_eq_real_def by simp
@@ -637,7 +655,8 @@
 
 definition "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
 
-instance by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2)
+instance
+  by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2)
 
 end
 
@@ -656,15 +675,16 @@
 
 instance real :: archimedean_field
 proof
-  fix x :: real
-  show "\<exists>z. x \<le> of_int z"
+  show "\<exists>z. x \<le> of_int z" for x :: real
     apply (induct x)
     apply (frule cauchy_imp_bounded, clarify)
     apply (rule_tac x="\<lceil>b\<rceil> + 1" in exI)
     apply (rule less_imp_le)
     apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
-    apply (rule_tac x=1 in exI, simp add: algebra_simps)
-    apply (rule_tac x=0 in exI, clarsimp)
+    apply (rule_tac x=1 in exI)
+    apply (simp add: algebra_simps)
+    apply (rule_tac x=0 in exI)
+    apply clarsimp
     apply (rule le_less_trans [OF abs_ge_self])
     apply (rule less_le_trans [OF _ le_of_int_ceiling])
     apply simp
@@ -687,26 +707,24 @@
 
 subsection \<open>Completeness\<close>
 
-lemma not_positive_Real:
-  assumes X: "cauchy X"
-  shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
-  unfolding positive_Real [OF X]
+lemma not_positive_Real: "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)" if "cauchy X"
+  apply (simp only: positive_Real [OF that])
   apply auto
-  apply (unfold not_less)
-  apply (erule obtain_pos_sum)
-  apply (drule_tac x=s in spec)
-  apply simp
-  apply (drule_tac r=t in cauchyD [OF X])
-  apply clarify
-  apply (drule_tac x=k in spec)
-  apply clarsimp
-  apply (rule_tac x=n in exI)
-  apply clarify
-  apply (rename_tac m)
-  apply (drule_tac x=m in spec)
-  apply simp
-  apply (drule_tac x=n in spec)
-  apply simp
+   apply (unfold not_less)
+   apply (erule obtain_pos_sum)
+   apply (drule_tac x=s in spec)
+   apply simp
+   apply (drule_tac r=t in cauchyD [OF that])
+   apply clarify
+   apply (drule_tac x=k in spec)
+   apply clarsimp
+   apply (rule_tac x=n in exI)
+   apply clarify
+   apply (rename_tac m)
+   apply (drule_tac x=m in spec)
+   apply simp
+   apply (drule_tac x=n in spec)
+   apply simp
   apply (drule spec)
   apply (drule (1) mp)
   apply clarify
@@ -743,9 +761,12 @@
     proof clarsimp
       fix n
       assume n: "i \<le> n" "j \<le> n"
-      have "X n \<le> Y i + t" using n j by simp
-      moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
-      ultimately show "X n \<le> Y n + r" unfolding r by simp
+      have "X n \<le> Y i + t"
+        using n j by simp
+      moreover have "\<bar>Y i - Y n\<bar> < s"
+        using n i by simp
+      ultimately show "X n \<le> Y n + r"
+        unfolding r by simp
     qed
     then show ?thesis ..
   qed
@@ -773,7 +794,7 @@
 
 lemma of_nat_less_two_power [simp]: "of_nat n < (2::'a::linordered_idom) ^ n"
   apply (induct n)
-  apply simp
+   apply simp
   apply (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
   done
 
@@ -821,7 +842,7 @@
 
   have width: "B n - A n = (b - a) / 2^n" for n
     apply (induct n)
-    apply (simp_all add: eq_divide_eq)
+     apply (simp_all add: eq_divide_eq)
     apply (simp_all add: C_def avg_def algebra_simps)
     done
 
@@ -833,13 +854,15 @@
     apply (rule_tac x=n in exI)
     apply (erule less_trans)
     apply (rule mult_strict_right_mono)
-    apply (rule le_less_trans [OF _ of_nat_less_two_power])
-    apply simp
+     apply (rule le_less_trans [OF _ of_nat_less_two_power])
+     apply simp
     apply assumption
     done
 
-  have PA: "\<not> P (A n)" for n by (induct n) (simp_all add: a)
-  have PB: "P (B n)" for n by (induct n) (simp_all add: b)
+  have PA: "\<not> P (A n)" for n
+    by (induct n) (simp_all add: a)
+  have PB: "P (B n)" for n
+    by (induct n) (simp_all add: b)
   have ab: "a < b"
     using a b unfolding P_def
     apply (clarsimp simp add: not_le)
@@ -847,21 +870,22 @@
     apply (drule (1) less_le_trans)
     apply (simp add: of_rat_less)
     done
-  have AB: "A n < B n" for n by (induct n) (simp_all add: ab C_def avg_def)
+  have AB: "A n < B n" for n
+    by (induct n) (simp_all add: ab C_def avg_def)
   have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
     apply (auto simp add: le_less [where 'a=nat])
     apply (erule less_Suc_induct)
-    apply (clarsimp simp add: C_def avg_def)
-    apply (simp add: add_divide_distrib [symmetric])
-    apply (rule AB [THEN less_imp_le])
+     apply (clarsimp simp add: C_def avg_def)
+     apply (simp add: add_divide_distrib [symmetric])
+     apply (rule AB [THEN less_imp_le])
     apply simp
     done
   have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
     apply (auto simp add: le_less [where 'a=nat])
     apply (erule less_Suc_induct)
-    apply (clarsimp simp add: C_def avg_def)
-    apply (simp add: add_divide_distrib [symmetric])
-    apply (rule AB [THEN less_imp_le])
+     apply (clarsimp simp add: C_def avg_def)
+     apply (simp add: add_divide_distrib [symmetric])
+     apply (rule AB [THEN less_imp_le])
     apply simp
     done
   have cauchy_lemma: "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
@@ -870,7 +894,7 @@
     apply (erule exE)
     apply (rule_tac x=n in exI, clarify, rename_tac i j)
     apply (rule_tac y="B n - A n" in le_less_trans) defer
-    apply (simp add: width)
+     apply (simp add: width)
     apply (drule_tac x=n in spec)
     apply (frule_tac x=i in spec, drule (1) mp)
     apply (frule_tac x=j in spec, drule (1) mp)
@@ -900,11 +924,13 @@
     apply clarify
     apply (erule contrapos_pp)
     apply (simp add: not_le)
-    apply (drule less_RealD [OF \<open>cauchy A\<close>], clarify)
+    apply (drule less_RealD [OF \<open>cauchy A\<close>])
+    apply clarify
     apply (subgoal_tac "\<not> P (A n)")
-    apply (simp add: P_def not_le, clarify)
-    apply (erule rev_bexI)
-    apply (erule (1) less_trans)
+     apply (simp add: P_def not_le)
+     apply clarify
+     apply (erule rev_bexI)
+     apply (erule (1) less_trans)
     apply (simp add: PA)
     done
   have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
@@ -945,13 +971,17 @@
 
 instance
 proof
-  show Sup_upper: "x \<le> Sup X" if "x \<in> X" "bdd_above X" for x :: real and X :: "real set"
+  show Sup_upper: "x \<le> Sup X"
+    if "x \<in> X" "bdd_above X"
+    for x :: real and X :: "real set"
   proof -
     from that obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
       using complete_real[of X] unfolding bdd_above_def by blast
-    then show ?thesis unfolding Sup_real_def by (rule LeastI2_order) (auto simp: that)
+    then show ?thesis
+      unfolding Sup_real_def by (rule LeastI2_order) (auto simp: that)
   qed
-  show Sup_least: "Sup X \<le> z" if "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
+  show Sup_least: "Sup X \<le> z"
+    if "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
     for z :: real and X :: "real set"
   proof -
     from that obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
@@ -962,9 +992,11 @@
       by blast
     finally show ?thesis .
   qed
-  show "Inf X \<le> x" if "x \<in> X" "bdd_below X" for x :: real and X :: "real set"
+  show "Inf X \<le> x" if "x \<in> X" "bdd_below X"
+    for x :: real and X :: "real set"
     using Sup_upper [of "-x" "uminus ` X"] by (auto simp: Inf_real_def that)
-  show "z \<le> Inf X" if "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" for z :: real and X :: "real set"
+  show "z \<le> Inf X" if "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x"
+    for z :: real and X :: "real set"
     using Sup_least [of "uminus ` X" "- z"] by (force simp: Inf_real_def that)
   show "\<exists>a b::real. a \<noteq> b"
     using zero_neq_one by blast
@@ -990,13 +1022,16 @@
 text \<open>BH: These lemmas should not be necessary; they should be
   covered by existing simp rules and simplification procedures.\<close>
 
-lemma real_mult_less_iff1 [simp]: "0 < z \<Longrightarrow> x * z < y * z \<longleftrightarrow> x < y" for x y z :: real
+lemma real_mult_less_iff1 [simp]: "0 < z \<Longrightarrow> x * z < y * z \<longleftrightarrow> x < y"
+  for x y z :: real
   by simp (* solved by linordered_ring_less_cancel_factor simproc *)
 
-lemma real_mult_le_cancel_iff1 [simp]: "0 < z \<Longrightarrow> x * z \<le> y * z \<longleftrightarrow> x \<le> y" for x y z :: real
+lemma real_mult_le_cancel_iff1 [simp]: "0 < z \<Longrightarrow> x * z \<le> y * z \<longleftrightarrow> x \<le> y"
+  for x y z :: real
   by simp (* solved by linordered_ring_le_cancel_factor simproc *)
 
-lemma real_mult_le_cancel_iff2 [simp]: "0 < z \<Longrightarrow> z * x \<le> z * y \<longleftrightarrow> x \<le> y" for x y z :: real
+lemma real_mult_le_cancel_iff2 [simp]: "0 < z \<Longrightarrow> z * x \<le> z * y \<longleftrightarrow> x \<le> y"
+  for x y z :: real
   by simp (* solved by linordered_ring_le_cancel_factor simproc *)
 
 
@@ -1072,7 +1107,7 @@
 
 lemma real_of_int_div2: "0 \<le> real_of_int n / real_of_int x - real_of_int (n div x)"
   apply (cases "x = 0")
-  apply simp
+   apply simp
   apply (cases "0 < x")
    apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
   apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
@@ -1096,7 +1131,8 @@
 lemma nat_less_real_le: "n < m \<longleftrightarrow> real n + 1 \<le> real m"
   by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)
 
-lemma nat_le_real_less: "n \<le> m \<longleftrightarrow> real n < real m + 1" for m n :: nat
+lemma nat_le_real_less: "n \<le> m \<longleftrightarrow> real n < real m + 1"
+  for m n :: nat
   by (meson nat_less_real_le not_le)
 
 lemma real_of_nat_div_aux: "real x / real d = real (x div d) + real (x mod d) / real d"
@@ -1122,7 +1158,7 @@
 
 lemma real_of_nat_div3: "real n / real x - real (n div x) \<le> 1" for n x :: nat
   apply (cases "x = 0")
-  apply simp
+   apply simp
   apply (simp add: algebra_simps)
   apply (subst real_of_nat_div_aux)
   apply simp
@@ -1138,10 +1174,8 @@
   using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
   by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc)
 
-lemma reals_Archimedean3:
-  assumes x_greater_zero: "0 < x"
-  shows "\<forall>y. \<exists>n. y < real n * x"
-  using \<open>0 < x\<close> by (auto intro: ex_less_of_nat_mult)
+lemma reals_Archimedean3: "0 < x \<Longrightarrow> \<forall>y. \<exists>n. y < real n * x"
+  by (auto intro: ex_less_of_nat_mult)
 
 lemma real_archimedian_rdiv_eq_0:
   assumes x0: "x \<ge> 0"
@@ -1250,7 +1284,7 @@
 
 text \<open>
   This density proof is due to Stefan Richter and was ported by TN.  The
-  original source is \emph{Real Analysis} by H.L. Royden.
+  original source is \<^emph>\<open>Real Analysis\<close> by H.L. Royden.
   It employs the Archimedean property of the reals.\<close>
 
 lemma Rats_dense_in_real:
@@ -1263,15 +1297,15 @@
     by blast
   define p where "p = \<lceil>y * real q\<rceil> - 1"
   define r where "r = of_int p / real q"
-  from q have "x < y - inverse (real q)" by simp
-  also have "y - inverse (real q) \<le> r"
-    unfolding r_def p_def
-    by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling \<open>0 < q\<close>)
+  from q have "x < y - inverse (real q)"
+    by simp
+  also from \<open>0 < q\<close> have "y - inverse (real q) \<le> r"
+    by (simp add: r_def p_def le_divide_eq left_diff_distrib)
   finally have "x < r" .
-  moreover have "r < y"
-    unfolding r_def p_def
-    by (simp add: divide_less_eq diff_less_eq \<open>0 < q\<close> less_ceiling_iff [symmetric])
-  moreover from r_def have "r \<in> \<rat>" by simp
+  moreover from \<open>0 < q\<close> have "r < y"
+    by (simp add: r_def p_def divide_less_eq diff_less_eq less_ceiling_iff [symmetric])
+  moreover have "r \<in> \<rat>"
+    by (simp add: r_def)
   ultimately show ?thesis by blast
 qed
 
@@ -1307,19 +1341,24 @@
 
 subsection \<open>Simprules combining \<open>x + y\<close> and \<open>0\<close>\<close> (* FIXME ARE THEY NEEDED? *)
 
-lemma real_add_minus_iff [simp]: "x + - a = 0 \<longleftrightarrow> x = a" for x a :: real
+lemma real_add_minus_iff [simp]: "x + - a = 0 \<longleftrightarrow> x = a"
+  for x a :: real
   by arith
 
-lemma real_add_less_0_iff: "x + y < 0 \<longleftrightarrow> y < - x" for x y :: real
+lemma real_add_less_0_iff: "x + y < 0 \<longleftrightarrow> y < - x"
+  for x y :: real
   by auto
 
-lemma real_0_less_add_iff: "0 < x + y \<longleftrightarrow> - x < y" for x y :: real
+lemma real_0_less_add_iff: "0 < x + y \<longleftrightarrow> - x < y"
+  for x y :: real
   by auto
 
-lemma real_add_le_0_iff: "x + y \<le> 0 \<longleftrightarrow> y \<le> - x" for x y :: real
+lemma real_add_le_0_iff: "x + y \<le> 0 \<longleftrightarrow> y \<le> - x"
+  for x y :: real
   by auto
 
-lemma real_0_le_add_iff: "0 \<le> x + y \<longleftrightarrow> - x \<le> y" for x y :: real
+lemma real_0_le_add_iff: "0 \<le> x + y \<longleftrightarrow> - x \<le> y"
+  for x y :: real
   by auto
 
 
@@ -1331,10 +1370,12 @@
 (* FIXME: declare this [simp] for all types, or not at all *)
 declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]
 
-lemma real_minus_mult_self_le [simp]: "- (u * u) \<le> x * x" for u x :: real
+lemma real_minus_mult_self_le [simp]: "- (u * u) \<le> x * x"
+  for u x :: real
   by (rule order_trans [where y = 0]) auto
 
-lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> x\<^sup>2" for u x :: real
+lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> x\<^sup>2"
+  for u x :: real
   by (auto simp add: power2_eq_square)
 
 lemma numeral_power_eq_real_of_int_cancel_iff [simp]:
@@ -1396,17 +1437,21 @@
 
 subsection \<open>Density of the Reals\<close>
 
-lemma real_lbound_gt_zero: "0 < d1 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> e < d2" for d1 d2 :: real
+lemma real_lbound_gt_zero: "0 < d1 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> e < d2"
+  for d1 d2 :: real
   by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def)
 
 text \<open>Similar results are proved in @{theory Fields}\<close>
-lemma real_less_half_sum: "x < y \<Longrightarrow> x < (x + y) / 2" for x y :: real
+lemma real_less_half_sum: "x < y \<Longrightarrow> x < (x + y) / 2"
+  for x y :: real
   by auto
 
-lemma real_gt_half_sum: "x < y \<Longrightarrow> (x + y) / 2 < y" for x y :: real
+lemma real_gt_half_sum: "x < y \<Longrightarrow> (x + y) / 2 < y"
+  for x y :: real
   by auto
 
-lemma real_sum_of_halves: "x / 2 + x / 2 = x" for x :: real
+lemma real_sum_of_halves: "x / 2 + x / 2 = x"
+  for x :: real
   by simp
 
 
@@ -1414,13 +1459,16 @@
 
 (* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)
 
-lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w \<longleftrightarrow> n < numeral w" for n :: nat
+lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w \<longleftrightarrow> n < numeral w"
+  for n :: nat
   by (metis of_nat_less_iff of_nat_numeral)
 
-lemma numeral_less_real_of_nat_iff [simp]: "numeral w < real n \<longleftrightarrow> numeral w < n" for n :: nat
+lemma numeral_less_real_of_nat_iff [simp]: "numeral w < real n \<longleftrightarrow> numeral w < n"
+  for n :: nat
   by (metis of_nat_less_iff of_nat_numeral)
 
-lemma numeral_le_real_of_nat_iff [simp]: "numeral n \<le> real m \<longleftrightarrow> numeral n \<le> m" for m :: nat
+lemma numeral_le_real_of_nat_iff [simp]: "numeral n \<le> real m \<longleftrightarrow> numeral n \<le> m"
+  for m :: nat
   by (metis not_le real_of_nat_less_numeral_iff)
 
 declare of_int_floor_le [simp]  (* FIXME duplicate!? *)
@@ -1531,7 +1579,8 @@
   and natceiling.
 \<close>
 
-lemma nat_floor_neg: "x \<le> 0 \<Longrightarrow> nat \<lfloor>x\<rfloor> = 0" for x :: real
+lemma nat_floor_neg: "x \<le> 0 \<Longrightarrow> nat \<lfloor>x\<rfloor> = 0"
+  for x :: real
   by linarith
 
 lemma le_nat_floor: "real x \<le> a \<Longrightarrow> x \<le> nat \<lfloor>a\<rfloor>"
@@ -1547,7 +1596,8 @@
 lemma real_nat_ceiling_ge: "x \<le> real (nat \<lceil>x\<rceil>)"
   by linarith
 
-lemma Rats_no_top_le: "\<exists>q \<in> \<rat>. x \<le> q" for x :: real
+lemma Rats_no_top_le: "\<exists>q \<in> \<rat>. x \<le> q"
+  for x :: real
   by (auto intro!: bexI[of _ "of_nat (nat \<lceil>x\<rceil>)"]) linarith
 
 lemma Rats_no_bot_less: "\<exists>q \<in> \<rat>. q < x" for x :: real
@@ -1616,10 +1666,10 @@
 text \<open>Operations\<close>
 
 lemma zero_real_code [code]: "0 = Ratreal 0"
-by simp
+  by simp
 
 lemma one_real_code [code]: "1 = Ratreal 1"
-by simp
+  by simp
 
 instantiation real :: equal
 begin
@@ -1631,7 +1681,8 @@
 lemma real_equal_code [code]: "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
   by (simp add: equal_real_def equal)
 
-lemma [code nbe]: "HOL.equal x x \<longleftrightarrow> True" for x :: real
+lemma [code nbe]: "HOL.equal x x \<longleftrightarrow> True"
+  for x :: real
   by (rule equal_refl)
 
 end
diff -r d51a0a772094 -r 7f0e36eb73b4 src/HOL/Topological_Spaces.thy
--- a/src/HOL/Topological_Spaces.thy	Thu Jul 14 16:49:22 2016 +0100
+++ b/src/HOL/Topological_Spaces.thy	Fri Jul 15 11:26:40 2016 +0200
@@ -6,7 +6,7 @@
 section \<open>Topological Spaces\<close>
 
 theory Topological_Spaces
-imports Main
+  imports Main
 begin
 
 named_theorems continuous_intros "structural introduction rules for continuity"
@@ -22,9 +22,8 @@
   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
 begin
 
-definition
-  closed :: "'a set \<Rightarrow> bool" where
-  "closed S \<longleftrightarrow> open (- S)"
+definition closed :: "'a set \<Rightarrow> bool"
+  where "closed S \<longleftrightarrow> open (- S)"
 
 lemma open_empty [continuous_intros, intro, simp]: "open {}"
   using open_Union [of "{}"] by simp
@@ -50,7 +49,7 @@
   ultimately show "open S" by simp
 qed
 
-lemma closed_empty [continuous_intros, intro, simp]:  "closed {}"
+lemma closed_empty [continuous_intros, intro, simp]: "closed {}"
   unfolding closed_def by simp
 
 lemma closed_Un [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
@@ -71,7 +70,8 @@
 lemma closed_Union [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
   by (induct set: finite) auto
 
-lemma closed_UN [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
+lemma closed_UN [continuous_intros, intro]:
+  "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
   using closed_Union [of "B ` A"] by simp
 
 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
@@ -95,10 +95,14 @@
 lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}"
   unfolding Collect_neg_eq by (rule open_Compl)
 
-lemma open_Collect_conj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<and> Q x}"
+lemma open_Collect_conj:
+  assumes "open {x. P x}" "open {x. Q x}"
+  shows "open {x. P x \<and> Q x}"
   using open_Int[OF assms] by (simp add: Int_def)
 
-lemma open_Collect_disj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<or> Q x}"
+lemma open_Collect_disj:
+  assumes "open {x. P x}" "open {x. Q x}"
+  shows "open {x. P x \<or> Q x}"
   using open_Un[OF assms] by (simp add: Un_def)
 
 lemma open_Collect_ex: "(\<And>i. open {x. P i x}) \<Longrightarrow> open {x. \<exists>i. P i x}"
@@ -113,14 +117,18 @@
 lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}"
   unfolding Collect_neg_eq by (rule closed_Compl)
 
-lemma closed_Collect_conj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<and> Q x}"
+lemma closed_Collect_conj:
+  assumes "closed {x. P x}" "closed {x. Q x}"
+  shows "closed {x. P x \<and> Q x}"
   using closed_Int[OF assms] by (simp add: Int_def)
 
-lemma closed_Collect_disj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<or> Q x}"
+lemma closed_Collect_disj:
+  assumes "closed {x. P x}" "closed {x. Q x}"
+  shows "closed {x. P x \<or> Q x}"
   using closed_Un[OF assms] by (simp add: Un_def)
 
 lemma closed_Collect_all: "(\<And>i. closed {x. P i x}) \<Longrightarrow> closed {x. \<forall>i. P i x}"
-  using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_all_eq by simp
+  using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] by (simp add: Collect_all_eq)
 
 lemma closed_Collect_imp: "open {x. P x} \<Longrightarrow> closed {x. Q x} \<Longrightarrow> closed {x. P x \<longrightarrow> Q x}"
   unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg)
@@ -130,7 +138,8 @@
 
 end
 
-subsection\<open>Hausdorff and other separation properties\<close>
+
+subsection \<open>Hausdorff and other separation properties\<close>
 
 class t0_space = topological_space +
   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
@@ -139,37 +148,39 @@
   assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
 
 instance t1_space \<subseteq> t0_space
-proof qed (fast dest: t1_space)
-
-lemma separation_t1:
-  fixes x y :: "'a::t1_space"
-  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
+  by standard (fast dest: t1_space)
+
+lemma separation_t1: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
+  for x y :: "'a::t1_space"
   using t1_space[of x y] by blast
 
-lemma closed_singleton [iff]:
-  fixes a :: "'a::t1_space"
-  shows "closed {a}"
+lemma closed_singleton [iff]: "closed {a}"
+  for a :: "'a::t1_space"
 proof -
   let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
-  have "open ?T" by (simp add: open_Union)
+  have "open ?T"
+    by (simp add: open_Union)
   also have "?T = - {a}"
-    by (simp add: set_eq_iff separation_t1, auto)
-  finally show "closed {a}" unfolding closed_def .
+    by (auto simp add: set_eq_iff separation_t1)
+  finally show "closed {a}"
+    by (simp only: closed_def)
 qed
 
 lemma closed_insert [continuous_intros, simp]:
   fixes a :: "'a::t1_space"
-  assumes "closed S" shows "closed (insert a S)"
+  assumes "closed S"
+  shows "closed (insert a S)"
 proof -
-  from closed_singleton assms
-  have "closed ({a} \<union> S)" by (rule closed_Un)
-  thus "closed (insert a S)" by simp
+  from closed_singleton assms have "closed ({a} \<union> S)"
+    by (rule closed_Un)
+  then show "closed (insert a S)"
+    by simp
 qed
 
-lemma finite_imp_closed:
-  fixes S :: "'a::t1_space set"
-  shows "finite S \<Longrightarrow> closed S"
-by (induct set: finite, simp_all)
+lemma finite_imp_closed: "finite S \<Longrightarrow> closed S"
+  for S :: "'a::t1_space set"
+  by (induct pred: finite) simp_all
+
 
 text \<open>T2 spaces are also known as Hausdorff spaces.\<close>
 
@@ -177,34 +188,35 @@
   assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
 
 instance t2_space \<subseteq> t1_space
-proof qed (fast dest: hausdorff)
-
-lemma separation_t2:
-  fixes x y :: "'a::t2_space"
-  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
-  using hausdorff[of x y] by blast
-
-lemma separation_t0:
-  fixes x y :: "'a::t0_space"
-  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
-  using t0_space[of x y] by blast
+  by standard (fast dest: hausdorff)
+
+lemma separation_t2: "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
+  for x y :: "'a::t2_space"
+  using hausdorff [of x y] by blast
+
+lemma separation_t0: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U))"
+  for x y :: "'a::t0_space"
+  using t0_space [of x y] by blast
+
 
 text \<open>A perfect space is a topological space with no isolated points.\<close>
 
 class perfect_space = topological_space +
   assumes not_open_singleton: "\<not> open {x}"
 
-lemma UNIV_not_singleton: "UNIV \<noteq> {x::'a::perfect_space}"
+lemma UNIV_not_singleton: "UNIV \<noteq> {x}"
+  for x :: "'a::perfect_space"
   by (metis open_UNIV not_open_singleton)
 
 
 subsection \<open>Generators for toplogies\<close>
 
-inductive generate_topology for S where
-  UNIV: "generate_topology S UNIV"
-| Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)"
-| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)"
-| Basis: "s \<in> S \<Longrightarrow> generate_topology S s"
+inductive generate_topology :: "'a set set \<Rightarrow> 'a set \<Rightarrow> bool" for S :: "'a set set"
+  where
+    UNIV: "generate_topology S UNIV"
+  | Int: "generate_topology S (a \<inter> b)" if "generate_topology S a" and "generate_topology S b"
+  | UN: "generate_topology S (\<Union>K)" if "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k)"
+  | Basis: "generate_topology S s" if "s \<in> S"
 
 hide_fact (open) UNIV Int UN Basis
 
@@ -212,10 +224,10 @@
   "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
   using generate_topology.UN [of "K ` I"] by auto
 
-lemma topological_space_generate_topology:
-  "class.topological_space (generate_topology S)"
+lemma topological_space_generate_topology: "class.topological_space (generate_topology S)"
   by standard (auto intro: generate_topology.intros)
 
+
 subsection \<open>Order topologies\<close>
 
 class order_topology = order + "open" +
@@ -239,13 +251,16 @@
 
 class linorder_topology = linorder + order_topology
 
-lemma closed_atMost [continuous_intros, simp]: "closed {.. a::'a::linorder_topology}"
+lemma closed_atMost [continuous_intros, simp]: "closed {..a}"
+  for a :: "'a::linorder_topology"
   by (simp add: closed_open)
 
-lemma closed_atLeast [continuous_intros, simp]: "closed {a::'a::linorder_topology ..}"
+lemma closed_atLeast [continuous_intros, simp]: "closed {a..}"
+  for a :: "'a::linorder_topology"
   by (simp add: closed_open)
 
-lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a::'a::linorder_topology .. b}"
+lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a..b}"
+  for a b :: "'a::linorder_topology"
 proof -
   have "{a .. b} = {a ..} \<inter> {.. b}"
     by auto
@@ -264,7 +279,7 @@
   then show ?thesis by blast
 next
   case False
-  with \<open>x < y\<close> have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}"
+  with \<open>x < y\<close> have "x \<in> {..< y}" "y \<in> {x <..}" "{x <..} \<inter> {..< y} = {}"
     by auto
   then show ?thesis by blast
 qed
@@ -272,56 +287,81 @@
 instance linorder_topology \<subseteq> t2_space
 proof
   fix x y :: 'a
-  from less_separate[of x y] less_separate[of y x]
   show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
-    by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+
+    using less_separate [of x y] less_separate [of y x]
+    by (elim neqE; metis open_lessThan open_greaterThan Int_commute)
 qed
 
 lemma (in linorder_topology) open_right:
-  assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S"
+  assumes "open S" "x \<in> S"
+    and gt_ex: "x < y"
+  shows "\<exists>b>x. {x ..< b} \<subseteq> S"
+  using assms unfolding open_generated_order
+proof induct
+  case UNIV
+  then show ?case by blast
+next
+  case (Int A B)
+  then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B"
+    by auto
+  then show ?case
+    by (auto intro!: exI[of _ "min a b"])
+next
+  case UN
+  then show ?case by blast
+next
+  case Basis
+  then show ?case
+    by (fastforce intro: exI[of _ y] gt_ex)
+qed
+
+lemma (in linorder_topology) open_left:
+  assumes "open S" "x \<in> S"
+    and lt_ex: "y < x"
+  shows "\<exists>b<x. {b <.. x} \<subseteq> S"
   using assms unfolding open_generated_order
 proof induction
+  case UNIV
+  then show ?case by blast
+next
   case (Int A B)
-  then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B" by auto
-  then show ?case by (auto intro!: exI[of _ "min a b"])
+  then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B"
+    by auto
+  then show ?case
+    by (auto intro!: exI[of _ "max a b"])
 next
-  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex)
-qed blast+
-
-lemma (in linorder_topology) open_left:
-  assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S"
-  using assms unfolding open_generated_order
-proof induction
-  case (Int A B)
-  then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B" by auto
-  then show ?case by (auto intro!: exI[of _ "max a b"])
+  case UN
+  then show ?case by blast
 next
-  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex)
-qed blast+
+  case Basis
+  then show ?case
+    by (fastforce intro: exI[of _ y] lt_ex)
+qed
+
 
 subsection \<open>Setup some topologies\<close>
 
 subsubsection \<open>Boolean is an order topology\<close>
 
-text \<open>It is a discrete topology, but don't have a type class for it (yet).\<close>
-
 class discrete_topology = topological_space +
   assumes open_discrete: "\<And>A. open A"
 
 instance discrete_topology < t2_space
 proof
-  fix x y :: 'a assume "x \<noteq> y" then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
+  fix x y :: 'a
+  assume "x \<noteq> y"
+  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
     by (intro exI[of _ "{_}"]) (auto intro!: open_discrete)
 qed
 
 instantiation bool :: linorder_topology
 begin
 
-definition open_bool :: "bool set \<Rightarrow> bool" where
-  "open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
+definition open_bool :: "bool set \<Rightarrow> bool"
+  where "open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
 
 instance
-  proof qed (rule open_bool_def)
+  by standard (rule open_bool_def)
 
 end
 
@@ -339,11 +379,11 @@
 instantiation nat :: linorder_topology
 begin
 
-definition open_nat :: "nat set \<Rightarrow> bool" where
-  "open_nat = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
+definition open_nat :: "nat set \<Rightarrow> bool"
+  where "open_nat = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
 
 instance
-  proof qed (rule open_nat_def)
+  by standard (rule open_nat_def)
 
 end
 
@@ -359,9 +399,9 @@
        by auto
   next
     case (Suc n')
-    moreover then have "{n} = {..<Suc n} \<inter> {n' <..}"
+    then have "{n} = {..<Suc n} \<inter> {n' <..}"
       by auto
-    ultimately show ?thesis
+    with Suc show ?thesis
       by (auto intro: open_lessThan open_greaterThan)
   qed
   then have "open (\<Union>a\<in>A. {a})"
@@ -373,11 +413,11 @@
 instantiation int :: linorder_topology
 begin
 
-definition open_int :: "int set \<Rightarrow> bool" where
-  "open_int = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
+definition open_int :: "int set \<Rightarrow> bool"
+  where "open_int = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
 
 instance
-  proof qed (rule open_int_def)
+  by standard (rule open_int_def)
 
 end
 
@@ -394,31 +434,34 @@
     by simp
 qed
 
+
 subsubsection \<open>Topological filters\<close>
 
 definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
   where "nhds a = (INF S:{S. open S \<and> a \<in> S}. principal S)"
 
-definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_)/ within (_)" [1000, 60] 60)
+definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter"
+    ("at (_)/ within (_)" [1000, 60] 60)
   where "at a within s = inf (nhds a) (principal (s - {a}))"
 
-abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
-  "at x \<equiv> at x within (CONST UNIV)"
-
-abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
-  "at_right x \<equiv> at x within {x <..}"
-
-abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
-  "at_left x \<equiv> at x within {..< x}"
+abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter"  ("at")
+  where "at x \<equiv> at x within (CONST UNIV)"
+
+abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter"
+  where "at_right x \<equiv> at x within {x <..}"
+
+abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter"
+  where "at_left x \<equiv> at x within {..< x}"
 
 lemma (in topological_space) nhds_generated_topology:
   "open = generate_topology T \<Longrightarrow> nhds x = (INF S:{S\<in>T. x \<in> S}. principal S)"
   unfolding nhds_def
 proof (safe intro!: antisym INF_greatest)
-  fix S assume "generate_topology T S" "x \<in> S"
+  fix S
+  assume "generate_topology T S" "x \<in> S"
   then show "(INF S:{S \<in> T. x \<in> S}. principal S) \<le> principal S"
-    by induction
-       (auto intro: INF_lower order_trans simp add: inf_principal[symmetric] simp del: inf_principal)
+    by induct
+      (auto intro: INF_lower order_trans simp: inf_principal[symmetric] simp del: inf_principal)
 qed (auto intro!: INF_lower intro: generate_topology.intros)
 
 lemma (in topological_space) eventually_nhds:
@@ -429,15 +472,13 @@
   "open s \<Longrightarrow> x \<in> s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
   by (subst eventually_nhds) blast
 
-lemma eventually_nhds_x_imp_x:
-  "eventually P (nhds x) \<Longrightarrow> P x"
+lemma eventually_nhds_x_imp_x: "eventually P (nhds x) \<Longrightarrow> P x"
   by (subst (asm) eventually_nhds) blast
 
 lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
-  unfolding trivial_limit_def eventually_nhds by simp
-
-lemma (in t1_space) t1_space_nhds:
-  "x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)"
+  by (simp add: trivial_limit_def eventually_nhds)
+
+lemma (in t1_space) t1_space_nhds: "x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)"
   by (drule t1_space) (auto simp: eventually_nhds)
 
 lemma (in topological_space) nhds_discrete_open: "open {x} \<Longrightarrow> nhds x = principal {x}"
@@ -450,24 +491,24 @@
   unfolding at_within_def nhds_discrete by simp
 
 lemma at_within_eq: "at x within s = (INF S:{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
-  unfolding nhds_def at_within_def by (subst INF_inf_const2[symmetric]) (auto simp add: Diff_Int_distrib)
+  unfolding nhds_def at_within_def
+  by (subst INF_inf_const2[symmetric]) (auto simp: Diff_Int_distrib)
 
 lemma eventually_at_filter:
   "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
-  unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
+  by (simp add: at_within_def eventually_inf_principal imp_conjL[symmetric] conj_commute)
 
 lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
   unfolding at_within_def by (intro inf_mono) auto
 
 lemma eventually_at_topological:
   "eventually P (at a within s) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
-  unfolding eventually_nhds eventually_at_filter by simp
+  by (simp add: eventually_nhds eventually_at_filter)
 
 lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
   unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
 
-lemma at_within_open_NO_MATCH:
-  "a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a"
+lemma at_within_open_NO_MATCH: "a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a"
   by (simp only: at_within_open)
 
 lemma at_within_nhd:
@@ -490,61 +531,70 @@
 
 lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
   unfolding trivial_limit_def eventually_at_topological
-  by (safe, case_tac "S = {a}", simp, fast, fast)
-
-lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
+  apply safe
+   apply (case_tac "S = {a}")
+    apply simp
+   apply fast
+  apply fast
+  done
+
+lemma at_neq_bot [simp]: "at a \<noteq> bot"
+  for a :: "'a::perfect_space"
   by (simp add: at_eq_bot_iff not_open_singleton)
 
-lemma (in order_topology) nhds_order: "nhds x =
-  inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})"
+lemma (in order_topology) nhds_order:
+  "nhds x = inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})"
 proof -
   have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} =
       (\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}"
     by auto
   show ?thesis
-    unfolding nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def ..
+    by (simp only: nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def)
 qed
 
 lemma filterlim_at_within_If:
   assumes "filterlim f G (at x within (A \<inter> {x. P x}))"
-  assumes "filterlim g G (at x within (A \<inter> {x. \<not>P x}))"
-  shows   "filterlim (\<lambda>x. if P x then f x else g x) G (at x within A)"
+    and "filterlim g G (at x within (A \<inter> {x. \<not>P x}))"
+  shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x within A)"
 proof (rule filterlim_If)
   note assms(1)
   also have "at x within (A \<inter> {x. P x}) = inf (nhds x) (principal (A \<inter> Collect P - {x}))"
     by (simp add: at_within_def)
-  also have "A \<inter> Collect P - {x} = (A - {x}) \<inter> Collect P" by blast
+  also have "A \<inter> Collect P - {x} = (A - {x}) \<inter> Collect P"
+    by blast
   also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal (Collect P))"
     by (simp add: at_within_def inf_assoc)
   finally show "filterlim f G (inf (at x within A) (principal (Collect P)))" .
 next
   note assms(2)
-  also have "at x within (A \<inter> {x. \<not>P x}) = inf (nhds x) (principal (A \<inter> {x. \<not>P x} - {x}))"
+  also have "at x within (A \<inter> {x. \<not> P x}) = inf (nhds x) (principal (A \<inter> {x. \<not> P x} - {x}))"
     by (simp add: at_within_def)
-  also have "A \<inter> {x. \<not>P x} - {x} = (A - {x}) \<inter> {x. \<not>P x}" by blast
-  also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal {x. \<not>P x})"
+  also have "A \<inter> {x. \<not> P x} - {x} = (A - {x}) \<inter> {x. \<not> P x}"
+    by blast
+  also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal {x. \<not> P x})"
     by (simp add: at_within_def inf_assoc)
-  finally show "filterlim g G (inf (at x within A) (principal {x. \<not>P x}))" .
+  finally show "filterlim g G (inf (at x within A) (principal {x. \<not> P x}))" .
 qed
 
 lemma filterlim_at_If:
   assumes "filterlim f G (at x within {x. P x})"
-  assumes "filterlim g G (at x within {x. \<not>P x})"
-  shows   "filterlim (\<lambda>x. if P x then f x else g x) G (at x)"
+    and "filterlim g G (at x within {x. \<not>P x})"
+  shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x)"
   using assms by (intro filterlim_at_within_If) simp_all
 
-lemma (in linorder_topology) at_within_order: "UNIV \<noteq> {x} \<Longrightarrow>
-  at x within s = inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x}))
-                      (INF a:{..< x}. principal ({a <..} \<inter> s - {x}))"
-proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split[case_product case_split])
-  assume "UNIV \<noteq> {x}" "{x<..} = {}" "{..< x} = {}"
-  moreover have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
+lemma (in linorder_topology) at_within_order:
+  assumes "UNIV \<noteq> {x}"
+  shows "at x within s =
+    inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x}))
+        (INF a:{..< x}. principal ({a <..} \<inter> s - {x}))"
+proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split [case_product case_split])
+  case True_True
+  have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
     by auto
-  ultimately show ?thesis
+  with assms True_True show ?thesis
     by auto
-qed (auto simp: at_within_def nhds_order Int_Diff inf_principal[symmetric] INF_inf_const2
-                inf_sup_aci[where 'a="'a filter"]
-          simp del: inf_principal)
+qed (auto simp del: inf_principal simp: at_within_def nhds_order Int_Diff
+      inf_principal[symmetric] INF_inf_const2 inf_sup_aci[where 'a="'a filter"])
 
 lemma (in linorder_topology) at_left_eq:
   "y < x \<Longrightarrow> at_left x = (INF a:{..< x}. principal {a <..< x})"
@@ -554,7 +604,8 @@
 
 lemma (in linorder_topology) eventually_at_left:
   "y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
-  unfolding at_left_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
+  unfolding at_left_eq
+  by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
 
 lemma (in linorder_topology) at_right_eq:
   "x < y \<Longrightarrow> at_right x = (INF a:{x <..}. principal {x <..< a})"
@@ -564,43 +615,47 @@
 
 lemma (in linorder_topology) eventually_at_right:
   "x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
-  unfolding at_right_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
+  unfolding at_right_eq
+  by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
 
 lemma eventually_at_right_less: "\<forall>\<^sub>F y in at_right (x::'a::{linorder_topology, no_top}). x < y"
   using gt_ex[of x] eventually_at_right[of x] by auto
 
-lemma trivial_limit_at_right_top: "at_right (top::_::{order_top, linorder_topology}) = bot"
-  unfolding filter_eq_iff eventually_at_topological by auto
-
-lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot, linorder_topology}) = bot"
-  unfolding filter_eq_iff eventually_at_topological by auto
-
-lemma trivial_limit_at_left_real [simp]:
-  "\<not> trivial_limit (at_left (x::'a::{no_bot, dense_order, linorder_topology}))"
-  using lt_ex[of x]
+lemma trivial_limit_at_right_top: "at_right (top::_::{order_top,linorder_topology}) = bot"
+  by (auto simp: filter_eq_iff eventually_at_topological)
+
+lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot,linorder_topology}) = bot"
+  by (auto simp: filter_eq_iff eventually_at_topological)
+
+lemma trivial_limit_at_left_real [simp]: "\<not> trivial_limit (at_left x)"
+  for x :: "'a::{no_bot,dense_order,linorder_topology}"
+  using lt_ex [of x]
   by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)
 
-lemma trivial_limit_at_right_real [simp]:
-  "\<not> trivial_limit (at_right (x::'a::{no_top, dense_order, linorder_topology}))"
+lemma trivial_limit_at_right_real [simp]: "\<not> trivial_limit (at_right x)"
+  for x :: "'a::{no_top,dense_order,linorder_topology}"
   using gt_ex[of x]
   by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense)
 
-lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
+lemma at_eq_sup_left_right: "at x = sup (at_left x) (at_right x)"
+  for x :: "'a::linorder_topology"
   by (auto simp: eventually_at_filter filter_eq_iff eventually_sup
-           elim: eventually_elim2 eventually_mono)
+      elim: eventually_elim2 eventually_mono)
 
 lemma eventually_at_split:
-  "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
+  "eventually P (at x) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
+  for x :: "'a::linorder_topology"
   by (subst at_eq_sup_left_right) (simp add: eventually_sup)
 
+
 subsubsection \<open>Tendsto\<close>
 
 abbreviation (in topological_space)
-  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "\<longlongrightarrow>" 55) where
-  "(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F"
-
-definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
-  "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
+  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool"  (infixr "\<longlongrightarrow>" 55)
+  where "(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F"
+
+definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a"
+  where "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
 
 lemma tendsto_eq_rhs: "(f \<longlongrightarrow> x) F \<Longrightarrow> x = y \<Longrightarrow> (f \<longlongrightarrow> y) F"
   by simp
@@ -617,11 +672,8 @@
    "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
    unfolding nhds_def filterlim_INF filterlim_principal by auto
 
-lemma tendsto_cong:
-  assumes "eventually (\<lambda>x. f x = g x) F"
-  shows   "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F"
-  by (rule filterlim_cong[OF refl refl assms])
-
+lemma tendsto_cong: "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F" if "eventually (\<lambda>x. f x = g x) F"
+  by (rule filterlim_cong [OF refl refl that])
 
 lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f \<longlongrightarrow> l) F' \<Longrightarrow> (f \<longlongrightarrow> l) F"
   unfolding tendsto_def le_filter_def by fast
@@ -630,46 +682,48 @@
   by (blast intro: tendsto_mono at_le)
 
 lemma filterlim_at:
-  "(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f \<longlongrightarrow> b) F)"
+  "(LIM x F. f x :> at b within s) \<longleftrightarrow> eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f \<longlongrightarrow> b) F"
   by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
 
 lemma (in topological_space) topological_tendstoI:
   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
-  unfolding tendsto_def by auto
+  by (auto simp: tendsto_def)
 
 lemma (in topological_space) topological_tendstoD:
   "(f \<longlongrightarrow> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
-  unfolding tendsto_def by auto
+  by (auto simp: tendsto_def)
 
 lemma (in order_topology) order_tendsto_iff:
   "(f \<longlongrightarrow> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"
-  unfolding nhds_order filterlim_inf filterlim_INF filterlim_principal by auto
+  by (auto simp: nhds_order filterlim_inf filterlim_INF filterlim_principal)
 
 lemma (in order_topology) order_tendstoI:
   "(\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F) \<Longrightarrow> (\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F) \<Longrightarrow>
     (f \<longlongrightarrow> y) F"
-  unfolding order_tendsto_iff by auto
+  by (auto simp: order_tendsto_iff)
 
 lemma (in order_topology) order_tendstoD:
   assumes "(f \<longlongrightarrow> y) F"
   shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
     and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
-  using assms unfolding order_tendsto_iff by auto
+  using assms by (auto simp: order_tendsto_iff)
 
 lemma tendsto_bot [simp]: "(f \<longlongrightarrow> a) bot"
-  unfolding tendsto_def by simp
+  by (simp add: tendsto_def)
 
 lemma (in linorder_topology) tendsto_max:
   assumes X: "(X \<longlongrightarrow> x) net"
-  assumes Y: "(Y \<longlongrightarrow> y) net"
+    and Y: "(Y \<longlongrightarrow> y) net"
   shows "((\<lambda>x. max (X x) (Y x)) \<longlongrightarrow> max x y) net"
 proof (rule order_tendstoI)
-  fix a assume "a < max x y"
+  fix a
+  assume "a < max x y"
   then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
     by (auto simp: less_max_iff_disj elim: eventually_mono)
 next
-  fix a assume "max x y < a"
+  fix a
+  assume "max x y < a"
   then show "eventually (\<lambda>x. max (X x) (Y x) < a) net"
     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
     by (auto simp: eventually_conj_iff)
@@ -677,28 +731,32 @@
 
 lemma (in linorder_topology) tendsto_min:
   assumes X: "(X \<longlongrightarrow> x) net"
-  assumes Y: "(Y \<longlongrightarrow> y) net"
+    and Y: "(Y \<longlongrightarrow> y) net"
   shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net"
 proof (rule order_tendstoI)
-  fix a assume "a < min x y"
+  fix a
+  assume "a < min x y"
   then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
     by (auto simp: eventually_conj_iff)
 next
-  fix a assume "min x y < a"
+  fix a
+  assume "min x y < a"
   then show "eventually (\<lambda>x. min (X x) (Y x) < a) net"
     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
     by (auto simp: min_less_iff_disj elim: eventually_mono)
 qed
 
 lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)"
-  unfolding tendsto_def eventually_at_topological by auto
+  by (auto simp: tendsto_def eventually_at_topological)
 
 lemma (in topological_space) tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) \<longlongrightarrow> k) F"
   by (simp add: tendsto_def)
 
 lemma (in t2_space) tendsto_unique:
-  assumes "F \<noteq> bot" and "(f \<longlongrightarrow> a) F" and "(f \<longlongrightarrow> b) F"
+  assumes "F \<noteq> bot"
+    and "(f \<longlongrightarrow> a) F"
+    and "(f \<longlongrightarrow> b) F"
   shows "a = b"
 proof (rule ccontr)
   assume "a \<noteq> b"
@@ -713,7 +771,7 @@
   have "eventually (\<lambda>x. False) F"
   proof eventually_elim
     case (elim x)
-    hence "f x \<in> U \<inter> V" by simp
+    then have "f x \<in> U \<inter> V" by simp
     with \<open>U \<inter> V = {}\<close> show ?case by simp
   qed
   with \<open>\<not> trivial_limit F\<close> show "False"
@@ -721,20 +779,22 @@
 qed
 
 lemma (in t2_space) tendsto_const_iff:
-  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b"
+  fixes a b :: 'a
+  assumes "\<not> trivial_limit F"
+  shows "((\<lambda>x. a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b"
   by (auto intro!: tendsto_unique [OF assms tendsto_const])
 
 lemma increasing_tendsto:
   fixes f :: "_ \<Rightarrow> 'a::order_topology"
   assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
-      and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
+    and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
   shows "(f \<longlongrightarrow> l) F"
   using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
 
 lemma decreasing_tendsto:
   fixes f :: "_ \<Rightarrow> 'a::order_topology"
   assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
-      and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
+    and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
   shows "(f \<longlongrightarrow> l) F"
   using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
 
@@ -744,40 +804,51 @@
   assumes lim: "(f \<longlongrightarrow> c) net" "(h \<longlongrightarrow> c) net"
   shows "(g \<longlongrightarrow> c) net"
 proof (rule order_tendstoI)
-  fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
+  fix a
+  show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
     using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
 next
-  fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
+  fix a
+  show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
     using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
 qed
 
 lemma limit_frequently_eq:
+  fixes c d :: "'a::t1_space"
   assumes "F \<noteq> bot"
-  assumes "frequently (\<lambda>x. f x = c) F"
-  assumes "(f \<longlongrightarrow> d) F"
-  shows   "d = (c :: 'a :: t1_space)"
+    and "frequently (\<lambda>x. f x = c) F"
+    and "(f \<longlongrightarrow> d) F"
+  shows "d = c"
 proof (rule ccontr)
   assume "d \<noteq> c"
-  from t1_space[OF this] obtain U where "open U" "d \<in> U" "c \<notin> U" by blast
-  from this assms have "eventually (\<lambda>x. f x \<in> U) F" unfolding tendsto_def by blast
-  hence "eventually (\<lambda>x. f x \<noteq> c) F" by eventually_elim (insert \<open>c \<notin> U\<close>, blast)
-  with assms(2) show False unfolding frequently_def by contradiction
+  from t1_space[OF this] obtain U where "open U" "d \<in> U" "c \<notin> U"
+    by blast
+  with assms have "eventually (\<lambda>x. f x \<in> U) F"
+    unfolding tendsto_def by blast
+  then have "eventually (\<lambda>x. f x \<noteq> c) F"
+    by eventually_elim (insert \<open>c \<notin> U\<close>, blast)
+  with assms(2) show False
+    unfolding frequently_def by contradiction
 qed
 
 lemma tendsto_imp_eventually_ne:
-  assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> (c' :: 'a :: t1_space)"
-  shows   "eventually (\<lambda>z. f z \<noteq> c') F"
+  fixes c :: "'a::t1_space"
+  assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> c'"
+  shows "eventually (\<lambda>z. f z \<noteq> c') F"
 proof (rule ccontr)
-  assume "\<not>eventually (\<lambda>z. f z \<noteq> c') F"
-  hence "frequently (\<lambda>z. f z = c') F" by (simp add: frequently_def)
-  from limit_frequently_eq[OF assms(1) this assms(2)] and assms(3) show False by contradiction
+  assume "\<not> eventually (\<lambda>z. f z \<noteq> c') F"
+  then have "frequently (\<lambda>z. f z = c') F"
+    by (simp add: frequently_def)
+  from limit_frequently_eq[OF assms(1) this assms(2)] and assms(3) show False
+    by contradiction
 qed
 
 lemma tendsto_le:
   fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
   assumes F: "\<not> trivial_limit F"
-  assumes x: "(f \<longlongrightarrow> x) F" and y: "(g \<longlongrightarrow> y) F"
-  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
+    and x: "(f \<longlongrightarrow> x) F"
+    and y: "(g \<longlongrightarrow> y) F"
+    and ev: "eventually (\<lambda>x. g x \<le> f x) F"
   shows "y \<le> x"
 proof (rule ccontr)
   assume "\<not> y \<le> x"
@@ -794,23 +865,24 @@
 lemma tendsto_le_const:
   fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
   assumes F: "\<not> trivial_limit F"
-  assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"
+    and x: "(f \<longlongrightarrow> x) F"
+    and ev: "eventually (\<lambda>i. a \<le> f i) F"
   shows "a \<le> x"
-  using F x tendsto_const a by (rule tendsto_le)
+  using F x tendsto_const ev by (rule tendsto_le)
 
 lemma tendsto_ge_const:
   fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
   assumes F: "\<not> trivial_limit F"
-  assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"
+    and x: "(f \<longlongrightarrow> x) F"
+    and ev: "eventually (\<lambda>i. a \<ge> f i) F"
   shows "a \<ge> x"
-  by (rule tendsto_le [OF F tendsto_const x a])
+  by (rule tendsto_le [OF F tendsto_const x ev])
 
 
 subsubsection \<open>Rules about @{const Lim}\<close>
 
-lemma tendsto_Lim:
-  "\<not>(trivial_limit net) \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l"
-  unfolding Lim_def using tendsto_unique[of net f] by auto
+lemma tendsto_Lim: "\<not> trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l"
+  unfolding Lim_def using tendsto_unique [of net f] by auto
 
 lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
   by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
@@ -818,20 +890,23 @@
 lemma filterlim_at_bot_at_right:
   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
-  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
-  assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
-  assumes P: "eventually P at_bot"
+    and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
+    and Q: "eventually Q (at_right a)"
+    and bound: "\<And>b. Q b \<Longrightarrow> a < b"
+    and P: "eventually P at_bot"
   shows "filterlim f at_bot (at_right a)"
 proof -
   from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
     unfolding eventually_at_bot_linorder by auto
   show ?thesis
   proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
-    fix z assume "z \<le> x"
+    fix z
+    assume "z \<le> x"
     with x have "P z" by auto
     have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
       using bound[OF bij(2)[OF \<open>P z\<close>]]
-      unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]] by (auto intro!: exI[of _ "g z"])
+      unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]]
+      by (auto intro!: exI[of _ "g z"])
     with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
       by eventually_elim (metis bij \<open>P z\<close> mono)
   qed
@@ -840,46 +915,55 @@
 lemma filterlim_at_top_at_left:
   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
-  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
-  assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
-  assumes P: "eventually P at_top"
+    and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
+    and Q: "eventually Q (at_left a)"
+    and bound: "\<And>b. Q b \<Longrightarrow> b < a"
+    and P: "eventually P at_top"
   shows "filterlim f at_top (at_left a)"
 proof -
   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
     unfolding eventually_at_top_linorder by auto
   show ?thesis
   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
-    fix z assume "x \<le> z"
+    fix z
+    assume "x \<le> z"
     with x have "P z" by auto
     have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
       using bound[OF bij(2)[OF \<open>P z\<close>]]
-      unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]] by (auto intro!: exI[of _ "g z"])
+      unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]]
+      by (auto intro!: exI[of _ "g z"])
     with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
       by eventually_elim (metis bij \<open>P z\<close> mono)
   qed
 qed
 
 lemma filterlim_split_at:
-  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
+  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow>
+    filterlim f F (at x)"
+  for x :: "'a::linorder_topology"
   by (subst at_eq_sup_left_right) (rule filterlim_sup)
 
 lemma filterlim_at_split:
-  "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
+  "filterlim f F (at x) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
+  for x :: "'a::linorder_topology"
   by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
 
 lemma eventually_nhds_top:
-  fixes P :: "'a :: {order_top, linorder_topology} \<Rightarrow> bool"
-  assumes "(b::'a) < top"
+  fixes P :: "'a :: {order_top,linorder_topology} \<Rightarrow> bool"
+    and b :: 'a
+  assumes "b < top"
   shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))"
   unfolding eventually_nhds
 proof safe
-  fix S :: "'a set" assume "open S" "top \<in> S"
+  fix S :: "'a set"
+  assume "open S" "top \<in> S"
   note open_left[OF this \<open>b < top\<close>]
   moreover assume "\<forall>s\<in>S. P s"
   ultimately show "\<exists>b<top. \<forall>z>b. P z"
     by (auto simp: subset_eq Ball_def)
 next
-  fix b assume "b < top" "\<forall>z>b. P z"
+  fix b
+  assume "b < top" "\<forall>z>b. P z"
   then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)"
     by (intro exI[of _ "{b <..}"]) auto
 qed
@@ -889,55 +973,54 @@
   unfolding tendsto_def eventually_at_filter eventually_inf_principal
   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
 
+
 subsection \<open>Limits on sequences\<close>
 
 abbreviation (in topological_space)
-  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
-    ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60) where
-  "X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially"
-
-abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
-  "lim X \<equiv> Lim sequentially X"
-
-definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
-  "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
+  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"  ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60)
+  where "X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially"
+
+abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"
+  where "lim X \<equiv> Lim sequentially X"
+
+definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"
+  where "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
 
 lemma lim_def: "lim X = (THE L. X \<longlonglongrightarrow> L)"
   unfolding Lim_def ..
 
+
 subsubsection \<open>Monotone sequences and subsequences\<close>
 
-definition
-  monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
-    \<comment>\<open>Definition of monotonicity.
-        The use of disjunction here complicates proofs considerably.
-        One alternative is to add a Boolean argument to indicate the direction.
-        Another is to develop the notions of increasing and decreasing first.\<close>
-  "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
-
-abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
-  "incseq X \<equiv> mono X"
+text \<open>
+  Definition of monotonicity.
+  The use of disjunction here complicates proofs considerably.
+  One alternative is to add a Boolean argument to indicate the direction.
+  Another is to develop the notions of increasing and decreasing first.
+\<close>
+definition monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
+  where "monoseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
+
+abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
+  where "incseq X \<equiv> mono X"
 
 lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"
   unfolding mono_def ..
 
-abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
-  "decseq X \<equiv> antimono X"
+abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
+  where "decseq X \<equiv> antimono X"
 
 lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
   unfolding antimono_def ..
 
-definition
-  subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
-    \<comment>\<open>Definition of subsequence\<close>
-  "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
-
-lemma incseq_SucI:
-  "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
-  using lift_Suc_mono_le[of X]
-  by (auto simp: incseq_def)
-
-lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
+text \<open>Definition of subsequence.\<close>
+definition subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool"
+  where "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
+
+lemma incseq_SucI: "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
+  using lift_Suc_mono_le[of X] by (auto simp: incseq_def)
+
+lemma incseqD: "incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
   by (auto simp: incseq_def)
 
 lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
@@ -949,12 +1032,10 @@
 lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
   unfolding incseq_def by auto
 
-lemma decseq_SucI:
-  "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
-  using order.lift_Suc_mono_le[OF dual_order, of X]
-  by (auto simp: decseq_def)
-
-lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
+lemma decseq_SucI: "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
+  using order.lift_Suc_mono_le[OF dual_order, of X] by (auto simp: decseq_def)
+
+lemma decseqD: "decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
   by (auto simp: decseq_def)
 
 lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
@@ -969,85 +1050,91 @@
 lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
   unfolding monoseq_def incseq_def decseq_def ..
 
-lemma monoseq_Suc:
-  "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
+lemma monoseq_Suc: "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
   unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
 
-lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
-by (simp add: monoseq_def)
-
-lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
-by (simp add: monoseq_def)
-
-lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
-by (simp add: monoseq_Suc)
-
-lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
-by (simp add: monoseq_Suc)
+lemma monoI1: "\<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> monoseq X"
+  by (simp add: monoseq_def)
+
+lemma monoI2: "\<forall>m. \<forall>n \<ge> m. X n \<le> X m \<Longrightarrow> monoseq X"
+  by (simp add: monoseq_def)
+
+lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) \<Longrightarrow> monoseq X"
+  by (simp add: monoseq_Suc)
+
+lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n \<Longrightarrow> monoseq X"
+  by (simp add: monoseq_Suc)
 
 lemma monoseq_minus:
   fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
   assumes "monoseq a"
   shows "monoseq (\<lambda> n. - a n)"
-proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
+proof (cases "\<forall>m. \<forall>n \<ge> m. a m \<le> a n")
   case True
-  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
-  thus ?thesis by (rule monoI2)
+  then have "\<forall>m. \<forall>n \<ge> m. - a n \<le> - a m" by auto
+  then show ?thesis by (rule monoI2)
 next
   case False
-  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using \<open>monoseq a\<close>[unfolded monoseq_def] by auto
-  thus ?thesis by (rule monoI1)
+  then have "\<forall>m. \<forall>n \<ge> m. - a m \<le> - a n"
+    using \<open>monoseq a\<close>[unfolded monoseq_def] by auto
+  then show ?thesis by (rule monoI1)
 qed
 
-text\<open>Subsequence (alternative definition, (e.g. Hoskins)\<close>
-
-lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
-apply (simp add: subseq_def)
-apply (auto dest!: less_imp_Suc_add)
-apply (induct_tac k)
-apply (auto intro: less_trans)
-done
+
+text \<open>Subsequence (alternative definition, (e.g. Hoskins)\<close>
+
+lemma subseq_Suc_iff: "subseq f \<longleftrightarrow> (\<forall>n. f n < f (Suc n))"
+  apply (simp add: subseq_def)
+  apply (auto dest!: less_imp_Suc_add)
+  apply (induct_tac k)
+   apply (auto intro: less_trans)
+  done
 
 lemma subseq_add: "subseq (\<lambda>n. n + k)"
   by (auto simp: subseq_Suc_iff)
 
-text\<open>for any sequence, there is a monotonic subsequence\<close>
+text \<open>For any sequence, there is a monotonic subsequence.\<close>
 lemma seq_monosub:
-  fixes s :: "nat => 'a::linorder"
+  fixes s :: "nat \<Rightarrow> 'a::linorder"
   shows "\<exists>f. subseq f \<and> monoseq (\<lambda>n. (s (f n)))"
-proof cases
-  assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p"
+proof (cases "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p")
+  case True
   then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)"
     by (intro dependent_nat_choice) (auto simp: conj_commute)
-  then obtain f where "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
+  then obtain f where f: "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
     by (auto simp: subseq_Suc_iff)
-  moreover
   then have "incseq f"
     unfolding subseq_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
   then have "monoseq (\<lambda>n. s (f n))"
     by (auto simp add: incseq_def intro!: mono monoI2)
-  ultimately show ?thesis
+  with f show ?thesis
     by auto
 next
-  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
-  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
+  case False
+  then obtain N where N: "p > N \<Longrightarrow> \<exists>m>p. s p < s m" for p
+    by (force simp: not_le le_less)
   have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))"
   proof (intro dependent_nat_choice)
-    fix x assume "N < x" with N[of x] show "\<exists>y>N. x < y \<and> s x \<le> s y"
+    fix x
+    assume "N < x" with N[of x]
+    show "\<exists>y>N. x < y \<and> s x \<le> s y"
       by (auto intro: less_trans)
   qed auto
   then show ?thesis
     by (auto simp: monoseq_iff incseq_Suc_iff subseq_Suc_iff)
 qed
 
-lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
-proof(induct n)
-  case 0 thus ?case by simp
+lemma seq_suble:
+  assumes sf: "subseq f"
+  shows "n \<le> f n"
+proof (induct n)
+  case 0
+  show ?case by simp
 next
   case (Suc n)
-  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
-  have "n < f (Suc n)" by arith
-  thus ?case by arith
+  with sf [unfolded subseq_Suc_iff, rule_format, of n] have "n < f (Suc n)"
+     by arith
+  then show ?case by arith
 qed
 
 lemma eventually_subseq:
@@ -1055,10 +1142,10 @@
   unfolding eventually_sequentially by (metis seq_suble le_trans)
 
 lemma not_eventually_sequentiallyD:
-  assumes P: "\<not> eventually P sequentially"
+  assumes "\<not> eventually P sequentially"
   shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
 proof -
-  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
+  from assms have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
     unfolding eventually_sequentially by (simp add: not_less)
   then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
     by (auto simp: choice_iff)
@@ -1073,13 +1160,14 @@
 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
   unfolding subseq_def by simp
 
-lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
-  using assms by (auto simp: subseq_def)
+lemma subseq_mono: "subseq r \<Longrightarrow> m < n \<Longrightarrow> r m < r n"
+  by (auto simp: subseq_def)
 
 lemma subseq_imp_inj_on: "subseq g \<Longrightarrow> inj_on g A"
 proof (rule inj_onI)
   assume g: "subseq g"
-  fix x y assume "g x = g y"
+  fix x y
+  assume "g x = g y"
   with subseq_mono[OF g, of x y] subseq_mono[OF g, of y x] show "x = y"
     by (cases x y rule: linorder_cases) simp_all
 qed
@@ -1093,15 +1181,16 @@
 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
   by (simp add: decseq_def monoseq_def)
 
-lemma decseq_eq_incseq:
-  fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)"
+lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"
+  for X :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
   by (simp add: decseq_def incseq_def)
 
 lemma INT_decseq_offset:
   assumes "decseq F"
   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
 proof safe
-  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
+  fix x i
+  assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
   show "x \<in> F i"
   proof cases
     from x have "x \<in> F n" by auto
@@ -1111,99 +1200,97 @@
   qed (insert x, simp)
 qed auto
 
-lemma LIMSEQ_const_iff:
-  fixes k l :: "'a::t2_space"
-  shows "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l"
+lemma LIMSEQ_const_iff: "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l"
+  for k l :: "'a::t2_space"
   using trivial_limit_sequentially by (rule tendsto_const_iff)
 
-lemma LIMSEQ_SUP:
-  "incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
+lemma LIMSEQ_SUP: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a::{complete_linorder,linorder_topology})"
   by (intro increasing_tendsto)
-     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
-
-lemma LIMSEQ_INF:
-  "decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
+    (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
+
+lemma LIMSEQ_INF: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a::{complete_linorder,linorder_topology})"
   by (intro decreasing_tendsto)
-     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
-
-lemma LIMSEQ_ignore_initial_segment:
-  "f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
-  unfolding tendsto_def
-  by (subst eventually_sequentially_seg[where k=k])
-
-lemma LIMSEQ_offset:
-  "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a"
+    (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
+
+lemma LIMSEQ_ignore_initial_segment: "f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
+  unfolding tendsto_def by (subst eventually_sequentially_seg[where k=k])
+
+lemma LIMSEQ_offset: "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a"
   unfolding tendsto_def
   by (subst (asm) eventually_sequentially_seg[where k=k])
 
 lemma LIMSEQ_Suc: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l"
-by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
+  by (drule LIMSEQ_ignore_initial_segment [where k="Suc 0"]) simp
 
 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l \<Longrightarrow> f \<longlonglongrightarrow> l"
-by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
+  by (rule LIMSEQ_offset [where k="Suc 0"]) simp
 
 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l = f \<longlonglongrightarrow> l"
-by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
-
-lemma LIMSEQ_unique:
-  fixes a b :: "'a::t2_space"
-  shows "\<lbrakk>X \<longlonglongrightarrow> a; X \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> a = b"
+  by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
+
+lemma LIMSEQ_unique: "X \<longlonglongrightarrow> a \<Longrightarrow> X \<longlonglongrightarrow> b \<Longrightarrow> a = b"
+  for a b :: "'a::t2_space"
   using trivial_limit_sequentially by (rule tendsto_unique)
 
-lemma LIMSEQ_le_const:
-  "\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
+lemma LIMSEQ_le_const: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. a \<le> X n \<Longrightarrow> a \<le> x"
+  for a x :: "'a::linorder_topology"
   using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
 
-lemma LIMSEQ_le:
-  "\<lbrakk>X \<longlonglongrightarrow> x; Y \<longlonglongrightarrow> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
+lemma LIMSEQ_le: "X \<longlonglongrightarrow> x \<Longrightarrow> Y \<longlonglongrightarrow> y \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X n \<le> Y n \<Longrightarrow> x \<le> y"
+  for x y :: "'a::linorder_topology"
   using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
 
-lemma LIMSEQ_le_const2:
-  "\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
+lemma LIMSEQ_le_const2: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X n \<le> a \<Longrightarrow> x \<le> a"
+  for a x :: "'a::linorder_topology"
   by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto
 
-lemma convergentD: "convergent X ==> \<exists>L. (X \<longlonglongrightarrow> L)"
-by (simp add: convergent_def)
-
-lemma convergentI: "(X \<longlonglongrightarrow> L) ==> convergent X"
-by (auto simp add: convergent_def)
-
-lemma convergent_LIMSEQ_iff: "convergent X = (X \<longlonglongrightarrow> lim X)"
-by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
+lemma convergentD: "convergent X \<Longrightarrow> \<exists>L. X \<longlonglongrightarrow> L"
+  by (simp add: convergent_def)
+
+lemma convergentI: "X \<longlonglongrightarrow> L \<Longrightarrow> convergent X"
+  by (auto simp add: convergent_def)
+
+lemma convergent_LIMSEQ_iff: "convergent X \<longleftrightarrow> X \<longlonglongrightarrow> lim X"
+  by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
 
 lemma convergent_const: "convergent (\<lambda>n. c)"
-  by (rule convergentI, rule tendsto_const)
+  by (rule convergentI) (rule tendsto_const)
 
 lemma monoseq_le:
-  "monoseq a \<Longrightarrow> a \<longlonglongrightarrow> (x::'a::linorder_topology) \<Longrightarrow>
-    ((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
+  "monoseq a \<Longrightarrow> a \<longlonglongrightarrow> x \<Longrightarrow>
+    (\<forall>n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n) \<or>
+    (\<forall>n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)"
+  for x :: "'a::linorder_topology"
   by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
 
-lemma LIMSEQ_subseq_LIMSEQ:
-  "\<lbrakk> X \<longlonglongrightarrow> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) \<longlonglongrightarrow> L"
-  unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
-
-lemma convergent_subseq_convergent:
-  "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
-  unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
-
-lemma limI: "X \<longlonglongrightarrow> L ==> lim X = L"
+lemma LIMSEQ_subseq_LIMSEQ: "X \<longlonglongrightarrow> L \<Longrightarrow> subseq f \<Longrightarrow> (X \<circ> f) \<longlonglongrightarrow> L"
+  unfolding comp_def by (rule filterlim_compose [of X, OF _ filterlim_subseq])
+
+lemma convergent_subseq_convergent: "convergent X \<Longrightarrow> subseq f \<Longrightarrow> convergent (X \<circ> f)"
+  by (auto simp: convergent_def intro: LIMSEQ_subseq_LIMSEQ)
+
+lemma limI: "X \<longlonglongrightarrow> L \<Longrightarrow> lim X = L"
   by (rule tendsto_Lim) (rule trivial_limit_sequentially)
 
-lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
+lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> x) \<Longrightarrow> lim f \<le> x"
+  for x :: "'a::linorder_topology"
   using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
 
 lemma lim_const [simp]: "lim (\<lambda>m. a) = a"
   by (simp add: limI)
 
-subsubsection\<open>Increasing and Decreasing Series\<close>
-
-lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
+
+subsubsection \<open>Increasing and Decreasing Series\<close>
+
+lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> L"
+  for L :: "'a::linorder_topology"
   by (metis incseq_def LIMSEQ_le_const)
 
-lemma decseq_le: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
+lemma decseq_le: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> L \<le> X n"
+  for L :: "'a::linorder_topology"
   by (metis decseq_def LIMSEQ_le_const2)
 
+
 subsection \<open>First countable topologies\<close>
 
 class first_countable_topology = topological_space +
@@ -1215,19 +1302,24 @@
     "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
 proof atomize_elim
-  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
-    nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
-    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
+  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set"
+    where nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
+      and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
+    by auto
   define F where "F n = (\<Inter>i\<le>n. A i)" for n
   show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
-      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
+    (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
   proof (safe intro!: exI[of _ F])
     fix i
-    show "open (F i)" using nhds(1) by (auto simp: F_def)
-    show "x \<in> F i" using nhds(2) by (auto simp: F_def)
+    show "open (F i)"
+      using nhds(1) by (auto simp: F_def)
+    show "x \<in> F i"
+      using nhds(2) by (auto simp: F_def)
   next
-    fix S assume "open S" "x \<in> S"
-    from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
+    fix S
+    assume "open S" "x \<in> S"
+    from incl[OF this] obtain i where "F i \<subseteq> S"
+      unfolding F_def by auto
     moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
       by (simp add: Inf_superset_mono F_def image_mono)
     ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
@@ -1240,22 +1332,23 @@
   where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))"
 proof -
   from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set"
-    where A: "\<And>n. x \<in> A n" "\<And>n. open (A n)" "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
+    where *: "\<And>n. x \<in> A n" "\<And>n. open (A n)" "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
     by metis
   show thesis
   proof
     show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)"
       by (simp add: antimono_iff_le_Suc atMost_Suc)
-    show "\<And>n. x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)"
-      using A by auto
+    show "x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)" for n
+      using * by auto
     show "nhds x = (INF n. principal (\<Inter>i\<le>n. A i))"
-      using A unfolding nhds_def
+      using *
+      unfolding nhds_def
       apply -
       apply (rule INF_eq)
-      apply simp_all
-      apply fastforce
+       apply simp_all
+       apply fastforce
       apply (intro exI [of _ "\<Inter>i\<le>n. A i" for n] conjI open_INT)
-      apply auto
+         apply auto
       done
   qed
 qed
@@ -1265,17 +1358,15 @@
     "\<And>i. open (A i)" "\<And>i. x \<in> A i"
     "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x"
 proof atomize_elim
-  obtain A :: "nat \<Rightarrow> 'a set" where A:
+  obtain A :: "nat \<Rightarrow> 'a set" where *:
     "\<And>i. open (A i)"
     "\<And>i. x \<in> A i"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by (rule countable_basis_at_decseq) blast
-  {
-    fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
-    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
-      by (auto elim: eventually_mono simp: subset_eq)
-  }
-  with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F \<longlonglongrightarrow> x)"
+  have "eventually (\<lambda>n. F n \<in> S) sequentially"
+    if "\<forall>n. F n \<in> A n" "open S" "x \<in> S" for F S
+    using *(3)[of S] that by (auto elim: eventually_mono simp: subset_eq)
+  with * show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F \<longlonglongrightarrow> x)"
     by (intro exI[of _ A]) (auto simp: tendsto_def)
 qed
 
@@ -1283,20 +1374,23 @@
   assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
   shows "eventually P (inf (nhds a) (principal s))"
 proof (rule ccontr)
-  obtain A :: "nat \<Rightarrow> 'a set" where A:
+  obtain A :: "nat \<Rightarrow> 'a set" where *:
     "\<And>i. open (A i)"
     "\<And>i. a \<in> A i"
     "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> a"
     by (rule countable_basis) blast
   assume "\<not> ?thesis"
-  with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
-    unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
-  then obtain F where F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
+  with * have "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
+    unfolding eventually_inf_principal eventually_nhds
+    by (intro choice) fastforce
+  then obtain F where F: "\<forall>n. F n \<in> s" and "\<forall>n. F n \<in> A n" and F': "\<forall>n. \<not> P (F n)"
     by blast
-  with A have "F \<longlonglongrightarrow> a" by auto
-  hence "eventually (\<lambda>n. P (F n)) sequentially"
-    using assms F0 by simp
-  thus "False" by (simp add: F3)
+  with * have "F \<longlonglongrightarrow> a"
+    by auto
+  then have "eventually (\<lambda>n. P (F n)) sequentially"
+    using assms F by simp
+  then show False
+    by (simp add: F')
 qed
 
 lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
@@ -1306,7 +1400,9 @@
   assume "eventually P (inf (nhds a) (principal s))"
   then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
     by (auto simp: eventually_inf_principal eventually_nhds)
-  moreover fix f assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a"
+  moreover
+  fix f
+  assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a"
   ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
     by (auto dest!: topological_tendstoD elim: eventually_mono)
 qed
@@ -1316,49 +1412,45 @@
   using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
 
 lemma tendsto_at_iff_sequentially:
-  fixes f :: "'a :: first_countable_topology \<Rightarrow> _"
-  shows "(f \<longlongrightarrow> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))"
-  unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def
+  "(f \<longlongrightarrow> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))"
+  for f :: "'a::first_countable_topology \<Rightarrow> _"
+  unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap
+    at_within_def eventually_nhds_within_iff_sequentially comp_def
   by metis
 
+
 subsection \<open>Function limit at a point\<close>
 
-abbreviation
-  LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
-        ("((_)/ \<midarrow>(_)/\<rightarrow> (_))" [60, 0, 60] 60) where
-  "f \<midarrow>a\<rightarrow> L \<equiv> (f \<longlongrightarrow> L) (at a)"
+abbreviation LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
+    ("((_)/ \<midarrow>(_)/\<rightarrow> (_))" [60, 0, 60] 60)
+  where "f \<midarrow>a\<rightarrow> L \<equiv> (f \<longlongrightarrow> L) (at a)"
 
 lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<midarrow>a\<rightarrow> l)"
-  unfolding tendsto_def by (simp add: at_within_open[where S=S])
+  by (simp add: tendsto_def at_within_open[where S = S])
 
 lemma tendsto_within_open_NO_MATCH:
-  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
-  shows "a \<in> S \<Longrightarrow> NO_MATCH UNIV S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l)(at a)"
-using tendsto_within_open by blast
-
-lemma LIM_const_not_eq[tendsto_intros]:
-  fixes a :: "'a::perfect_space"
-  fixes k L :: "'b::t2_space"
-  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L"
+  "a \<in> S \<Longrightarrow> NO_MATCH UNIV S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l)(at a)"
+  for f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
+  using tendsto_within_open by blast
+
+lemma LIM_const_not_eq[tendsto_intros]: "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L"
+  for a :: "'a::perfect_space" and k L :: "'b::t2_space"
   by (simp add: tendsto_const_iff)
 
 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
 
-lemma LIM_const_eq:
-  fixes a :: "'a::perfect_space"
-  fixes k L :: "'b::t2_space"
-  shows "(\<lambda>x. k) \<midarrow>a\<rightarrow> L \<Longrightarrow> k = L"
+lemma LIM_const_eq: "(\<lambda>x. k) \<midarrow>a\<rightarrow> L \<Longrightarrow> k = L"
+  for a :: "'a::perfect_space" and k L :: "'b::t2_space"
   by (simp add: tendsto_const_iff)
 
-lemma LIM_unique:
-  fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"
-  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> M \<Longrightarrow> L = M"
+lemma LIM_unique: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> M \<Longrightarrow> L = M"
+  for a :: "'a::perfect_space" and L M :: "'b::t2_space"
   using at_neq_bot by (rule tendsto_unique)
 
-text \<open>Limits are equal for functions equal except at limit point\<close>
-
-lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>a\<rightarrow> l)"
-  unfolding tendsto_def eventually_at_topological by simp
+
+text \<open>Limits are equal for functions equal except at limit point.\<close>
+lemma LIM_equal: "\<forall>x. x \<noteq> a \<longrightarrow> f x = g x \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>a\<rightarrow> l)"
+  by (simp add: tendsto_def eventually_at_topological)
 
 lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>b\<rightarrow> m)"
   by (simp add: LIM_equal)
@@ -1366,16 +1458,14 @@
 lemma LIM_cong_limit: "f \<midarrow>x\<rightarrow> L \<Longrightarrow> K = L \<Longrightarrow> f \<midarrow>x\<rightarrow> K"
   by simp
 
-lemma tendsto_at_iff_tendsto_nhds:
-  "g \<midarrow>l\<rightarrow> g l \<longleftrightarrow> (g \<longlongrightarrow> g l) (nhds l)"
+lemma tendsto_at_iff_tendsto_nhds: "g \<midarrow>l\<rightarrow> g l \<longleftrightarrow> (g \<longlongrightarrow> g l) (nhds l)"
   unfolding tendsto_def eventually_at_filter
   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
 
-lemma tendsto_compose:
-  "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
+lemma tendsto_compose: "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
   unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
 
-lemma LIM_o: "\<lbrakk>g \<midarrow>l\<rightarrow> g l; f \<midarrow>a\<rightarrow> l\<rbrakk> \<Longrightarrow> (g \<circ> f) \<midarrow>a\<rightarrow> g l"
+lemma LIM_o: "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> f \<midarrow>a\<rightarrow> l \<Longrightarrow> (g \<circ> f) \<midarrow>a\<rightarrow> g l"
   unfolding o_def by (rule tendsto_compose)
 
 lemma tendsto_compose_eventually:
@@ -1383,16 +1473,17 @@
   by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
 
 lemma LIM_compose_eventually:
-  assumes f: "f \<midarrow>a\<rightarrow> b"
-  assumes g: "g \<midarrow>b\<rightarrow> c"
-  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
+  assumes "f \<midarrow>a\<rightarrow> b"
+    and "g \<midarrow>b\<rightarrow> c"
+    and "eventually (\<lambda>x. f x \<noteq> b) (at a)"
   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
-  using g f inj by (rule tendsto_compose_eventually)
+  using assms(2,1,3) by (rule tendsto_compose_eventually)
 
 lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)"
   by (simp add: filterlim_def filtermap_filtermap comp_def)
 
-subsubsection \<open>Relation of LIM and LIMSEQ\<close>
+
+subsubsection \<open>Relation of \<open>LIM\<close> and \<open>LIMSEQ\<close>\<close>
 
 lemma (in first_countable_topology) sequentially_imp_eventually_within:
   "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
@@ -1416,22 +1507,24 @@
   shows "f \<midarrow>a\<rightarrow> l"
   using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
 
-lemma LIMSEQ_SEQ_conv:
-  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L) =
-   (X \<midarrow>a\<rightarrow> (L::'b::topological_space))"
+lemma LIMSEQ_SEQ_conv: "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L) \<longleftrightarrow> X \<midarrow>a\<rightarrow> L"
+  for a :: "'a::first_countable_topology" and L :: "'b::topological_space"
   using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
 
 lemma sequentially_imp_eventually_at_left:
-  fixes a :: "'a :: {linorder_topology, first_countable_topology}"
+  fixes a :: "'a::{linorder_topology,first_countable_topology}"
   assumes b[simp]: "b < a"
-  assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
+    and *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow>
+      eventually (\<lambda>n. P (f n)) sequentially"
   shows "eventually P (at_left a)"
 proof (safe intro!: sequentially_imp_eventually_within)
-  fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
+  fix X
+  assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
   show "eventually (\<lambda>n. P (X n)) sequentially"
   proof (rule ccontr)
-    assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
+    assume neg: "\<not> ?thesis"
     have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> b < X (s n)) \<and> (X (s n) \<le> X (s (Suc n)) \<and> Suc (s n) \<le> s (Suc n))"
+      (is "\<exists>s. ?P s")
     proof (rule dependent_nat_choice)
       have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially"
         by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
@@ -1440,36 +1533,47 @@
     next
       fix x n
       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially"
-        using X by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
+        using X
+        by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
       then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)"
         by (auto dest!: not_eventuallyD)
     qed
-    then guess s ..
-    then have "\<And>n. b < X (s n)" "\<And>n. X (s n) < a" "incseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))"
-      using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
-    from *[OF this(1,2,3,4)] this(5) show False by auto
+    then obtain s where "?P s" ..
+    with X have "b < X (s n)"
+      and "X (s n) < a"
+      and "incseq (\<lambda>n. X (s n))"
+      and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a"
+      and "\<not> P (X (s n))"
+      for n
+      by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff
+          intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
+    from *[OF this(1,2,3,4)] this(5) show False
+      by auto
   qed
 qed
 
 lemma tendsto_at_left_sequentially:
-  fixes a :: "_ :: {linorder_topology, first_countable_topology}"
+  fixes a b :: "'b::{linorder_topology,first_countable_topology}"
   assumes "b < a"
-  assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
+  assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow>
+    (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
   shows "(X \<longlongrightarrow> L) (at_left a)"
-  using assms unfolding tendsto_def [where l=L]
-  by (simp add: sequentially_imp_eventually_at_left)
+  using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_left)
 
 lemma sequentially_imp_eventually_at_right:
-  fixes a :: "'a :: {linorder_topology, first_countable_topology}"
+  fixes a b :: "'a::{linorder_topology,first_countable_topology}"
   assumes b[simp]: "a < b"
-  assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
+  assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow>
+    eventually (\<lambda>n. P (f n)) sequentially"
   shows "eventually P (at_right a)"
 proof (safe intro!: sequentially_imp_eventually_within)
-  fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
+  fix X
+  assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
   show "eventually (\<lambda>n. P (X n)) sequentially"
   proof (rule ccontr)
-    assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
+    assume neg: "\<not> ?thesis"
     have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> X (s n) < b) \<and> (X (s (Suc n)) \<le> X (s n) \<and> Suc (s n) \<le> s (Suc n))"
+      (is "\<exists>s. ?P s")
     proof (rule dependent_nat_choice)
       have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially"
         by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
@@ -1478,35 +1582,45 @@
     next
       fix x n
       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially"
-        using X by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
+        using X
+        by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
       then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)"
         by (auto dest!: not_eventuallyD)
     qed
-    then guess s ..
-    then have "\<And>n. a < X (s n)" "\<And>n. X (s n) < b" "decseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))"
-      using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
-    from *[OF this(1,2,3,4)] this(5) show False by auto
+    then obtain s where "?P s" ..
+    with X have "a < X (s n)"
+      and "X (s n) < b"
+      and "decseq (\<lambda>n. X (s n))"
+      and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a"
+      and "\<not> P (X (s n))"
+      for n
+      by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff
+          intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
+    from *[OF this(1,2,3,4)] this(5) show False
+      by auto
   qed
 qed
 
 lemma tendsto_at_right_sequentially:
   fixes a :: "_ :: {linorder_topology, first_countable_topology}"
   assumes "a < b"
-  assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
+    and *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow>
+      (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
   shows "(X \<longlongrightarrow> L) (at_right a)"
-  using assms unfolding tendsto_def [where l=L]
-  by (simp add: sequentially_imp_eventually_at_right)
+  using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_right)
+
 
 subsection \<open>Continuity\<close>
 
 subsubsection \<open>Continuity on a set\<close>
 
-definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
-  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
+definition continuous_on :: "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
+  where "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
 
 lemma continuous_on_cong [cong]:
   "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
-  unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
+  unfolding continuous_on_def
+  by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
 
 lemma continuous_on_topological:
   "continuous_on s f \<longleftrightarrow>
@@ -1516,7 +1630,8 @@
 lemma continuous_on_open_invariant:
   "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
 proof safe
-  fix B :: "'b set" assume "continuous_on s f" "open B"
+  fix B :: "'b set"
+  assume "continuous_on s f" "open B"
   then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)"
     by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
   then obtain A where "\<forall>x\<in>f -` B \<inter> s. open (A x) \<and> x \<in> A x \<and> s \<inter> A x \<subseteq> f -` B"
@@ -1528,8 +1643,10 @@
   show "continuous_on s f"
     unfolding continuous_on_topological
   proof safe
-    fix x B assume "x \<in> s" "open B" "f x \<in> B"
-    with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto
+    fix x B
+    assume "x \<in> s" "open B" "f x \<in> B"
+    with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s"
+      by auto
     with \<open>x \<in> s\<close> \<open>f x \<in> B\<close> show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
       by (intro exI[of _ A]) auto
   qed
@@ -1542,22 +1659,24 @@
 
 corollary continuous_imp_open_vimage:
   assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"
-    shows "open (f -` B)"
-by (metis assms continuous_on_open_vimage le_iff_inf)
+  shows "open (f -` B)"
+  by (metis assms continuous_on_open_vimage le_iff_inf)
 
 corollary open_vimage[continuous_intros]:
-  assumes "open s" and "continuous_on UNIV f"
+  assumes "open s"
+    and "continuous_on UNIV f"
   shows "open (f -` s)"
-  using assms unfolding continuous_on_open_vimage [OF open_UNIV]
-  by simp
+  using assms by (simp add: continuous_on_open_vimage [OF open_UNIV])
 
 lemma continuous_on_closed_invariant:
   "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
 proof -
-  have *: "\<And>P Q::'b set\<Rightarrow>bool. (\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"
+  have *: "(\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"
+    for P Q :: "'b set \<Rightarrow> bool"
     by (metis double_compl)
   show ?thesis
-    unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])
+    unfolding continuous_on_open_invariant
+    by (intro *) (auto simp: open_closed[symmetric])
 qed
 
 lemma continuous_on_closed_vimage:
@@ -1566,12 +1685,15 @@
   by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
 
 corollary closed_vimage_Int[continuous_intros]:
-  assumes "closed s" and "continuous_on t f" and t: "closed t"
+  assumes "closed s"
+    and "continuous_on t f"
+    and t: "closed t"
   shows "closed (f -` s \<inter> t)"
-  using assms unfolding continuous_on_closed_vimage [OF t]  by simp
+  using assms by (simp add: continuous_on_closed_vimage [OF t])
 
 corollary closed_vimage[continuous_intros]:
-  assumes "closed s" and "continuous_on UNIV f"
+  assumes "closed s"
+    and "continuous_on UNIV f"
   shows "closed (f -` s)"
   using closed_vimage_Int [OF assms] by simp
 
@@ -1583,10 +1705,12 @@
 
 lemma continuous_on_open_Union:
   "(\<And>s. s \<in> S \<Longrightarrow> open s) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on s f) \<Longrightarrow> continuous_on (\<Union>S) f"
-  unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)
+  unfolding continuous_on_def
+  by safe (metis open_Union at_within_open UnionI)
 
 lemma continuous_on_open_UN:
-  "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow> continuous_on (\<Union>s\<in>S. A s) f"
+  "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow>
+    continuous_on (\<Union>s\<in>S. A s) f"
   by (rule continuous_on_open_Union) auto
 
 lemma continuous_on_open_Un:
@@ -1598,9 +1722,11 @@
   by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
 
 lemma continuous_on_If:
-  assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"
+  assumes closed: "closed s" "closed t"
+    and cont: "continuous_on s f" "continuous_on t g"
     and P: "\<And>x. x \<in> s \<Longrightarrow> \<not> P x \<Longrightarrow> f x = g x" "\<And>x. x \<in> t \<Longrightarrow> P x \<Longrightarrow> f x = g x"
-  shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")
+  shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
+    (is "continuous_on _ ?h")
 proof-
   from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
     by auto
@@ -1623,7 +1749,7 @@
   unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
 
 lemma continuous_on_compose[continuous_intros]:
-  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
+  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g \<circ> f)"
   unfolding continuous_on_topological by simp metis
 
 lemma continuous_on_compose2:
@@ -1632,13 +1758,15 @@
 
 lemma continuous_on_generate_topology:
   assumes *: "open = generate_topology X"
-  assumes **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
+    and **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
   shows "continuous_on A f"
   unfolding continuous_on_open_invariant
 proof safe
-  fix B :: "'a set" assume "open B" then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
+  fix B :: "'a set"
+  assume "open B"
+  then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
     unfolding *
-  proof induction
+  proof induct
     case (UN K)
     then obtain C where "\<And>k. k \<in> K \<Longrightarrow> open (C k)" "\<And>k. k \<in> K \<Longrightarrow> C k \<inter> A = f -` k \<inter> A"
       by metis
@@ -1648,47 +1776,53 @@
 qed
 
 lemma continuous_onI_mono:
-  fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order, linorder_topology}"
+  fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order,linorder_topology}"
   assumes "open (f`A)"
-  assumes mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
+    and mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
   shows "continuous_on A f"
 proof (rule continuous_on_generate_topology[OF open_generated_order], safe)
   have monoD: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x < f y \<Longrightarrow> x < y"
     by (auto simp: not_le[symmetric] mono)
-
-  { fix a b assume "a \<in> A" "f a < b"
-    moreover
-    with open_right[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A"
+  have "\<exists>x. x \<in> A \<and> f x < b \<and> a < x" if a: "a \<in> A" and fa: "f a < b" for a b
+  proof -
+    obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A"
+      using open_right[OF \<open>open (f`A)\<close>, of "f a" b] a fa
       by auto
-    moreover then obtain z where "f a < z" "z < min b y"
+    obtain z where z: "f a < z" "z < min b y"
       using dense[of "f a" "min b y"] \<open>f a < y\<close> \<open>f a < b\<close> by auto
-    moreover then obtain c where "z = f c" "c \<in> A"
+    then obtain c where "z = f c" "c \<in> A"
       using \<open>{f a ..< y} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
-    ultimately have "\<exists>x. x \<in> A \<and> f x < b \<and> a < x"
-      by (auto intro!: exI[of _ c] simp: monoD) }
+    with a z show ?thesis
+      by (auto intro!: exI[of _ c] simp: monoD)
+  qed
   then show "\<exists>C. open C \<and> C \<inter> A = f -` {..<b} \<inter> A" for b
     by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. f x < b}. {..< x})"])
        (auto intro: le_less_trans[OF mono] less_imp_le)
 
-  { fix a b assume "a \<in> A" "b < f a"
+  have "\<exists>x. x \<in> A \<and> b < f x \<and> x < a" if a: "a \<in> A" and fa: "b < f a" for a b
+  proof -
+    note a fa
     moreover
-    with open_left[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A"
+    obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A"
+      using open_left[OF \<open>open (f`A)\<close>, of "f a" b]  a fa
       by auto
-    moreover then obtain z where "max b y < z" "z < f a"
+    then obtain z where z: "max b y < z" "z < f a"
       using dense[of "max b y" "f a"] \<open>y < f a\<close> \<open>b < f a\<close> by auto
-    moreover then obtain c where "z = f c" "c \<in> A"
+    then obtain c where "z = f c" "c \<in> A"
       using \<open>{y <.. f a} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
-    ultimately have "\<exists>x. x \<in> A \<and> b < f x \<and> x < a"
-      by (auto intro!: exI[of _ c] simp: monoD) }
+    with a z show ?thesis
+      by (auto intro!: exI[of _ c] simp: monoD)
+  qed
   then show "\<exists>C. open C \<and> C \<inter> A = f -` {b <..} \<inter> A" for b
     by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. b < f x}. {x <..})"])
        (auto intro: less_le_trans[OF _ mono] less_imp_le)
 qed
 
+
 subsubsection \<open>Continuity at a point\<close>
 
-definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
-  "continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
+definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
+  where "continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
 
 lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
   unfolding continuous_def by auto
@@ -1706,12 +1840,12 @@
 
 lemma continuous_within_compose[continuous_intros]:
   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
-  continuous (at x within s) (g o f)"
+    continuous (at x within s) (g \<circ> f)"
   by (simp add: continuous_within_topological) metis
 
 lemma continuous_within_compose2:
   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
-  continuous (at x within s) (\<lambda>x. g (f x))"
+    continuous (at x within s) (\<lambda>x. g (f x))"
   using continuous_within_compose[of x s f g] by (simp add: comp_def)
 
 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x"
@@ -1727,17 +1861,18 @@
   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
   unfolding continuous_on_def continuous_within ..
 
-abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where
-  "isCont f a \<equiv> continuous (at a) f"
+abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool"
+  where "isCont f a \<equiv> continuous (at a) f"
 
 lemma isCont_def: "isCont f a \<longleftrightarrow> f \<midarrow>a\<rightarrow> f a"
   by (rule continuous_at)
 
 lemma isCont_cong:
   assumes "eventually (\<lambda>x. f x = g x) (nhds x)"
-  shows   "isCont f x \<longleftrightarrow> isCont g x"
+  shows "isCont f x \<longleftrightarrow> isCont g x"
 proof -
-  from assms have [simp]: "f x = g x" by (rule eventually_nhds_x_imp_x)
+  from assms have [simp]: "f x = g x"
+    by (rule eventually_nhds_x_imp_x)
   from assms have "eventually (\<lambda>x. f x = g x) (at x)"
     by (auto simp: eventually_at_filter elim!: eventually_mono)
   with assms have "isCont f x \<longleftrightarrow> isCont g x" unfolding isCont_def
@@ -1768,9 +1903,9 @@
 
 lemma continuous_on_tendsto_compose:
   assumes f_cont: "continuous_on s f"
-  assumes g: "(g \<longlongrightarrow> l) F"
-  assumes l: "l \<in> s"
-  assumes ev: "\<forall>\<^sub>F x in F. g x \<in> s"
+    and g: "(g \<longlongrightarrow> l) F"
+    and l: "l \<in> s"
+    and ev: "\<forall>\<^sub>Fx in F. g x \<in> s"
   shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
 proof -
   from f_cont l have f: "(f \<longlongrightarrow> f l) (at l within s)"
@@ -1788,27 +1923,35 @@
   using continuous_at_imp_continuous_at_within continuous_within_compose2 by blast
 
 lemma filtermap_nhds_open_map:
-  assumes cont: "isCont f a" and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
+  assumes cont: "isCont f a"
+    and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
   shows "filtermap f (nhds a) = nhds (f a)"
   unfolding filter_eq_iff
 proof safe
-  fix P assume "eventually P (filtermap f (nhds a))"
-  then guess S unfolding eventually_filtermap eventually_nhds ..
+  fix P
+  assume "eventually P (filtermap f (nhds a))"
+  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. P (f x)"
+    by (auto simp: eventually_filtermap eventually_nhds)
   then show "eventually P (nhds (f a))"
     unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
 qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)
 
 lemma continuous_at_split:
-  "continuous (at (x::'a::linorder_topology)) f = (continuous (at_left x) f \<and> continuous (at_right x) f)"
+  "continuous (at x) f \<longleftrightarrow> continuous (at_left x) f \<and> continuous (at_right x) f"
+  for x :: "'a::linorder_topology"
   by (simp add: continuous_within filterlim_at_split)
 
-(* The following open/closed Collect lemmas are ported from Sébastien Gouëzel's Ergodic_Theory *)
+text \<open>
+  The following open/closed Collect lemmas are ported from
+  Sébastien Gouëzel's \<open>Ergodic_Theory\<close>.
+\<close>
 lemma open_Collect_neq:
-  fixes f g :: "'a :: topological_space \<Rightarrow> 'b::t2_space"
+  fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
   shows "open {x. f x \<noteq> g x}"
 proof (rule openI)
-  fix t assume "t \<in> {x. f x \<noteq> g x}"
+  fix t
+  assume "t \<in> {x. f x \<noteq> g x}"
   then obtain U V where *: "open U" "open V" "f t \<in> U" "g t \<in> V" "U \<inter> V = {}"
     by (auto simp add: separation_t2)
   with open_vimage[OF \<open>open U\<close> f] open_vimage[OF \<open>open V\<close> g]
@@ -1817,26 +1960,27 @@
 qed
 
 lemma closed_Collect_eq:
-  fixes f g :: "'a :: topological_space \<Rightarrow> 'b::t2_space"
+  fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
   shows "closed {x. f x = g x}"
   using open_Collect_neq[OF f g] by (simp add: closed_def Collect_neg_eq)
 
 lemma open_Collect_less:
-  fixes f g :: "'a :: topological_space \<Rightarrow> 'b::linorder_topology"
+  fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
   shows "open {x. f x < g x}"
 proof (rule openI)
-  fix t assume t: "t \<in> {x. f x < g x}"
+  fix t
+  assume t: "t \<in> {x. f x < g x}"
   show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x. f x < g x}"
-  proof (cases)
-    assume "\<exists>z. f t < z \<and> z < g t"
-    then obtain z where z: "f t < z \<and> z < g t" by blast
+  proof (cases "\<exists>z. f t < z \<and> z < g t")
+    case True
+    then obtain z where "f t < z \<and> z < g t" by blast
     then show ?thesis
       using open_vimage[OF _ f, of "{..< z}"] open_vimage[OF _ g, of "{z <..}"]
       by (intro exI[of _ "f -` {..<z} \<inter> g -` {z<..}"]) auto
   next
-    assume "\<not>(\<exists>z. f t < z \<and> z < g t)"
+    case False
     then have *: "{g t ..} = {f t <..}" "{..< g t} = {.. f t}"
       using t by (auto intro: leI)
     show ?thesis
@@ -1850,17 +1994,20 @@
 
 lemma closed_Collect_le:
   fixes f g :: "'a :: topological_space \<Rightarrow> 'b::linorder_topology"
-  assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
+  assumes f: "continuous_on UNIV f"
+    and g: "continuous_on UNIV g"
   shows "closed {x. f x \<le> g x}"
-  using open_Collect_less[OF g f] by (simp add: closed_def Collect_neg_eq[symmetric] not_le)
+  using open_Collect_less [OF g f]
+  by (simp add: closed_def Collect_neg_eq[symmetric] not_le)
+
 
 subsubsection \<open>Open-cover compactness\<close>
 
 context topological_space
 begin
 
-definition compact :: "'a set \<Rightarrow> bool" where
-  compact_eq_heine_borel: \<comment> "This name is used for backwards compatibility"
+definition compact :: "'a set \<Rightarrow> bool"
+  where compact_eq_heine_borel:  (* This name is used for backwards compatibility *)
     "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
 
 lemma compactI:
@@ -1872,24 +2019,33 @@
   by (auto intro!: compactI)
 
 lemma compactE:
-  assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
+  assumes "compact s"
+    and "\<forall>t\<in>C. open t"
+    and "s \<subseteq> \<Union>C"
   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
   using assms unfolding compact_eq_heine_borel by metis
 
 lemma compactE_image:
-  assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
+  assumes "compact s"
+    and "\<forall>t\<in>C. open (f t)"
+    and "s \<subseteq> (\<Union>c\<in>C. f c)"
   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
   using assms unfolding ball_simps [symmetric]
   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
 
 lemma compact_Int_closed [intro]:
-  assumes "compact s" and "closed t"
+  assumes "compact s"
+    and "closed t"
   shows "compact (s \<inter> t)"
 proof (rule compactI)
-  fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
-  from C \<open>closed t\<close> have "\<forall>c\<in>C \<union> {-t}. open c" by auto
-  moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
-  ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
+  fix C
+  assume C: "\<forall>c\<in>C. open c"
+  assume cover: "s \<inter> t \<subseteq> \<Union>C"
+  from C \<open>closed t\<close> have "\<forall>c\<in>C \<union> {- t}. open c"
+    by auto
+  moreover from cover have "s \<subseteq> \<Union>(C \<union> {- t})"
+    by auto
+  ultimately have "\<exists>D\<subseteq>C \<union> {- t}. finite D \<and> s \<subseteq> \<Union>D"
     using \<open>compact s\<close> unfolding compact_eq_heine_borel by auto
   then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" ..
   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
@@ -1899,7 +2055,8 @@
 lemma inj_setminus: "inj_on uminus (A::'a set set)"
   by (auto simp: inj_on_def)
 
-subsection\<open> Finite intersection property\<close>
+
+subsection \<open>Finite intersection property\<close>
 
 lemma compact_fip:
   "compact U \<longleftrightarrow>
@@ -1908,13 +2065,13 @@
 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
   fix A
   assume "compact U"
-    and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
-    and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
+  assume A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
+  assume fin: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
   from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
     by auto
   with \<open>compact U\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
     unfolding compact_eq_heine_borel by (metis subset_image_iff)
-  with fi[THEN spec, of B] show False
+  with fin[THEN spec, of B] show False
     by (auto dest: finite_imageD intro: inj_setminus)
 next
   fix A
@@ -1925,15 +2082,15 @@
   with \<open>?R\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
     by (metis subset_image_iff)
   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
-    by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
+    by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
 qed
 
 lemma compact_imp_fip:
-    "\<lbrakk>compact S;
-      \<And>T. T \<in> F \<Longrightarrow> closed T;
-      \<And>F'. \<lbrakk>finite F'; F' \<subseteq> F\<rbrakk> \<Longrightarrow> S \<inter> (\<Inter>F') \<noteq> {}\<rbrakk>
-     \<Longrightarrow> S \<inter> (\<Inter>F) \<noteq> {}"
-  unfolding compact_fip by auto
+  assumes "compact S"
+    and "\<And>T. T \<in> F \<Longrightarrow> closed T"
+    and "\<And>F'. finite F' \<Longrightarrow> F' \<subseteq> F \<Longrightarrow> S \<inter> (\<Inter>F') \<noteq> {}"
+  shows "S \<inter> (\<Inter>F) \<noteq> {}"
+  using assms unfolding compact_fip by auto
 
 lemma compact_imp_fip_image:
   assumes "compact s"
@@ -1942,55 +2099,66 @@
   shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"
 proof -
   note \<open>compact s\<close>
-  moreover from P have "\<forall>i \<in> f ` I. closed i" by blast
+  moreover from P have "\<forall>i \<in> f ` I. closed i"
+    by blast
   moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"
-  proof (rule, rule, erule conjE)
+    apply rule
+    apply rule
+    apply (erule conjE)
+  proof -
     fix A :: "'a set set"
-    assume "finite A"
-    moreover assume "A \<subseteq> f ` I"
-    ultimately obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"
+    assume "finite A" and "A \<subseteq> f ` I"
+    then obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"
       using finite_subset_image [of A f I] by blast
-    with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}" by simp
+    with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}"
+      by simp
   qed
-  ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}" by (metis compact_imp_fip)
+  ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}"
+    by (metis compact_imp_fip)
   then show ?thesis by simp
 qed
 
 end
 
 lemma (in t2_space) compact_imp_closed:
-  assumes "compact s" shows "closed s"
-unfolding closed_def
+  assumes "compact s"
+  shows "closed s"
+  unfolding closed_def
 proof (rule openI)
-  fix y assume "y \<in> - s"
+  fix y
+  assume "y \<in> - s"
   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
   note \<open>compact s\<close>
   moreover have "\<forall>u\<in>?C. open u" by simp
   moreover have "s \<subseteq> \<Union>?C"
   proof
-    fix x assume "x \<in> s"
+    fix x
+    assume "x \<in> s"
     with \<open>y \<in> - s\<close> have "x \<noteq> y" by clarsimp
-    hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
+    then have "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
       by (rule hausdorff)
     with \<open>x \<in> s\<close> show "x \<in> \<Union>?C"
       unfolding eventually_nhds by auto
   qed
   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
     by (rule compactE)
-  from \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
+  from \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)"
+    by auto
   with \<open>finite D\<close> have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
     by (simp add: eventually_ball_finite)
   with \<open>s \<subseteq> \<Union>D\<close> have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
     by (auto elim!: eventually_mono)
-  thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
+  then show "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
     by (simp add: eventually_nhds subset_eq)
 qed
 
 lemma compact_continuous_image:
-  assumes f: "continuous_on s f" and s: "compact s"
+  assumes f: "continuous_on s f"
+    and s: "compact s"
   shows "compact (f ` s)"
 proof (rule compactI)
-  fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
+  fix C
+  assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
   with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
     unfolding continuous_on_open_invariant by blast
   then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s"
@@ -2004,9 +2172,11 @@
 
 lemma continuous_on_inv:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
-  assumes "continuous_on s f"  "compact s"  "\<forall>x\<in>s. g (f x) = x"
+  assumes "continuous_on s f"
+    and "compact s"
+    and "\<forall>x\<in>s. g (f x) = x"
   shows "continuous_on (f ` s) g"
-unfolding continuous_on_topological
+  unfolding continuous_on_topological
 proof (clarsimp simp add: assms(3))
   fix x :: 'a and B :: "'a set"
   assume "x \<in> s" and "open B" and "x \<in> B"
@@ -2020,9 +2190,9 @@
     unfolding Diff_eq by (intro compact_Int_closed closed_Compl)
   ultimately have "compact (f ` (s - B))"
     by (rule compact_continuous_image)
-  hence "closed (f ` (s - B))"
+  then have "closed (f ` (s - B))"
     by (rule compact_imp_closed)
-  hence "open (- f ` (s - B))"
+  then have "open (- f ` (s - B))"
     by (rule open_Compl)
   moreover have "f x \<in> - f ` (s - B)"
     using \<open>x \<in> s\<close> and \<open>x \<in> B\<close> by (simp add: 1)
@@ -2034,7 +2204,8 @@
 
 lemma continuous_on_inv_into:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
-  assumes s: "continuous_on s f" "compact s" and f: "inj_on f s"
+  assumes s: "continuous_on s f" "compact s"
+    and f: "inj_on f s"
   shows "continuous_on (f ` s) (the_inv_into s f)"
   by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
 
@@ -2086,6 +2257,7 @@
   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y)"
   using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
 
+
 subsection \<open>Connectedness\<close>
 
 context topological_space
@@ -2112,28 +2284,29 @@
 end
 
 lemma connected_closed:
-    "connected s \<longleftrightarrow>
-     ~ (\<exists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
-apply (simp add: connected_def del: ex_simps, safe)
-apply (drule_tac x="-A" in spec)
-apply (drule_tac x="-B" in spec)
-apply (fastforce simp add: closed_def [symmetric])
-apply (drule_tac x="-A" in spec)
-apply (drule_tac x="-B" in spec)
-apply (fastforce simp add: open_closed [symmetric])
-done
+  "connected s \<longleftrightarrow>
+    \<not> (\<exists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
+  apply (simp add: connected_def del: ex_simps, safe)
+   apply (drule_tac x="-A" in spec)
+   apply (drule_tac x="-B" in spec)
+   apply (fastforce simp add: closed_def [symmetric])
+  apply (drule_tac x="-A" in spec)
+  apply (drule_tac x="-B" in spec)
+  apply (fastforce simp add: open_closed [symmetric])
+  done
 
 lemma connected_closedD:
-    "\<lbrakk>connected s; A \<inter> B \<inter> s = {}; s \<subseteq> A \<union> B; closed A; closed B\<rbrakk> \<Longrightarrow> A \<inter> s = {} \<or> B \<inter> s = {}"
-by (simp add: connected_closed)
+  "\<lbrakk>connected s; A \<inter> B \<inter> s = {}; s \<subseteq> A \<union> B; closed A; closed B\<rbrakk> \<Longrightarrow> A \<inter> s = {} \<or> B \<inter> s = {}"
+  by (simp add: connected_closed)
 
 lemma connected_Union:
-  assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s" and ne: "\<Inter>S \<noteq> {}"
-    shows "connected(\<Union>S)"
+  assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s"
+    and ne: "\<Inter>S \<noteq> {}"
+  shows "connected(\<Union>S)"
 proof (rule connectedI)
   fix A B
   assume A: "open A" and B: "open B" and Alap: "A \<inter> \<Union>S \<noteq> {}" and Blap: "B \<inter> \<Union>S \<noteq> {}"
-     and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B"
+    and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B"
   have disjs:"\<And>s. s \<in> S \<Longrightarrow> A \<inter> B \<inter> s = {}"
     using disj by auto
   obtain sa where sa: "sa \<in> S" "A \<inter> sa \<noteq> {}"
@@ -2151,38 +2324,41 @@
     by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans)
 qed
 
-lemma connected_Un: "\<lbrakk>connected s; connected t; s \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> connected (s \<union> t)"
+lemma connected_Un: "connected s \<Longrightarrow> connected t \<Longrightarrow> s \<inter> t \<noteq> {} \<Longrightarrow> connected (s \<union> t)"
   using connected_Union [of "{s,t}"] by auto
 
 lemma connected_diff_open_from_closed:
-  assumes st: "s \<subseteq> t" and tu: "t \<subseteq> u" and s: "open s"
-      and t: "closed t" and u: "connected u" and ts: "connected (t - s)"
+  assumes st: "s \<subseteq> t"
+    and tu: "t \<subseteq> u"
+    and s: "open s"
+    and t: "closed t"
+    and u: "connected u"
+    and ts: "connected (t - s)"
   shows "connected(u - s)"
 proof (rule connectedI)
   fix A B
   assume AB: "open A" "open B" "A \<inter> (u - s) \<noteq> {}" "B \<inter> (u - s) \<noteq> {}"
-     and disj: "A \<inter> B \<inter> (u - s) = {}" and cover: "u - s \<subseteq> A \<union> B"
+    and disj: "A \<inter> B \<inter> (u - s) = {}"
+    and cover: "u - s \<subseteq> A \<union> B"
   then consider "A \<inter> (t - s) = {}" | "B \<inter> (t - s) = {}"
-    using st ts tu connectedD [of "t-s" "A" "B"]
-    by auto
+    using st ts tu connectedD [of "t-s" "A" "B"] by auto
   then show False
   proof cases
     case 1
     then have "(A - t) \<inter> (B \<union> s) \<inter> u = {}"
       using disj st by auto
-    moreover have  "u \<subseteq> (A - t) \<union> (B \<union> s)" using 1 cover by auto
+    moreover have "u \<subseteq> (A - t) \<union> (B \<union> s)"
+      using 1 cover by auto
     ultimately show False
-      using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u
-      by auto
+      using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u by auto
   next
     case 2
     then have "(A \<union> s) \<inter> (B - t) \<inter> u = {}"
-      using disj st
-      by auto
-    moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)" using 2 cover by auto
+      using disj st by auto
+    moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)"
+      using 2 cover by auto
     ultimately show False
-      using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u
-      by auto
+      using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u by auto
   qed
 qed
 
@@ -2190,7 +2366,8 @@
   fixes S :: "'a::topological_space set"
   shows "connected S \<longleftrightarrow> (\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))"
 proof safe
-  fix P :: "'a \<Rightarrow> bool" assume "connected S" "continuous_on S P"
+  fix P :: "'a \<Rightarrow> bool"
+  assume "connected S" "continuous_on S P"
   then have "\<And>b. \<exists>A. open A \<and> A \<inter> S = P -` {b} \<inter> S"
     unfolding continuous_on_open_invariant by (simp add: open_discrete)
   from this[of True] this[of False]
@@ -2200,17 +2377,20 @@
     by (intro connectedD[OF \<open>connected S\<close>])  auto
   then show "\<exists>c. \<forall>s\<in>S. P s = c"
   proof (rule disjE)
-    assume "t \<inter> S = {}" then show ?thesis
+    assume "t \<inter> S = {}"
+    then show ?thesis
       unfolding * by (intro exI[of _ False]) auto
   next
-    assume "f \<inter> S = {}" then show ?thesis
+    assume "f \<inter> S = {}"
+    then show ?thesis
       unfolding * by (intro exI[of _ True]) auto
   qed
 next
   assume P: "\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c)"
   show "connected S"
   proof (rule connectedI)
-    fix A B assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
+    fix A B
+    assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
     have "continuous_on S (\<lambda>x. x \<in> A)"
       unfolding continuous_on_open_invariant
     proof safe
@@ -2220,30 +2400,29 @@
       with * show "\<exists>T. open T \<and> T \<inter> S = (\<lambda>x. x \<in> A) -` C \<inter> S"
         by (intro exI[of _ "(if True \<in> C then A else {}) \<union> (if False \<in> C then B else {})"]) auto
     qed
-    from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c" by blast
+    from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c"
+      by blast
     with * show False
       by (cases c) auto
   qed
 qed
 
-lemma connectedD_const:
-  fixes P :: "'a::topological_space \<Rightarrow> bool"
-  shows "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c"
-  unfolding connected_iff_const by auto
+lemma connectedD_const: "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c"
+  for P :: "'a::topological_space \<Rightarrow> bool"
+  by (auto simp: connected_iff_const)
 
 lemma connectedI_const:
   "(\<And>P::'a::topological_space \<Rightarrow> bool. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c) \<Longrightarrow> connected S"
-  unfolding connected_iff_const by auto
+  by (auto simp: connected_iff_const)
 
 lemma connected_local_const:
   assumes "connected A" "a \<in> A" "b \<in> A"
-  assumes *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"
+    and *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"
   shows "f a = f b"
 proof -
   obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)"
     "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x"
     using * unfolding eventually_at_topological by metis
-
   let ?P = "\<Union>b\<in>{b\<in>A. f a = f b}. S b" and ?N = "\<Union>b\<in>{b\<in>A. f a \<noteq> f b}. S b"
   have "?P \<inter> A = {} \<or> ?N \<inter> A = {}"
     using \<open>connected A\<close> S \<open>a\<in>A\<close>
@@ -2261,28 +2440,31 @@
 qed
 
 lemma (in linorder_topology) connectedD_interval:
-  assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y"
+  assumes "connected U"
+    and xy: "x \<in> U" "y \<in> U"
+    and "x \<le> z" "z \<le> y"
   shows "z \<in> U"
 proof -
   have eq: "{..<z} \<union> {z<..} = - {z}"
     by auto
-  { assume "z \<notin> U" "x < z" "z < y"
-    with xy have "\<not> connected U"
-      unfolding connected_def simp_thms
-      apply (rule_tac exI[of _ "{..< z}"])
-      apply (rule_tac exI[of _ "{z <..}"])
-      apply (auto simp add: eq)
-      done }
+  have "\<not> connected U" if "z \<notin> U" "x < z" "z < y"
+    using xy that
+    apply (simp only: connected_def simp_thms)
+    apply (rule_tac exI[of _ "{..< z}"])
+    apply (rule_tac exI[of _ "{z <..}"])
+    apply (auto simp add: eq)
+    done
   with assms show "z \<in> U"
     by (metis less_le)
 qed
 
 lemma connected_continuous_image:
   assumes *: "continuous_on s f"
-  assumes "connected s"
+    and "connected s"
   shows "connected (f ` s)"
 proof (rule connectedI_const)
-  fix P :: "'b \<Rightarrow> bool" assume "continuous_on (f ` s) P"
+  fix P :: "'b \<Rightarrow> bool"
+  assume "continuous_on (f ` s) P"
   then have "continuous_on s (P \<circ> f)"
     by (rule continuous_on_compose[OF *])
   from connectedD_const[OF \<open>connected s\<close> this] show "\<exists>c. \<forall>s\<in>f ` s. P s = c"
@@ -2296,7 +2478,8 @@
 begin
 
 lemma Inf_notin_open:
-  assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a"
+  assumes A: "open A"
+    and bnd: "\<forall>a\<in>A. x < a"
   shows "Inf A \<notin> A"
 proof
   assume "Inf A \<in> A"
@@ -2305,20 +2488,23 @@
   with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
     by (auto simp: subset_eq)
   then show False
-    using cInf_lower[OF \<open>c \<in> A\<close>] bnd by (metis not_le less_imp_le bdd_belowI)
+    using cInf_lower[OF \<open>c \<in> A\<close>] bnd
+    by (metis not_le less_imp_le bdd_belowI)
 qed
 
 lemma Sup_notin_open:
-  assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x"
+  assumes A: "open A"
+    and bnd: "\<forall>a\<in>A. a < x"
   shows "Sup A \<notin> A"
 proof
   assume "Sup A \<in> A"
-  then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
-    using open_right[of A "Sup A" x] assms by auto
+  with assms obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
+    using open_right[of A "Sup A" x] by auto
   with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
     by (auto simp: subset_eq)
   then show False
-    using cSup_upper[OF \<open>c \<in> A\<close>] bnd by (metis less_imp_le not_le bdd_aboveI)
+    using cSup_upper[OF \<open>c \<in> A\<close>] bnd
+    by (metis less_imp_le not_le bdd_aboveI)
 qed
 
 end
@@ -2328,8 +2514,7 @@
   fix x :: 'a
   obtain y where "x < y \<or> y < x"
     using ex_gt_or_lt [of x] ..
-  with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y]
-  show "\<not> open {x}"
+  with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y] show "\<not> open {x}"
     by auto
 qed
 
@@ -2338,8 +2523,11 @@
   assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U"
   shows "connected U"
 proof (rule connectedI)
-  { fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
-    fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
+  {
+    fix A B
+    assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
+    fix x y
+    assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
 
     let ?z = "Inf (B \<inter> {x <..})"
 
@@ -2351,8 +2539,8 @@
       using \<open>open B\<close> by (intro Inf_notin_open) auto
     ultimately have "?z \<in> A"
       using \<open>x \<le> ?z\<close> \<open>A \<inter> B \<inter> U = {}\<close> \<open>x \<in> A\<close> \<open>U \<subseteq> A \<union> B\<close> by auto
-
-    { assume "?z < y"
+    have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U" if "?z < y"
+    proof -
       obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
         using open_right[OF \<open>open A\<close> \<open>?z \<in> A\<close> \<open>?z < y\<close>] by auto
       moreover obtain b where "b \<in> B" "x < b" "b < min a y"
@@ -2364,71 +2552,86 @@
       moreover have "b \<in> U"
         using \<open>x \<le> ?z\<close> \<open>?z \<le> b\<close> \<open>b < min a y\<close>
         by (intro *[OF \<open>x \<in> U\<close> \<open>y \<in> U\<close>]) (auto simp: less_imp_le)
-      ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U"
-        by (intro bexI[of _ b]) auto }
+      ultimately show ?thesis
+        by (intro bexI[of _ b]) auto
+    qed
     then have False
-      using \<open>?z \<le> y\<close> \<open>?z \<in> A\<close> \<open>y \<in> B\<close> \<open>y \<in> U\<close> \<open>A \<inter> B \<inter> U = {}\<close> unfolding le_less by blast }
+      using \<open>?z \<le> y\<close> \<open>?z \<in> A\<close> \<open>y \<in> B\<close> \<open>y \<in> U\<close> \<open>A \<inter> B \<inter> U = {}\<close>
+      unfolding le_less by blast
+  }
   note not_disjoint = this
 
   fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
   moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
   moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
   moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
-  ultimately show False by (cases x y rule: linorder_cases) auto
+  ultimately show False
+    by (cases x y rule: linorder_cases) auto
 qed
 
-lemma connected_iff_interval:
-  fixes U :: "'a :: linear_continuum_topology set"
-  shows "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)"
+lemma connected_iff_interval: "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)"
+  for U :: "'a::linear_continuum_topology set"
   by (auto intro: connectedI_interval dest: connectedD_interval)
 
 lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
-  unfolding connected_iff_interval by auto
-
-lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}"
-  unfolding connected_iff_interval by auto
-
-lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}"
-  unfolding connected_iff_interval by auto
-
-lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}"
-  unfolding connected_iff_interval by auto
-
-lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}"
+  by (simp add: connected_iff_interval)
+
+lemma connected_Ioi[simp]: "connected {a<..}"
+  for a :: "'a::linear_continuum_topology"
+  by (auto simp: connected_iff_interval)
+
+lemma connected_Ici[simp]: "connected {a..}"
+  for a :: "'a::linear_continuum_topology"
+  by (auto simp: connected_iff_interval)
+
+lemma connected_Iio[simp]: "connected {..<a}"
+  for a :: "'a::linear_continuum_topology"
+  by (auto simp: connected_iff_interval)
+
+lemma connected_Iic[simp]: "connected {..a}"
+  for a :: "'a::linear_continuum_topology"
+  by (auto simp: connected_iff_interval)
+
+lemma connected_Ioo[simp]: "connected {a<..<b}"
+  for a b :: "'a::linear_continuum_topology"
   unfolding connected_iff_interval by auto
 
-lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}"
-  unfolding connected_iff_interval by auto
-
-lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}"
-  unfolding connected_iff_interval by auto
-
-lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}"
-  unfolding connected_iff_interval by auto
-
-lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}"
-  unfolding connected_iff_interval by auto
+lemma connected_Ioc[simp]: "connected {a<..b}"
+  for a b :: "'a::linear_continuum_topology"
+  by (auto simp: connected_iff_interval)
+
+lemma connected_Ico[simp]: "connected {a..<b}"
+  for a b :: "'a::linear_continuum_topology"
+  by (auto simp: connected_iff_interval)
+
+lemma connected_Icc[simp]: "connected {a..b}"
+  for a b :: "'a::linear_continuum_topology"
+  by (auto simp: connected_iff_interval)
 
 lemma connected_contains_Ioo:
   fixes A :: "'a :: linorder_topology set"
-  assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
-  using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le)
+  assumes "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
+  using connectedD_interval[OF assms] by (simp add: subset_eq Ball_def less_imp_le)
 
 lemma connected_contains_Icc:
-  assumes "connected (A :: ('a :: {linorder_topology}) set)" "a \<in> A" "b \<in> A"
-  shows   "{a..b} \<subseteq> A"
+  fixes A :: "'a::linorder_topology set"
+  assumes "connected A" "a \<in> A" "b \<in> A"
+  shows "{a..b} \<subseteq> A"
 proof
   fix x assume "x \<in> {a..b}"
-  hence "x = a \<or> x = b \<or> x \<in> {a<..<b}" by auto
-  thus "x \<in> A" using assms connected_contains_Ioo[of A a b] by auto
+  then have "x = a \<or> x = b \<or> x \<in> {a<..<b}"
+    by auto
+  then show "x \<in> A"
+    using assms connected_contains_Ioo[of A a b] by auto
 qed
 
+
 subsection \<open>Intermediate Value Theorem\<close>
 
 lemma IVT':
-  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
+  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
   assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
-  assumes *: "continuous_on {a .. b} f"
+    and *: "continuous_on {a .. b} f"
   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
 proof -
   have "connected {a..b}"
@@ -2441,7 +2644,7 @@
 lemma IVT2':
   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
   assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"
-  assumes *: "continuous_on {a .. b} f"
+    and *: "continuous_on {a .. b} f"
   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
 proof -
   have "connected {a..b}"
@@ -2452,51 +2655,59 @@
 qed
 
 lemma IVT:
-  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
-  shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
+  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
+  shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow>
+    \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
   by (rule IVT') (auto intro: continuous_at_imp_continuous_on)
 
 lemma IVT2:
-  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
-  shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
+  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
+  shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow>
+    \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
   by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)
 
 lemma continuous_inj_imp_mono:
-  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
+  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
   assumes x: "a < x" "x < b"
-  assumes cont: "continuous_on {a..b} f"
-  assumes inj: "inj_on f {a..b}"
+    and cont: "continuous_on {a..b} f"
+    and inj: "inj_on f {a..b}"
   shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"
 proof -
   note I = inj_on_eq_iff[OF inj]
-  { assume "f x < f a" "f x < f b"
+  {
+    assume "f x < f a" "f x < f b"
     then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s"
       using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
-    with x I have False by auto }
+    with x I have False by auto
+  }
   moreover
-  { assume "f a < f x" "f b < f x"
+  {
+    assume "f a < f x" "f b < f x"
     then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x"
       using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
-    with x I have False by auto }
+    with x I have False by auto
+  }
   ultimately show ?thesis
-    using I[of a x] I[of x b] x less_trans[OF x] by (auto simp add: le_less less_imp_neq neq_iff)
+    using I[of a x] I[of x b] x less_trans[OF x]
+    by (auto simp add: le_less less_imp_neq neq_iff)
 qed
 
 lemma continuous_at_Sup_mono:
-  fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
+  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
+    'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "mono f"
-  assumes cont: "continuous (at_left (Sup S)) f"
-  assumes S: "S \<noteq> {}" "bdd_above S"
+    and cont: "continuous (at_left (Sup S)) f"
+    and S: "S \<noteq> {}" "bdd_above S"
   shows "f (Sup S) = (SUP s:S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
     using cont unfolding continuous_within .
-
   show "f (Sup S) \<le> (SUP s:S. f s)"
   proof cases
-    assume "Sup S \<in> S" then show ?thesis
+    assume "Sup S \<in> S"
+    then show ?thesis
       by (rule cSUP_upper) (auto intro: bdd_above_image_mono S \<open>mono f\<close>)
   next
     assume "Sup S \<notin> S"
@@ -2522,18 +2733,19 @@
 qed (intro cSUP_least \<open>mono f\<close>[THEN monoD] cSup_upper S)
 
 lemma continuous_at_Sup_antimono:
-  fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
+  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
+    'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "antimono f"
-  assumes cont: "continuous (at_left (Sup S)) f"
-  assumes S: "S \<noteq> {}" "bdd_above S"
+    and cont: "continuous (at_left (Sup S)) f"
+    and S: "S \<noteq> {}" "bdd_above S"
   shows "f (Sup S) = (INF s:S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
     using cont unfolding continuous_within .
-
   show "(INF s:S. f s) \<le> f (Sup S)"
   proof cases
-    assume "Sup S \<in> S" then show ?thesis
+    assume "Sup S \<in> S"
+    then show ?thesis
       by (intro cINF_lower) (auto intro: bdd_below_image_antimono S \<open>antimono f\<close>)
   next
     assume "Sup S \<notin> S"
@@ -2559,18 +2771,19 @@
 qed (intro cINF_greatest \<open>antimono f\<close>[THEN antimonoD] cSup_upper S)
 
 lemma continuous_at_Inf_mono:
-  fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
+  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
+    'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "mono f"
-  assumes cont: "continuous (at_right (Inf S)) f"
-  assumes S: "S \<noteq> {}" "bdd_below S"
+    and cont: "continuous (at_right (Inf S)) f"
+    and S: "S \<noteq> {}" "bdd_below S"
   shows "f (Inf S) = (INF s:S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
     using cont unfolding continuous_within .
-
   show "(INF s:S. f s) \<le> f (Inf S)"
   proof cases
-    assume "Inf S \<in> S" then show ?thesis
+    assume "Inf S \<in> S"
+    then show ?thesis
       by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S \<open>mono f\<close>)
   next
     assume "Inf S \<notin> S"
@@ -2596,18 +2809,19 @@
 qed (intro cINF_greatest \<open>mono f\<close>[THEN monoD] cInf_lower \<open>bdd_below S\<close> \<open>S \<noteq> {}\<close>)
 
 lemma continuous_at_Inf_antimono:
-  fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
+  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
+    'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "antimono f"
-  assumes cont: "continuous (at_right (Inf S)) f"
-  assumes S: "S \<noteq> {}" "bdd_below S"
+    and cont: "continuous (at_right (Inf S)) f"
+    and S: "S \<noteq> {}" "bdd_below S"
   shows "f (Inf S) = (SUP s:S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
     using cont unfolding continuous_within .
-
   show "f (Inf S) \<le> (SUP s:S. f s)"
   proof cases
-    assume "Inf S \<in> S" then show ?thesis
+    assume "Inf S \<in> S"
+    then show ?thesis
       by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S \<open>antimono f\<close>)
   next
     assume "Inf S \<notin> S"
@@ -2632,14 +2846,15 @@
   qed
 qed (intro cSUP_least \<open>antimono f\<close>[THEN antimonoD] cInf_lower S)
 
+
 subsection \<open>Uniform spaces\<close>
 
 class uniformity =
   fixes uniformity :: "('a \<times> 'a) filter"
 begin
 
-abbreviation uniformity_on :: "'a set \<Rightarrow> ('a \<times> 'a) filter" where
-  "uniformity_on s \<equiv> inf uniformity (principal (s\<times>s))"
+abbreviation uniformity_on :: "'a set \<Rightarrow> ('a \<times> 'a) filter"
+  where "uniformity_on s \<equiv> inf uniformity (principal (s\<times>s))"
 
 end
 
@@ -2649,48 +2864,58 @@
   by simp
 
 class open_uniformity = "open" + uniformity +
-  assumes open_uniformity: "\<And>U. open U \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
+  assumes open_uniformity:
+    "\<And>U. open U \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
 
 class uniform_space = open_uniformity +
   assumes uniformity_refl: "eventually E uniformity \<Longrightarrow> E (x, x)"
-  assumes uniformity_sym: "eventually E uniformity \<Longrightarrow> eventually (\<lambda>(x, y). E (y, x)) uniformity"
-  assumes uniformity_trans: "eventually E uniformity \<Longrightarrow> \<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))"
+    and uniformity_sym: "eventually E uniformity \<Longrightarrow> eventually (\<lambda>(x, y). E (y, x)) uniformity"
+    and uniformity_trans:
+      "eventually E uniformity \<Longrightarrow>
+        \<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))"
 begin
 
 subclass topological_space
-  proof qed (force elim: eventually_mono eventually_elim2 simp: split_beta' open_uniformity)+
+  by standard (force elim: eventually_mono eventually_elim2 simp: split_beta' open_uniformity)+
 
 lemma uniformity_bot: "uniformity \<noteq> bot"
   using uniformity_refl by auto
 
 lemma uniformity_trans':
-  "eventually E uniformity \<Longrightarrow> eventually (\<lambda>((x, y), (y', z)). y = y' \<longrightarrow> E (x, z)) (uniformity \<times>\<^sub>F uniformity)"
+  "eventually E uniformity \<Longrightarrow>
+    eventually (\<lambda>((x, y), (y', z)). y = y' \<longrightarrow> E (x, z)) (uniformity \<times>\<^sub>F uniformity)"
   by (drule uniformity_trans) (auto simp add: eventually_prod_same)
 
 lemma uniformity_transE:
-  assumes E: "eventually E uniformity"
+  assumes "eventually E uniformity"
   obtains D where "eventually D uniformity" "\<And>x y z. D (x, y) \<Longrightarrow> D (y, z) \<Longrightarrow> E (x, z)"
-  using uniformity_trans[OF E] by auto
+  using uniformity_trans [OF assms] by auto
 
 lemma eventually_nhds_uniformity:
-  "eventually P (nhds x) \<longleftrightarrow> eventually (\<lambda>(x', y). x' = x \<longrightarrow> P y) uniformity" (is "_ \<longleftrightarrow> ?N P x")
+  "eventually P (nhds x) \<longleftrightarrow> eventually (\<lambda>(x', y). x' = x \<longrightarrow> P y) uniformity"
+  (is "_ \<longleftrightarrow> ?N P x")
   unfolding eventually_nhds
 proof safe
   assume *: "?N P x"
-  { fix x assume "?N P x"
-    then guess D by (rule uniformity_transE) note D = this
-    from D(1) have "?N (?N P) x"
-      by eventually_elim (insert D, force elim: eventually_mono split: prod.split) }
+  have "?N (?N P) x" if "?N P x" for x
+  proof -
+    from that obtain D where ev: "eventually D uniformity"
+      and D: "D (a, b) \<Longrightarrow> D (b, c) \<Longrightarrow> case (a, c) of (x', y) \<Rightarrow> x' = x \<longrightarrow> P y" for a b c
+      by (rule uniformity_transE) simp
+    from ev show ?thesis
+      by eventually_elim (insert ev D, force elim: eventually_mono split: prod.split)
+  qed
   then have "open {x. ?N P x}"
     by (simp add: open_uniformity)
   then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>x\<in>S. P x)"
     by (intro exI[of _ "{x. ?N P x}"]) (auto dest: uniformity_refl simp: *)
 qed (force simp add: open_uniformity elim: eventually_mono)
 
+
 subsubsection \<open>Totally bounded sets\<close>
 
-definition totally_bounded :: "'a set \<Rightarrow> bool" where
-  "totally_bounded S \<longleftrightarrow>
+definition totally_bounded :: "'a set \<Rightarrow> bool"
+  where "totally_bounded S \<longleftrightarrow>
     (\<forall>E. eventually E uniformity \<longrightarrow> (\<exists>X. finite X \<and> (\<forall>s\<in>S. \<exists>x\<in>X. E (x, s))))"
 
 lemma totally_bounded_empty[iff]: "totally_bounded {}"
@@ -2700,23 +2925,26 @@
   by (fastforce simp add: totally_bounded_def)
 
 lemma totally_bounded_Union[intro]:
-  assumes M: "finite M" "\<And>S. S \<in> M \<Longrightarrow> totally_bounded S" shows "totally_bounded (\<Union>M)"
+  assumes M: "finite M" "\<And>S. S \<in> M \<Longrightarrow> totally_bounded S"
+  shows "totally_bounded (\<Union>M)"
   unfolding totally_bounded_def
 proof safe
-  fix E assume "eventually E uniformity"
+  fix E
+  assume "eventually E uniformity"
   with M obtain X where "\<forall>S\<in>M. finite (X S) \<and> (\<forall>s\<in>S. \<exists>x\<in>X S. E (x, s))"
     by (metis totally_bounded_def)
   with \<open>finite M\<close> show "\<exists>X. finite X \<and> (\<forall>s\<in>\<Union>M. \<exists>x\<in>X. E (x, s))"
     by (intro exI[of _ "\<Union>S\<in>M. X S"]) force
 qed
 
+
 subsubsection \<open>Cauchy filter\<close>
 
-definition cauchy_filter :: "'a filter \<Rightarrow> bool" where
-  "cauchy_filter F \<longleftrightarrow> F \<times>\<^sub>F F \<le> uniformity"
-
-definition Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
-  Cauchy_uniform: "Cauchy X = cauchy_filter (filtermap X sequentially)"
+definition cauchy_filter :: "'a filter \<Rightarrow> bool"
+  where "cauchy_filter F \<longleftrightarrow> F \<times>\<^sub>F F \<le> uniformity"
+
+definition Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"
+  where Cauchy_uniform: "Cauchy X = cauchy_filter (filtermap X sequentially)"
 
 lemma Cauchy_uniform_iff:
   "Cauchy X \<longleftrightarrow> (\<forall>P. eventually P uniformity \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)))"
@@ -2724,12 +2952,16 @@
     eventually_filtermap eventually_sequentially
 proof safe
   let ?U = "\<lambda>P. eventually P uniformity"
-  { fix P assume "?U P" "\<forall>P. ?U P \<longrightarrow> (\<exists>Q. (\<exists>N. \<forall>n\<ge>N. Q (X n)) \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
+  {
+    fix P
+    assume "?U P" "\<forall>P. ?U P \<longrightarrow> (\<exists>Q. (\<exists>N. \<forall>n\<ge>N. Q (X n)) \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
     then obtain Q N where "\<And>n. n \<ge> N \<Longrightarrow> Q (X n)" "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> P (x, y)"
       by metis
     then show "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)"
-      by blast }
-  { fix P assume "?U P" and P: "\<forall>P. ?U P \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m))"
+      by blast
+  next
+    fix P
+    assume "?U P" and P: "\<forall>P. ?U P \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m))"
     then obtain Q where "?U Q" and Q: "\<And>x y z. Q (x, y) \<Longrightarrow> Q (y, z) \<Longrightarrow> P (x, z)"
       by (auto elim: uniformity_transE)
     then have "?U (\<lambda>x. Q x \<and> (\<lambda>(x, y). Q (y, x)) x)"
@@ -2739,25 +2971,32 @@
       by auto
     show "\<exists>Q. (\<exists>N. \<forall>n\<ge>N. Q (X n)) \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))"
     proof (safe intro!: exI[of _ "\<lambda>x. \<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)"] exI[of _ N] N)
-      fix x y assume "\<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)" "\<forall>n\<ge>N. Q (y, X n) \<and> Q (X n, y)"
+      fix x y
+      assume "\<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)" "\<forall>n\<ge>N. Q (y, X n) \<and> Q (X n, y)"
       then have "Q (x, X N)" "Q (X N, y)" by auto
       then show "P (x, y)"
         by (rule Q)
-    qed }
+    qed
+  }
 qed
 
 lemma nhds_imp_cauchy_filter:
-  assumes *: "F \<le> nhds x" shows "cauchy_filter F"
+  assumes *: "F \<le> nhds x"
+  shows "cauchy_filter F"
 proof -
   have "F \<times>\<^sub>F F \<le> nhds x \<times>\<^sub>F nhds x"
     by (intro prod_filter_mono *)
   also have "\<dots> \<le> uniformity"
     unfolding le_filter_def eventually_nhds_uniformity eventually_prod_same
   proof safe
-    fix P assume "eventually P uniformity"
-    then guess Ql by (rule uniformity_transE) note Ql = this
-    moreover note Ql(1)[THEN uniformity_sym]
-    ultimately show "\<exists>Q. eventually (\<lambda>(x', y). x' = x \<longrightarrow> Q y) uniformity \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))"
+    fix P
+    assume "eventually P uniformity"
+    then obtain Ql where ev: "eventually Ql uniformity"
+      and "Ql (x, y) \<Longrightarrow> Ql (y, z) \<Longrightarrow> P (x, z)" for x y z
+      by (rule uniformity_transE) simp
+    with ev[THEN uniformity_sym]
+    show "\<exists>Q. eventually (\<lambda>(x', y). x' = x \<longrightarrow> Q y) uniformity \<and>
+        (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))"
       by (rule_tac exI[of _ "\<lambda>y. Ql (y, x) \<and> Ql (x, y)"]) (fastforce elim: eventually_elim2)
   qed
   finally show ?thesis
@@ -2767,7 +3006,9 @@
 lemma LIMSEQ_imp_Cauchy: "X \<longlonglongrightarrow> x \<Longrightarrow> Cauchy X"
   unfolding Cauchy_uniform filterlim_def by (intro nhds_imp_cauchy_filter)
 
-lemma Cauchy_subseq_Cauchy: assumes "Cauchy X" "subseq f" shows "Cauchy (X \<circ> f)"
+lemma Cauchy_subseq_Cauchy:
+  assumes "Cauchy X" "subseq f"
+  shows "Cauchy (X \<circ> f)"
   unfolding Cauchy_uniform comp_def filtermap_filtermap[symmetric] cauchy_filter_def
   by (rule order_trans[OF _ \<open>Cauchy X\<close>[unfolded Cauchy_uniform cauchy_filter_def]])
      (intro prod_filter_mono filtermap_mono filterlim_subseq[OF \<open>subseq f\<close>, unfolded filterlim_def])
@@ -2775,21 +3016,24 @@
 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
   unfolding convergent_def by (erule exE, erule LIMSEQ_imp_Cauchy)
 
-definition complete :: "'a set \<Rightarrow> bool" where
-  complete_uniform: "complete S \<longleftrightarrow> (\<forall>F \<le> principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x))"
+definition complete :: "'a set \<Rightarrow> bool"
+  where complete_uniform: "complete S \<longleftrightarrow>
+    (\<forall>F \<le> principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x))"
 
 end
 
+
 subsubsection \<open>Uniformly continuous functions\<close>
 
-definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::uniform_space \<Rightarrow> 'b::uniform_space) \<Rightarrow> bool" where
-  uniformly_continuous_on_uniformity: "uniformly_continuous_on s f \<longleftrightarrow>
+definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::uniform_space \<Rightarrow> 'b::uniform_space) \<Rightarrow> bool"
+  where uniformly_continuous_on_uniformity: "uniformly_continuous_on s f \<longleftrightarrow>
     (LIM (x, y) (uniformity_on s). (f x, f y) :> uniformity)"
 
 lemma uniformly_continuous_onD:
-  "uniformly_continuous_on s f \<Longrightarrow> eventually E uniformity
-    \<Longrightarrow> eventually (\<lambda>(x, y). x \<in> s \<longrightarrow> y \<in> s \<longrightarrow> E (f x, f y)) uniformity"
-  by (simp add: uniformly_continuous_on_uniformity filterlim_iff eventually_inf_principal split_beta' mem_Times_iff imp_conjL)
+  "uniformly_continuous_on s f \<Longrightarrow> eventually E uniformity \<Longrightarrow>
+    eventually (\<lambda>(x, y). x \<in> s \<longrightarrow> y \<in> s \<longrightarrow> E (f x, f y)) uniformity"
+  by (simp add: uniformly_continuous_on_uniformity filterlim_iff
+      eventually_inf_principal split_beta' mem_Times_iff imp_conjL)
 
 lemma uniformly_continuous_on_const[continuous_intros]: "uniformly_continuous_on s (\<lambda>x. c)"
   by (auto simp: uniformly_continuous_on_uniformity filterlim_iff uniformity_refl)
@@ -2798,17 +3042,22 @@
   by (auto simp: uniformly_continuous_on_uniformity filterlim_def)
 
 lemma uniformly_continuous_on_compose[continuous_intros]:
-  "uniformly_continuous_on s g \<Longrightarrow> uniformly_continuous_on (g`s) f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f (g x))"
-  using filterlim_compose[of "\<lambda>(x, y). (f x, f y)" uniformity "uniformity_on (g`s)"  "\<lambda>(x, y). (g x, g y)" "uniformity_on s"]
-  by (simp add: split_beta' uniformly_continuous_on_uniformity filterlim_inf filterlim_principal eventually_inf_principal mem_Times_iff)
-
-lemma uniformly_continuous_imp_continuous: assumes f: "uniformly_continuous_on s f" shows "continuous_on s f"
+  "uniformly_continuous_on s g \<Longrightarrow> uniformly_continuous_on (g`s) f \<Longrightarrow>
+    uniformly_continuous_on s (\<lambda>x. f (g x))"
+  using filterlim_compose[of "\<lambda>(x, y). (f x, f y)" uniformity
+      "uniformity_on (g`s)"  "\<lambda>(x, y). (g x, g y)" "uniformity_on s"]
+  by (simp add: split_beta' uniformly_continuous_on_uniformity
+      filterlim_inf filterlim_principal eventually_inf_principal mem_Times_iff)
+
+lemma uniformly_continuous_imp_continuous:
+  assumes f: "uniformly_continuous_on s f"
+  shows "continuous_on s f"
   by (auto simp: filterlim_iff eventually_at_filter eventually_nhds_uniformity continuous_on_def
            elim: eventually_mono dest!: uniformly_continuous_onD[OF f])
 
+
 section \<open>Product Topology\<close>
 
-
 subsection \<open>Product is a topological space\<close>
 
 instantiation prod :: (topological_space, topological_space) topological_space
@@ -2821,12 +3070,12 @@
 lemma open_prod_elim:
   assumes "open S" and "x \<in> S"
   obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
-using assms unfolding open_prod_def by fast
+  using assms unfolding open_prod_def by fast
 
 lemma open_prod_intro:
   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
   shows "open S"
-using assms unfolding open_prod_def by fast
+  using assms unfolding open_prod_def by fast
 
 instance
 proof
@@ -2837,7 +3086,8 @@
   assume "open S" "open T"
   show "open (S \<inter> T)"
   proof (rule open_prod_intro)
-    fix x assume x: "x \<in> S \<inter> T"
+    fix x
+    assume x: "x \<in> S \<inter> T"
     from x have "x \<in> S" by simp
     obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
       using \<open>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim)
@@ -2847,46 +3097,48 @@
     let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
     have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
       using A B by (auto simp add: open_Int)
-    thus "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
+    then show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
       by fast
   qed
 next
   fix K :: "('a \<times> 'b) set set"
-  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
+  assume "\<forall>S\<in>K. open S"
+  then show "open (\<Union>K)"
     unfolding open_prod_def by fast
 qed
 
 end
 
-declare [[code abort: "open::('a::topological_space*'b::topological_space) set \<Rightarrow> bool"]]
+declare [[code abort: "open :: ('a::topological_space \<times> 'b::topological_space) set \<Rightarrow> bool"]]
 
 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
-unfolding open_prod_def by auto
+  unfolding open_prod_def by auto
 
 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
-by auto
+  by auto
 
 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
-by auto
+  by auto
 
 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
-by (simp add: fst_vimage_eq_Times open_Times)
+  by (simp add: fst_vimage_eq_Times open_Times)
 
 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
-by (simp add: snd_vimage_eq_Times open_Times)
+  by (simp add: snd_vimage_eq_Times open_Times)
 
 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
-unfolding closed_open vimage_Compl [symmetric]
-by (rule open_vimage_fst)
+  unfolding closed_open vimage_Compl [symmetric]
+  by (rule open_vimage_fst)
 
 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
-unfolding closed_open vimage_Compl [symmetric]
-by (rule open_vimage_snd)
+  unfolding closed_open vimage_Compl [symmetric]
+  by (rule open_vimage_snd)
 
 lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
 proof -
-  have "S \<times> T = (fst -` S) \<inter> (snd -` T)" by auto
-  thus "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
+  have "S \<times> T = (fst -` S) \<inter> (snd -` T)"
+    by auto
+  then show "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
 qed
 
@@ -2896,49 +3148,66 @@
 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
   unfolding image_def subset_eq by force
 
-lemma open_image_fst: assumes "open S" shows "open (fst ` S)"
+lemma open_image_fst:
+  assumes "open S"
+  shows "open (fst ` S)"
 proof (rule openI)
-  fix x assume "x \<in> fst ` S"
-  then obtain y where "(x, y) \<in> S" by auto
+  fix x
+  assume "x \<in> fst ` S"
+  then obtain y where "(x, y) \<in> S"
+    by auto
   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
     using \<open>open S\<close> unfolding open_prod_def by auto
-  from \<open>A \<times> B \<subseteq> S\<close> \<open>y \<in> B\<close> have "A \<subseteq> fst ` S" by (rule subset_fst_imageI)
-  with \<open>open A\<close> \<open>x \<in> A\<close> have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S" by simp
-  then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" by - (rule exI)
+  from \<open>A \<times> B \<subseteq> S\<close> \<open>y \<in> B\<close> have "A \<subseteq> fst ` S"
+    by (rule subset_fst_imageI)
+  with \<open>open A\<close> \<open>x \<in> A\<close> have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S"
+    by simp
+  then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" ..
 qed
 
-lemma open_image_snd: assumes "open S" shows "open (snd ` S)"
+lemma open_image_snd:
+  assumes "open S"
+  shows "open (snd ` S)"
 proof (rule openI)
-  fix y assume "y \<in> snd ` S"
-  then obtain x where "(x, y) \<in> S" by auto
+  fix y
+  assume "y \<in> snd ` S"
+  then obtain x where "(x, y) \<in> S"
+    by auto
   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
     using \<open>open S\<close> unfolding open_prod_def by auto
-  from \<open>A \<times> B \<subseteq> S\<close> \<open>x \<in> A\<close> have "B \<subseteq> snd ` S" by (rule subset_snd_imageI)
-  with \<open>open B\<close> \<open>y \<in> B\<close> have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S" by simp
-  then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" by - (rule exI)
+  from \<open>A \<times> B \<subseteq> S\<close> \<open>x \<in> A\<close> have "B \<subseteq> snd ` S"
+    by (rule subset_snd_imageI)
+  with \<open>open B\<close> \<open>y \<in> B\<close> have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S"
+    by simp
+  then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" ..
 qed
 
 lemma nhds_prod: "nhds (a, b) = nhds a \<times>\<^sub>F nhds b"
   unfolding nhds_def
 proof (subst prod_filter_INF, auto intro!: antisym INF_greatest simp: principal_prod_principal)
-  fix S T assume "open S" "a \<in> S" "open T" "b \<in> T"
+  fix S T
+  assume "open S" "a \<in> S" "open T" "b \<in> T"
   then show "(INF x : {S. open S \<and> (a, b) \<in> S}. principal x) \<le> principal (S \<times> T)"
     by (intro INF_lower) (auto intro!: open_Times)
 next
-  fix S' assume "open S'" "(a, b) \<in> S'"
+  fix S'
+  assume "open S'" "(a, b) \<in> S'"
   then obtain S T where "open S" "a \<in> S" "open T" "b \<in> T" "S \<times> T \<subseteq> S'"
     by (auto elim: open_prod_elim)
-  then show "(INF x : {S. open S \<and> a \<in> S}. INF y : {S. open S \<and> b \<in> S}. principal (x \<times> y)) \<le> principal S'"
+  then show "(INF x : {S. open S \<and> a \<in> S}. INF y : {S. open S \<and> b \<in> S}.
+      principal (x \<times> y)) \<le> principal S'"
     by (auto intro!: INF_lower2)
 qed
 
+
 subsubsection \<open>Continuity of operations\<close>
 
 lemma tendsto_fst [tendsto_intros]:
   assumes "(f \<longlongrightarrow> a) F"
   shows "((\<lambda>x. fst (f x)) \<longlongrightarrow> fst a) F"
 proof (rule topological_tendstoI)
-  fix S assume "open S" and "fst a \<in> S"
+  fix S
+  assume "open S" and "fst a \<in> S"
   then have "open (fst -` S)" and "a \<in> fst -` S"
     by (simp_all add: open_vimage_fst)
   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
@@ -2951,7 +3220,8 @@
   assumes "(f \<longlongrightarrow> a) F"
   shows "((\<lambda>x. snd (f x)) \<longlongrightarrow> snd a) F"
 proof (rule topological_tendstoI)
-  fix S assume "open S" and "snd a \<in> S"
+  fix S
+  assume "open S" and "snd a \<in> S"
   then have "open (snd -` S)" and "a \<in> snd -` S"
     by (simp_all add: open_vimage_snd)
   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
@@ -2964,7 +3234,8 @@
   assumes "(f \<longlongrightarrow> a) F" and "(g \<longlongrightarrow> b) F"
   shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F"
 proof (rule topological_tendstoI)
-  fix S assume "open S" and "(a, b) \<in> S"
+  fix S
+  assume "open S" and "(a, b) \<in> S"
   then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
     unfolding open_prod_def by fast
   have "eventually (\<lambda>x. f x \<in> A) F"
@@ -2986,26 +3257,31 @@
 lemma continuous_snd[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))"
   unfolding continuous_def by (rule tendsto_snd)
 
-lemma continuous_Pair[continuous_intros]: "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
+lemma continuous_Pair[continuous_intros]:
+  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
   unfolding continuous_def by (rule tendsto_Pair)
 
-lemma continuous_on_fst[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))"
+lemma continuous_on_fst[continuous_intros]:
+  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))"
   unfolding continuous_on_def by (auto intro: tendsto_fst)
 
-lemma continuous_on_snd[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))"
+lemma continuous_on_snd[continuous_intros]:
+  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))"
   unfolding continuous_on_def by (auto intro: tendsto_snd)
 
-lemma continuous_on_Pair[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))"
+lemma continuous_on_Pair[continuous_intros]:
+  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))"
   unfolding continuous_on_def by (auto intro: tendsto_Pair)
 
 lemma continuous_on_swap[continuous_intros]: "continuous_on A prod.swap"
-  by (simp add: prod.swap_def continuous_on_fst continuous_on_snd continuous_on_Pair continuous_on_id)
+  by (simp add: prod.swap_def continuous_on_fst continuous_on_snd
+      continuous_on_Pair continuous_on_id)
 
 lemma continuous_on_swap_args:
   assumes "continuous_on (A\<times>B) (\<lambda>(x,y). d x y)"
     shows "continuous_on (B\<times>A) (\<lambda>(x,y). d y x)"
 proof -
-  have "(\<lambda>(x,y). d y x) = (\<lambda>(x,y). d x y) o prod.swap"
+  have "(\<lambda>(x,y). d y x) = (\<lambda>(x,y). d x y) \<circ> prod.swap"
     by force
   then show ?thesis
     apply (rule ssubst)
@@ -3024,32 +3300,36 @@
 lemma isCont_Pair [simp]: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
   by (fact continuous_Pair)
 
+
 subsubsection \<open>Separation axioms\<close>
 
 instance prod :: (t0_space, t0_space) t0_space
 proof
-  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
-  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
+  fix x y :: "'a \<times> 'b"
+  assume "x \<noteq> y"
+  then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
     by (simp add: prod_eq_iff)
-  thus "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
+  then show "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
     by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd)
 qed
 
 instance prod :: (t1_space, t1_space) t1_space
 proof
-  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
-  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
+  fix x y :: "'a \<times> 'b"
+  assume "x \<noteq> y"
+  then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
     by (simp add: prod_eq_iff)
-  thus "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
+  then show "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
     by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd)
 qed
 
 instance prod :: (t2_space, t2_space) t2_space
 proof
-  fix x y :: "'a \<times> 'b" assume "x \<noteq> y"
-  hence "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
+  fix x y :: "'a \<times> 'b"
+  assume "x \<noteq> y"
+  then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
     by (simp add: prod_eq_iff)
-  thus "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
+  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
     by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd)
 qed