diff --git a/src/HOL/Complete_Partial_Order.thy b/src/HOL/Complete_Partial_Order.thy
--- a/src/HOL/Complete_Partial_Order.thy
+++ b/src/HOL/Complete_Partial_Order.thy
@@ -1,381 +1,372 @@
 (*  Title:      HOL/Complete_Partial_Order.thy
     Author:     Brian Huffman, Portland State University
     Author:     Alexander Krauss, TU Muenchen
 *)
 
 section \<open>Chain-complete partial orders and their fixpoints\<close>
 
 theory Complete_Partial_Order
   imports Product_Type
 begin
 
 subsection \<open>Monotone functions\<close>
 
 text \<open>Dictionary-passing version of \<^const>\<open>Orderings.mono\<close>.\<close>
 
 definition monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
   where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
 
 lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)) \<Longrightarrow> monotone orda ordb f"
   unfolding monotone_def by iprover
 
 lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"
   unfolding monotone_def by iprover
 
 
 subsection \<open>Chains\<close>
 
 text \<open>
   A chain is a totally-ordered set. Chains are parameterized over
   the order for maximal flexibility, since type classes are not enough.
 \<close>
 
 definition chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
   where "chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
 
 lemma chainI:
   assumes "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> ord x y \<or> ord y x"
   shows "chain ord S"
   using assms unfolding chain_def by fast
 
 lemma chainD:
   assumes "chain ord S" and "x \<in> S" and "y \<in> S"
   shows "ord x y \<or> ord y x"
   using assms unfolding chain_def by fast
 
 lemma chainE:
   assumes "chain ord S" and "x \<in> S" and "y \<in> S"
   obtains "ord x y" | "ord y x"
   using assms unfolding chain_def by fast
 
 lemma chain_empty: "chain ord {}"
   by (simp add: chain_def)
 
 lemma chain_equality: "chain (=) A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x = y)"
   by (auto simp add: chain_def)
 
 lemma chain_subset: "chain ord A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> chain ord B"
   by (rule chainI) (blast dest: chainD)
 
 lemma chain_imageI:
   assumes chain: "chain le_a Y"
     and mono: "\<And>x y. x \<in> Y \<Longrightarrow> y \<in> Y \<Longrightarrow> le_a x y \<Longrightarrow> le_b (f x) (f y)"
   shows "chain le_b (f ` Y)"
   by (blast intro: chainI dest: chainD[OF chain] mono)
 
 
 subsection \<open>Chain-complete partial orders\<close>
 
 text \<open>
   A \<open>ccpo\<close> has a least upper bound for any chain.  In particular, the
   empty set is a chain, so every \<open>ccpo\<close> must have a bottom element.
 \<close>
 
 class ccpo = order + Sup +
   assumes ccpo_Sup_upper: "chain (\<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> Sup A"
   assumes ccpo_Sup_least: "chain (\<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup A \<le> z"
 begin
 
 lemma chain_singleton: "Complete_Partial_Order.chain (\<le>) {x}"
   by (rule chainI) simp
 
 lemma ccpo_Sup_singleton [simp]: "\<Squnion>{x} = x"
   by (rule antisym) (auto intro: ccpo_Sup_least ccpo_Sup_upper simp add: chain_singleton)
 
 
 subsection \<open>Transfinite iteration of a function\<close>
 
 context notes [[inductive_internals]]
 begin
 
 inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set"
   for f :: "'a \<Rightarrow> 'a"
   where
     step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
   | Sup: "chain (\<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f"
 
 end
 
 lemma iterates_le_f: "x \<in> iterates f \<Longrightarrow> monotone (\<le>) (\<le>) f \<Longrightarrow> x \<le> f x"
   by (induct x rule: iterates.induct)
     (force dest: monotoneD intro!: ccpo_Sup_upper ccpo_Sup_least)+
 
 lemma chain_iterates:
   assumes f: "monotone (\<le>) (\<le>) f"
   shows "chain (\<le>) (iterates f)" (is "chain _ ?C")
 proof (rule chainI)
   fix x y
   assume "x \<in> ?C" "y \<in> ?C"
   then show "x \<le> y \<or> y \<le> x"
   proof (induct x arbitrary: y rule: iterates.induct)
     fix x y
     assume y: "y \<in> ?C"
       and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"
     from y show "f x \<le> y \<or> y \<le> f x"
     proof (induct y rule: iterates.induct)
       case (step y)
       with IH f show ?case by (auto dest: monotoneD)
     next
       case (Sup M)
       then have chM: "chain (\<le>) M"
         and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto
       show "f x \<le> Sup M \<or> Sup M \<le> f x"
       proof (cases "\<exists>z\<in>M. f x \<le> z")
         case True
         then have "f x \<le> Sup M"
-          apply rule
-          apply (erule order_trans)
-          apply (rule ccpo_Sup_upper[OF chM])
-          apply assumption
-          done
+          by (blast intro: ccpo_Sup_upper[OF chM] order_trans)
         then show ?thesis ..
       next
         case False
         with IH' show ?thesis
           by (auto intro: ccpo_Sup_least[OF chM])
       qed
     qed
   next
     case (Sup M y)
     show ?case
     proof (cases "\<exists>x\<in>M. y \<le> x")
       case True
       then have "y \<le> Sup M"
-        apply rule
-        apply (erule order_trans)
-        apply (rule ccpo_Sup_upper[OF Sup(1)])
-        apply assumption
-        done
+        by (blast intro: ccpo_Sup_upper[OF Sup(1)] order_trans)
       then show ?thesis ..
     next
       case False with Sup
       show ?thesis by (auto intro: ccpo_Sup_least)
     qed
   qed
 qed
 
 lemma bot_in_iterates: "Sup {} \<in> iterates f"
   by (auto intro: iterates.Sup simp add: chain_empty)
 
 
 subsection \<open>Fixpoint combinator\<close>
 
 definition fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
   where "fixp f = Sup (iterates f)"
 
 lemma iterates_fixp:
   assumes f: "monotone (\<le>) (\<le>) f"
   shows "fixp f \<in> iterates f"
   unfolding fixp_def
   by (simp add: iterates.Sup chain_iterates f)
 
 lemma fixp_unfold:
   assumes f: "monotone (\<le>) (\<le>) f"
   shows "fixp f = f (fixp f)"
 proof (rule antisym)
   show "fixp f \<le> f (fixp f)"
     by (intro iterates_le_f iterates_fixp f)
   have "f (fixp f) \<le> Sup (iterates f)"
     by (intro ccpo_Sup_upper chain_iterates f iterates.step iterates_fixp)
   then show "f (fixp f) \<le> fixp f"
     by (simp only: fixp_def)
 qed
 
 lemma fixp_lowerbound:
   assumes f: "monotone (\<le>) (\<le>) f"
     and z: "f z \<le> z"
   shows "fixp f \<le> z"
   unfolding fixp_def
 proof (rule ccpo_Sup_least[OF chain_iterates[OF f]])
   fix x
   assume "x \<in> iterates f"
   then show "x \<le> z"
   proof (induct x rule: iterates.induct)
     case (step x)
     from f \<open>x \<le> z\<close> have "f x \<le> f z" by (rule monotoneD)
     also note z
     finally show "f x \<le> z" .
   next
     case (Sup M)
     then show ?case
       by (auto intro: ccpo_Sup_least)
   qed
 qed
 
 end
 
 
 subsection \<open>Fixpoint induction\<close>
 
 setup \<open>Sign.map_naming (Name_Space.mandatory_path "ccpo")\<close>
 
 definition admissible :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   where "admissible lub ord P \<longleftrightarrow> (\<forall>A. chain ord A \<longrightarrow> A \<noteq> {} \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
 
 lemma admissibleI:
   assumes "\<And>A. chain ord A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (lub A)"
   shows "ccpo.admissible lub ord P"
   using assms unfolding ccpo.admissible_def by fast
 
 lemma admissibleD:
   assumes "ccpo.admissible lub ord P"
   assumes "chain ord A"
   assumes "A \<noteq> {}"
   assumes "\<And>x. x \<in> A \<Longrightarrow> P x"
   shows "P (lub A)"
   using assms by (auto simp: ccpo.admissible_def)
 
 setup \<open>Sign.map_naming Name_Space.parent_path\<close>
 
 lemma (in ccpo) fixp_induct:
   assumes adm: "ccpo.admissible Sup (\<le>) P"
   assumes mono: "monotone (\<le>) (\<le>) f"
   assumes bot: "P (Sup {})"
   assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"
   shows "P (fixp f)"
   unfolding fixp_def
   using adm chain_iterates[OF mono]
 proof (rule ccpo.admissibleD)
   show "iterates f \<noteq> {}"
     using bot_in_iterates by auto
 next
   fix x
   assume "x \<in> iterates f"
   then show "P x"
   proof (induct rule: iterates.induct)
     case prems: (step x)
     from this(2) show ?case by (rule step)
   next
     case (Sup M)
     then show ?case by (cases "M = {}") (auto intro: step bot ccpo.admissibleD adm)
   qed
 qed
 
 lemma admissible_True: "ccpo.admissible lub ord (\<lambda>x. True)"
   unfolding ccpo.admissible_def by simp
 
 (*lemma admissible_False: "\<not> ccpo.admissible lub ord (\<lambda>x. False)"
 unfolding ccpo.admissible_def chain_def by simp
 *)
 lemma admissible_const: "ccpo.admissible lub ord (\<lambda>x. t)"
   by (auto intro: ccpo.admissibleI)
 
 lemma admissible_conj:
   assumes "ccpo.admissible lub ord (\<lambda>x. P x)"
   assumes "ccpo.admissible lub ord (\<lambda>x. Q x)"
   shows "ccpo.admissible lub ord (\<lambda>x. P x \<and> Q x)"
   using assms unfolding ccpo.admissible_def by simp
 
 lemma admissible_all:
   assumes "\<And>y. ccpo.admissible lub ord (\<lambda>x. P x y)"
   shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y. P x y)"
   using assms unfolding ccpo.admissible_def by fast
 
 lemma admissible_ball:
   assumes "\<And>y. y \<in> A \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x y)"
   shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y\<in>A. P x y)"
   using assms unfolding ccpo.admissible_def by fast
 
 lemma chain_compr: "chain ord A \<Longrightarrow> chain ord {x \<in> A. P x}"
   unfolding chain_def by fast
 
 context ccpo
 begin
 
 lemma admissible_disj:
   fixes P Q :: "'a \<Rightarrow> bool"
   assumes P: "ccpo.admissible Sup (\<le>) (\<lambda>x. P x)"
   assumes Q: "ccpo.admissible Sup (\<le>) (\<lambda>x. Q x)"
   shows "ccpo.admissible Sup (\<le>) (\<lambda>x. P x \<or> Q x)"
 proof (rule ccpo.admissibleI)
   fix A :: "'a set"
   assume chain: "chain (\<le>) A"
   assume A: "A \<noteq> {}" and P_Q: "\<forall>x\<in>A. P x \<or> Q x"
   have "(\<exists>x\<in>A. P x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<exists>x\<in>A. Q x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)"
     (is "?P \<or> ?Q" is "?P1 \<and> ?P2 \<or> _")
   proof (rule disjCI)
     assume "\<not> ?Q"
     then consider "\<forall>x\<in>A. \<not> Q x" | a where "a \<in> A" "\<forall>y\<in>A. a \<le> y \<longrightarrow> \<not> Q y"
       by blast
     then show ?P
     proof cases
       case 1
       with P_Q have "\<forall>x\<in>A. P x" by blast
       with A show ?P by blast
     next
       case 2
       note a = \<open>a \<in> A\<close>
       show ?P
       proof
         from P_Q 2 have *: "\<forall>y\<in>A. a \<le> y \<longrightarrow> P y" by blast
         with a have "P a" by blast
         with a show ?P1 by blast
         show ?P2
         proof
           fix x
           assume x: "x \<in> A"
           with chain a show "\<exists>y\<in>A. x \<le> y \<and> P y"
           proof (rule chainE)
             assume le: "a \<le> x"
             with * a x have "P x" by blast
             with x le show ?thesis by blast
           next
             assume "a \<ge> x"
             with a \<open>P a\<close> show ?thesis by blast
           qed
         qed
       qed
     qed
   qed
   moreover
-  have "Sup A = Sup {x \<in> A. P x}" if "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y" for P
+  have "Sup A = Sup {x \<in> A. P x}" if "\<And>x. x\<in>A \<Longrightarrow> \<exists>y\<in>A. x \<le> y \<and> P y" for P
   proof (rule antisym)
     have chain_P: "chain (\<le>) {x \<in> A. P x}"
       by (rule chain_compr [OF chain])
     show "Sup A \<le> Sup {x \<in> A. P x}"
-      apply (rule ccpo_Sup_least [OF chain])
-      apply (drule that [rule_format])
-      apply clarify
-      apply (erule order_trans)
-      apply (simp add: ccpo_Sup_upper [OF chain_P])
-      done
+    proof (rule ccpo_Sup_least [OF chain])
+      show "\<And>x. x \<in> A \<Longrightarrow> x \<le> \<Squnion> {x \<in> A. P x}"
+          by (blast intro: ccpo_Sup_upper[OF chain_P] order_trans dest: that)
+      qed
     show "Sup {x \<in> A. P x} \<le> Sup A"
       apply (rule ccpo_Sup_least [OF chain_P])
-      apply clarify
       apply (simp add: ccpo_Sup_upper [OF chain])
       done
   qed
   ultimately
   consider "\<exists>x. x \<in> A \<and> P x" "Sup A = Sup {x \<in> A. P x}"
     | "\<exists>x. x \<in> A \<and> Q x" "Sup A = Sup {x \<in> A. Q x}"
     by blast
   then show "P (Sup A) \<or> Q (Sup A)"
-    apply cases
-     apply simp_all
-     apply (rule disjI1)
-     apply (rule ccpo.admissibleD [OF P chain_compr [OF chain]]; simp)
-    apply (rule disjI2)
-    apply (rule ccpo.admissibleD [OF Q chain_compr [OF chain]]; simp)
-    done
+  proof cases
+    case 1
+    then show ?thesis
+      using ccpo.admissibleD [OF P chain_compr [OF chain]] by force
+  next
+    case 2
+    then show ?thesis 
+      using ccpo.admissibleD [OF Q chain_compr [OF chain]] by force
+  qed
 qed
 
 end
 
 instance complete_lattice \<subseteq> ccpo
   by standard (fast intro: Sup_upper Sup_least)+
 
 lemma lfp_eq_fixp:
   assumes mono: "mono f"
   shows "lfp f = fixp f"
 proof (rule antisym)
   from mono have f': "monotone (\<le>) (\<le>) f"
     unfolding mono_def monotone_def .
   show "lfp f \<le> fixp f"
     by (rule lfp_lowerbound, subst fixp_unfold [OF f'], rule order_refl)
   show "fixp f \<le> lfp f"
     by (rule fixp_lowerbound [OF f']) (simp add: lfp_fixpoint [OF mono])
 qed
 
 hide_const (open) iterates fixp
 
 end