src/HOL/Multivariate_Analysis/Path_Connected.thy

 begin
 
 subsection {* Paths. *}
 
 definition path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
-  where "path g \<longleftrightarrow> continuous_on {0 .. 1} g"
+  where "path g \<longleftrightarrow> continuous_on {0..1} g"
 
 definition pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
   where "pathstart g = g 0"
 
 definition pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
   where "pathfinish g = g 1"
 
 definition path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
   where "path_image g = g ` {0 .. 1}"
 
-definition reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a)"
+definition reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
   where "reversepath g = (\<lambda>x. g(1 - x))"
 
-definition joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a)"
+definition joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a"
     (infixr "+++" 75)
   where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
 
 definition simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
   where "simple_path g \<longleftrightarrow>

 
 
 subsection {* Some lemmas about these concepts. *}
 
 lemma injective_imp_simple_path: "injective_path g \<Longrightarrow> simple_path g"
-  unfolding injective_path_def simple_path_def by auto
+  unfolding injective_path_def simple_path_def
+  by auto
 
 lemma path_image_nonempty: "path_image g \<noteq> {}"
-  unfolding path_image_def image_is_empty interval_eq_empty by auto 
-
-lemma pathstart_in_path_image[intro]: "(pathstart g) \<in> path_image g"
-  unfolding pathstart_def path_image_def by auto
-
-lemma pathfinish_in_path_image[intro]: "(pathfinish g) \<in> path_image g"
-  unfolding pathfinish_def path_image_def by auto
-
-lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)"
+  unfolding path_image_def image_is_empty interval_eq_empty
+  by auto
+
+lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g"
+  unfolding pathstart_def path_image_def
+  by auto
+
+lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g"
+  unfolding pathfinish_def path_image_def
+  by auto
+
+lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)"
   unfolding path_def path_image_def
   apply (erule connected_continuous_image)
   apply (rule convex_connected, rule convex_real_interval)
   done
 
-lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)"
+lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)"
   unfolding path_def path_image_def
-  by (erule compact_continuous_image, rule compact_interval)
-
-lemma reversepath_reversepath[simp]: "reversepath(reversepath g) = g"
-  unfolding reversepath_def by auto
-
-lemma pathstart_reversepath[simp]: "pathstart(reversepath g) = pathfinish g"
-  unfolding pathstart_def reversepath_def pathfinish_def by auto
-
-lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g"
-  unfolding pathstart_def reversepath_def pathfinish_def by auto
+  apply (erule compact_continuous_image)
+  apply (rule compact_interval)
+  done
+
+lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
+  unfolding reversepath_def
+  by auto
+
+lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
+  unfolding pathstart_def reversepath_def pathfinish_def
+  by auto
+
+lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
+  unfolding pathstart_def reversepath_def pathfinish_def
+  by auto
 
 lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
-  unfolding pathstart_def joinpaths_def pathfinish_def by auto
+  unfolding pathstart_def joinpaths_def pathfinish_def
+  by auto
 
 lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
-  unfolding pathstart_def joinpaths_def pathfinish_def by auto
-
-lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g"
+  unfolding pathstart_def joinpaths_def pathfinish_def
+  by auto
+
+lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
 proof -
-  have *: "\<And>g. path_image(reversepath g) \<subseteq> path_image g"
+  have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g"
     unfolding path_image_def subset_eq reversepath_def Ball_def image_iff
-    apply(rule,rule,erule bexE)
-    apply(rule_tac x="1 - xa" in bexI)
+    apply rule
+    apply rule
+    apply (erule bexE)
+    apply (rule_tac x="1 - xa" in bexI)
     apply auto
     done
   show ?thesis
     using *[of g] *[of "reversepath g"]
-    unfolding reversepath_reversepath by auto
-qed
-
-lemma path_reversepath[simp]: "path (reversepath g) \<longleftrightarrow> path g"
+    unfolding reversepath_reversepath
+    by auto
+qed
+
+lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g"
 proof -
   have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)"
     unfolding path_def reversepath_def
     apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
     apply (intro continuous_on_intros)
-    apply (rule continuous_on_subset[of "{0..1}"], assumption)
+    apply (rule continuous_on_subset[of "{0..1}"])
+    apply assumption
     apply auto
     done
   show ?thesis
     using *[of "reversepath g"] *[of g]
     unfolding reversepath_reversepath

 proof safe
   assume cont: "continuous_on {0..1} (g1 +++ g2)"
   have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))"
     by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
   have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))"
-    using assms by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
-  show "continuous_on {0..1} g1" "continuous_on {0..1} g2"
+    using assms
+    by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
+  show "continuous_on {0..1} g1" and "continuous_on {0..1} g2"
     unfolding g1 g2
     by (auto intro!: continuous_on_intros continuous_on_subset[OF cont] simp del: o_apply)
 next
   assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
   have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}"
     by auto
-  { fix x :: real assume "0 \<le> x" "x \<le> 1" then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
-      by (intro image_eqI[where x="x/2"]) auto }
+  {
+    fix x :: real
+    assume "0 \<le> x" and "x \<le> 1"
+    then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
+      by (intro image_eqI[where x="x/2"]) auto
+  }
   note 1 = this
-  { fix x :: real assume "0 \<le> x" "x \<le> 1" then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
-      by (intro image_eqI[where x="x/2 + 1/2"]) auto }
+  {
+    fix x :: real
+    assume "0 \<le> x" and "x \<le> 1"
+    then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
+      by (intro image_eqI[where x="x/2 + 1/2"]) auto
+  }
   note 2 = this
   show "continuous_on {0..1} (g1 +++ g2)"
-    using assms unfolding joinpaths_def 01
-    by (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_on_intros)
-       (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
-qed
-
-lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)"
-  unfolding path_image_def joinpaths_def by auto
+    using assms
+    unfolding joinpaths_def 01
+    apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_on_intros)
+    apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
+    done
+qed
+
+lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2"
+  unfolding path_image_def joinpaths_def
+  by auto
 
 lemma subset_path_image_join:
-  assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s"
-  shows "path_image(g1 +++ g2) \<subseteq> s"
-  using path_image_join_subset[of g1 g2] and assms by auto
+  assumes "path_image g1 \<subseteq> s"
+    and "path_image g2 \<subseteq> s"
+  shows "path_image (g1 +++ g2) \<subseteq> s"
+  using path_image_join_subset[of g1 g2] and assms
+  by auto
 
 lemma path_image_join:
   assumes "pathfinish g1 = pathstart g2"
-  shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)"
-  apply (rule, rule path_image_join_subset, rule)
+  shows "path_image (g1 +++ g2) = path_image g1 \<union> path_image g2"
+  apply rule
+  apply (rule path_image_join_subset)
+  apply rule
   unfolding Un_iff
 proof (erule disjE)
   fix x
   assume "x \<in> path_image g1"
-  then obtain y where y: "y\<in>{0..1}" "x = g1 y"
+  then obtain y where y: "y \<in> {0..1}" "x = g1 y"
     unfolding path_image_def image_iff by auto
   then show "x \<in> path_image (g1 +++ g2)"
     unfolding joinpaths_def path_image_def image_iff
     apply (rule_tac x="(1/2) *\<^sub>R y" in bexI)
     apply auto
     done
 next
   fix x
   assume "x \<in> path_image g2"
-  then obtain y where y: "y\<in>{0..1}" "x = g2 y"
+  then obtain y where y: "y \<in> {0..1}" "x = g2 y"
     unfolding path_image_def image_iff by auto
   then show "x \<in> path_image (g1 +++ g2)"
     unfolding joinpaths_def path_image_def image_iff
-    apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI)
+    apply (rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI)
     using assms(1)[unfolded pathfinish_def pathstart_def]
-    apply (auto simp add: add_divide_distrib) 
+    apply (auto simp add: add_divide_distrib)
     done
 qed
 
 lemma not_in_path_image_join:
-  assumes "x \<notin> path_image g1" "x \<notin> path_image g2"
-  shows "x \<notin> path_image(g1 +++ g2)"
-  using assms and path_image_join_subset[of g1 g2] by auto
+  assumes "x \<notin> path_image g1"
+    and "x \<notin> path_image g2"
+  shows "x \<notin> path_image (g1 +++ g2)"
+  using assms and path_image_join_subset[of g1 g2]
+  by auto
 
 lemma simple_path_reversepath:
   assumes "simple_path g"
   shows "simple_path (reversepath g)"
   using assms
   unfolding simple_path_def reversepath_def
   apply -
   apply (rule ballI)+
-  apply (erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)
+  apply (erule_tac x="1-x" in ballE)
+  apply (erule_tac x="1-y" in ballE)
   apply auto
   done
 
 lemma simple_path_join_loop:
-  assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"
-    "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"
-  shows "simple_path(g1 +++ g2)"
+  assumes "injective_path g1"
+    and "injective_path g2"
+    and "pathfinish g2 = pathstart g1"
+    and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
+  shows "simple_path (g1 +++ g2)"
   unfolding simple_path_def
-proof ((rule ballI)+, rule impI)
+proof (intro ballI impI)
   let ?g = "g1 +++ g2"
   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
   fix x y :: real
   assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"
   show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
-  proof (case_tac "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)
+  proof (cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)
     assume as: "x \<le> 1 / 2" "y \<le> 1 / 2"
     then have "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)"
-      using xy(3) unfolding joinpaths_def by auto
-    moreover
-    have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as
-      by auto
-    ultimately
-    show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto
+      using xy(3)
+      unfolding joinpaths_def
+      by auto
+    moreover have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}"
+      using xy(1,2) as
+      by auto
+    ultimately show ?thesis
+      using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"]
+      by auto
   next
-    assume as:"x > 1 / 2" "y > 1 / 2"
+    assume as: "x > 1 / 2" "y > 1 / 2"
     then have "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)"
-      using xy(3) unfolding joinpaths_def by auto
-    moreover
-    have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}"
-      using xy(1,2) as by auto
-    ultimately
-    show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
+      using xy(3)
+      unfolding joinpaths_def
+      by auto
+    moreover have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}"
+      using xy(1,2) as
+      by auto
+    ultimately show ?thesis
+      using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto
   next
-    assume as:"x \<le> 1 / 2" "y > 1 / 2"
+    assume as: "x \<le> 1 / 2" "y > 1 / 2"
     then have "?g x \<in> path_image g1" "?g y \<in> path_image g2"
       unfolding path_image_def joinpaths_def
       using xy(1,2) by auto
-    moreover
-      have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def
+    moreover have "?g y \<noteq> pathstart g2"
+      using as(2)
+      unfolding pathstart_def joinpaths_def
       using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2)
       by (auto simp add: field_simps)
-    ultimately
-    have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto
-    then have "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)
-      using inj(1)[of "2 *\<^sub>R x" 0] by auto
-    moreover
-    have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]
-      unfolding joinpaths_def pathfinish_def using as(2) and xy(2)
-      using inj(2)[of "2 *\<^sub>R y - 1" 1] by auto
-    ultimately show ?thesis by auto
+    ultimately have *: "?g x = pathstart g1"
+      using assms(4)
+      unfolding xy(3)
+      by auto
+    then have "x = 0"
+      unfolding pathstart_def joinpaths_def
+      using as(1) and xy(1)
+      using inj(1)[of "2 *\<^sub>R x" 0]
+      by auto
+    moreover have "y = 1"
+      using *
+      unfolding xy(3) assms(3)[symmetric]
+      unfolding joinpaths_def pathfinish_def
+      using as(2) and xy(2)
+      using inj(2)[of "2 *\<^sub>R y - 1" 1]
+      by auto
+    ultimately show ?thesis
+      by auto
   next
     assume as: "x > 1 / 2" "y \<le> 1 / 2"
-    then have "?g x \<in> path_image g2" "?g y \<in> path_image g1"
+    then have "?g x \<in> path_image g2" and "?g y \<in> path_image g1"
       unfolding path_image_def joinpaths_def
       using xy(1,2) by auto
-    moreover
-      have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def
+    moreover have "?g x \<noteq> pathstart g2"
+      using as(1)
+      unfolding pathstart_def joinpaths_def
       using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1)
       by (auto simp add: field_simps)
-    ultimately
-    have *: "?g y = pathstart g1" using assms(4) unfolding xy(3) by auto
-    then have "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)
-      using inj(1)[of "2 *\<^sub>R y" 0] by auto
-    moreover
-    have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]
+    ultimately have *: "?g y = pathstart g1"
+      using assms(4)
+      unfolding xy(3)
+      by auto
+    then have "y = 0"
+      unfolding pathstart_def joinpaths_def
+      using as(2) and xy(2)
+      using inj(1)[of "2 *\<^sub>R y" 0]
+      by auto
+    moreover have "x = 1"
+      using *
+      unfolding xy(3)[symmetric] assms(3)[symmetric]
       unfolding joinpaths_def pathfinish_def using as(1) and xy(1)
-      using inj(2)[of "2 *\<^sub>R x - 1" 1] by auto
-    ultimately show ?thesis by auto
+      using inj(2)[of "2 *\<^sub>R x - 1" 1]
+      by auto
+    ultimately show ?thesis
+      by auto
   qed
 qed
 
 lemma injective_path_join:
-  assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"
-    "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"
-  shows "injective_path(g1 +++ g2)"
+  assumes "injective_path g1"
+    and "injective_path g2"
+    and "pathfinish g1 = pathstart g2"
+    and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
+  shows "injective_path (g1 +++ g2)"
   unfolding injective_path_def
 proof (rule, rule, rule)
   let ?g = "g1 +++ g2"
   note inj = assms(1,2)[unfolded injective_path_def, rule_format]
   fix x y
   assume xy: "x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"
   show "x = y"
   proof (cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)
-    assume "x \<le> 1 / 2" "y \<le> 1 / 2"
-    then show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy
+    assume "x \<le> 1 / 2" and "y \<le> 1 / 2"
+    then show ?thesis
+      using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy
       unfolding joinpaths_def by auto
   next
-    assume "x > 1 / 2" "y > 1 / 2"
-    then show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy
+    assume "x > 1 / 2" and "y > 1 / 2"
+    then show ?thesis
+      using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy
       unfolding joinpaths_def by auto
   next
     assume as: "x \<le> 1 / 2" "y > 1 / 2"
-    then have "?g x \<in> path_image g1" "?g y \<in> path_image g2"
+    then have "?g x \<in> path_image g1" and "?g y \<in> path_image g2"
       unfolding path_image_def joinpaths_def
-      using xy(1,2) by auto
-    then have "?g x = pathfinish g1" "?g y = pathstart g2"
-      using assms(4) unfolding assms(3) xy(3) by auto
+      using xy(1,2)
+      by auto
+    then have "?g x = pathfinish g1" and "?g y = pathstart g2"
+      using assms(4)
+      unfolding assms(3) xy(3)
+      by auto
     then show ?thesis
       using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2)
       unfolding pathstart_def pathfinish_def joinpaths_def
       by auto
   next
-    assume as:"x > 1 / 2" "y \<le> 1 / 2" 
-    then have "?g x \<in> path_image g2" "?g y \<in> path_image g1"
+    assume as:"x > 1 / 2" "y \<le> 1 / 2"
+    then have "?g x \<in> path_image g2" and "?g y \<in> path_image g1"
       unfolding path_image_def joinpaths_def
-      using xy(1,2) by auto
-    then have "?g x = pathstart g2" "?g y = pathfinish g1"
-      using assms(4) unfolding assms(3) xy(3) by auto
+      using xy(1,2)
+      by auto
+    then have "?g x = pathstart g2" and "?g y = pathfinish g1"
+      using assms(4)
+      unfolding assms(3) xy(3)
+      by auto
     then show ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2)
       unfolding pathstart_def pathfinish_def joinpaths_def
       by auto
   qed
 qed
 
 lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
- 
-
-subsection {* Reparametrizing a closed curve to start at some chosen point. *}
-
-definition "shiftpath a (f::real \<Rightarrow> 'a::topological_space) =
-  (\<lambda>x. if (a + x) \<le> 1 then f(a + x) else f(a + x - 1))"
-
-lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a"
+
+
+subsection {* Reparametrizing a closed curve to start at some chosen point *}
+
+definition shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
+  where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))"
+
+lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a"
   unfolding pathstart_def shiftpath_def by auto
 
 lemma pathfinish_shiftpath:
-  assumes "0 \<le> a" "pathfinish g = pathstart g"
-  shows "pathfinish(shiftpath a g) = g a"
-  using assms unfolding pathstart_def pathfinish_def shiftpath_def
+  assumes "0 \<le> a"
+    and "pathfinish g = pathstart g"
+  shows "pathfinish (shiftpath a g) = g a"
+  using assms
+  unfolding pathstart_def pathfinish_def shiftpath_def
   by auto
 
 lemma endpoints_shiftpath:
-  assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}" 
-  shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a"
-  using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath)
+  assumes "pathfinish g = pathstart g"
+    and "a \<in> {0 .. 1}"
+  shows "pathfinish (shiftpath a g) = g a"
+    and "pathstart (shiftpath a g) = g a"
+  using assms
+  by (auto intro!: pathfinish_shiftpath pathstart_shiftpath)
 
 lemma closed_shiftpath:
-  assumes "pathfinish g = pathstart g" "a \<in> {0..1}"
-  shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)"
-  using endpoints_shiftpath[OF assms] by auto
+  assumes "pathfinish g = pathstart g"
+    and "a \<in> {0..1}"
+  shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
+  using endpoints_shiftpath[OF assms]
+  by auto
 
 lemma path_shiftpath:
-  assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
-  shows "path(shiftpath a g)"
+  assumes "path g"
+    and "pathfinish g = pathstart g"
+    and "a \<in> {0..1}"
+  shows "path (shiftpath a g)"
 proof -
-  have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by auto
+  have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}"
+    using assms(3) by auto
   have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
-    using assms(2)[unfolded pathfinish_def pathstart_def] by auto
+    using assms(2)[unfolded pathfinish_def pathstart_def]
+    by auto
   show ?thesis
     unfolding path_def shiftpath_def *
     apply (rule continuous_on_union)
     apply (rule closed_real_atLeastAtMost)+
-    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3
-    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3
-    apply (rule continuous_on_intros)+ prefer 2
+    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"])
+    prefer 3
+    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"])
+    defer
+    prefer 3
+    apply (rule continuous_on_intros)+
+    prefer 2
     apply (rule continuous_on_intros)+
     apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
     using assms(3) and **
-    apply (auto, auto simp add: field_simps)
+    apply auto
+    apply (auto simp add: field_simps)
     done
 qed
 
 lemma shiftpath_shiftpath:
-  assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}" 
+  assumes "pathfinish g = pathstart g"
+    and "a \<in> {0..1}"
+    and "x \<in> {0..1}"
   shows "shiftpath (1 - a) (shiftpath a g) x = g x"
-  using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto
+  using assms
+  unfolding pathfinish_def pathstart_def shiftpath_def
+  by auto
 
 lemma path_image_shiftpath:
-  assumes "a \<in> {0..1}" "pathfinish g = pathstart g"
-  shows "path_image(shiftpath a g) = path_image g"
+  assumes "a \<in> {0..1}"
+    and "pathfinish g = pathstart g"
+  shows "path_image (shiftpath a g) = path_image g"
 proof -
   { fix x
-    assume as:"g 1 = g 0" "x \<in> {0..1::real}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1}. g x \<noteq> g (a + y - 1)" 
+    assume as: "g 1 = g 0" "x \<in> {0..1::real}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1}. g x \<noteq> g (a + y - 1)"
     then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)"
     proof (cases "a \<le> x")
       case False
       then show ?thesis
         apply (rule_tac x="1 + x - a" in bexI)
         using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
         apply (auto simp add: field_simps atomize_not)
         done
     next
       case True
-      then show ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)
-        by(auto simp add: field_simps)
+      then show ?thesis
+        using as(1-2) and assms(1)
+        apply (rule_tac x="x - a" in bexI)
+        apply (auto simp add: field_simps)
+        done
     qed
   }
   then show ?thesis
-    using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
-    by(auto simp add: image_iff)
-qed
-
-
-subsection {* Special case of straight-line paths. *}
+    using assms
+    unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
+    by (auto simp add: image_iff)
+qed
+
+
+subsection {* Special case of straight-line paths *}
 
 definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
   where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
 
-lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"
-  unfolding pathstart_def linepath_def by auto
-
-lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b"
-  unfolding pathfinish_def linepath_def by auto
+lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
+  unfolding pathstart_def linepath_def
+  by auto
+
+lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
+  unfolding pathfinish_def linepath_def
+  by auto
 
 lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
-  unfolding linepath_def by (intro continuous_intros)
+  unfolding linepath_def
+  by (intro continuous_intros)
 
 lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
-  using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
-
-lemma path_linepath[intro]: "path(linepath a b)"
-  unfolding path_def by(rule continuous_on_linepath)
-
-lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"
+  using continuous_linepath_at
+  by (auto intro!: continuous_at_imp_continuous_on)
+
+lemma path_linepath[intro]: "path (linepath a b)"
+  unfolding path_def
+  by (rule continuous_on_linepath)
+
+lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
   unfolding path_image_def segment linepath_def
-  apply (rule set_eqI, rule) defer
+  apply (rule set_eqI)
+  apply rule
+  defer
   unfolding mem_Collect_eq image_iff
-  apply(erule exE)
-  apply(rule_tac x="u *\<^sub>R 1" in bexI)
+  apply (erule exE)
+  apply (rule_tac x="u *\<^sub>R 1" in bexI)
   apply auto
   done
 
-lemma reversepath_linepath[simp]: "reversepath(linepath a b) = linepath b a"
+lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
   unfolding reversepath_def linepath_def
   by auto
 
 lemma injective_path_linepath:
   assumes "a \<noteq> b"
   shows "injective_path (linepath a b)"
 proof -
-  { fix x y :: "real"
+  {
+    fix x y :: "real"
     assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
-    then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps)
-    with assms have "x = y" by simp }
+    then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b"
+      by (simp add: algebra_simps)
+    with assms have "x = y"
+      by simp
+  }
   then show ?thesis
     unfolding injective_path_def linepath_def
     by (auto simp add: algebra_simps)
 qed
 
-lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)"
-  by(auto intro!: injective_imp_simple_path injective_path_linepath)
-
-
-subsection {* Bounding a point away from a path. *}
+lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)"
+  by (auto intro!: injective_imp_simple_path injective_path_linepath)
+
+
+subsection {* Bounding a point away from a path *}
 
 lemma not_on_path_ball:
   fixes g :: "real \<Rightarrow> 'a::heine_borel"
-  assumes "path g" "z \<notin> path_image g"
-  shows "\<exists>e > 0. ball z e \<inter> (path_image g) = {}"
+  assumes "path g"
+    and "z \<notin> path_image g"
+  shows "\<exists>e > 0. ball z e \<inter> path_image g = {}"
 proof -
   obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
     using distance_attains_inf[OF _ path_image_nonempty, of g z]
     using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
   then show ?thesis

     done
 qed
 
 lemma not_on_path_cball:
   fixes g :: "real \<Rightarrow> 'a::heine_borel"
-  assumes "path g" "z \<notin> path_image g"
+  assumes "path g"
+    and "z \<notin> path_image g"
   shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
 proof -
-  obtain e where "ball z e \<inter> path_image g = {}" "e>0"
+  obtain e where "ball z e \<inter> path_image g = {}" "e > 0"
     using not_on_path_ball[OF assms] by auto
-  moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto
-  ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto
-qed
-
-
-subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}
+  moreover have "cball z (e/2) \<subseteq> ball z e"
+    using `e > 0` by auto
+  ultimately show ?thesis
+    apply (rule_tac x="e/2" in exI)
+    apply auto
+    done
+qed
+
+
+subsection {* Path component, considered as a "joinability" relation (from Tom Hales) *}
 
 definition "path_component s x y \<longleftrightarrow>
   (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
 
-lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def 
+lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
 
 lemma path_component_mem:
   assumes "path_component s x y"
-  shows "x \<in> s" "y \<in> s"
-  using assms unfolding path_defs by auto
+  shows "x \<in> s" and "y \<in> s"
+  using assms
+  unfolding path_defs
+  by auto
 
 lemma path_component_refl:
   assumes "x \<in> s"
   shows "path_component s x x"
   unfolding path_defs
   apply (rule_tac x="\<lambda>u. x" in exI)
-  using assms apply (auto intro!:continuous_on_intros) done
+  using assms
+  apply (auto intro!: continuous_on_intros)
+  done
 
 lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
   by (auto intro!: path_component_mem path_component_refl)
 
 lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"

   apply (rule_tac x="reversepath g" in exI)
   apply auto
   done
 
 lemma path_component_trans:
-  assumes "path_component s x y" "path_component s y z"
+  assumes "path_component s x y"
+    and "path_component s y z"
   shows "path_component s x z"
   using assms
   unfolding path_component_def
-  apply -
-  apply (erule exE)+
+  apply (elim exE)
   apply (rule_tac x="g +++ ga" in exI)
   apply (auto simp add: path_image_join)
   done
 
-lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow>  path_component s x y \<Longrightarrow> path_component t x y"
+lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y"
   unfolding path_component_def by auto
 
 
-subsection {* Can also consider it as a set, as the name suggests. *}
+text {* Can also consider it as a set, as the name suggests. *}
 
 lemma path_component_set:
   "{y. path_component s x y} =
     {y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)}"
   apply (rule set_eqI)

   unfolding path_component_def
   apply auto
   done
 
 lemma path_component_subset: "{y. path_component s x y} \<subseteq> s"
-  apply (rule, rule path_component_mem(2))
+  apply rule
+  apply (rule path_component_mem(2))
   apply auto
   done
 
 lemma path_component_eq_empty: "{y. path_component s x y} = {} \<longleftrightarrow> x \<notin> s"
   apply rule
-  apply (drule equals0D[of _ x]) defer
+  apply (drule equals0D[of _ x])
+  defer
   apply (rule equals0I)
   unfolding mem_Collect_eq
   apply (drule path_component_mem(1))
   using path_component_refl
   apply auto
   done
 
 
-subsection {* Path connectedness of a space. *}
+subsection {* Path connectedness of a space *}
 
 definition "path_connected s \<longleftrightarrow>
-  (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> (path_image g) \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
+  (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
 
 lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
   unfolding path_connected_def path_component_def by auto
 
-lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)" 
+lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)"
   unfolding path_connected_component
-  apply (rule, rule, rule, rule path_component_subset) 
+  apply rule
+  apply rule
+  apply rule
+  apply (rule path_component_subset)
   unfolding subset_eq mem_Collect_eq Ball_def
   apply auto
   done
 
 
-subsection {* Some useful lemmas about path-connectedness. *}
+subsection {* Some useful lemmas about path-connectedness *}
 
 lemma convex_imp_path_connected:
   fixes s :: "'a::real_normed_vector set"
-  assumes "convex s" shows "path_connected s"
+  assumes "convex s"
+  shows "path_connected s"
   unfolding path_connected_def
-  apply (rule, rule, rule_tac x = "linepath x y" in exI)
+  apply rule
+  apply rule
+  apply (rule_tac x = "linepath x y" in exI)
   unfolding path_image_linepath
   using assms [unfolded convex_contains_segment]
   apply auto
   done
 
 lemma path_connected_imp_connected:
   assumes "path_connected s"
   shows "connected s"
   unfolding connected_def not_ex
-  apply (rule, rule, rule ccontr)
+  apply rule
+  apply rule
+  apply (rule ccontr)
   unfolding not_not
-  apply (erule conjE)+
+  apply (elim conjE)
 proof -
   fix e1 e2
   assume as: "open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
-  then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto
-  then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
+  then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> s" "x2 \<in> e2 \<inter> s"
+    by auto
+  then obtain g where g: "path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
     using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
   have *: "connected {0..1::real}"
     by (auto intro!: convex_connected convex_real_interval)
   have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}"
     using as(3) g(2)[unfolded path_defs] by blast
   moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}"
-    using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto 
+    using as(4) g(2)[unfolded path_defs]
+    unfolding subset_eq
+    by auto
   moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}"
-    using g(3,4)[unfolded path_defs] using obt
+    using g(3,4)[unfolded path_defs]
+    using obt
     by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl)
   ultimately show False
-    using *[unfolded connected_local not_ex, rule_format, of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]
+    using *[unfolded connected_local not_ex, rule_format,
+      of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]
     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)]
     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)]
     by auto
 qed
 

     apply -
     apply (rule path_component_mem(2))
     unfolding mem_Collect_eq
     apply auto
     done
-  then obtain e where e:"e>0" "ball y e \<subseteq> s"
-    using assms[unfolded open_contains_ball] by auto
+  then obtain e where e: "e > 0" "ball y e \<subseteq> s"
+    using assms[unfolded open_contains_ball]
+    by auto
   show "\<exists>e > 0. ball y e \<subseteq> {y. path_component s x y}"
     apply (rule_tac x=e in exI)
-    apply (rule,rule `e>0`, rule)
+    apply (rule,rule `e>0`)
+    apply rule
     unfolding mem_ball mem_Collect_eq
   proof -
     fix z
     assume "dist y z < e"
     then show "path_component s x z"
-      apply (rule_tac path_component_trans[of _ _ y]) defer
+      apply (rule_tac path_component_trans[of _ _ y])
+      defer
       apply (rule path_component_of_subset[OF e(2)])
       apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
-      using `e>0` as
+      using `e > 0` as
       apply auto
       done
   qed
 qed
 
 lemma open_non_path_component:
   fixes s :: "'a::real_normed_vector set"
   assumes "open s"
-  shows "open(s - {y. path_component s x y})"
+  shows "open (s - {y. path_component s x y})"
   unfolding open_contains_ball
 proof
   fix y
-  assume as: "y\<in>s - {y. path_component s x y}"
-  then obtain e where e:"e>0" "ball y e \<subseteq> s"
-    using assms [unfolded open_contains_ball] by auto
+  assume as: "y \<in> s - {y. path_component s x y}"
+  then obtain e where e: "e > 0" "ball y e \<subseteq> s"
+    using assms [unfolded open_contains_ball]
+    by auto
   show "\<exists>e>0. ball y e \<subseteq> s - {y. path_component s x y}"
     apply (rule_tac x=e in exI)
-    apply (rule, rule `e>0`, rule, rule) defer
+    apply rule
+    apply (rule `e>0`)
+    apply rule
+    apply rule
+    defer
   proof (rule ccontr)
     fix z
     assume "z \<in> ball y e" "\<not> z \<notin> {y. path_component s x y}"
     then have "y \<in> {y. path_component s x y}"
       unfolding not_not mem_Collect_eq using `e>0`

       apply (rule path_component_trans, assumption)
       apply (rule path_component_of_subset[OF e(2)])
       apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
       apply auto
       done
-    then show False using as by auto
+    then show False
+      using as by auto
   qed (insert e(2), auto)
 qed
 
 lemma connected_open_path_connected:
   fixes s :: "'a::real_normed_vector set"
-  assumes "open s" "connected s"
+  assumes "open s"
+    and "connected s"
   shows "path_connected s"
   unfolding path_connected_component_set
 proof (rule, rule, rule path_component_subset, rule)
   fix x y
-  assume "x \<in> s" "y \<in> s"
+  assume "x \<in> s" and "y \<in> s"
   show "y \<in> {y. path_component s x y}"
   proof (rule ccontr)
-    assume "y \<notin> {y. path_component s x y}"
-    moreover
-    have "{y. path_component s x y} \<inter> s \<noteq> {}"
-      using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto
+    assume "\<not> ?thesis"
+    moreover have "{y. path_component s x y} \<inter> s \<noteq> {}"
+      using `x \<in> s` path_component_eq_empty path_component_subset[of s x]
+      by auto
     ultimately
     show False
-      using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
-      using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s - {y. path_component s x y}"]
+      using `y \<in> s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
+      using assms(2)[unfolded connected_def not_ex, rule_format,
+        of"{y. path_component s x y}" "s - {y. path_component s x y}"]
       by auto
   qed
 qed
 
 lemma path_connected_continuous_image:
-  assumes "continuous_on s f" "path_connected s"
+  assumes "continuous_on s f"
+    and "path_connected s"
   shows "path_connected (f ` s)"
   unfolding path_connected_def
 proof (rule, rule)
   fix x' y'
   assume "x' \<in> f ` s" "y' \<in> f ` s"
-  then obtain x y where xy: "x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto
-  guess g using assms(2)[unfolded path_connected_def, rule_format, OF xy(1,2)] ..
+  then obtain x y where x: "x \<in> s" and y: "y \<in> s" and x': "x' = f x" and y': "y' = f y"
+    by auto
+  from x y obtain g where "path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y"
+    using assms(2)[unfolded path_connected_def] by fast
   then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'"
-    unfolding xy
+    unfolding x' y'
     apply (rule_tac x="f \<circ> g" in exI)
     unfolding path_defs
     apply (intro conjI continuous_on_compose continuous_on_subset[OF assms(1)])
     apply auto
     done
 qed
 
 lemma homeomorphic_path_connectedness:
-  "s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)"
+  "s homeomorphic t \<Longrightarrow> path_connected s \<longleftrightarrow> path_connected t"
   unfolding homeomorphic_def homeomorphism_def
-  apply (erule exE|erule conjE)+  
-  apply rule
-  apply (drule_tac f=f in path_connected_continuous_image) prefer 3
+  apply (erule exE|erule conjE)+
+  apply rule
+  apply (drule_tac f=f in path_connected_continuous_image)
+  prefer 3
   apply (drule_tac f=g in path_connected_continuous_image)
   apply auto
   done
 
 lemma path_connected_empty: "path_connected {}"
   unfolding path_connected_def by auto
 
 lemma path_connected_singleton: "path_connected {a}"
   unfolding path_connected_def pathstart_def pathfinish_def path_image_def
-  apply (clarify, rule_tac x="\<lambda>x. a" in exI, simp add: image_constant_conv)
+  apply clarify
+  apply (rule_tac x="\<lambda>x. a" in exI)
+  apply (simp add: image_constant_conv)
   apply (simp add: path_def continuous_on_const)
   done
 
 lemma path_connected_Un:
-  assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"
+  assumes "path_connected s"
+    and "path_connected t"
+    and "s \<inter> t \<noteq> {}"
   shows "path_connected (s \<union> t)"
   unfolding path_connected_component
 proof (rule, rule)
   fix x y
   assume as: "x \<in> s \<union> t" "y \<in> s \<union> t"
-  from assms(3) obtain z where "z \<in> s \<inter> t" by auto
+  from assms(3) obtain z where "z \<in> s \<inter> t"
+    by auto
   then show "path_component (s \<union> t) x y"
     using as and assms(1-2)[unfolded path_connected_component]
-    apply - 
+    apply -
     apply (erule_tac[!] UnE)+
     apply (rule_tac[2-3] path_component_trans[of _ _ z])
     apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2])
     done
 qed

   then show "path_component (\<Union>i\<in>A. S i) x y"
     by (rule path_component_trans)
 qed
 
 
-subsection {* sphere is path-connected. *}
+subsection {* Sphere is path-connected *}
 
 lemma path_connected_punctured_universe:
   assumes "2 \<le> DIM('a::euclidean_space)"
-  shows "path_connected((UNIV::'a::euclidean_space set) - {a})"
+  shows "path_connected ((UNIV::'a set) - {a})"
 proof -
   let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}"
   let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}"
 
   have A: "path_connected ?A"
     unfolding Collect_bex_eq
   proof (rule path_connected_UNION)
     fix i :: 'a
     assume "i \<in> Basis"
-    then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" by simp
+    then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}"
+      by simp
     show "path_connected {x. x \<bullet> i < a \<bullet> i}"
       using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"]
       by (simp add: inner_commute)
   qed
-  have B: "path_connected ?B" unfolding Collect_bex_eq
+  have B: "path_connected ?B"
+    unfolding Collect_bex_eq
   proof (rule path_connected_UNION)
     fix i :: 'a
     assume "i \<in> Basis"
-    then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" by simp
+    then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}"
+      by simp
     show "path_connected {x. a \<bullet> i < x \<bullet> i}"
       using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i]
       by (simp add: inner_commute)
   qed
-  obtain S :: "'a set" where "S \<subseteq> Basis" "card S = Suc (Suc 0)"
-    using ex_card[OF assms] by auto
-  then obtain b0 b1 :: 'a where "b0 \<in> Basis" "b1 \<in> Basis" "b0 \<noteq> b1"
+  obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)"
+    using ex_card[OF assms]
+    by auto
+  then obtain b0 b1 :: 'a where "b0 \<in> Basis" and "b1 \<in> Basis" and "b0 \<noteq> b1"
     unfolding card_Suc_eq by auto
-  then have "a + b0 - b1 \<in> ?A \<inter> ?B" by (auto simp: inner_simps inner_Basis)
-  then have "?A \<inter> ?B \<noteq> {}" by fast
+  then have "a + b0 - b1 \<in> ?A \<inter> ?B"
+    by (auto simp: inner_simps inner_Basis)
+  then have "?A \<inter> ?B \<noteq> {}"
+    by fast
   with A B have "path_connected (?A \<union> ?B)"
     by (rule path_connected_Un)
   also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}"
     unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
   also have "\<dots> = {x. x \<noteq> a}"
-    unfolding euclidean_eq_iff [where 'a='a] by (simp add: Bex_def)
-  also have "\<dots> = UNIV - {a}" by auto
+    unfolding euclidean_eq_iff [where 'a='a]
+    by (simp add: Bex_def)
+  also have "\<dots> = UNIV - {a}"
+    by auto
   finally show ?thesis .
 qed
 
 lemma path_connected_sphere:
   assumes "2 \<le> DIM('a::euclidean_space)"
-  shows "path_connected {x::'a::euclidean_space. norm(x - a) = r}"
+  shows "path_connected {x::'a. norm (x - a) = r}"
 proof (rule linorder_cases [of r 0])
   assume "r < 0"
-  then have "{x::'a. norm(x - a) = r} = {}" by auto
-  then show ?thesis using path_connected_empty by simp
+  then have "{x::'a. norm(x - a) = r} = {}"
+    by auto
+  then show ?thesis
+    using path_connected_empty by simp
 next
   assume "r = 0"
-  then show ?thesis using path_connected_singleton by simp
+  then show ?thesis
+    using path_connected_singleton by simp
 next
   assume r: "0 < r"
-  then have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}"
-    apply -
-    apply (rule set_eqI, rule)
+  have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}"
+    apply (rule set_eqI)
+    apply rule
     unfolding image_iff
     apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI)
     unfolding mem_Collect_eq norm_scaleR
+    using r
     apply (auto simp add: scaleR_right_diff_distrib)
     done
   have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})"
-    apply (rule set_eqI,rule)
+    apply (rule set_eqI)
+    apply rule
     unfolding image_iff
     apply (rule_tac x=x in bexI)
     unfolding mem_Collect_eq
-    apply (auto split:split_if_asm)
+    apply (auto split: split_if_asm)
     done
   have "continuous_on (UNIV - {0}) (\<lambda>x::'a. 1 / norm x)"
-    unfolding field_divide_inverse by (simp add: continuous_on_intros)
-  then show ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
+    unfolding field_divide_inverse
+    by (simp add: continuous_on_intros)
+  then show ?thesis
+    unfolding * **
+    using path_connected_punctured_universe[OF assms]
     by (auto intro!: path_connected_continuous_image continuous_on_intros)
 qed
 
-lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm(x - a) = r}"
-  using path_connected_sphere path_connected_imp_connected by auto
+lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm (x - a) = r}"
+  using path_connected_sphere path_connected_imp_connected
+  by auto
 
 end