
diff -r a962f349c8c9 -r a95e7432d86c src/HOL/Transitive_Closure.thy
--- a/src/HOL/Transitive_Closure.thy	Wed Jul 06 14:09:13 2016 +0200
+++ b/src/HOL/Transitive_Closure.thy	Wed Jul 06 20:19:51 2016 +0200
@@ -65,67 +65,65 @@
 
 subsection \<open>Reflexive closure\<close>
 
-lemma refl_reflcl[simp]: "refl(r^=)"
-by(simp add:refl_on_def)
+lemma refl_reflcl[simp]: "refl (r\<^sup>=)"
+  by (simp add: refl_on_def)
 
-lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"
-by(simp add:antisym_def)
+lemma antisym_reflcl[simp]: "antisym (r\<^sup>=) = antisym r"
+  by (simp add: antisym_def)
 
-lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"
-unfolding trans_def by blast
+lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans (r\<^sup>=)"
+  unfolding trans_def by blast
 
-lemma reflclp_idemp [simp]: "(P^==)^==  =  P^=="
-by blast
+lemma reflclp_idemp [simp]: "(P\<^sup>=\<^sup>=)\<^sup>=\<^sup>= = P\<^sup>=\<^sup>="
+  by blast
+
 
 subsection \<open>Reflexive-transitive closure\<close>
 
 lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"
   by (auto simp add: fun_eq_iff)
 
-lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
+lemma r_into_rtrancl [intro]: "\<And>p. p \<in> r \<Longrightarrow> p \<in> r\<^sup>*"
   \<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
   apply (simp only: split_tupled_all)
   apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
   done
 
-lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"
+lemma r_into_rtranclp [intro]: "r x y \<Longrightarrow> r\<^sup>*\<^sup>* x y"
   \<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
   by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
 
-lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"
+lemma rtranclp_mono: "r \<le> s \<Longrightarrow> r\<^sup>*\<^sup>* \<le> s\<^sup>*\<^sup>*"
   \<comment> \<open>monotonicity of \<open>rtrancl\<close>\<close>
   apply (rule predicate2I)
   apply (erule rtranclp.induct)
    apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
   done
 
-lemma mono_rtranclp[mono]:
-   "(\<And>a b. x a b \<longrightarrow> y a b) \<Longrightarrow> x^** a b \<longrightarrow> y^** a b"
+lemma mono_rtranclp[mono]: "(\<And>a b. x a b \<longrightarrow> y a b) \<Longrightarrow> x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
    using rtranclp_mono[of x y] by auto
 
 lemmas rtrancl_mono = rtranclp_mono [to_set]
 
 theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
-  assumes a: "r^** a b"
-    and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
-  shows "P b" using a
-  by (induct x\<equiv>a b) (rule cases)+
+  assumes a: "r\<^sup>*\<^sup>* a b"
+    and cases: "P a" "\<And>y z. r\<^sup>*\<^sup>* a y \<Longrightarrow> r y z \<Longrightarrow> P y \<Longrightarrow> P z"
+  shows "P b"
+  using a by (induct x\<equiv>a b) (rule cases)+
 
 lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]
 
 lemmas rtranclp_induct2 =
-  rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,
-                 consumes 1, case_names refl step]
+  rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names refl step]
 
 lemmas rtrancl_induct2 =
-  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
-                 consumes 1, case_names refl step]
+  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), consumes 1, case_names refl step]
 
-lemma refl_rtrancl: "refl (r^*)"
-by (unfold refl_on_def) fast
+lemma refl_rtrancl: "refl (r\<^sup>*)"
+  unfolding refl_on_def by fast
 
 text \<open>Transitivity of transitive closure.\<close>
-lemma trans_rtrancl: "trans (r^*)"
+lemma trans_rtrancl: "trans (r\<^sup>*)"
 proof (rule transI)
   fix x y z
   assume "(x, y) \<in> r\<^sup>*"
@@ -144,42 +142,40 @@
 lemmas rtrancl_trans = trans_rtrancl [THEN transD]
 
 lemma rtranclp_trans:
-  assumes xy: "r^** x y"
-  and yz: "r^** y z"
-  shows "r^** x z" using yz xy
-  by induct iprover+
+  assumes "r\<^sup>*\<^sup>* x y"
+    and "r\<^sup>*\<^sup>* y z"
+  shows "r\<^sup>*\<^sup>* x z"
+  using assms(2,1) by induct iprover+
 
 lemma rtranclE [cases set: rtrancl]:
-  assumes major: "(a::'a, b) : r^*"
+  fixes a b :: 'a
+  assumes major: "(a, b) \<in> r\<^sup>*"
   obtains
     (base) "a = b"
-  | (step) y where "(a, y) : r^*" and "(y, b) : r"
+  | (step) y where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
   \<comment> \<open>elimination of \<open>rtrancl\<close> -- by induction on a special formula\<close>
-  apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
+  apply (subgoal_tac "a = b \<or> (\<exists>y. (a, y) \<in> r\<^sup>* \<and> (y, b) \<in> r)")
    apply (rule_tac [2] major [THEN rtrancl_induct])
     prefer 2 apply blast
    prefer 2 apply blast
   apply (erule asm_rl exE disjE conjE base step)+
   done
 
-lemma rtrancl_Int_subset: "[| Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s|] ==> r^* \<subseteq> s"
+lemma rtrancl_Int_subset: "Id \<subseteq> s \<Longrightarrow> (r\<^sup>* \<inter> s) O r \<subseteq> s \<Longrightarrow> r\<^sup>* \<subseteq> s"
   apply (rule subsetI)
   apply auto
   apply (erule rtrancl_induct)
   apply auto
   done
 
-lemma converse_rtranclp_into_rtranclp:
-  "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
+lemma converse_rtranclp_into_rtranclp: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
   by (rule rtranclp_trans) iprover+
 
 lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
 
-text \<open>
-  \medskip More @{term "r^*"} equations and inclusions.
-\<close>
+text \<open>\<^medskip> More @{term "r\<^sup>*"} equations and inclusions.\<close>
 
-lemma rtranclp_idemp [simp]: "(r^**)^** = r^**"
+lemma rtranclp_idemp [simp]: "(r\<^sup>*\<^sup>*)\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
   apply (auto intro!: order_antisym)
   apply (erule rtranclp_induct)
    apply (rule rtranclp.rtrancl_refl)
@@ -188,18 +184,18 @@
 
 lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
 
-lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
+lemma rtrancl_idemp_self_comp [simp]: "R\<^sup>* O R\<^sup>* = R\<^sup>*"
   apply (rule set_eqI)
   apply (simp only: split_tupled_all)
   apply (blast intro: rtrancl_trans)
   done
 
-lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
+lemma rtrancl_subset_rtrancl: "r \<subseteq> s\<^sup>* \<Longrightarrow> r\<^sup>* \<subseteq> s\<^sup>*"
   apply (drule rtrancl_mono)
   apply simp
   done
 
-lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**"
+lemma rtranclp_subset: "R \<le> S \<Longrightarrow> S \<le> R\<^sup>*\<^sup>* \<Longrightarrow> S\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
   apply (drule rtranclp_mono)
   apply (drule rtranclp_mono)
   apply simp
@@ -207,17 +203,17 @@
 
 lemmas rtrancl_subset = rtranclp_subset [to_set]
 
-lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**"
-by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
+lemma rtranclp_sup_rtranclp: "(sup (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*))\<^sup>*\<^sup>* = (sup R S)\<^sup>*\<^sup>*"
+  by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
 
 lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
 
-lemma rtranclp_reflclp [simp]: "(R^==)^** = R^**"
-by (blast intro!: rtranclp_subset)
+lemma rtranclp_reflclp [simp]: "(R\<^sup>=\<^sup>=)\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
+  by (blast intro!: rtranclp_subset)
 
 lemmas rtrancl_reflcl [simp] = rtranclp_reflclp [to_set]
 
-lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
+lemma rtrancl_r_diff_Id: "(r - Id)\<^sup>* = r\<^sup>*"
   apply (rule sym)
   apply (rule rtrancl_subset, blast, clarify)
   apply (rename_tac a b)
@@ -226,39 +222,35 @@
   apply blast
   done
 
-lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"
+lemma rtranclp_r_diff_Id: "(inf r op \<noteq>)\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
   apply (rule sym)
   apply (rule rtranclp_subset)
    apply blast+
   done
 
 theorem rtranclp_converseD:
-  assumes r: "(r^--1)^** x y"
-  shows "r^** y x"
-proof -
-  from r show ?thesis
-    by induct (iprover intro: rtranclp_trans dest!: conversepD)+
-qed
+  assumes "(r\<inverse>\<inverse>)\<^sup>*\<^sup>* x y"
+  shows "r\<^sup>*\<^sup>* y x"
+  using assms by induct (iprover intro: rtranclp_trans dest!: conversepD)+
 
 lemmas rtrancl_converseD = rtranclp_converseD [to_set]
 
 theorem rtranclp_converseI:
-  assumes "r^** y x"
-  shows "(r^--1)^** x y"
-  using assms
-  by induct (iprover intro: rtranclp_trans conversepI)+
+  assumes "r\<^sup>*\<^sup>* y x"
+  shows "(r\<inverse>\<inverse>)\<^sup>*\<^sup>* x y"
+  using assms by induct (iprover intro: rtranclp_trans conversepI)+
 
 lemmas rtrancl_converseI = rtranclp_converseI [to_set]
 
-lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
+lemma rtrancl_converse: "(r^-1)\<^sup>* = (r\<^sup>*)^-1"
   by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
 
-lemma sym_rtrancl: "sym r ==> sym (r^*)"
+lemma sym_rtrancl: "sym r \<Longrightarrow> sym (r\<^sup>*)"
   by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
 
 theorem converse_rtranclp_induct [consumes 1, case_names base step]:
-  assumes major: "r^** a b"
-    and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"
+  assumes major: "r\<^sup>*\<^sup>* a b"
+    and cases: "P b" "\<And>y z. r y z \<Longrightarrow> r\<^sup>*\<^sup>* z b \<Longrightarrow> P z \<Longrightarrow> P y"
   shows "P a"
   using rtranclp_converseI [OF major]
   by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
@@ -266,19 +258,17 @@
 lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
 
 lemmas converse_rtranclp_induct2 =
-  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
-                 consumes 1, case_names refl step]
+  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names refl step]
 
 lemmas converse_rtrancl_induct2 =
   converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
-                 consumes 1, case_names refl step]
+    consumes 1, case_names refl step]
 
 lemma converse_rtranclpE [consumes 1, case_names base step]:
-  assumes major: "r^** x z"
-    and cases: "x=z ==> P"
-      "!!y. [| r x y; r^** y z |] ==> P"
+  assumes major: "r\<^sup>*\<^sup>* x z"
+    and cases: "x = z \<Longrightarrow> P" "\<And>y. r x y \<Longrightarrow> r\<^sup>*\<^sup>* y z \<Longrightarrow> P"
   shows P
-  apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")
+  apply (subgoal_tac "x = z \<or> (\<exists>y. r x y \<and> r\<^sup>*\<^sup>* y z)")
    apply (rule_tac [2] major [THEN converse_rtranclp_induct])
     prefer 2 apply iprover
    prefer 2 apply iprover
@@ -291,41 +281,42 @@
 
 lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
 
-lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
+lemma r_comp_rtrancl_eq: "r O r\<^sup>* = r\<^sup>* O r"
   by (blast elim: rtranclE converse_rtranclE
     intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
 
-lemma rtrancl_unfold: "r^* = Id Un r^* O r"
+lemma rtrancl_unfold: "r\<^sup>* = Id \<union> r\<^sup>* O r"
   by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
 
 lemma rtrancl_Un_separatorE:
-  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*"
-apply (induct rule:rtrancl.induct)
- apply blast
-apply (blast intro:rtrancl_trans)
-done
+  "(a, b) \<in> (P \<union> Q)\<^sup>* \<Longrightarrow> \<forall>x y. (a, x) \<in> P\<^sup>* \<longrightarrow> (x, y) \<in> Q \<longrightarrow> x = y \<Longrightarrow> (a, b) \<in> P\<^sup>*"
+  apply (induct rule:rtrancl.induct)
+   apply blast
+  apply (blast intro:rtrancl_trans)
+  done
 
 lemma rtrancl_Un_separator_converseE:
-  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*"
-apply (induct rule:converse_rtrancl_induct)
- apply blast
-apply (blast intro:rtrancl_trans)
-done
+  "(a, b) \<in> (P \<union> Q)\<^sup>* \<Longrightarrow> \<forall>x y. (x, b) \<in> P\<^sup>* \<longrightarrow> (y, x) \<in> Q \<longrightarrow> y = x \<Longrightarrow> (a, b) \<in> P\<^sup>*"
+  apply (induct rule:converse_rtrancl_induct)
+   apply blast
+  apply (blast intro:rtrancl_trans)
+  done
 
 lemma Image_closed_trancl:
-  assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
+  assumes "r `` X \<subseteq> X"
+  shows "r\<^sup>* `` X = X"
 proof -
-  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto
-  have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"
+  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X"
+    by auto
+  have "x \<in> X" if 1: "(y, x) \<in> r\<^sup>*" and 2: "y \<in> X" for x y
   proof -
-    fix x y
-    assume *: "y \<in> X"
-    assume "(y, x) \<in> r\<^sup>*"
-    then show "x \<in> X"
+    from 1 show "x \<in> X"
     proof induct
-      case base show ?case by (fact *)
+      case base
+      show ?case by (fact 2)
     next
-      case step with ** show ?case by auto
+      case step
+      with ** show ?case by auto
     qed
   qed
   then show ?thesis by auto
@@ -334,31 +325,30 @@
 
 subsection \<open>Transitive closure\<close>
 
-lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
+lemma trancl_mono: "\<And>p. p \<in> r\<^sup>+ \<Longrightarrow> r \<subseteq> s \<Longrightarrow> p \<in> s\<^sup>+"
   apply (simp add: split_tupled_all)
   apply (erule trancl.induct)
    apply (iprover dest: subsetD)+
   done
 
-lemma r_into_trancl': "!!p. p : r ==> p : r^+"
+lemma r_into_trancl': "\<And>p. p \<in> r \<Longrightarrow> p \<in> r\<^sup>+"
   by (simp only: split_tupled_all) (erule r_into_trancl)
 
-text \<open>
-  \medskip Conversions between \<open>trancl\<close> and \<open>rtrancl\<close>.
-\<close>
+text \<open>\<^medskip> Conversions between \<open>trancl\<close> and \<open>rtrancl\<close>.\<close>
 
-lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
+lemma tranclp_into_rtranclp: "r\<^sup>+\<^sup>+ a b \<Longrightarrow> r\<^sup>*\<^sup>* a b"
   by (erule tranclp.induct) iprover+
 
 lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
 
-lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
-  shows "!!c. r b c ==> r^++ a c" using r
-  by induct iprover+
+lemma rtranclp_into_tranclp1:
+  assumes "r\<^sup>*\<^sup>* a b"
+  shows "r b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
+  using assms by (induct arbitrary: c) iprover+
 
 lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
 
-lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
+lemma rtranclp_into_tranclp2: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
   \<comment> \<open>intro rule from \<open>r\<close> and \<open>rtrancl\<close>\<close>
   apply (erule rtranclp.cases)
    apply iprover
@@ -370,26 +360,23 @@
 
 text \<open>Nice induction rule for \<open>trancl\<close>\<close>
 lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
-  assumes a: "r^++ a b"
-  and cases: "!!y. r a y ==> P y"
-    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
-  shows "P b" using a
-  by (induct x\<equiv>a b) (iprover intro: cases)+
+  assumes a: "r\<^sup>+\<^sup>+ a b"
+    and cases: "\<And>y. r a y \<Longrightarrow> P y" "\<And>y z. r\<^sup>+\<^sup>+ a y \<Longrightarrow> r y z \<Longrightarrow> P y \<Longrightarrow> P z"
+  shows "P b"
+  using a by (induct x\<equiv>a b) (iprover intro: cases)+
 
 lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
 
 lemmas tranclp_induct2 =
-  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
-    consumes 1, case_names base step]
+  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names base step]
 
 lemmas trancl_induct2 =
   trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
     consumes 1, case_names base step]
 
 lemma tranclp_trans_induct:
-  assumes major: "r^++ x y"
-    and cases: "!!x y. r x y ==> P x y"
-      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
+  assumes major: "r\<^sup>+\<^sup>+ x y"
+    and cases: "\<And>x y. r x y \<Longrightarrow> P x y" "\<And>x y z. r\<^sup>+\<^sup>+ x y \<Longrightarrow> P x y \<Longrightarrow> r\<^sup>+\<^sup>+ y z \<Longrightarrow> P y z \<Longrightarrow> P x z"
   shows "P x y"
   \<comment> \<open>Another induction rule for trancl, incorporating transitivity\<close>
   by (iprover intro: major [THEN tranclp_induct] cases)
@@ -397,49 +384,49 @@
 lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
 
 lemma tranclE [cases set: trancl]:
-  assumes "(a, b) : r^+"
+  assumes "(a, b) \<in> r\<^sup>+"
   obtains
-    (base) "(a, b) : r"
-  | (step) c where "(a, c) : r^+" and "(c, b) : r"
+    (base) "(a, b) \<in> r"
+  | (step) c where "(a, c) \<in> r\<^sup>+" and "(c, b) \<in> r"
   using assms by cases simp_all
 
-lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
+lemma trancl_Int_subset: "r \<subseteq> s \<Longrightarrow> (r\<^sup>+ \<inter> s) O r \<subseteq> s \<Longrightarrow> r\<^sup>+ \<subseteq> s"
   apply (rule subsetI)
   apply auto
   apply (erule trancl_induct)
   apply auto
   done
 
-lemma trancl_unfold: "r^+ = r Un r^+ O r"
+lemma trancl_unfold: "r\<^sup>+ = r \<union> r\<^sup>+ O r"
   by (auto intro: trancl_into_trancl elim: tranclE)
 
-text \<open>Transitivity of @{term "r^+"}\<close>
-lemma trans_trancl [simp]: "trans (r^+)"
+text \<open>Transitivity of @{term "r\<^sup>+"}\<close>
+lemma trans_trancl [simp]: "trans (r\<^sup>+)"
 proof (rule transI)
   fix x y z
-  assume "(x, y) \<in> r^+"
-  assume "(y, z) \<in> r^+"
-  then show "(x, z) \<in> r^+"
+  assume "(x, y) \<in> r\<^sup>+"
+  assume "(y, z) \<in> r\<^sup>+"
+  then show "(x, z) \<in> r\<^sup>+"
   proof induct
     case (base u)
-    from \<open>(x, y) \<in> r^+\<close> and \<open>(y, u) \<in> r\<close>
-    show "(x, u) \<in> r^+" ..
+    from \<open>(x, y) \<in> r\<^sup>+\<close> and \<open>(y, u) \<in> r\<close>
+    show "(x, u) \<in> r\<^sup>+" ..
   next
     case (step u v)
-    from \<open>(x, u) \<in> r^+\<close> and \<open>(u, v) \<in> r\<close>
-    show "(x, v) \<in> r^+" ..
+    from \<open>(x, u) \<in> r\<^sup>+\<close> and \<open>(u, v) \<in> r\<close>
+    show "(x, v) \<in> r\<^sup>+" ..
   qed
 qed
 
 lemmas trancl_trans = trans_trancl [THEN transD]
 
 lemma tranclp_trans:
-  assumes xy: "r^++ x y"
-  and yz: "r^++ y z"
-  shows "r^++ x z" using yz xy
-  by induct iprover+
+  assumes "r\<^sup>+\<^sup>+ x y"
+    and "r\<^sup>+\<^sup>+ y z"
+  shows "r\<^sup>+\<^sup>+ x z"
+  using assms(2,1) by induct iprover+
 
-lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
+lemma trancl_id [simp]: "trans r \<Longrightarrow> r\<^sup>+ = r"
   apply auto
   apply (erule trancl_induct)
    apply assumption
@@ -448,18 +435,18 @@
   done
 
 lemma rtranclp_tranclp_tranclp:
-  assumes "r^** x y"
-  shows "!!z. r^++ y z ==> r^++ x z" using assms
-  by induct (iprover intro: tranclp_trans)+
+  assumes "r\<^sup>*\<^sup>* x y"
+  shows "\<And>z. r\<^sup>+\<^sup>+ y z \<Longrightarrow> r\<^sup>+\<^sup>+ x z"
+  using assms by induct (iprover intro: tranclp_trans)+
 
 lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
 
-lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
+lemma tranclp_into_tranclp2: "r a b \<Longrightarrow> r\<^sup>+\<^sup>+ b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
   by (erule tranclp_trans [OF tranclp.r_into_trancl])
 
 lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
 
-lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
+lemma tranclp_converseI: "(r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y \<Longrightarrow> (r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y"
   apply (drule conversepD)
   apply (erule tranclp_induct)
   apply (iprover intro: conversepI tranclp_trans)+
@@ -467,7 +454,7 @@
 
 lemmas trancl_converseI = tranclp_converseI [to_set]
 
-lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
+lemma tranclp_converseD: "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y \<Longrightarrow> (r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y"
   apply (rule conversepI)
   apply (erule tranclp_induct)
   apply (iprover dest: conversepD intro: tranclp_trans)+
@@ -475,19 +462,17 @@
 
 lemmas trancl_converseD = tranclp_converseD [to_set]
 
-lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
-  by (fastforce simp add: fun_eq_iff
-    intro!: tranclp_converseI dest!: tranclp_converseD)
+lemma tranclp_converse: "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ = (r\<^sup>+\<^sup>+)\<inverse>\<inverse>"
+  by (fastforce simp add: fun_eq_iff intro!: tranclp_converseI dest!: tranclp_converseD)
 
 lemmas trancl_converse = tranclp_converse [to_set]
 
-lemma sym_trancl: "sym r ==> sym (r^+)"
+lemma sym_trancl: "sym r \<Longrightarrow> sym (r\<^sup>+)"
   by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
 
 lemma converse_tranclp_induct [consumes 1, case_names base step]:
-  assumes major: "r^++ a b"
-    and cases: "!!y. r y b ==> P(y)"
-      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
+  assumes major: "r\<^sup>+\<^sup>+ a b"
+    and cases: "\<And>y. r y b \<Longrightarrow> P y" "\<And>y z. r y z \<Longrightarrow> r\<^sup>+\<^sup>+ z b \<Longrightarrow> P z \<Longrightarrow> P y"
   shows "P a"
   apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
    apply (rule cases)
@@ -497,7 +482,7 @@
 
 lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
 
-lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
+lemma tranclpD: "R\<^sup>+\<^sup>+ x y \<Longrightarrow> \<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
   apply (erule converse_tranclp_induct)
    apply auto
   apply (blast intro: rtranclp_trans)
@@ -507,48 +492,48 @@
 
 lemma converse_tranclpE:
   assumes major: "tranclp r x z"
-  assumes base: "r x z ==> P"
-  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
+    and base: "r x z \<Longrightarrow> P"
+    and step: "\<And> y. r x y \<Longrightarrow> tranclp r y z \<Longrightarrow> P"
   shows P
 proof -
-  from tranclpD[OF major]
-  obtain y where "r x y" and "rtranclp r y z" by iprover
+  from tranclpD [OF major] obtain y where "r x y" and "rtranclp r y z"
+    by iprover
   from this(2) show P
   proof (cases rule: rtranclp.cases)
     case rtrancl_refl
-    with \<open>r x y\<close> base show P by iprover
+    with \<open>r x y\<close> base show P
+      by iprover
   next
     case rtrancl_into_rtrancl
     from this have "tranclp r y z"
       by (iprover intro: rtranclp_into_tranclp1)
-    with \<open>r x y\<close> step show P by iprover
+    with \<open>r x y\<close> step show P
+      by iprover
   qed
 qed
 
 lemmas converse_tranclE = converse_tranclpE [to_set]
 
-lemma tranclD2:
-  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
+lemma tranclD2: "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
   by (blast elim: tranclE intro: trancl_into_rtrancl)
 
-lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
+lemma irrefl_tranclI: "r\<inverse> \<inter> r\<^sup>* = {} \<Longrightarrow> (x, x) \<notin> r\<^sup>+"
   by (blast elim: tranclE dest: trancl_into_rtrancl)
 
-lemma irrefl_trancl_rD: "\<forall>x. (x, x) \<notin> r^+ \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> x \<noteq> y"
+lemma irrefl_trancl_rD: "\<forall>x. (x, x) \<notin> r\<^sup>+ \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> x \<noteq> y"
   by (blast dest: r_into_trancl)
 
-lemma trancl_subset_Sigma_aux:
-    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
+lemma trancl_subset_Sigma_aux: "(a, b) \<in> r\<^sup>* \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> a = b \<or> a \<in> A"
   by (induct rule: rtrancl_induct) auto
 
-lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
+lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A \<Longrightarrow> r\<^sup>+ \<subseteq> A \<times> A"
   apply (rule subsetI)
   apply (simp only: split_tupled_all)
   apply (erule tranclE)
    apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
   done
 
-lemma reflclp_tranclp [simp]: "(r^++)^== = r^**"
+lemma reflclp_tranclp [simp]: "(r\<^sup>+\<^sup>+)\<^sup>=\<^sup>= = r\<^sup>*\<^sup>*"
   apply (safe intro!: order_antisym)
    apply (erule tranclp_into_rtranclp)
   apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
@@ -556,7 +541,7 @@
 
 lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
 
-lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
+lemma trancl_reflcl [simp]: "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
   apply safe
    apply (drule trancl_into_rtrancl, simp)
   apply (erule rtranclE, safe)
@@ -565,32 +550,30 @@
    apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
   done
 
-lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
+lemma rtrancl_trancl_reflcl [code]: "r\<^sup>* = (r\<^sup>+)\<^sup>="
   by simp
 
-lemma trancl_empty [simp]: "{}^+ = {}"
+lemma trancl_empty [simp]: "{}\<^sup>+ = {}"
   by (auto elim: trancl_induct)
 
-lemma rtrancl_empty [simp]: "{}^* = Id"
+lemma rtrancl_empty [simp]: "{}\<^sup>* = Id"
   by (rule subst [OF reflcl_trancl]) simp
 
-lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
-by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
+lemma rtranclpD: "R\<^sup>*\<^sup>* a b \<Longrightarrow> a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
+  by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
 
 lemmas rtranclD = rtranclpD [to_set]
 
-lemma rtrancl_eq_or_trancl:
-  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
+lemma rtrancl_eq_or_trancl: "(x,y) \<in> R\<^sup>* \<longleftrightarrow> x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+"
   by (fast elim: trancl_into_rtrancl dest: rtranclD)
 
-lemma trancl_unfold_right: "r^+ = r^* O r"
-by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
+lemma trancl_unfold_right: "r\<^sup>+ = r\<^sup>* O r"
+  by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
 
-lemma trancl_unfold_left: "r^+ = r O r^*"
-by (auto dest: tranclD intro: rtrancl_into_trancl2)
+lemma trancl_unfold_left: "r\<^sup>+ = r O r\<^sup>*"
+  by (auto dest: tranclD intro: rtrancl_into_trancl2)
 
-lemma trancl_insert:
-  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
+lemma trancl_insert: "(insert (y, x) r)\<^sup>+ = r\<^sup>+ \<union> {(a, b). (a, y) \<in> r\<^sup>* \<and> (x, b) \<in> r\<^sup>*}"
   \<comment> \<open>primitive recursion for \<open>trancl\<close> over finite relations\<close>
   apply (rule equalityI)
    apply (rule subsetI)
@@ -603,62 +586,60 @@
   done
 
 lemma trancl_insert2:
-  "(insert (a,b) r)^+ = r^+ \<union> {(x,y). ((x,a) : r^+ \<or> x=a) \<and> ((b,y) \<in> r^+ \<or> y=b)}"
-by(auto simp add: trancl_insert rtrancl_eq_or_trancl)
+  "(insert (a, b) r)\<^sup>+ = r\<^sup>+ \<union> {(x, y). ((x, a) \<in> r\<^sup>+ \<or> x = a) \<and> ((b, y) \<in> r\<^sup>+ \<or> y = b)}"
+  by (auto simp add: trancl_insert rtrancl_eq_or_trancl)
 
-lemma rtrancl_insert:
-  "(insert (a,b) r)^* = r^* \<union> {(x,y). (x,a) : r^* \<and> (b,y) \<in> r^*}"
-using trancl_insert[of a b r]
-by(simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast
+lemma rtrancl_insert: "(insert (a,b) r)\<^sup>* = r\<^sup>* \<union> {(x, y). (x, a) \<in> r\<^sup>* \<and> (b, y) \<in> r\<^sup>*}"
+  using trancl_insert[of a b r]
+  by (simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast
 
 
 text \<open>Simplifying nested closures\<close>
 
-lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
-by (simp add: trans_rtrancl)
+lemma rtrancl_trancl_absorb[simp]: "(R\<^sup>*)\<^sup>+ = R\<^sup>*"
+  by (simp add: trans_rtrancl)
 
-lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
-by (subst reflcl_trancl[symmetric]) simp
+lemma trancl_rtrancl_absorb[simp]: "(R\<^sup>+)\<^sup>* = R\<^sup>*"
+  by (subst reflcl_trancl[symmetric]) simp
 
-lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
-by auto
+lemma rtrancl_reflcl_absorb[simp]: "(R\<^sup>*)\<^sup>= = R\<^sup>*"
+  by auto
 
 
 text \<open>\<open>Domain\<close> and \<open>Range\<close>\<close>
 
-lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
+lemma Domain_rtrancl [simp]: "Domain (R\<^sup>*) = UNIV"
   by blast
 
-lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
+lemma Range_rtrancl [simp]: "Range (R\<^sup>*) = UNIV"
   by blast
 
-lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
+lemma rtrancl_Un_subset: "(R\<^sup>* \<union> S\<^sup>*) \<subseteq> (R \<union> S)\<^sup>*"
   by (rule rtrancl_Un_rtrancl [THEN subst]) fast
 
-lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
+lemma in_rtrancl_UnI: "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>* \<Longrightarrow> x \<in> (R \<union> S)\<^sup>*"
   by (blast intro: subsetD [OF rtrancl_Un_subset])
 
-lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
+lemma trancl_domain [simp]: "Domain (r\<^sup>+) = Domain r"
   by (unfold Domain_unfold) (blast dest: tranclD)
 
-lemma trancl_range [simp]: "Range (r^+) = Range r"
+lemma trancl_range [simp]: "Range (r\<^sup>+) = Range r"
   unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
 
-lemma Not_Domain_rtrancl:
-    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
+lemma Not_Domain_rtrancl: "x \<notin> Domain R \<Longrightarrow> (x, y) \<in> R\<^sup>* \<longleftrightarrow> x = y"
   apply auto
   apply (erule rev_mp)
   apply (erule rtrancl_induct)
    apply auto
   done
 
-lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
+lemma trancl_subset_Field2: "r\<^sup>+ \<subseteq> Field r \<times> Field r"
   apply clarify
   apply (erule trancl_induct)
    apply (auto simp add: Field_def)
   done
 
-lemma finite_trancl[simp]: "finite (r^+) = finite r"
+lemma finite_trancl[simp]: "finite (r\<^sup>+) = finite r"
   apply auto
    prefer 2
    apply (rule trancl_subset_Field2 [THEN finite_subset])
@@ -672,8 +653,7 @@
   be merged with main body.\<close>
 
 lemma single_valued_confluent:
-  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
-  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
+  "single_valued r \<Longrightarrow> (x, y) \<in> r\<^sup>* \<Longrightarrow> (x, z) \<in> r\<^sup>* \<Longrightarrow> (y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
   apply (erule rtrancl_induct)
   apply simp
   apply (erule disjE)
@@ -681,18 +661,16 @@
   apply(blast intro:rtrancl_trans)
   done
 
-lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
+lemma r_r_into_trancl: "(a, b) \<in> R \<Longrightarrow> (b, c) \<in> R \<Longrightarrow> (a, c) \<in> R\<^sup>+"
   by (fast intro: trancl_trans)
 
-lemma trancl_into_trancl [rule_format]:
-    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
-  apply (erule trancl_induct)
+lemma trancl_into_trancl: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
+  apply (induct rule: trancl_induct)
    apply (fast intro: r_r_into_trancl)
   apply (fast intro: r_r_into_trancl trancl_trans)
   done
 
-lemma tranclp_rtranclp_tranclp:
-    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
+lemma tranclp_rtranclp_tranclp: "r\<^sup>+\<^sup>+ a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
   apply (drule tranclpD)
   apply (elim exE conjE)
   apply (drule rtranclp_trans, assumption)
@@ -715,37 +693,39 @@
 
 declare trancl_into_rtrancl [elim]
 
+
 subsection \<open>The power operation on relations\<close>
 
-text \<open>\<open>R ^^ n = R O ... O R\<close>, the n-fold composition of \<open>R\<close>\<close>
+text \<open>\<open>R ^^ n = R O \<dots> O R\<close>, the n-fold composition of \<open>R\<close>\<close>
 
 overloading
-  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
-  relpowp == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
+  relpow \<equiv> "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
+  relpowp \<equiv> "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
 begin
 
-primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
-    "relpow 0 R = Id"
-  | "relpow (Suc n) R = (R ^^ n) O R"
+primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
+where
+  "relpow 0 R = Id"
+| "relpow (Suc n) R = (R ^^ n) O R"
 
-primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
-    "relpowp 0 R = HOL.eq"
-  | "relpowp (Suc n) R = (R ^^ n) OO R"
+primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
+where
+  "relpowp 0 R = HOL.eq"
+| "relpowp (Suc n) R = (R ^^ n) OO R"
 
 end
 
 lemma relpowp_relpow_eq [pred_set_conv]:
-  fixes R :: "'a rel"
-  shows "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)"
+  "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)" for R :: "'a rel"
   by (induct n) (simp_all add: relcompp_relcomp_eq)
 
-text \<open>for code generation\<close>
+text \<open>For code generation:\<close>
 
-definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
-  relpow_code_def [code_abbrev]: "relpow = compow"
+definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
+  where relpow_code_def [code_abbrev]: "relpow = compow"
 
-definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
-  relpowp_code_def [code_abbrev]: "relpowp = compow"
+definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
+  where relpowp_code_def [code_abbrev]: "relpowp = compow"
 
 lemma [code]:
   "relpow (Suc n) R = (relpow n R) O R"
@@ -760,54 +740,40 @@
 hide_const (open) relpow
 hide_const (open) relpowp
 
-lemma relpow_1 [simp]:
-  fixes R :: "('a \<times> 'a) set"
-  shows "R ^^ 1 = R"
+lemma relpow_1 [simp]: "R ^^ 1 = R" for R :: "('a \<times> 'a) set"
   by simp
 
-lemma relpowp_1 [simp]:
-  fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
-  shows "P ^^ 1 = P"
+lemma relpowp_1 [simp]: "P ^^ 1 = P" for P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
   by (fact relpow_1 [to_pred])
 
-lemma relpow_0_I: 
-  "(x, x) \<in> R ^^ 0"
+lemma relpow_0_I: "(x, x) \<in> R ^^ 0"
   by simp
 
-lemma relpowp_0_I:
-  "(P ^^ 0) x x"
+lemma relpowp_0_I: "(P ^^ 0) x x"
   by (fact relpow_0_I [to_pred])
 
-lemma relpow_Suc_I:
-  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
+lemma relpow_Suc_I: "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
   by auto
 
-lemma relpowp_Suc_I:
-  "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
+lemma relpowp_Suc_I: "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
   by (fact relpow_Suc_I [to_pred])
 
-lemma relpow_Suc_I2:
-  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
+lemma relpow_Suc_I2: "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
   by (induct n arbitrary: z) (simp, fastforce)
 
-lemma relpowp_Suc_I2:
-  "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
+lemma relpowp_Suc_I2: "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
   by (fact relpow_Suc_I2 [to_pred])
 
-lemma relpow_0_E:
-  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
+lemma relpow_0_E: "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
   by simp
 
-lemma relpowp_0_E:
-  "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
+lemma relpowp_0_E: "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
   by (fact relpow_0_E [to_pred])
 
-lemma relpow_Suc_E:
-  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
+lemma relpow_Suc_E: "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
   by auto
 
-lemma relpowp_Suc_E:
-  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
+lemma relpowp_Suc_E: "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
   by (fact relpow_Suc_E [to_pred])
 
 lemma relpow_E:
@@ -822,31 +788,25 @@
   \<Longrightarrow> Q"
   by (fact relpow_E [to_pred])
 
-lemma relpow_Suc_D2:
-  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
+lemma relpow_Suc_D2: "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
   apply (induct n arbitrary: x z)
    apply (blast intro: relpow_0_I elim: relpow_0_E relpow_Suc_E)
   apply (blast intro: relpow_Suc_I elim: relpow_0_E relpow_Suc_E)
   done
 
-lemma relpowp_Suc_D2:
-  "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
+lemma relpowp_Suc_D2: "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
   by (fact relpow_Suc_D2 [to_pred])
 
-lemma relpow_Suc_E2:
-  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
+lemma relpow_Suc_E2: "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
   by (blast dest: relpow_Suc_D2)
 
-lemma relpowp_Suc_E2:
-  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
+lemma relpowp_Suc_E2: "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
   by (fact relpow_Suc_E2 [to_pred])
 
-lemma relpow_Suc_D2':
-  "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
+lemma relpow_Suc_D2': "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
   by (induct n) (simp_all, blast)
 
-lemma relpowp_Suc_D2':
-  "\<forall>x y z. (P ^^ n) x y \<and> P y z \<longrightarrow> (\<exists>w. P x w \<and> (P ^^ n) w z)"
+lemma relpowp_Suc_D2': "\<forall>x y z. (P ^^ n) x y \<and> P y z \<longrightarrow> (\<exists>w. P x w \<and> (P ^^ n) w z)"
   by (fact relpow_Suc_D2' [to_pred])
 
 lemma relpow_E2:
@@ -864,83 +824,78 @@
   \<Longrightarrow> Q"
   by (fact relpow_E2 [to_pred])
 
-lemma relpow_add: "R ^^ (m+n) = R^^m O R^^n"
+lemma relpow_add: "R ^^ (m + n) = R^^m O R^^n"
   by (induct n) auto
 
 lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
   by (fact relpow_add [to_pred])
 
 lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
-  by (induct n) (simp, simp add: O_assoc [symmetric])
+  by (induct n) (simp_all add: O_assoc [symmetric])
 
 lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
   by (fact relpow_commute [to_pred])
 
-lemma relpow_empty:
-  "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
+lemma relpow_empty: "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
   by (cases n) auto
 
-lemma relpowp_bot:
-  "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
+lemma relpowp_bot: "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
   by (fact relpow_empty [to_pred])
 
 lemma rtrancl_imp_UN_relpow:
-  assumes "p \<in> R^*"
+  assumes "p \<in> R\<^sup>*"
   shows "p \<in> (\<Union>n. R ^^ n)"
 proof (cases p)
   case (Pair x y)
-  with assms have "(x, y) \<in> R^*" by simp
+  with assms have "(x, y) \<in> R\<^sup>*" by simp
   then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
-    case base show ?case by (blast intro: relpow_0_I)
+    case base
+    show ?case by (blast intro: relpow_0_I)
   next
-    case step then show ?case by (blast intro: relpow_Suc_I)
+    case step
+    then show ?case by (blast intro: relpow_Suc_I)
   qed
   with Pair show ?thesis by simp
 qed
 
 lemma rtranclp_imp_Sup_relpowp:
-  assumes "(P^**) x y"
+  assumes "(P\<^sup>*\<^sup>*) x y"
   shows "(\<Squnion>n. P ^^ n) x y"
   using assms and rtrancl_imp_UN_relpow [of "(x, y)", to_pred] by simp
 
 lemma relpow_imp_rtrancl:
   assumes "p \<in> R ^^ n"
-  shows "p \<in> R^*"
+  shows "p \<in> R\<^sup>*"
 proof (cases p)
   case (Pair x y)
   with assms have "(x, y) \<in> R ^^ n" by simp
-  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
-    case 0 then show ?case by simp
+  then have "(x, y) \<in> R\<^sup>*" proof (induct n arbitrary: x y)
+    case 0
+    then show ?case by simp
   next
-    case Suc then show ?case
+    case Suc
+    then show ?case
       by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
   qed
   with Pair show ?thesis by simp
 qed
 
-lemma relpowp_imp_rtranclp:
-  assumes "(P ^^ n) x y"
-  shows "(P^**) x y"
-  using assms and relpow_imp_rtrancl [of "(x, y)", to_pred] by simp
+lemma relpowp_imp_rtranclp: "(P ^^ n) x y \<Longrightarrow> (P\<^sup>*\<^sup>*) x y"
+  using relpow_imp_rtrancl [of "(x, y)", to_pred] by simp
 
-lemma rtrancl_is_UN_relpow:
-  "R^* = (\<Union>n. R ^^ n)"
+lemma rtrancl_is_UN_relpow: "R\<^sup>* = (\<Union>n. R ^^ n)"
   by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
 
-lemma rtranclp_is_Sup_relpowp:
-  "P^** = (\<Squnion>n. P ^^ n)"
+lemma rtranclp_is_Sup_relpowp: "P\<^sup>*\<^sup>* = (\<Squnion>n. P ^^ n)"
   using rtrancl_is_UN_relpow [to_pred, of P] by auto
 
-lemma rtrancl_power:
-  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
+lemma rtrancl_power: "p \<in> R\<^sup>* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
   by (simp add: rtrancl_is_UN_relpow)
 
-lemma rtranclp_power:
-  "(P^**) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
+lemma rtranclp_power: "(P\<^sup>*\<^sup>*) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
   by (simp add: rtranclp_is_Sup_relpowp)
 
-lemma trancl_power:
-  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
+lemma trancl_power: "p \<in> R\<^sup>+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
   apply (cases p)
   apply simp
   apply (rule iffI)
@@ -956,187 +911,204 @@
   apply (drule rtrancl_into_trancl1) apply auto
   done
 
-lemma tranclp_power:
-  "(P^++) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
+lemma tranclp_power: "(P\<^sup>+\<^sup>+) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
   using trancl_power [to_pred, of P "(x, y)"] by simp
 
-lemma rtrancl_imp_relpow:
-  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
+lemma rtrancl_imp_relpow: "p \<in> R\<^sup>* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
   by (auto dest: rtrancl_imp_UN_relpow)
 
-lemma rtranclp_imp_relpowp:
-  "(P^**) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
+lemma rtranclp_imp_relpowp: "(P\<^sup>*\<^sup>*) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
   by (auto dest: rtranclp_imp_Sup_relpowp)
 
-text\<open>By Sternagel/Thiemann:\<close>
-lemma relpow_fun_conv:
-  "((a,b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f(Suc i)) \<in> R))"
+text \<open>By Sternagel/Thiemann:\<close>
+lemma relpow_fun_conv: "(a, b) \<in> R ^^ n \<longleftrightarrow> (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> R))"
 proof (induct n arbitrary: b)
-  case 0 show ?case by auto
+  case 0
+  show ?case by auto
 next
   case (Suc n)
   show ?case
   proof (simp add: relcomp_unfold Suc)
-    show "(\<exists>y. (\<exists>f. f 0 = a \<and> f n = y \<and> (\<forall>i<n. (f i,f(Suc i)) \<in> R)) \<and> (y,b) \<in> R)
-     = (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
+    show "(\<exists>y. (\<exists>f. f 0 = a \<and> f n = y \<and> (\<forall>i<n. (f i,f(Suc i)) \<in> R)) \<and> (y,b) \<in> R) \<longleftrightarrow>
+      (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
     (is "?l = ?r")
     proof
       assume ?l
-      then obtain c f where 1: "f 0 = a"  "f n = c"  "\<And>i. i < n \<Longrightarrow> (f i, f (Suc i)) \<in> R"  "(c,b) \<in> R" by auto
+      then obtain c f
+        where 1: "f 0 = a"  "f n = c"  "\<And>i. i < n \<Longrightarrow> (f i, f (Suc i)) \<in> R"  "(c,b) \<in> R"
+        by auto
       let ?g = "\<lambda> m. if m = Suc n then b else f m"
-      show ?r by (rule exI[of _ ?g], simp add: 1)
+      show ?r by (rule exI[of _ ?g]) (simp add: 1)
     next
       assume ?r
-      then obtain f where 1: "f 0 = a"  "b = f (Suc n)"  "\<And>i. i < Suc n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
+      then obtain f where 1: "f 0 = a"  "b = f (Suc n)"  "\<And>i. i < Suc n \<Longrightarrow> (f i, f (Suc i)) \<in> R"
+        by auto
       show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
     qed
   qed
 qed
 
-lemma relpowp_fun_conv:
-  "(P ^^ n) x y \<longleftrightarrow> (\<exists>f. f 0 = x \<and> f n = y \<and> (\<forall>i<n. P (f i) (f (Suc i))))"
+lemma relpowp_fun_conv: "(P ^^ n) x y \<longleftrightarrow> (\<exists>f. f 0 = x \<and> f n = y \<and> (\<forall>i<n. P (f i) (f (Suc i))))"
   by (fact relpow_fun_conv [to_pred])
 
 lemma relpow_finite_bounded1:
-assumes "finite(R :: ('a*'a)set)" and "k>0"
-shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")
-proof-
-  { fix a b k
-    have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"
-    proof(induct k arbitrary: b)
-      case 0
-      hence "R \<noteq> {}" by auto
-      with card_0_eq[OF \<open>finite R\<close>] have "card R >= Suc 0" by auto
-      thus ?case using 0 by force
+  fixes R :: "('a \<times> 'a) set"
+  assumes "finite R" and "k > 0"
+  shows "R^^k \<subseteq> (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)" (is "_ \<subseteq> ?r")
+proof -
+  have "(a, b) \<in> R^^(Suc k) \<Longrightarrow> \<exists>n. 0 < n \<and> n \<le> card R \<and> (a, b) \<in> R^^n" for a b k
+  proof (induct k arbitrary: b)
+    case 0
+    then have "R \<noteq> {}" by auto
+    with card_0_eq[OF \<open>finite R\<close>] have "card R \<ge> Suc 0" by auto
+    then show ?case using 0 by force
+  next
+    case (Suc k)
+    then obtain a' where "(a, a') \<in> R^^(Suc k)" and "(a', b) \<in> R"
+      by auto
+    from Suc(1)[OF \<open>(a, a') \<in> R^^(Suc k)\<close>] obtain n where "n \<le> card R" and "(a, a') \<in> R ^^ n"
+      by auto
+    have "(a, b) \<in> R^^(Suc n)"
+      using \<open>(a, a') \<in> R^^n\<close> and \<open>(a', b)\<in> R\<close> by auto
+    from \<open>n \<le> card R\<close> consider "n < card R" | "n = card R" by force
+    then show ?case
+    proof cases
+      case 1
+      then show ?thesis
+        using \<open>(a, b) \<in> R^^(Suc n)\<close> Suc_leI[OF \<open>n < card R\<close>] by blast
     next
-      case (Suc k)
-      then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto
-      from Suc(1)[OF \<open>(a,a') : R^^(Suc k)\<close>]
-      obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto
-      have "(a,b) : R^^(Suc n)" using \<open>(a,a') \<in> R^^n\<close> and \<open>(a',b)\<in> R\<close> by auto
-      { assume "n < card R"
-        hence ?case using \<open>(a,b): R^^(Suc n)\<close> Suc_leI[OF \<open>n < card R\<close>] by blast
-      } moreover
-      { assume "n = card R"
-        from \<open>(a,b) \<in> R ^^ (Suc n)\<close>[unfolded relpow_fun_conv]
-        obtain f where "f 0 = a" and "f(Suc n) = b"
-          and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
-        let ?p = "%i. (f i, f(Suc i))"
-        let ?N = "{i. i \<le> n}"
-        have "?p ` ?N <= R" using steps by auto
-        from card_mono[OF assms(1) this]
-        have "card(?p ` ?N) <= card R" .
-        also have "\<dots> < card ?N" using \<open>n = card R\<close> by simp
-        finally have "~ inj_on ?p ?N" by(rule pigeonhole)
-        then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and
-          pij: "?p i = ?p j" by(auto simp: inj_on_def)
-        let ?i = "min i j" let ?j = "max i j"
-        have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 
-          and ij: "?i < ?j"
-          using i j ij pij unfolding min_def max_def by auto
-        from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"
-          and pij: "?p i = ?p j" by blast
-        let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"
-        let ?n = "Suc(n - (j - i))"
-        have abl: "(a,b) \<in> R ^^ ?n" unfolding relpow_fun_conv
-        proof (rule exI[of _ ?g], intro conjI impI allI)
-          show "?g ?n = b" using \<open>f(Suc n) = b\<close> j ij by auto
+      case 2
+      from \<open>(a, b) \<in> R ^^ (Suc n)\<close> [unfolded relpow_fun_conv]
+      obtain f where "f 0 = a" and "f (Suc n) = b"
+        and steps: "\<And>i. i \<le> n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
+      let ?p = "\<lambda>i. (f i, f(Suc i))"
+      let ?N = "{i. i \<le> n}"
+      have "?p ` ?N \<subseteq> R"
+        using steps by auto
+      from card_mono[OF assms(1) this] have "card (?p ` ?N) \<le> card R" .
+      also have "\<dots> < card ?N"
+        using \<open>n = card R\<close> by simp
+      finally have "\<not> inj_on ?p ?N"
+        by (rule pigeonhole)
+      then obtain i j where i: "i \<le> n" and j: "j \<le> n" and ij: "i \<noteq> j" and pij: "?p i = ?p j"
+        by (auto simp: inj_on_def)
+      let ?i = "min i j"
+      let ?j = "max i j"
+      have i: "?i \<le> n" and j: "?j \<le> n" and pij: "?p ?i = ?p ?j" and ij: "?i < ?j"
+        using i j ij pij unfolding min_def max_def by auto
+      from i j pij ij obtain i j where i: "i \<le> n" and j: "j \<le> n" and ij: "i < j"
+        and pij: "?p i = ?p j"
+        by blast
+      let ?g = "\<lambda>l. if l \<le> i then f l else f (l + (j - i))"
+      let ?n = "Suc (n - (j - i))"
+      have abl: "(a, b) \<in> R ^^ ?n"
+        unfolding relpow_fun_conv
+      proof (rule exI[of _ ?g], intro conjI impI allI)
+        show "?g ?n = b"
+          using \<open>f(Suc n) = b\<close> j ij by auto
+      next
+        fix k
+        assume "k < ?n"
+        show "(?g k, ?g (Suc k)) \<in> R"
+        proof (cases "k < i")
+          case True
+          with i have "k \<le> n"
+            by auto
+          from steps[OF this] show ?thesis
+            using True by simp
         next
-          fix k assume "k < ?n"
-          show "(?g k, ?g (Suc k)) \<in> R"
-          proof (cases "k < i")
+          case False
+          then have "i \<le> k" by auto
+          show ?thesis
+          proof (cases "k = i")
             case True
-            with i have "k <= n" by auto
-            from steps[OF this] show ?thesis using True by simp
+            then show ?thesis
+              using ij pij steps[OF i] by simp
           next
             case False
-            hence "i \<le> k" by auto
+            with \<open>i \<le> k\<close> have "i < k" by auto
+            then have small: "k + (j - i) \<le> n"
+              using \<open>k<?n\<close> by arith
             show ?thesis
-            proof (cases "k = i")
-              case True
-              thus ?thesis using ij pij steps[OF i] by simp
-            next
-              case False
-              with \<open>i \<le> k\<close> have "i < k" by auto
-              hence small: "k + (j - i) <= n" using \<open>k<?n\<close> by arith
-              show ?thesis using steps[OF small] \<open>i<k\<close> by auto
-            qed
+              using steps[OF small] \<open>i<k\<close> by auto
           qed
-        qed (simp add: \<open>f 0 = a\<close>)
-        moreover have "?n <= n" using i j ij by arith
-        ultimately have ?case using \<open>n = card R\<close> by blast
-      }
-      ultimately show ?case using \<open>n \<le> card R\<close> by force
+        qed
+      qed (simp add: \<open>f 0 = a\<close>)
+      moreover have "?n \<le> n"
+        using i j ij by arith
+      ultimately show ?thesis
+        using \<open>n = card R\<close> by blast
     qed
-  }
-  thus ?thesis using gr0_implies_Suc[OF \<open>k>0\<close>] by auto
+  qed
+  then show ?thesis
+    using gr0_implies_Suc[OF \<open>k > 0\<close>] by auto
 qed
 
 lemma relpow_finite_bounded:
-assumes "finite(R :: ('a*'a)set)"
-shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"
-apply(cases k)
- apply force
-using relpow_finite_bounded1[OF assms, of k] by auto
+  fixes R :: "('a \<times> 'a) set"
+  assumes "finite R"
+  shows "R^^k \<subseteq> (UN n:{n. n \<le> card R}. R^^n)"
+  apply (cases k)
+   apply force
+  using relpow_finite_bounded1[OF assms, of k]
+  apply auto
+  done
 
-lemma rtrancl_finite_eq_relpow:
-  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
-by(fastforce simp: rtrancl_power dest: relpow_finite_bounded)
+lemma rtrancl_finite_eq_relpow: "finite R \<Longrightarrow> R\<^sup>* = (\<Union>n\<in>{n. n \<le> card R}. R^^n)"
+  by (fastforce simp: rtrancl_power dest: relpow_finite_bounded)
 
-lemma trancl_finite_eq_relpow:
-  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
-apply(auto simp add: trancl_power)
-apply(auto dest: relpow_finite_bounded1)
-done
+lemma trancl_finite_eq_relpow: "finite R \<Longrightarrow> R\<^sup>+ = (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)"
+  apply (auto simp: trancl_power)
+  apply (auto dest: relpow_finite_bounded1)
+  done
 
 lemma finite_relcomp[simp,intro]:
-assumes "finite R" and "finite S"
-shows "finite(R O S)"
+  assumes "finite R" and "finite S"
+  shows "finite (R O S)"
 proof-
   have "R O S = (\<Union>(x, y)\<in>R. \<Union>(u, v)\<in>S. if u = y then {(x, v)} else {})"
     by (force simp add: split_def image_constant_conv split: if_splits)
-  then show ?thesis using assms by clarsimp
+  then show ?thesis
+    using assms by clarsimp
 qed
 
-lemma finite_relpow[simp,intro]:
-  assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"
-apply(induct n)
- apply simp
-apply(case_tac n)
- apply(simp_all add: assms)
-done
+lemma finite_relpow [simp, intro]:
+  fixes R :: "('a \<times> 'a) set"
+  assumes "finite R"
+  shows "n > 0 \<Longrightarrow> finite (R^^n)"
+  apply (induct n)
+   apply simp
+  apply (case_tac n)
+   apply (simp_all add: assms)
+  done
 
 lemma single_valued_relpow:
-  fixes R :: "('a * 'a) set"
+  fixes R :: "('a \<times> 'a) set"
   shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
-apply (induct n arbitrary: R)
-apply simp_all
-apply (rule single_valuedI)
-apply (fast dest: single_valuedD elim: relpow_Suc_E)
-done
+  apply (induct n arbitrary: R)
+  apply simp_all
+  apply (rule single_valuedI)
+  apply (fast dest: single_valuedD elim: relpow_Suc_E)
+  done
 
 
 subsection \<open>Bounded transitive closure\<close>
 
 definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
-where
-  "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
+  where "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
 
-lemma ntrancl_Zero [simp, code]:
-  "ntrancl 0 R = R"
+lemma ntrancl_Zero [simp, code]: "ntrancl 0 R = R"
 proof
   show "R \<subseteq> ntrancl 0 R"
     unfolding ntrancl_def by fastforce
 next
-  { 
-    fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
-  }
-  from this show "ntrancl 0 R \<le> R"
+  have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" for i
+    by auto
+  then show "ntrancl 0 R \<le> R"
     unfolding ntrancl_def by auto
 qed
 
-lemma ntrancl_Suc [simp]:
-  "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
+lemma ntrancl_Suc [simp]: "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
 proof
   {
     fix a b
@@ -1159,75 +1131,67 @@
       from this c2 show ?thesis by fastforce
     qed
   }
-  from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
+  then show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
     by auto
-next
   show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
     unfolding ntrancl_def by fastforce
 qed
 
-lemma [code]:
-  "ntrancl (Suc n) r = (let r' = ntrancl n r in r' Un r' O r)"
-unfolding Let_def by auto
+lemma [code]: "ntrancl (Suc n) r = (let r' = ntrancl n r in r' \<union> r' O r)"
+  by (auto simp: Let_def)
 
-lemma finite_trancl_ntranl:
-  "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
+lemma finite_trancl_ntranl: "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
   by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)
 
 
 subsection \<open>Acyclic relations\<close>
 
-definition acyclic :: "('a * 'a) set => bool" where
-  "acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)"
+definition acyclic :: "('a \<times> 'a) set \<Rightarrow> bool"
+  where "acyclic r \<longleftrightarrow> (\<forall>x. (x,x) \<notin> r\<^sup>+)"
 
-abbreviation acyclicP :: "('a => 'a => bool) => bool" where
-  "acyclicP r \<equiv> acyclic {(x, y). r x y}"
+abbreviation acyclicP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+  where "acyclicP r \<equiv> acyclic {(x, y). r x y}"
 
-lemma acyclic_irrefl [code]:
-  "acyclic r \<longleftrightarrow> irrefl (r^+)"
+lemma acyclic_irrefl [code]: "acyclic r \<longleftrightarrow> irrefl (r\<^sup>+)"
   by (simp add: acyclic_def irrefl_def)
 
-lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
+lemma acyclicI: "\<forall>x. (x, x) \<notin> r\<^sup>+ \<Longrightarrow> acyclic r"
   by (simp add: acyclic_def)
 
 lemma (in order) acyclicI_order:
   assumes *: "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b < f a"
   shows "acyclic r"
 proof -
-  { fix a b assume "(a, b) \<in> r\<^sup>+"
-    then have "f b < f a"
-      by induct (auto intro: * less_trans) }
+  have "f b < f a" if "(a, b) \<in> r\<^sup>+" for a b
+    using that by induct (auto intro: * less_trans)
   then show ?thesis
     by (auto intro!: acyclicI)
 qed
 
-lemma acyclic_insert [iff]:
-     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
-apply (simp add: acyclic_def trancl_insert)
-apply (blast intro: rtrancl_trans)
-done
+lemma acyclic_insert [iff]: "acyclic (insert (y, x) r) \<longleftrightarrow> acyclic r \<and> (x, y) \<notin> r\<^sup>*"
+  apply (simp add: acyclic_def trancl_insert)
+  apply (blast intro: rtrancl_trans)
+  done
 
-lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
-by (simp add: acyclic_def trancl_converse)
+lemma acyclic_converse [iff]: "acyclic (r\<inverse>) \<longleftrightarrow> acyclic r"
+  by (simp add: acyclic_def trancl_converse)
 
 lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
 
-lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
-apply (simp add: acyclic_def antisym_def)
-apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
-done
+lemma acyclic_impl_antisym_rtrancl: "acyclic r \<Longrightarrow> antisym (r\<^sup>*)"
+  apply (simp add: acyclic_def antisym_def)
+  apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
+  done
 
 (* Other direction:
 acyclic = no loops
 antisym = only self loops
-Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
-==> antisym( r^* ) = acyclic(r - Id)";
+Goalw [acyclic_def,antisym_def] "antisym( r\<^sup>* ) \<Longrightarrow> acyclic(r - Id)
+\<Longrightarrow> antisym( r\<^sup>* ) = acyclic(r - Id)";
 *)
 
-lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
-apply (simp add: acyclic_def)
-apply (blast intro: trancl_mono)
-done
+lemma acyclic_subset: "acyclic s \<Longrightarrow> r \<subseteq> s \<Longrightarrow> acyclic r"
+  unfolding acyclic_def by (blast intro: trancl_mono)
 
 
 subsection \<open>Setup of transitivity reasoner\<close>
@@ -1246,14 +1210,16 @@
   val rtrancl_trans = @{thm rtrancl_trans};
 
   fun decomp (@{const Trueprop} $ t) =
-    let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) =
-        let fun decr (Const (@{const_name rtrancl}, _ ) $ r) = (r,"r*")
-              | decr (Const (@{const_name trancl}, _ ) $ r)  = (r,"r+")
-              | decr r = (r,"r");
-            val (rel,r) = decr (Envir.beta_eta_contract rel);
-        in SOME (a,b,rel,r) end
-      | dec _ =  NONE
-    in dec t end
+        let
+          fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel) =
+              let
+                fun decr (Const (@{const_name rtrancl}, _ ) $ r) = (r,"r*")
+                  | decr (Const (@{const_name trancl}, _ ) $ r)  = (r,"r+")
+                  | decr r = (r,"r");
+                val (rel,r) = decr (Envir.beta_eta_contract rel);
+              in SOME (a,b,rel,r) end
+          | dec _ =  NONE
+        in dec t end
     | decomp _ = NONE;
 );
 
@@ -1269,14 +1235,16 @@
   val rtrancl_trans = @{thm rtranclp_trans};
 
   fun decomp (@{const Trueprop} $ t) =
-    let fun dec (rel $ a $ b) =
-        let fun decr (Const (@{const_name rtranclp}, _ ) $ r) = (r,"r*")
-              | decr (Const (@{const_name tranclp}, _ ) $ r)  = (r,"r+")
-              | decr r = (r,"r");
-            val (rel,r) = decr rel;
-        in SOME (a, b, rel, r) end
-      | dec _ =  NONE
-    in dec t end
+        let
+          fun dec (rel $ a $ b) =
+            let
+              fun decr (Const (@{const_name rtranclp}, _ ) $ r) = (r,"r*")
+                | decr (Const (@{const_name tranclp}, _ ) $ r)  = (r,"r+")
+                | decr r = (r,"r");
+              val (rel,r) = decr rel;
+            in SOME (a, b, rel, r) end
+          | dec _ =  NONE
+        in dec t end
     | decomp _ = NONE;
 );
 \<close>




