diff --git a/src/HOL/Library/Complete_Partial_Order2.thy b/src/HOL/Library/Complete_Partial_Order2.thy
--- a/src/HOL/Library/Complete_Partial_Order2.thy
+++ b/src/HOL/Library/Complete_Partial_Order2.thy
@@ -1,1753 +1,1765 @@
 (*  Title:      HOL/Library/Complete_Partial_Order2.thy
     Author:     Andreas Lochbihler, ETH Zurich
 *)
 
 section \<open>Formalisation of chain-complete partial orders, continuity and admissibility\<close>
 
 theory Complete_Partial_Order2 imports 
   Main
 begin
 
 unbundle lattice_syntax
 
 lemma chain_transfer [transfer_rule]:
   includes lifting_syntax
   shows "((A ===> A ===> (=)) ===> rel_set A ===> (=)) Complete_Partial_Order.chain Complete_Partial_Order.chain"
 unfolding chain_def[abs_def] by transfer_prover
                              
 lemma linorder_chain [simp, intro!]:
   fixes Y :: "_ :: linorder set"
   shows "Complete_Partial_Order.chain (\<le>) Y"
 by(auto intro: chainI)
 
 lemma fun_lub_apply: "\<And>Sup. fun_lub Sup Y x = Sup ((\<lambda>f. f x) ` Y)"
 by(simp add: fun_lub_def image_def)
 
 lemma fun_lub_empty [simp]: "fun_lub lub {} = (\<lambda>_. lub {})"
 by(rule ext)(simp add: fun_lub_apply)
 
 lemma chain_fun_ordD: 
   assumes "Complete_Partial_Order.chain (fun_ord le) Y"
   shows "Complete_Partial_Order.chain le ((\<lambda>f. f x) ` Y)"
 by(rule chainI)(auto dest: chainD[OF assms] simp add: fun_ord_def)
 
 lemma chain_Diff:
   "Complete_Partial_Order.chain ord A
   \<Longrightarrow> Complete_Partial_Order.chain ord (A - B)"
 by(erule chain_subset) blast
 
 lemma chain_rel_prodD1:
   "Complete_Partial_Order.chain (rel_prod orda ordb) Y
   \<Longrightarrow> Complete_Partial_Order.chain orda (fst ` Y)"
 by(auto 4 3 simp add: chain_def)
 
 lemma chain_rel_prodD2:
   "Complete_Partial_Order.chain (rel_prod orda ordb) Y
   \<Longrightarrow> Complete_Partial_Order.chain ordb (snd ` Y)"
 by(auto 4 3 simp add: chain_def)
 
 
 context ccpo begin
 
 lemma ccpo_fun: "class.ccpo (fun_lub Sup) (fun_ord (\<le>)) (mk_less (fun_ord (\<le>)))"
   by standard (auto 4 3 simp add: mk_less_def fun_ord_def fun_lub_apply
     intro: order.trans order.antisym chain_imageI ccpo_Sup_upper ccpo_Sup_least)
 
 lemma ccpo_Sup_below_iff: "Complete_Partial_Order.chain (\<le>) Y \<Longrightarrow> Sup Y \<le> x \<longleftrightarrow> (\<forall>y\<in>Y. y \<le> x)"
 by(fast intro: order_trans[OF ccpo_Sup_upper] ccpo_Sup_least)
 
 lemma Sup_minus_bot: 
   assumes chain: "Complete_Partial_Order.chain (\<le>) A"
   shows "\<Squnion>(A - {\<Squnion>{}}) = \<Squnion>A"
     (is "?lhs = ?rhs")
 proof (rule order.antisym)
   show "?lhs \<le> ?rhs"
     by (blast intro: ccpo_Sup_least chain_Diff[OF chain] ccpo_Sup_upper[OF chain])
   show "?rhs \<le> ?lhs"
   proof (rule ccpo_Sup_least [OF chain])
     show "x \<in> A \<Longrightarrow> x \<le> ?lhs" for x
       by (cases "x = \<Squnion>{}")
         (blast intro: ccpo_Sup_least chain_empty ccpo_Sup_upper[OF chain_Diff[OF chain]])+
   qed
 qed
 
 lemma mono_lub:
   fixes le_b (infix "\<sqsubseteq>" 60)
   assumes chain: "Complete_Partial_Order.chain (fun_ord (\<le>)) Y"
   and mono: "\<And>f. f \<in> Y \<Longrightarrow> monotone le_b (\<le>) f"
   shows "monotone (\<sqsubseteq>) (\<le>) (fun_lub Sup Y)"
 proof(rule monotoneI)
   fix x y
   assume "x \<sqsubseteq> y"
 
   have chain'': "\<And>x. Complete_Partial_Order.chain (\<le>) ((\<lambda>f. f x) ` Y)"
     using chain by(rule chain_imageI)(simp add: fun_ord_def)
   then show "fun_lub Sup Y x \<le> fun_lub Sup Y y" unfolding fun_lub_apply
   proof(rule ccpo_Sup_least)
     fix x'
     assume "x' \<in> (\<lambda>f. f x) ` Y"
     then obtain f where "f \<in> Y" "x' = f x" by blast
     note \<open>x' = f x\<close> also
     from \<open>f \<in> Y\<close> \<open>x \<sqsubseteq> y\<close> have "f x \<le> f y" by(blast dest: mono monotoneD)
     also have "\<dots> \<le> \<Squnion>((\<lambda>f. f y) ` Y)" using chain''
       by(rule ccpo_Sup_upper)(simp add: \<open>f \<in> Y\<close>)
     finally show "x' \<le> \<Squnion>((\<lambda>f. f y) ` Y)" .
   qed
 qed
 
 context
   fixes le_b (infix "\<sqsubseteq>" 60) and Y f
   assumes chain: "Complete_Partial_Order.chain le_b Y" 
   and mono1: "\<And>y. y \<in> Y \<Longrightarrow> monotone le_b (\<le>) (\<lambda>x. f x y)"
   and mono2: "\<And>x a b. \<lbrakk> x \<in> Y; a \<sqsubseteq> b; a \<in> Y; b \<in> Y \<rbrakk> \<Longrightarrow> f x a \<le> f x b"
 begin
 
 lemma Sup_mono: 
   assumes le: "x \<sqsubseteq> y" and x: "x \<in> Y" and y: "y \<in> Y"
   shows "\<Squnion>(f x ` Y) \<le> \<Squnion>(f y ` Y)" (is "_ \<le> ?rhs")
 proof(rule ccpo_Sup_least)
   from chain show chain': "Complete_Partial_Order.chain (\<le>) (f x ` Y)" when "x \<in> Y" for x
     by(rule chain_imageI) (insert that, auto dest: mono2)
 
   fix x'
   assume "x' \<in> f x ` Y"
   then obtain y' where "y' \<in> Y" "x' = f x y'" by blast note this(2)
   also from mono1[OF \<open>y' \<in> Y\<close>] le have "\<dots> \<le> f y y'" by(rule monotoneD)
   also have "\<dots> \<le> ?rhs" using chain'[OF y]
     by (auto intro!: ccpo_Sup_upper simp add: \<open>y' \<in> Y\<close>)
   finally show "x' \<le> ?rhs" .
 qed(rule x)
 
 lemma diag_Sup: "\<Squnion>((\<lambda>x. \<Squnion>(f x ` Y)) ` Y) = \<Squnion>((\<lambda>x. f x x) ` Y)" (is "?lhs = ?rhs")
 proof(rule order.antisym)
   have chain1: "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. \<Squnion>(f x ` Y)) ` Y)"
     using chain by(rule chain_imageI)(rule Sup_mono)
   have chain2: "\<And>y'. y' \<in> Y \<Longrightarrow> Complete_Partial_Order.chain (\<le>) (f y' ` Y)" using chain
     by(rule chain_imageI)(auto dest: mono2)
   have chain3: "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. f x x) ` Y)"
     using chain by(rule chain_imageI)(auto intro: monotoneD[OF mono1] mono2 order.trans)
 
   show "?lhs \<le> ?rhs" using chain1
   proof(rule ccpo_Sup_least)
     fix x'
     assume "x' \<in> (\<lambda>x. \<Squnion>(f x ` Y)) ` Y"
     then obtain y' where "y' \<in> Y" "x' = \<Squnion>(f y' ` Y)" by blast note this(2)
     also have "\<dots> \<le> ?rhs" using chain2[OF \<open>y' \<in> Y\<close>]
     proof(rule ccpo_Sup_least)
       fix x
       assume "x \<in> f y' ` Y"
       then obtain y where "y \<in> Y" and x: "x = f y' y" by blast
       define y'' where "y'' = (if y \<sqsubseteq> y' then y' else y)"
       from chain \<open>y \<in> Y\<close> \<open>y' \<in> Y\<close> have "y \<sqsubseteq> y' \<or> y' \<sqsubseteq> y" by(rule chainD)
       hence "f y' y \<le> f y'' y''" using \<open>y \<in> Y\<close> \<open>y' \<in> Y\<close>
         by(auto simp add: y''_def intro: mono2 monotoneD[OF mono1])
       also from \<open>y \<in> Y\<close> \<open>y' \<in> Y\<close> have "y'' \<in> Y" by(simp add: y''_def)
       from chain3 have "f y'' y'' \<le> ?rhs" by(rule ccpo_Sup_upper)(simp add: \<open>y'' \<in> Y\<close>)
       finally show "x \<le> ?rhs" by(simp add: x)
     qed
     finally show "x' \<le> ?rhs" .
   qed
 
   show "?rhs \<le> ?lhs" using chain3
   proof(rule ccpo_Sup_least)
     fix y
     assume "y \<in> (\<lambda>x. f x x) ` Y"
     then obtain x where "x \<in> Y" and "y = f x x" by blast note this(2)
     also from chain2[OF \<open>x \<in> Y\<close>] have "\<dots> \<le> \<Squnion>(f x ` Y)"
       by(rule ccpo_Sup_upper)(simp add: \<open>x \<in> Y\<close>)
     also have "\<dots> \<le> ?lhs" by(rule ccpo_Sup_upper[OF chain1])(simp add: \<open>x \<in> Y\<close>)
     finally show "y \<le> ?lhs" .
   qed
 qed
 
 end
 
 lemma Sup_image_mono_le:
   fixes le_b (infix "\<sqsubseteq>" 60) and Sup_b ("\<Or>")
   assumes ccpo: "class.ccpo Sup_b (\<sqsubseteq>) lt_b"
   assumes chain: "Complete_Partial_Order.chain (\<sqsubseteq>) Y"
   and mono: "\<And>x y. \<lbrakk> x \<sqsubseteq> y; x \<in> Y \<rbrakk> \<Longrightarrow> f x \<le> f y"
   shows "Sup (f ` Y) \<le> f (\<Or>Y)"
 proof(rule ccpo_Sup_least)
   show "Complete_Partial_Order.chain (\<le>) (f ` Y)"
     using chain by(rule chain_imageI)(rule mono)
 
   fix x
   assume "x \<in> f ` Y"
   then obtain y where "y \<in> Y" and "x = f y" by blast note this(2)
   also have "y \<sqsubseteq> \<Or>Y" using ccpo chain \<open>y \<in> Y\<close> by(rule ccpo.ccpo_Sup_upper)
   hence "f y \<le> f (\<Or>Y)" using \<open>y \<in> Y\<close> by(rule mono)
   finally show "x \<le> \<dots>" .
 qed
 
 lemma swap_Sup:
   fixes le_b (infix "\<sqsubseteq>" 60)
   assumes Y: "Complete_Partial_Order.chain (\<sqsubseteq>) Y"
   and Z: "Complete_Partial_Order.chain (fun_ord (\<le>)) Z"
   and mono: "\<And>f. f \<in> Z \<Longrightarrow> monotone (\<sqsubseteq>) (\<le>) f"
   shows "\<Squnion>((\<lambda>x. \<Squnion>(x ` Y)) ` Z) = \<Squnion>((\<lambda>x. \<Squnion>((\<lambda>f. f x) ` Z)) ` Y)"
   (is "?lhs = ?rhs")
 proof(cases "Y = {}")
   case True
   then show ?thesis
     by (simp add: image_constant_conv cong del: SUP_cong_simp)
 next
   case False
   have chain1: "\<And>f. f \<in> Z \<Longrightarrow> Complete_Partial_Order.chain (\<le>) (f ` Y)"
     by(rule chain_imageI[OF Y])(rule monotoneD[OF mono])
   have chain2: "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. \<Squnion>(x ` Y)) ` Z)" using Z
   proof(rule chain_imageI)
     fix f g
     assume "f \<in> Z" "g \<in> Z"
       and "fun_ord (\<le>) f g"
     from chain1[OF \<open>f \<in> Z\<close>] show "\<Squnion>(f ` Y) \<le> \<Squnion>(g ` Y)"
     proof(rule ccpo_Sup_least)
       fix x
       assume "x \<in> f ` Y"
       then obtain y where "y \<in> Y" "x = f y" by blast note this(2)
       also have "\<dots> \<le> g y" using \<open>fun_ord (\<le>) f g\<close> by(simp add: fun_ord_def)
       also have "\<dots> \<le> \<Squnion>(g ` Y)" using chain1[OF \<open>g \<in> Z\<close>]
         by(rule ccpo_Sup_upper)(simp add: \<open>y \<in> Y\<close>)
       finally show "x \<le> \<Squnion>(g ` Y)" .
     qed
   qed
   have chain3: "\<And>x. Complete_Partial_Order.chain (\<le>) ((\<lambda>f. f x) ` Z)"
     using Z by(rule chain_imageI)(simp add: fun_ord_def)
   have chain4: "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. \<Squnion>((\<lambda>f. f x) ` Z)) ` Y)"
     using Y
   proof(rule chain_imageI)
     fix f x y
     assume "x \<sqsubseteq> y"
     show "\<Squnion>((\<lambda>f. f x) ` Z) \<le> \<Squnion>((\<lambda>f. f y) ` Z)" (is "_ \<le> ?rhs") using chain3
     proof(rule ccpo_Sup_least)
       fix x'
       assume "x' \<in> (\<lambda>f. f x) ` Z"
       then obtain f where "f \<in> Z" "x' = f x" by blast note this(2)
       also have "f x \<le> f y" using \<open>f \<in> Z\<close> \<open>x \<sqsubseteq> y\<close> by(rule monotoneD[OF mono])
       also have "f y \<le> ?rhs" using chain3
         by(rule ccpo_Sup_upper)(simp add: \<open>f \<in> Z\<close>)
       finally show "x' \<le> ?rhs" .
     qed
   qed
 
   from chain2 have "?lhs \<le> ?rhs"
   proof(rule ccpo_Sup_least)
     fix x
     assume "x \<in> (\<lambda>x. \<Squnion>(x ` Y)) ` Z"
     then obtain f where "f \<in> Z" "x = \<Squnion>(f ` Y)" by blast note this(2)
     also have "\<dots> \<le> ?rhs" using chain1[OF \<open>f \<in> Z\<close>]
     proof(rule ccpo_Sup_least)
       fix x'
       assume "x' \<in> f ` Y"
       then obtain y where "y \<in> Y" "x' = f y" by blast note this(2)
       also have "f y \<le> \<Squnion>((\<lambda>f. f y) ` Z)" using chain3
         by(rule ccpo_Sup_upper)(simp add: \<open>f \<in> Z\<close>)
       also have "\<dots> \<le> ?rhs" using chain4 by(rule ccpo_Sup_upper)(simp add: \<open>y \<in> Y\<close>)
       finally show "x' \<le> ?rhs" .
     qed
     finally show "x \<le> ?rhs" .
   qed
   moreover
   have "?rhs \<le> ?lhs" using chain4
   proof(rule ccpo_Sup_least)
     fix x
     assume "x \<in> (\<lambda>x. \<Squnion>((\<lambda>f. f x) ` Z)) ` Y"
     then obtain y where "y \<in> Y" "x = \<Squnion>((\<lambda>f. f y) ` Z)" by blast note this(2)
     also have "\<dots> \<le> ?lhs" using chain3
     proof(rule ccpo_Sup_least)
       fix x'
       assume "x' \<in> (\<lambda>f. f y) ` Z"
       then obtain f where "f \<in> Z" "x' = f y" by blast note this(2)
       also have "f y \<le> \<Squnion>(f ` Y)" using chain1[OF \<open>f \<in> Z\<close>]
         by(rule ccpo_Sup_upper)(simp add: \<open>y \<in> Y\<close>)
       also have "\<dots> \<le> ?lhs" using chain2
         by(rule ccpo_Sup_upper)(simp add: \<open>f \<in> Z\<close>)
       finally show "x' \<le> ?lhs" .
     qed
     finally show "x \<le> ?lhs" .
   qed
   ultimately show "?lhs = ?rhs"
     by (rule order.antisym)
 qed
 
 lemma fixp_mono:
   assumes fg: "fun_ord (\<le>) f g"
   and f: "monotone (\<le>) (\<le>) f"
   and g: "monotone (\<le>) (\<le>) g"
   shows "ccpo_class.fixp f \<le> ccpo_class.fixp g"
 unfolding fixp_def
 proof(rule ccpo_Sup_least)
   fix x
   assume "x \<in> ccpo_class.iterates f"
   thus "x \<le> \<Squnion>ccpo_class.iterates g"
   proof induction
     case (step x)
     from f step.IH have "f x \<le> f (\<Squnion>ccpo_class.iterates g)" by(rule monotoneD)
     also have "\<dots> \<le> g (\<Squnion>ccpo_class.iterates g)" using fg by(simp add: fun_ord_def)
     also have "\<dots> = \<Squnion>ccpo_class.iterates g" by(fold fixp_def fixp_unfold[OF g]) simp
     finally show ?case .
   qed(blast intro: ccpo_Sup_least)
 qed(rule chain_iterates[OF f])
 
 context fixes ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool" (infix "\<sqsubseteq>" 60) begin
 
 lemma iterates_mono:
   assumes f: "f \<in> ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F"
   and mono: "\<And>f. monotone (\<sqsubseteq>) (\<le>) f \<Longrightarrow> monotone (\<sqsubseteq>) (\<le>) (F f)"
   shows "monotone (\<sqsubseteq>) (\<le>) f"
 using f
 by(induction rule: ccpo.iterates.induct[OF ccpo_fun, consumes 1, case_names step Sup])(blast intro: mono mono_lub)+
 
 lemma fixp_preserves_mono:
   assumes mono: "\<And>x. monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. F f x)"
   and mono2: "\<And>f. monotone (\<sqsubseteq>) (\<le>) f \<Longrightarrow> monotone (\<sqsubseteq>) (\<le>) (F f)"
   shows "monotone (\<sqsubseteq>) (\<le>) (ccpo.fixp (fun_lub Sup) (fun_ord (\<le>)) F)"
   (is "monotone _ _ ?fixp")
 proof(rule monotoneI)
   have mono: "monotone (fun_ord (\<le>)) (fun_ord (\<le>)) F"
     by(rule monotoneI)(auto simp add: fun_ord_def intro: monotoneD[OF mono])
   let ?iter = "ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F"
   have chain: "\<And>x. Complete_Partial_Order.chain (\<le>) ((\<lambda>f. f x) ` ?iter)"
     by(rule chain_imageI[OF ccpo.chain_iterates[OF ccpo_fun mono]])(simp add: fun_ord_def)
 
   fix x y
   assume "x \<sqsubseteq> y"
   show "?fixp x \<le> ?fixp y"
     apply (simp only: ccpo.fixp_def[OF ccpo_fun] fun_lub_apply)
     using chain
   proof(rule ccpo_Sup_least)
     fix x'
     assume "x' \<in> (\<lambda>f. f x) ` ?iter"
     then obtain f where "f \<in> ?iter" "x' = f x" by blast note this(2)
     also have "f x \<le> f y"
       by(rule monotoneD[OF iterates_mono[OF \<open>f \<in> ?iter\<close> mono2]])(blast intro: \<open>x \<sqsubseteq> y\<close>)+
     also have "f y \<le> \<Squnion>((\<lambda>f. f y) ` ?iter)" using chain
       by(rule ccpo_Sup_upper)(simp add: \<open>f \<in> ?iter\<close>)
     finally show "x' \<le> \<dots>" .
   qed
 qed
 
 end
 
 end
 
 lemma monotone2monotone:
   assumes 2: "\<And>x. monotone ordb ordc (\<lambda>y. f x y)"
   and t: "monotone orda ordb (\<lambda>x. t x)"
   and 1: "\<And>y. monotone orda ordc (\<lambda>x. f x y)"
   and trans: "transp ordc"
   shows "monotone orda ordc (\<lambda>x. f x (t x))"
 by(blast intro: monotoneI transpD[OF trans] monotoneD[OF t] monotoneD[OF 2] monotoneD[OF 1])
 
 subsection \<open>Continuity\<close>
 
 definition cont :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b set \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
 where
   "cont luba orda lubb ordb f \<longleftrightarrow> 
   (\<forall>Y. Complete_Partial_Order.chain orda Y \<longrightarrow> Y \<noteq> {} \<longrightarrow> f (luba Y) = lubb (f ` Y))"
 
 definition mcont :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b set \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
 where
   "mcont luba orda lubb ordb f \<longleftrightarrow>
    monotone orda ordb f \<and> cont luba orda lubb ordb f"
 
 subsubsection \<open>Theorem collection \<open>cont_intro\<close>\<close>
 
 named_theorems cont_intro "continuity and admissibility intro rules"
 ML \<open>
 (* apply cont_intro rules as intro and try to solve 
    the remaining of the emerging subgoals with simp *)
 fun cont_intro_tac ctxt =
   REPEAT_ALL_NEW (resolve_tac ctxt (rev (Named_Theorems.get ctxt \<^named_theorems>\<open>cont_intro\<close>)))
   THEN_ALL_NEW (SOLVED' (simp_tac ctxt))
 
 fun cont_intro_simproc ctxt ct =
   let
     fun mk_stmt t = t
       |> HOLogic.mk_Trueprop
       |> Thm.cterm_of ctxt
       |> Goal.init
     fun mk_thm t =
-      case SINGLE (cont_intro_tac ctxt 1) (mk_stmt t) of
-        SOME thm => SOME (Goal.finish ctxt thm RS @{thm Eq_TrueI})
-      | NONE => NONE
+      if exists_subterm Term.is_Var t then
+        NONE
+      else
+        case SINGLE (cont_intro_tac ctxt 1) (mk_stmt t) of
+          SOME thm => SOME (Goal.finish ctxt thm RS @{thm Eq_TrueI})
+        | NONE => NONE
   in
     case Thm.term_of ct of
       t as \<^Const_>\<open>ccpo.admissible _ for _ _ _\<close> => mk_thm t
     | t as \<^Const_>\<open>mcont _ _ for _ _ _ _ _\<close> => mk_thm t
-    | t as \<^Const_>\<open>monotone_on _ _ for \<^Const_>\<open>Orderings.top _\<close> _ _ _\<close> => mk_thm t
+    | t as \<^Const_>\<open>monotone_on _ _ for _ _ _ _\<close> => mk_thm t
     | _ => NONE
   end
   handle THM _ => NONE 
   | TYPE _ => NONE
 \<close>
 
 simproc_setup "cont_intro"
   ( "ccpo.admissible lub ord P"
   | "mcont lub ord lub' ord' f"
   | "monotone ord ord' f"
   ) = \<open>K cont_intro_simproc\<close>
 
 lemmas [cont_intro] =
   call_mono
   let_mono
   if_mono
   option.const_mono
   tailrec.const_mono
   bind_mono
 
+experiment begin
+
+text \<open>The following proof by simplification diverges if variables are not handled properly.\<close>
+
+lemma "(\<And>f. monotone R S f \<Longrightarrow> thesis) \<Longrightarrow> monotone R S g \<Longrightarrow> thesis"
+  by simp
+
+end
+
 declare if_mono[simp]
 
 lemma monotone_id' [cont_intro]: "monotone ord ord (\<lambda>x. x)"
 by(simp add: monotone_def)
 
 lemma monotone_applyI:
   "monotone orda ordb F \<Longrightarrow> monotone (fun_ord orda) ordb (\<lambda>f. F (f x))"
 by(rule monotoneI)(auto simp add: fun_ord_def dest: monotoneD)
 
 lemma monotone_if_fun [partial_function_mono]:
   "\<lbrakk> monotone (fun_ord orda) (fun_ord ordb) F; monotone (fun_ord orda) (fun_ord ordb) G \<rbrakk>
   \<Longrightarrow> monotone (fun_ord orda) (fun_ord ordb) (\<lambda>f n. if c n then F f n else G f n)"
 by(simp add: monotone_def fun_ord_def)
 
 lemma monotone_fun_apply_fun [partial_function_mono]: 
   "monotone (fun_ord (fun_ord ord)) (fun_ord ord) (\<lambda>f n. f t (g n))"
 by(rule monotoneI)(simp add: fun_ord_def)
 
 lemma monotone_fun_ord_apply: 
   "monotone orda (fun_ord ordb) f \<longleftrightarrow> (\<forall>x. monotone orda ordb (\<lambda>y. f y x))"
 by(auto simp add: monotone_def fun_ord_def)
 
 context preorder begin
 
 declare transp_le[cont_intro]
 
 lemma monotone_const [simp, cont_intro]: "monotone ord (\<le>) (\<lambda>_. c)"
 by(rule monotoneI) simp
 
 end
 
 lemma transp_le [cont_intro, simp]:
   "class.preorder ord (mk_less ord) \<Longrightarrow> transp ord"
 by(rule preorder.transp_le)
 
 context partial_function_definitions begin
 
 declare const_mono [cont_intro, simp]
 
 lemma transp_le [cont_intro, simp]: "transp leq"
 by(rule transpI)(rule leq_trans)
 
 lemma preorder [cont_intro, simp]: "class.preorder leq (mk_less leq)"
 by(unfold_locales)(auto simp add: mk_less_def intro: leq_refl leq_trans)
 
 declare ccpo[cont_intro, simp]
 
 end
 
 lemma contI [intro?]:
   "(\<And>Y. \<lbrakk> Complete_Partial_Order.chain orda Y; Y \<noteq> {} \<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)) 
   \<Longrightarrow> cont luba orda lubb ordb f"
 unfolding cont_def by blast
 
 lemma contD:
   "\<lbrakk> cont luba orda lubb ordb f; Complete_Partial_Order.chain orda Y; Y \<noteq> {} \<rbrakk> 
   \<Longrightarrow> f (luba Y) = lubb (f ` Y)"
 unfolding cont_def by blast
 
 lemma cont_id [simp, cont_intro]: "\<And>Sup. cont Sup ord Sup ord id"
 by(rule contI) simp
 
 lemma cont_id' [simp, cont_intro]: "\<And>Sup. cont Sup ord Sup ord (\<lambda>x. x)"
 using cont_id[unfolded id_def] .
 
 lemma cont_applyI [cont_intro]:
   assumes cont: "cont luba orda lubb ordb g"
   shows "cont (fun_lub luba) (fun_ord orda) lubb ordb (\<lambda>f. g (f x))"
 by(rule contI)(drule chain_fun_ordD[where x=x], simp add: fun_lub_apply image_image contD[OF cont])
 
 lemma call_cont: "cont (fun_lub lub) (fun_ord ord) lub ord (\<lambda>f. f t)"
 by(simp add: cont_def fun_lub_apply)
 
 lemma cont_if [cont_intro]:
   "\<lbrakk> cont luba orda lubb ordb f; cont luba orda lubb ordb g \<rbrakk>
   \<Longrightarrow> cont luba orda lubb ordb (\<lambda>x. if c then f x else g x)"
 by(cases c) simp_all
 
 lemma mcontI [intro?]:
    "\<lbrakk> monotone orda ordb f; cont luba orda lubb ordb f \<rbrakk> \<Longrightarrow> mcont luba orda lubb ordb f"
 by(simp add: mcont_def)
 
 lemma mcont_mono: "mcont luba orda lubb ordb f \<Longrightarrow> monotone orda ordb f"
 by(simp add: mcont_def)
 
 lemma mcont_cont [simp]: "mcont luba orda lubb ordb f \<Longrightarrow> cont luba orda lubb ordb f"
 by(simp add: mcont_def)
 
 lemma mcont_monoD:
   "\<lbrakk> mcont luba orda lubb ordb f; orda x y \<rbrakk> \<Longrightarrow> ordb (f x) (f y)"
 by(auto simp add: mcont_def dest: monotoneD)
 
 lemma mcont_contD:
   "\<lbrakk> mcont luba orda lubb ordb f; Complete_Partial_Order.chain orda Y; Y \<noteq> {} \<rbrakk>
   \<Longrightarrow> f (luba Y) = lubb (f ` Y)"
 by(auto simp add: mcont_def dest: contD)
 
 lemma mcont_call [cont_intro, simp]:
   "mcont (fun_lub lub) (fun_ord ord) lub ord (\<lambda>f. f t)"
 by(simp add: mcont_def call_mono call_cont)
 
 lemma mcont_id' [cont_intro, simp]: "mcont lub ord lub ord (\<lambda>x. x)"
 by(simp add: mcont_def monotone_id')
 
 lemma mcont_applyI:
   "mcont luba orda lubb ordb (\<lambda>x. F x) \<Longrightarrow> mcont (fun_lub luba) (fun_ord orda) lubb ordb (\<lambda>f. F (f x))"
 by(simp add: mcont_def monotone_applyI cont_applyI)
 
 lemma mcont_if [cont_intro, simp]:
   "\<lbrakk> mcont luba orda lubb ordb (\<lambda>x. f x); mcont luba orda lubb ordb (\<lambda>x. g x) \<rbrakk>
   \<Longrightarrow> mcont luba orda lubb ordb (\<lambda>x. if c then f x else g x)"
 by(simp add: mcont_def cont_if)
 
 lemma cont_fun_lub_apply: 
   "cont luba orda (fun_lub lubb) (fun_ord ordb) f \<longleftrightarrow> (\<forall>x. cont luba orda lubb ordb (\<lambda>y. f y x))"
 by(simp add: cont_def fun_lub_def fun_eq_iff)(auto simp add: image_def)
 
 lemma mcont_fun_lub_apply: 
   "mcont luba orda (fun_lub lubb) (fun_ord ordb) f \<longleftrightarrow> (\<forall>x. mcont luba orda lubb ordb (\<lambda>y. f y x))"
 by(auto simp add: monotone_fun_ord_apply cont_fun_lub_apply mcont_def)
 
 context ccpo begin
 
 lemma cont_const [simp, cont_intro]: "cont luba orda Sup (\<le>) (\<lambda>x. c)"
 by (rule contI) (simp add: image_constant_conv cong del: SUP_cong_simp)
 
 lemma mcont_const [cont_intro, simp]:
   "mcont luba orda Sup (\<le>) (\<lambda>x. c)"
 by(simp add: mcont_def)
 
 lemma cont_apply:
   assumes 2: "\<And>x. cont lubb ordb Sup (\<le>) (\<lambda>y. f x y)"
   and t: "cont luba orda lubb ordb (\<lambda>x. t x)"
   and 1: "\<And>y. cont luba orda Sup (\<le>) (\<lambda>x. f x y)"
   and mono: "monotone orda ordb (\<lambda>x. t x)"
   and mono2: "\<And>x. monotone ordb (\<le>) (\<lambda>y. f x y)"
   and mono1: "\<And>y. monotone orda (\<le>) (\<lambda>x. f x y)"
   shows "cont luba orda Sup (\<le>) (\<lambda>x. f x (t x))"
 proof
   fix Y
   assume chain: "Complete_Partial_Order.chain orda Y" and "Y \<noteq> {}"
   moreover from chain have chain': "Complete_Partial_Order.chain ordb (t ` Y)"
     by(rule chain_imageI)(rule monotoneD[OF mono])
   ultimately show "f (luba Y) (t (luba Y)) = \<Squnion>((\<lambda>x. f x (t x)) ` Y)"
     by(simp add: contD[OF 1] contD[OF t] contD[OF 2] image_image)
       (rule diag_Sup[OF chain], auto intro: monotone2monotone[OF mono2 mono monotone_const transpI] monotoneD[OF mono1])
 qed
 
 lemma mcont2mcont':
   "\<lbrakk> \<And>x. mcont lub' ord' Sup (\<le>) (\<lambda>y. f x y);
      \<And>y. mcont lub ord Sup (\<le>) (\<lambda>x. f x y);
      mcont lub ord lub' ord' (\<lambda>y. t y) \<rbrakk>
   \<Longrightarrow> mcont lub ord Sup (\<le>) (\<lambda>x. f x (t x))"
 unfolding mcont_def by(blast intro: transp_le monotone2monotone cont_apply)
 
 lemma mcont2mcont:
   "\<lbrakk>mcont lub' ord' Sup (\<le>) (\<lambda>x. f x); mcont lub ord lub' ord' (\<lambda>x. t x)\<rbrakk> 
   \<Longrightarrow> mcont lub ord Sup (\<le>) (\<lambda>x. f (t x))"
 by(rule mcont2mcont'[OF _ mcont_const]) 
 
 context
   fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool" (infix "\<sqsubseteq>" 60) 
   and lub :: "'b set \<Rightarrow> 'b" ("\<Or>")
 begin
 
 lemma cont_fun_lub_Sup:
   assumes chainM: "Complete_Partial_Order.chain (fun_ord (\<le>)) M"
   and mcont [rule_format]: "\<forall>f\<in>M. mcont lub (\<sqsubseteq>) Sup (\<le>) f"
   shows "cont lub (\<sqsubseteq>) Sup (\<le>) (fun_lub Sup M)"
 proof(rule contI)
   fix Y
   assume chain: "Complete_Partial_Order.chain (\<sqsubseteq>) Y"
     and Y: "Y \<noteq> {}"
   from swap_Sup[OF chain chainM mcont[THEN mcont_mono]]
   show "fun_lub Sup M (\<Or>Y) = \<Squnion>(fun_lub Sup M ` Y)"
     by(simp add: mcont_contD[OF mcont chain Y] fun_lub_apply cong: image_cong)
 qed
 
 lemma mcont_fun_lub_Sup:
   "\<lbrakk> Complete_Partial_Order.chain (fun_ord (\<le>)) M;
     \<forall>f\<in>M. mcont lub ord Sup (\<le>) f \<rbrakk>
   \<Longrightarrow> mcont lub (\<sqsubseteq>) Sup (\<le>) (fun_lub Sup M)"
 by(simp add: mcont_def cont_fun_lub_Sup mono_lub)
 
 lemma iterates_mcont:
   assumes f: "f \<in> ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F"
   and mono: "\<And>f. mcont lub (\<sqsubseteq>) Sup (\<le>) f \<Longrightarrow> mcont lub (\<sqsubseteq>) Sup (\<le>) (F f)"
   shows "mcont lub (\<sqsubseteq>) Sup (\<le>) f"
 using f
 by(induction rule: ccpo.iterates.induct[OF ccpo_fun, consumes 1, case_names step Sup])(blast intro: mono mcont_fun_lub_Sup)+
 
 lemma fixp_preserves_mcont:
   assumes mono: "\<And>x. monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. F f x)"
   and mcont: "\<And>f. mcont lub (\<sqsubseteq>) Sup (\<le>) f \<Longrightarrow> mcont lub (\<sqsubseteq>) Sup (\<le>) (F f)"
   shows "mcont lub (\<sqsubseteq>) Sup (\<le>) (ccpo.fixp (fun_lub Sup) (fun_ord (\<le>)) F)"
   (is "mcont _ _ _ _ ?fixp")
 unfolding mcont_def
 proof(intro conjI monotoneI contI)
   have mono: "monotone (fun_ord (\<le>)) (fun_ord (\<le>)) F"
     by(rule monotoneI)(auto simp add: fun_ord_def intro: monotoneD[OF mono])
   let ?iter = "ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F"
   have chain: "\<And>x. Complete_Partial_Order.chain (\<le>) ((\<lambda>f. f x) ` ?iter)"
     by(rule chain_imageI[OF ccpo.chain_iterates[OF ccpo_fun mono]])(simp add: fun_ord_def)
 
   {
     fix x y
     assume "x \<sqsubseteq> y"
     show "?fixp x \<le> ?fixp y"
       apply (simp only: ccpo.fixp_def[OF ccpo_fun] fun_lub_apply)
       using chain
     proof(rule ccpo_Sup_least)
       fix x'
       assume "x' \<in> (\<lambda>f. f x) ` ?iter"
       then obtain f where "f \<in> ?iter" "x' = f x" by blast note this(2)
       also from _ \<open>x \<sqsubseteq> y\<close> have "f x \<le> f y"
         by(rule mcont_monoD[OF iterates_mcont[OF \<open>f \<in> ?iter\<close> mcont]])
       also have "f y \<le> \<Squnion>((\<lambda>f. f y) ` ?iter)" using chain
         by(rule ccpo_Sup_upper)(simp add: \<open>f \<in> ?iter\<close>)
       finally show "x' \<le> \<dots>" .
     qed
   next
     fix Y
     assume chain: "Complete_Partial_Order.chain (\<sqsubseteq>) Y"
       and Y: "Y \<noteq> {}"
     { fix f
       assume "f \<in> ?iter"
       hence "f (\<Or>Y) = \<Squnion>(f ` Y)"
         using mcont chain Y by(rule mcont_contD[OF iterates_mcont]) }
     moreover have "\<Squnion>((\<lambda>f. \<Squnion>(f ` Y)) ` ?iter) = \<Squnion>((\<lambda>x. \<Squnion>((\<lambda>f. f x) ` ?iter)) ` Y)"
       using chain ccpo.chain_iterates[OF ccpo_fun mono]
       by(rule swap_Sup)(rule mcont_mono[OF iterates_mcont[OF _ mcont]])
     ultimately show "?fixp (\<Or>Y) = \<Squnion>(?fixp ` Y)" unfolding ccpo.fixp_def[OF ccpo_fun]
       by(simp add: fun_lub_apply cong: image_cong)
   }
 qed
 
 end
 
 context
   fixes F :: "'c \<Rightarrow> 'c" and U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" and C :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c" and f
   assumes mono: "\<And>x. monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. U (F (C f)) x)"
   and eq: "f \<equiv> C (ccpo.fixp (fun_lub Sup) (fun_ord (\<le>)) (\<lambda>f. U (F (C f))))"
   and inverse: "\<And>f. U (C f) = f"
 begin
 
 lemma fixp_preserves_mono_uc:
   assumes mono2: "\<And>f. monotone ord (\<le>) (U f) \<Longrightarrow> monotone ord (\<le>) (U (F f))"
   shows "monotone ord (\<le>) (U f)"
 using fixp_preserves_mono[OF mono mono2] by(subst eq)(simp add: inverse)
 
 lemma fixp_preserves_mcont_uc:
   assumes mcont: "\<And>f. mcont lubb ordb Sup (\<le>) (U f) \<Longrightarrow> mcont lubb ordb Sup (\<le>) (U (F f))"
   shows "mcont lubb ordb Sup (\<le>) (U f)"
 using fixp_preserves_mcont[OF mono mcont] by(subst eq)(simp add: inverse)
 
 end
 
 lemmas fixp_preserves_mono1 = fixp_preserves_mono_uc[of "\<lambda>x. x" _ "\<lambda>x. x", OF _ _ refl]
 lemmas fixp_preserves_mono2 =
   fixp_preserves_mono_uc[of "case_prod" _ "curry", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
 lemmas fixp_preserves_mono3 =
   fixp_preserves_mono_uc[of "\<lambda>f. case_prod (case_prod f)" _ "\<lambda>f. curry (curry f)", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
 lemmas fixp_preserves_mono4 =
   fixp_preserves_mono_uc[of "\<lambda>f. case_prod (case_prod (case_prod f))" _ "\<lambda>f. curry (curry (curry f))", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
 
 lemmas fixp_preserves_mcont1 = fixp_preserves_mcont_uc[of "\<lambda>x. x" _ "\<lambda>x. x", OF _ _ refl]
 lemmas fixp_preserves_mcont2 =
   fixp_preserves_mcont_uc[of "case_prod" _ "curry", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
 lemmas fixp_preserves_mcont3 =
   fixp_preserves_mcont_uc[of "\<lambda>f. case_prod (case_prod f)" _ "\<lambda>f. curry (curry f)", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
 lemmas fixp_preserves_mcont4 =
   fixp_preserves_mcont_uc[of "\<lambda>f. case_prod (case_prod (case_prod f))" _ "\<lambda>f. curry (curry (curry f))", unfolded case_prod_curry curry_case_prod, OF _ _ refl]
 
 end
 
 lemma (in preorder) monotone_if_bot:
   fixes bot
   assumes mono: "\<And>x y. \<lbrakk> x \<le> y; \<not> (x \<le> bound) \<rbrakk> \<Longrightarrow> ord (f x) (f y)"
   and bot: "\<And>x. \<not> x \<le> bound \<Longrightarrow> ord bot (f x)" "ord bot bot"
   shows "monotone (\<le>) ord (\<lambda>x. if x \<le> bound then bot else f x)"
 by(rule monotoneI)(auto intro: bot intro: mono order_trans)
 
 lemma (in ccpo) mcont_if_bot:
   fixes bot and lub ("\<Or>") and ord (infix "\<sqsubseteq>" 60)
   assumes ccpo: "class.ccpo lub (\<sqsubseteq>) lt"
   and mono: "\<And>x y. \<lbrakk> x \<le> y; \<not> x \<le> bound \<rbrakk> \<Longrightarrow> f x \<sqsubseteq> f y"
   and cont: "\<And>Y. \<lbrakk> Complete_Partial_Order.chain (\<le>) Y; Y \<noteq> {}; \<And>x. x \<in> Y \<Longrightarrow> \<not> x \<le> bound \<rbrakk> \<Longrightarrow> f (\<Squnion>Y) = \<Or>(f ` Y)"
   and bot: "\<And>x. \<not> x \<le> bound \<Longrightarrow> bot \<sqsubseteq> f x"
   shows "mcont Sup (\<le>) lub (\<sqsubseteq>) (\<lambda>x. if x \<le> bound then bot else f x)" (is "mcont _ _ _ _ ?g")
 proof(intro mcontI contI)
   interpret c: ccpo lub "(\<sqsubseteq>)" lt by(fact ccpo)
   show "monotone (\<le>) (\<sqsubseteq>) ?g" by(rule monotone_if_bot)(simp_all add: mono bot)
 
   fix Y
   assume chain: "Complete_Partial_Order.chain (\<le>) Y" and Y: "Y \<noteq> {}"
   show "?g (\<Squnion>Y) = \<Or>(?g ` Y)"
   proof(cases "Y \<subseteq> {x. x \<le> bound}")
     case True
     hence "\<Squnion>Y \<le> bound" using chain by(auto intro: ccpo_Sup_least)
     moreover have "Y \<inter> {x. \<not> x \<le> bound} = {}" using True by auto
     ultimately show ?thesis using True Y
       by (auto simp add: image_constant_conv cong del: c.SUP_cong_simp)
   next
     case False
     let ?Y = "Y \<inter> {x. \<not> x \<le> bound}"
     have chain': "Complete_Partial_Order.chain (\<le>) ?Y"
       using chain by(rule chain_subset) simp
 
     from False obtain y where ybound: "\<not> y \<le> bound" and y: "y \<in> Y" by blast
     hence "\<not> \<Squnion>Y \<le> bound" by (metis ccpo_Sup_upper chain order.trans)
     hence "?g (\<Squnion>Y) = f (\<Squnion>Y)" by simp
     also have "\<Squnion>Y \<le> \<Squnion>?Y" using chain
     proof(rule ccpo_Sup_least)
       fix x
       assume x: "x \<in> Y"
       show "x \<le> \<Squnion>?Y"
       proof(cases "x \<le> bound")
         case True
         with chainD[OF chain x y] have "x \<le> y" using ybound by(auto intro: order_trans)
         thus ?thesis by(rule order_trans)(auto intro: ccpo_Sup_upper[OF chain'] simp add: y ybound)
       qed(auto intro: ccpo_Sup_upper[OF chain'] simp add: x)
     qed
     hence "\<Squnion>Y = \<Squnion>?Y" by(rule order.antisym)(blast intro: ccpo_Sup_least[OF chain'] ccpo_Sup_upper[OF chain])
     hence "f (\<Squnion>Y) = f (\<Squnion>?Y)" by simp
     also have "f (\<Squnion>?Y) = \<Or>(f ` ?Y)" using chain' by(rule cont)(insert y ybound, auto)
     also have "\<Or>(f ` ?Y) = \<Or>(?g ` Y)"
     proof(cases "Y \<inter> {x. x \<le> bound} = {}")
       case True
       hence "f ` ?Y = ?g ` Y" by auto
       thus ?thesis by(rule arg_cong)
     next
       case False
       have chain'': "Complete_Partial_Order.chain (\<sqsubseteq>) (insert bot (f ` ?Y))"
         using chain by(auto intro!: chainI bot dest: chainD intro: mono)
       hence chain''': "Complete_Partial_Order.chain (\<sqsubseteq>) (f ` ?Y)" by(rule chain_subset) blast
       have "bot \<sqsubseteq> \<Or>(f ` ?Y)" using y ybound by(blast intro: c.order_trans[OF bot] c.ccpo_Sup_upper[OF chain'''])
       hence "\<Or>(insert bot (f ` ?Y)) \<sqsubseteq> \<Or>(f ` ?Y)" using chain''
         by(auto intro: c.ccpo_Sup_least c.ccpo_Sup_upper[OF chain''']) 
       with _ have "\<dots> = \<Or>(insert bot (f ` ?Y))"
         by(rule c.order.antisym)(blast intro: c.ccpo_Sup_least[OF chain'''] c.ccpo_Sup_upper[OF chain''])
       also have "insert bot (f ` ?Y) = ?g ` Y" using False by auto
       finally show ?thesis .
     qed
     finally show ?thesis .
   qed
 qed
 
 context partial_function_definitions begin
 
 lemma mcont_const [cont_intro, simp]:
   "mcont luba orda lub leq (\<lambda>x. c)"
 by(rule ccpo.mcont_const)(rule Partial_Function.ccpo[OF partial_function_definitions_axioms])
 
 lemmas [cont_intro, simp] =
   ccpo.cont_const[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 
 lemma mono2mono:
   assumes "monotone ordb leq (\<lambda>y. f y)" "monotone orda ordb (\<lambda>x. t x)"
   shows "monotone orda leq (\<lambda>x. f (t x))"
 using assms by(rule monotone2monotone) simp_all
 
 lemmas mcont2mcont' = ccpo.mcont2mcont'[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas mcont2mcont = ccpo.mcont2mcont[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 
 lemmas fixp_preserves_mono1 = ccpo.fixp_preserves_mono1[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas fixp_preserves_mono2 = ccpo.fixp_preserves_mono2[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas fixp_preserves_mono3 = ccpo.fixp_preserves_mono3[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas fixp_preserves_mono4 = ccpo.fixp_preserves_mono4[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas fixp_preserves_mcont1 = ccpo.fixp_preserves_mcont1[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas fixp_preserves_mcont2 = ccpo.fixp_preserves_mcont2[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas fixp_preserves_mcont3 = ccpo.fixp_preserves_mcont3[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 lemmas fixp_preserves_mcont4 = ccpo.fixp_preserves_mcont4[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 
 lemma monotone_if_bot:
   fixes bot
   assumes g: "\<And>x. g x = (if leq x bound then bot else f x)"
   and mono: "\<And>x y. \<lbrakk> leq x y; \<not> leq x bound \<rbrakk> \<Longrightarrow> ord (f x) (f y)"
   and bot: "\<And>x. \<not> leq x bound \<Longrightarrow> ord bot (f x)" "ord bot bot"
   shows "monotone leq ord g"
 unfolding g[abs_def] using preorder mono bot by(rule preorder.monotone_if_bot)
 
 lemma mcont_if_bot:
   fixes bot
   assumes ccpo: "class.ccpo lub' ord (mk_less ord)"
   and bot: "\<And>x. \<not> leq x bound \<Longrightarrow> ord bot (f x)"
   and g: "\<And>x. g x = (if leq x bound then bot else f x)"
   and mono: "\<And>x y. \<lbrakk> leq x y; \<not> leq x bound \<rbrakk> \<Longrightarrow> ord (f x) (f y)"
   and cont: "\<And>Y. \<lbrakk> Complete_Partial_Order.chain leq Y; Y \<noteq> {}; \<And>x. x \<in> Y \<Longrightarrow> \<not> leq x bound \<rbrakk> \<Longrightarrow> f (lub Y) = lub' (f ` Y)"
   shows "mcont lub leq lub' ord g"
 unfolding g[abs_def] using ccpo mono cont bot by(rule ccpo.mcont_if_bot[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]])
 
 end
 
 subsection \<open>Admissibility\<close>
 
 lemma admissible_subst:
   assumes adm: "ccpo.admissible luba orda (\<lambda>x. P x)"
   and mcont: "mcont lubb ordb luba orda f"
   shows "ccpo.admissible lubb ordb (\<lambda>x. P (f x))"
 apply(rule ccpo.admissibleI)
 apply(frule (1) mcont_contD[OF mcont])
 apply(auto intro: ccpo.admissibleD[OF adm] chain_imageI dest: mcont_monoD[OF mcont])
 done
 
 lemmas [simp, cont_intro] = 
   admissible_all
   admissible_ball
   admissible_const
   admissible_conj
 
 lemma admissible_disj' [simp, cont_intro]:
   "\<lbrakk> class.ccpo lub ord (mk_less ord); ccpo.admissible lub ord P; ccpo.admissible lub ord Q \<rbrakk>
   \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x \<or> Q x)"
 by(rule ccpo.admissible_disj)
 
 lemma admissible_imp' [cont_intro]:
   "\<lbrakk> class.ccpo lub ord (mk_less ord);
      ccpo.admissible lub ord (\<lambda>x. \<not> P x);
      ccpo.admissible lub ord (\<lambda>x. Q x) \<rbrakk>
   \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x \<longrightarrow> Q x)"
 unfolding imp_conv_disj by(rule ccpo.admissible_disj)
 
 lemma admissible_imp [cont_intro]:
   "(Q \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x))
   \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. Q \<longrightarrow> P x)"
 by(rule ccpo.admissibleI)(auto dest: ccpo.admissibleD)
 
 lemma admissible_not_mem' [THEN admissible_subst, cont_intro, simp]:
   shows admissible_not_mem: "ccpo.admissible Union (\<subseteq>) (\<lambda>A. x \<notin> A)"
 by(rule ccpo.admissibleI) auto
 
 lemma admissible_eqI:
   assumes f: "cont luba orda lub ord (\<lambda>x. f x)"
   and g: "cont luba orda lub ord (\<lambda>x. g x)"
   shows "ccpo.admissible luba orda (\<lambda>x. f x = g x)"
 apply(rule ccpo.admissibleI)
 apply(simp_all add: contD[OF f] contD[OF g] cong: image_cong)
 done
 
 corollary admissible_eq_mcontI [cont_intro]:
   "\<lbrakk> mcont luba orda lub ord (\<lambda>x. f x); 
     mcont luba orda lub ord (\<lambda>x. g x) \<rbrakk>
   \<Longrightarrow> ccpo.admissible luba orda (\<lambda>x. f x = g x)"
 by(rule admissible_eqI)(auto simp add: mcont_def)
 
 lemma admissible_iff [cont_intro, simp]:
   "\<lbrakk> ccpo.admissible lub ord (\<lambda>x. P x \<longrightarrow> Q x); ccpo.admissible lub ord (\<lambda>x. Q x \<longrightarrow> P x) \<rbrakk>
   \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x \<longleftrightarrow> Q x)"
 by(subst iff_conv_conj_imp)(rule admissible_conj)
 
 context ccpo begin
 
 lemma admissible_leI:
   assumes f: "mcont luba orda Sup (\<le>) (\<lambda>x. f x)"
   and g: "mcont luba orda Sup (\<le>) (\<lambda>x. g x)"
   shows "ccpo.admissible luba orda (\<lambda>x. f x \<le> g x)"
 proof(rule ccpo.admissibleI)
   fix A
   assume chain: "Complete_Partial_Order.chain orda A"
     and le: "\<forall>x\<in>A. f x \<le> g x"
     and False: "A \<noteq> {}"
   have "f (luba A) = \<Squnion>(f ` A)" by(simp add: mcont_contD[OF f] chain False)
   also have "\<dots> \<le> \<Squnion>(g ` A)"
   proof(rule ccpo_Sup_least)
     from chain show "Complete_Partial_Order.chain (\<le>) (f ` A)"
       by(rule chain_imageI)(rule mcont_monoD[OF f])
     
     fix x
     assume "x \<in> f ` A"
     then obtain y where "y \<in> A" "x = f y" by blast note this(2)
     also have "f y \<le> g y" using le \<open>y \<in> A\<close> by simp
     also have "Complete_Partial_Order.chain (\<le>) (g ` A)"
       using chain by(rule chain_imageI)(rule mcont_monoD[OF g])
     hence "g y \<le> \<Squnion>(g ` A)" by(rule ccpo_Sup_upper)(simp add: \<open>y \<in> A\<close>)
     finally show "x \<le> \<dots>" .
   qed
   also have "\<dots> = g (luba A)" by(simp add: mcont_contD[OF g] chain False)
   finally show "f (luba A) \<le> g (luba A)" .
 qed
 
 end
 
 lemma admissible_leI:
   fixes ord (infix "\<sqsubseteq>" 60) and lub ("\<Or>")
   assumes "class.ccpo lub (\<sqsubseteq>) (mk_less (\<sqsubseteq>))"
   and "mcont luba orda lub (\<sqsubseteq>) (\<lambda>x. f x)"
   and "mcont luba orda lub (\<sqsubseteq>) (\<lambda>x. g x)"
   shows "ccpo.admissible luba orda (\<lambda>x. f x \<sqsubseteq> g x)"
 using assms by(rule ccpo.admissible_leI)
 
 declare ccpo_class.admissible_leI[cont_intro]
 
 context ccpo begin
 
 lemma admissible_not_below: "ccpo.admissible Sup (\<le>) (\<lambda>x. \<not> (\<le>) x y)"
 by(rule ccpo.admissibleI)(simp add: ccpo_Sup_below_iff)
 
 end
 
 lemma (in preorder) preorder [cont_intro, simp]: "class.preorder (\<le>) (mk_less (\<le>))"
 by(unfold_locales)(auto simp add: mk_less_def intro: order_trans)
 
 context partial_function_definitions begin
 
 lemmas [cont_intro, simp] =
   admissible_leI[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
   ccpo.admissible_not_below[THEN admissible_subst, OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 
 end
 
 setup \<open>Sign.map_naming (Name_Space.mandatory_path "ccpo")\<close>
 
 inductive compact :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
   for lub ord x 
 where compact:
   "\<lbrakk> ccpo.admissible lub ord (\<lambda>y. \<not> ord x y);
      ccpo.admissible lub ord (\<lambda>y. x \<noteq> y) \<rbrakk>
   \<Longrightarrow> compact lub ord x"
 
 setup \<open>Sign.map_naming Name_Space.parent_path\<close>
 
 context ccpo begin
 
 lemma compactI:
   assumes "ccpo.admissible Sup (\<le>) (\<lambda>y. \<not> x \<le> y)"
   shows "ccpo.compact Sup (\<le>) x"
 using assms
 proof(rule ccpo.compact.intros)
   have neq: "(\<lambda>y. x \<noteq> y) = (\<lambda>y. \<not> x \<le> y \<or> \<not> y \<le> x)" by(auto)
   show "ccpo.admissible Sup (\<le>) (\<lambda>y. x \<noteq> y)"
     by(subst neq)(rule admissible_disj admissible_not_below assms)+
 qed
 
 lemma compact_bot:
   assumes "x = Sup {}"
   shows "ccpo.compact Sup (\<le>) x"
 proof(rule compactI)
   show "ccpo.admissible Sup (\<le>) (\<lambda>y. \<not> x \<le> y)" using assms
     by(auto intro!: ccpo.admissibleI intro: ccpo_Sup_least chain_empty)
 qed
 
 end
 
 lemma admissible_compact_neq' [THEN admissible_subst, cont_intro, simp]:
   shows admissible_compact_neq: "ccpo.compact lub ord k \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. k \<noteq> x)"
 by(simp add: ccpo.compact.simps)
 
 lemma admissible_neq_compact' [THEN admissible_subst, cont_intro, simp]:
   shows admissible_neq_compact: "ccpo.compact lub ord k \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. x \<noteq> k)"
 by(subst eq_commute)(rule admissible_compact_neq)
 
 context partial_function_definitions begin
 
 lemmas [cont_intro, simp] = ccpo.compact_bot[OF Partial_Function.ccpo[OF partial_function_definitions_axioms]]
 
 end
 
 context ccpo begin
 
 lemma fixp_strong_induct:
   assumes [cont_intro]: "ccpo.admissible Sup (\<le>) P"
   and mono: "monotone (\<le>) (\<le>) f"
   and bot: "P (\<Squnion>{})"
   and step: "\<And>x. \<lbrakk> x \<le> ccpo_class.fixp f; P x \<rbrakk> \<Longrightarrow> P (f x)"
   shows "P (ccpo_class.fixp f)"
 proof(rule fixp_induct[where P="\<lambda>x. x \<le> ccpo_class.fixp f \<and> P x", THEN conjunct2])
   note [cont_intro] = admissible_leI
   show "ccpo.admissible Sup (\<le>) (\<lambda>x. x \<le> ccpo_class.fixp f \<and> P x)" by simp
 next
   show "\<Squnion>{} \<le> ccpo_class.fixp f \<and> P (\<Squnion>{})"
     by(auto simp add: bot intro: ccpo_Sup_least chain_empty)
 next
   fix x
   assume "x \<le> ccpo_class.fixp f \<and> P x"
   thus "f x \<le> ccpo_class.fixp f \<and> P (f x)"
     by(subst fixp_unfold[OF mono])(auto dest: monotoneD[OF mono] intro: step)
 qed(rule mono)
 
 end
 
 context partial_function_definitions begin
 
 lemma fixp_strong_induct_uc:
   fixes F :: "'c \<Rightarrow> 'c"
     and U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"
     and C :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c"
     and P :: "('b \<Rightarrow> 'a) \<Rightarrow> bool"
   assumes mono: "\<And>x. mono_body (\<lambda>f. U (F (C f)) x)"
     and eq: "f \<equiv> C (fixp_fun (\<lambda>f. U (F (C f))))"
     and inverse: "\<And>f. U (C f) = f"
     and adm: "ccpo.admissible lub_fun le_fun P"
     and bot: "P (\<lambda>_. lub {})"
     and step: "\<And>f'. \<lbrakk> P (U f'); le_fun (U f') (U f) \<rbrakk> \<Longrightarrow> P (U (F f'))"
   shows "P (U f)"
 unfolding eq inverse
 apply (rule ccpo.fixp_strong_induct[OF ccpo adm])
 apply (insert mono, auto simp: monotone_def fun_ord_def bot fun_lub_def)[2]
 apply (rule_tac f'5="C x" in step)
 apply (simp_all add: inverse eq)
 done
 
 end
 
 subsection \<open>\<^term>\<open>(=)\<close> as order\<close>
 
 definition lub_singleton :: "('a set \<Rightarrow> 'a) \<Rightarrow> bool"
 where "lub_singleton lub \<longleftrightarrow> (\<forall>a. lub {a} = a)"
 
 definition the_Sup :: "'a set \<Rightarrow> 'a"
 where "the_Sup A = (THE a. a \<in> A)"
 
 lemma lub_singleton_the_Sup [cont_intro, simp]: "lub_singleton the_Sup"
 by(simp add: lub_singleton_def the_Sup_def)
 
 lemma (in ccpo) lub_singleton: "lub_singleton Sup"
 by(simp add: lub_singleton_def)
 
 lemma (in partial_function_definitions) lub_singleton [cont_intro, simp]: "lub_singleton lub"
 by(rule ccpo.lub_singleton)(rule Partial_Function.ccpo[OF partial_function_definitions_axioms])
 
 lemma preorder_eq [cont_intro, simp]:
   "class.preorder (=) (mk_less (=))"
 by(unfold_locales)(simp_all add: mk_less_def)
 
 lemma monotone_eqI [cont_intro]:
   assumes "class.preorder ord (mk_less ord)"
   shows "monotone (=) ord f"
 proof -
   interpret preorder ord "mk_less ord" by fact
   show ?thesis by(simp add: monotone_def)
 qed
 
 lemma cont_eqI [cont_intro]: 
   fixes f :: "'a \<Rightarrow> 'b"
   assumes "lub_singleton lub"
   shows "cont the_Sup (=) lub ord f"
 proof(rule contI)
   fix Y :: "'a set"
   assume "Complete_Partial_Order.chain (=) Y" "Y \<noteq> {}"
   then obtain a where "Y = {a}" by(auto simp add: chain_def)
   thus "f (the_Sup Y) = lub (f ` Y)" using assms
     by(simp add: the_Sup_def lub_singleton_def)
 qed
 
 lemma mcont_eqI [cont_intro, simp]:
   "\<lbrakk> class.preorder ord (mk_less ord); lub_singleton lub \<rbrakk>
   \<Longrightarrow> mcont the_Sup (=) lub ord f"
 by(simp add: mcont_def cont_eqI monotone_eqI)
 
 subsection \<open>ccpo for products\<close>
 
 definition prod_lub :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('b set \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'b) set \<Rightarrow> 'a \<times> 'b"
 where "prod_lub Sup_a Sup_b Y = (Sup_a (fst ` Y), Sup_b (snd ` Y))"
 
 lemma lub_singleton_prod_lub [cont_intro, simp]:
   "\<lbrakk> lub_singleton luba; lub_singleton lubb \<rbrakk> \<Longrightarrow> lub_singleton (prod_lub luba lubb)"
 by(simp add: lub_singleton_def prod_lub_def)
 
 lemma prod_lub_empty [simp]: "prod_lub luba lubb {} = (luba {}, lubb {})"
 by(simp add: prod_lub_def)
 
 lemma preorder_rel_prodI [cont_intro, simp]:
   assumes "class.preorder orda (mk_less orda)"
   and "class.preorder ordb (mk_less ordb)"
   shows "class.preorder (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
 proof -
   interpret a: preorder orda "mk_less orda" by fact
   interpret b: preorder ordb "mk_less ordb" by fact
   show ?thesis by(unfold_locales)(auto simp add: mk_less_def intro: a.order_trans b.order_trans)
 qed
 
 lemma order_rel_prodI:
   assumes a: "class.order orda (mk_less orda)"
   and b: "class.order ordb (mk_less ordb)"
   shows "class.order (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
   (is "class.order ?ord ?ord'")
 proof(intro class.order.intro class.order_axioms.intro)
   interpret a: order orda "mk_less orda" by(fact a)
   interpret b: order ordb "mk_less ordb" by(fact b)
   show "class.preorder ?ord ?ord'" by(rule preorder_rel_prodI) unfold_locales
 
   fix x y
   assume "?ord x y" "?ord y x"
   thus "x = y" by(cases x y rule: prod.exhaust[case_product prod.exhaust]) auto
 qed
 
 lemma monotone_rel_prodI:
   assumes mono2: "\<And>a. monotone ordb ordc (\<lambda>b. f (a, b))"
   and mono1: "\<And>b. monotone orda ordc (\<lambda>a. f (a, b))"
   and a: "class.preorder orda (mk_less orda)"
   and b: "class.preorder ordb (mk_less ordb)"
   and c: "class.preorder ordc (mk_less ordc)"
   shows "monotone (rel_prod orda ordb) ordc f"
 proof -
   interpret a: preorder orda "mk_less orda" by(rule a)
   interpret b: preorder ordb "mk_less ordb" by(rule b)
   interpret c: preorder ordc "mk_less ordc" by(rule c)
   show ?thesis using mono2 mono1
     by(auto 7 2 simp add: monotone_def intro: c.order_trans)
 qed
 
 lemma monotone_rel_prodD1:
   assumes mono: "monotone (rel_prod orda ordb) ordc f"
   and preorder: "class.preorder ordb (mk_less ordb)"
   shows "monotone orda ordc (\<lambda>a. f (a, b))"
 proof -
   interpret preorder ordb "mk_less ordb" by(rule preorder)
   show ?thesis using mono by(simp add: monotone_def)
 qed
 
 lemma monotone_rel_prodD2:
   assumes mono: "monotone (rel_prod orda ordb) ordc f"
   and preorder: "class.preorder orda (mk_less orda)"
   shows "monotone ordb ordc (\<lambda>b. f (a, b))"
 proof -
   interpret preorder orda "mk_less orda" by(rule preorder)
   show ?thesis using mono by(simp add: monotone_def)
 qed
 
 lemma monotone_case_prodI:
   "\<lbrakk> \<And>a. monotone ordb ordc (f a); \<And>b. monotone orda ordc (\<lambda>a. f a b);
     class.preorder orda (mk_less orda); class.preorder ordb (mk_less ordb);
     class.preorder ordc (mk_less ordc) \<rbrakk>
   \<Longrightarrow> monotone (rel_prod orda ordb) ordc (case_prod f)"
 by(rule monotone_rel_prodI) simp_all
 
 lemma monotone_case_prodD1:
   assumes mono: "monotone (rel_prod orda ordb) ordc (case_prod f)"
   and preorder: "class.preorder ordb (mk_less ordb)"
   shows "monotone orda ordc (\<lambda>a. f a b)"
 using monotone_rel_prodD1[OF assms] by simp
 
 lemma monotone_case_prodD2:
   assumes mono: "monotone (rel_prod orda ordb) ordc (case_prod f)"
   and preorder: "class.preorder orda (mk_less orda)"
   shows "monotone ordb ordc (f a)"
 using monotone_rel_prodD2[OF assms] by simp
 
 context 
   fixes orda ordb ordc
   assumes a: "class.preorder orda (mk_less orda)"
   and b: "class.preorder ordb (mk_less ordb)"
   and c: "class.preorder ordc (mk_less ordc)"
 begin
 
 lemma monotone_rel_prod_iff:
   "monotone (rel_prod orda ordb) ordc f \<longleftrightarrow>
    (\<forall>a. monotone ordb ordc (\<lambda>b. f (a, b))) \<and> 
    (\<forall>b. monotone orda ordc (\<lambda>a. f (a, b)))"
 using a b c by(blast intro: monotone_rel_prodI dest: monotone_rel_prodD1 monotone_rel_prodD2)
 
 lemma monotone_case_prod_iff [simp]:
   "monotone (rel_prod orda ordb) ordc (case_prod f) \<longleftrightarrow>
    (\<forall>a. monotone ordb ordc (f a)) \<and> (\<forall>b. monotone orda ordc (\<lambda>a. f a b))"
 by(simp add: monotone_rel_prod_iff)
 
 end
 
 lemma monotone_case_prod_apply_iff:
   "monotone orda ordb (\<lambda>x. (case_prod f x) y) \<longleftrightarrow> monotone orda ordb (case_prod (\<lambda>a b. f a b y))"
 by(simp add: monotone_def)
 
 lemma monotone_case_prod_applyD:
   "monotone orda ordb (\<lambda>x. (case_prod f x) y)
   \<Longrightarrow> monotone orda ordb (case_prod (\<lambda>a b. f a b y))"
 by(simp add: monotone_case_prod_apply_iff)
 
 lemma monotone_case_prod_applyI:
   "monotone orda ordb (case_prod (\<lambda>a b. f a b y))
   \<Longrightarrow> monotone orda ordb (\<lambda>x. (case_prod f x) y)"
 by(simp add: monotone_case_prod_apply_iff)
 
 
 lemma cont_case_prod_apply_iff:
   "cont luba orda lubb ordb (\<lambda>x. (case_prod f x) y) \<longleftrightarrow> cont luba orda lubb ordb (case_prod (\<lambda>a b. f a b y))"
 by(simp add: cont_def split_def)
 
 lemma cont_case_prod_applyI:
   "cont luba orda lubb ordb (case_prod (\<lambda>a b. f a b y))
   \<Longrightarrow> cont luba orda lubb ordb (\<lambda>x. (case_prod f x) y)"
 by(simp add: cont_case_prod_apply_iff)
 
 lemma cont_case_prod_applyD:
   "cont luba orda lubb ordb (\<lambda>x. (case_prod f x) y)
   \<Longrightarrow> cont luba orda lubb ordb (case_prod (\<lambda>a b. f a b y))"
 by(simp add: cont_case_prod_apply_iff)
 
 lemma mcont_case_prod_apply_iff [simp]:
   "mcont luba orda lubb ordb (\<lambda>x. (case_prod f x) y) \<longleftrightarrow> 
    mcont luba orda lubb ordb (case_prod (\<lambda>a b. f a b y))"
 by(simp add: mcont_def monotone_case_prod_apply_iff cont_case_prod_apply_iff)
 
 lemma cont_prodD1: 
   assumes cont: "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc f"
   and "class.preorder orda (mk_less orda)"
   and luba: "lub_singleton luba"
   shows "cont lubb ordb lubc ordc (\<lambda>y. f (x, y))"
 proof(rule contI)
   interpret preorder orda "mk_less orda" by fact
 
   fix Y :: "'b set"
   let ?Y = "{x} \<times> Y"
   assume "Complete_Partial_Order.chain ordb Y" "Y \<noteq> {}"
   hence "Complete_Partial_Order.chain (rel_prod orda ordb) ?Y" "?Y \<noteq> {}" 
     by(simp_all add: chain_def)
   with cont have "f (prod_lub luba lubb ?Y) = lubc (f ` ?Y)" by(rule contD)
   moreover have "f ` ?Y = (\<lambda>y. f (x, y)) ` Y" by auto
   ultimately show "f (x, lubb Y) = lubc ((\<lambda>y. f (x, y)) ` Y)" using luba
     by(simp add: prod_lub_def \<open>Y \<noteq> {}\<close> lub_singleton_def)
 qed
 
 lemma cont_prodD2: 
   assumes cont: "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc f"
   and "class.preorder ordb (mk_less ordb)"
   and lubb: "lub_singleton lubb"
   shows "cont luba orda lubc ordc (\<lambda>x. f (x, y))"
 proof(rule contI)
   interpret preorder ordb "mk_less ordb" by fact
 
   fix Y
   assume Y: "Complete_Partial_Order.chain orda Y" "Y \<noteq> {}"
   let ?Y = "Y \<times> {y}"
   have "f (luba Y, y) = f (prod_lub luba lubb ?Y)"
     using lubb by(simp add: prod_lub_def Y lub_singleton_def)
   also from Y have "Complete_Partial_Order.chain (rel_prod orda ordb) ?Y" "?Y \<noteq> {}"
     by(simp_all add: chain_def)
   with cont have "f (prod_lub luba lubb ?Y) = lubc (f ` ?Y)" by(rule contD)
   also have "f ` ?Y = (\<lambda>x. f (x, y)) ` Y" by auto
   finally show "f (luba Y, y) = lubc \<dots>" .
 qed
 
 lemma cont_case_prodD1:
   assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc (case_prod f)"
   and "class.preorder orda (mk_less orda)"
   and "lub_singleton luba"
   shows "cont lubb ordb lubc ordc (f x)"
 using cont_prodD1[OF assms] by simp
 
 lemma cont_case_prodD2:
   assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc (case_prod f)"
   and "class.preorder ordb (mk_less ordb)"
   and "lub_singleton lubb"
   shows "cont luba orda lubc ordc (\<lambda>x. f x y)"
 using cont_prodD2[OF assms] by simp
 
 context ccpo begin
 
 lemma cont_prodI: 
   assumes mono: "monotone (rel_prod orda ordb) (\<le>) f"
   and cont1: "\<And>x. cont lubb ordb Sup (\<le>) (\<lambda>y. f (x, y))"
   and cont2: "\<And>y. cont luba orda Sup (\<le>) (\<lambda>x. f (x, y))"
   and "class.preorder orda (mk_less orda)"
   and "class.preorder ordb (mk_less ordb)"
   shows "cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (\<le>) f"
 proof(rule contI)
   interpret a: preorder orda "mk_less orda" by fact 
   interpret b: preorder ordb "mk_less ordb" by fact
   
   fix Y
   assume chain: "Complete_Partial_Order.chain (rel_prod orda ordb) Y"
     and "Y \<noteq> {}"
   have "f (prod_lub luba lubb Y) = f (luba (fst ` Y), lubb (snd ` Y))"
     by(simp add: prod_lub_def)
   also from cont2 have "f (luba (fst ` Y), lubb (snd ` Y)) = \<Squnion>((\<lambda>x. f (x, lubb (snd ` Y))) ` fst ` Y)"
     by(rule contD)(simp_all add: chain_rel_prodD1[OF chain] \<open>Y \<noteq> {}\<close>)
   also from cont1 have "\<And>x. f (x, lubb (snd ` Y)) = \<Squnion>((\<lambda>y. f (x, y)) ` snd ` Y)"
     by(rule contD)(simp_all add: chain_rel_prodD2[OF chain] \<open>Y \<noteq> {}\<close>)
   hence "\<Squnion>((\<lambda>x. f (x, lubb (snd ` Y))) ` fst ` Y) = \<Squnion>((\<lambda>x. \<dots> x) ` fst ` Y)" by simp
   also have "\<dots> = \<Squnion>((\<lambda>x. f (fst x, snd x)) ` Y)"
     unfolding image_image split_def using chain
     apply(rule diag_Sup)
     using monotoneD[OF mono]
     by(auto intro: monotoneI)
   finally show "f (prod_lub luba lubb Y) = \<Squnion>(f ` Y)" by simp
 qed
 
 lemma cont_case_prodI:
   assumes "monotone (rel_prod orda ordb) (\<le>) (case_prod f)"
   and "\<And>x. cont lubb ordb Sup (\<le>) (\<lambda>y. f x y)"
   and "\<And>y. cont luba orda Sup (\<le>) (\<lambda>x. f x y)"
   and "class.preorder orda (mk_less orda)"
   and "class.preorder ordb (mk_less ordb)"
   shows "cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (\<le>) (case_prod f)"
 by(rule cont_prodI)(simp_all add: assms)
 
 lemma cont_case_prod_iff:
   "\<lbrakk> monotone (rel_prod orda ordb) (\<le>) (case_prod f);
      class.preorder orda (mk_less orda); lub_singleton luba;
      class.preorder ordb (mk_less ordb); lub_singleton lubb \<rbrakk>
   \<Longrightarrow> cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (\<le>) (case_prod f) \<longleftrightarrow>
    (\<forall>x. cont lubb ordb Sup (\<le>) (\<lambda>y. f x y)) \<and> (\<forall>y. cont luba orda Sup (\<le>) (\<lambda>x. f x y))"
 by(blast dest: cont_case_prodD1 cont_case_prodD2 intro: cont_case_prodI)
 
 end
 
 context partial_function_definitions begin
 
 lemma mono2mono2:
   assumes f: "monotone (rel_prod ordb ordc) leq (\<lambda>(x, y). f x y)"
   and t: "monotone orda ordb (\<lambda>x. t x)"
   and t': "monotone orda ordc (\<lambda>x. t' x)"
   shows "monotone orda leq (\<lambda>x. f (t x) (t' x))"
 proof(rule monotoneI)
   fix x y
   assume "orda x y"
   hence "rel_prod ordb ordc (t x, t' x) (t y, t' y)"
     using t t' by(auto dest: monotoneD)
   from monotoneD[OF f this] show "leq (f (t x) (t' x)) (f (t y) (t' y))" by simp
 qed
 
 lemma cont_case_prodI [cont_intro]:
   "\<lbrakk> monotone (rel_prod orda ordb) leq (case_prod f);
     \<And>x. cont lubb ordb lub leq (\<lambda>y. f x y);
     \<And>y. cont luba orda lub leq (\<lambda>x. f x y);
     class.preorder orda (mk_less orda);
     class.preorder ordb (mk_less ordb) \<rbrakk>
   \<Longrightarrow> cont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (case_prod f)"
 by(rule ccpo.cont_case_prodI)(rule Partial_Function.ccpo[OF partial_function_definitions_axioms])
 
 lemma cont_case_prod_iff:
   "\<lbrakk> monotone (rel_prod orda ordb) leq (case_prod f);
      class.preorder orda (mk_less orda); lub_singleton luba;
      class.preorder ordb (mk_less ordb); lub_singleton lubb \<rbrakk>
   \<Longrightarrow> cont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (case_prod f) \<longleftrightarrow>
    (\<forall>x. cont lubb ordb lub leq (\<lambda>y. f x y)) \<and> (\<forall>y. cont luba orda lub leq (\<lambda>x. f x y))"
 by(blast dest: cont_case_prodD1 cont_case_prodD2 intro: cont_case_prodI)
 
 lemma mcont_case_prod_iff [simp]:
   "\<lbrakk> class.preorder orda (mk_less orda); lub_singleton luba;
      class.preorder ordb (mk_less ordb); lub_singleton lubb \<rbrakk>
   \<Longrightarrow> mcont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (case_prod f) \<longleftrightarrow>
    (\<forall>x. mcont lubb ordb lub leq (\<lambda>y. f x y)) \<and> (\<forall>y. mcont luba orda lub leq (\<lambda>x. f x y))"
 unfolding mcont_def by(auto simp add: cont_case_prod_iff)
 
 end
 
 lemma mono2mono_case_prod [cont_intro]:
   assumes "\<And>x y. monotone orda ordb (\<lambda>f. pair f x y)"
   shows "monotone orda ordb (\<lambda>f. case_prod (pair f) x)"
 by(rule monotoneI)(auto split: prod.split dest: monotoneD[OF assms])
 
 subsection \<open>Complete lattices as ccpo\<close>
 
 context complete_lattice begin
 
 lemma complete_lattice_ccpo: "class.ccpo Sup (\<le>) (<)"
 by(unfold_locales)(fast intro: Sup_upper Sup_least)+
 
 lemma complete_lattice_ccpo': "class.ccpo Sup (\<le>) (mk_less (\<le>))"
 by(unfold_locales)(auto simp add: mk_less_def intro: Sup_upper Sup_least)
 
 lemma complete_lattice_partial_function_definitions: 
   "partial_function_definitions (\<le>) Sup"
 by(unfold_locales)(auto intro: Sup_least Sup_upper)
 
 lemma complete_lattice_partial_function_definitions_dual:
   "partial_function_definitions (\<ge>) Inf"
 by(unfold_locales)(auto intro: Inf_lower Inf_greatest)
 
 lemmas [cont_intro, simp] =
   Partial_Function.ccpo[OF complete_lattice_partial_function_definitions]
   Partial_Function.ccpo[OF complete_lattice_partial_function_definitions_dual]
 
 lemma mono2mono_inf:
   assumes f: "monotone ord (\<le>) (\<lambda>x. f x)" 
   and g: "monotone ord (\<le>) (\<lambda>x. g x)"
   shows "monotone ord (\<le>) (\<lambda>x. f x \<sqinter> g x)"
 by(auto 4 3 dest: monotoneD[OF f] monotoneD[OF g] intro: le_infI1 le_infI2 intro!: monotoneI)
 
 lemma mcont_const [simp]: "mcont lub ord Sup (\<le>) (\<lambda>_. c)"
 by(rule ccpo.mcont_const[OF complete_lattice_ccpo])
 
 lemma mono2mono_sup:
   assumes f: "monotone ord (\<le>) (\<lambda>x. f x)"
   and g: "monotone ord (\<le>) (\<lambda>x. g x)"
   shows "monotone ord (\<le>) (\<lambda>x. f x \<squnion> g x)"
 by(auto 4 3 intro!: monotoneI intro: sup.coboundedI1 sup.coboundedI2 dest: monotoneD[OF f] monotoneD[OF g])
 
 lemma Sup_image_sup: 
   assumes "Y \<noteq> {}"
   shows "\<Squnion>((\<squnion>) x ` Y) = x \<squnion> \<Squnion>Y"
 proof(rule Sup_eqI)
   fix y
   assume "y \<in> (\<squnion>) x ` Y"
   then obtain z where "y = x \<squnion> z" and "z \<in> Y" by blast
   from \<open>z \<in> Y\<close> have "z \<le> \<Squnion>Y" by(rule Sup_upper)
   with _ show "y \<le> x \<squnion> \<Squnion>Y" unfolding \<open>y = x \<squnion> z\<close> by(rule sup_mono) simp
 next
   fix y
   assume upper: "\<And>z. z \<in> (\<squnion>) x ` Y \<Longrightarrow> z \<le> y"
   show "x \<squnion> \<Squnion>Y \<le> y" unfolding Sup_insert[symmetric]
   proof(rule Sup_least)
     fix z
     assume "z \<in> insert x Y"
     from assms obtain z' where "z' \<in> Y" by blast
     let ?z = "if z \<in> Y then x \<squnion> z else x \<squnion> z'"
     have "z \<le> x \<squnion> ?z" using \<open>z' \<in> Y\<close> \<open>z \<in> insert x Y\<close> by auto
     also have "\<dots> \<le> y" by(rule upper)(auto split: if_split_asm intro: \<open>z' \<in> Y\<close>)
     finally show "z \<le> y" .
   qed
 qed
 
 lemma mcont_sup1: "mcont Sup (\<le>) Sup (\<le>) (\<lambda>y. x \<squnion> y)"
 by(auto 4 3 simp add: mcont_def sup.coboundedI1 sup.coboundedI2 intro!: monotoneI contI intro: Sup_image_sup[symmetric])
 
 lemma mcont_sup2: "mcont Sup (\<le>) Sup (\<le>) (\<lambda>x. x \<squnion> y)"
 by(subst sup_commute)(rule mcont_sup1)
 
 lemma mcont2mcont_sup [cont_intro, simp]:
   "\<lbrakk> mcont lub ord Sup (\<le>) (\<lambda>x. f x);
      mcont lub ord Sup (\<le>) (\<lambda>x. g x) \<rbrakk>
   \<Longrightarrow> mcont lub ord Sup (\<le>) (\<lambda>x. f x \<squnion> g x)"
 by(best intro: ccpo.mcont2mcont'[OF complete_lattice_ccpo] mcont_sup1 mcont_sup2 ccpo.mcont_const[OF complete_lattice_ccpo])
 
 end
 
 lemmas [cont_intro] = admissible_leI[OF complete_lattice_ccpo']
 
 context complete_distrib_lattice begin
 
 lemma mcont_inf1: "mcont Sup (\<le>) Sup (\<le>) (\<lambda>y. x \<sqinter> y)"
 by(auto intro: monotoneI contI simp add: le_infI2 inf_Sup mcont_def)
 
 lemma mcont_inf2: "mcont Sup (\<le>) Sup (\<le>) (\<lambda>x. x \<sqinter> y)"
 by(auto intro: monotoneI contI simp add: le_infI1 Sup_inf mcont_def)
 
 lemma mcont2mcont_inf [cont_intro, simp]:
   "\<lbrakk> mcont lub ord Sup (\<le>) (\<lambda>x. f x);
     mcont lub ord Sup (\<le>) (\<lambda>x. g x) \<rbrakk>
   \<Longrightarrow> mcont lub ord Sup (\<le>) (\<lambda>x. f x \<sqinter> g x)"
 by(best intro: ccpo.mcont2mcont'[OF complete_lattice_ccpo] mcont_inf1 mcont_inf2 ccpo.mcont_const[OF complete_lattice_ccpo])
 
 end
 
 interpretation lfp: partial_function_definitions "(\<le>) :: _ :: complete_lattice \<Rightarrow> _" Sup
 by(rule complete_lattice_partial_function_definitions)
 
 declaration \<open>Partial_Function.init "lfp" \<^term>\<open>lfp.fixp_fun\<close> \<^term>\<open>lfp.mono_body\<close>
   @{thm lfp.fixp_rule_uc} @{thm lfp.fixp_induct_uc} NONE\<close>
 
 interpretation gfp: partial_function_definitions "(\<ge>) :: _ :: complete_lattice \<Rightarrow> _" Inf
 by(rule complete_lattice_partial_function_definitions_dual)
 
 declaration \<open>Partial_Function.init "gfp" \<^term>\<open>gfp.fixp_fun\<close> \<^term>\<open>gfp.mono_body\<close>
   @{thm gfp.fixp_rule_uc} @{thm gfp.fixp_induct_uc} NONE\<close>
 
 lemma insert_mono [partial_function_mono]:
    "monotone (fun_ord (\<subseteq>)) (\<subseteq>) A \<Longrightarrow> monotone (fun_ord (\<subseteq>)) (\<subseteq>) (\<lambda>y. insert x (A y))"
 by(rule monotoneI)(auto simp add: fun_ord_def dest: monotoneD)
 
 lemma mono2mono_insert [THEN lfp.mono2mono, cont_intro, simp]:
   shows monotone_insert: "monotone (\<subseteq>) (\<subseteq>) (insert x)"
 by(rule monotoneI) blast
 
 lemma mcont2mcont_insert[THEN lfp.mcont2mcont, cont_intro, simp]:
   shows mcont_insert: "mcont Union (\<subseteq>) Union (\<subseteq>) (insert x)"
 by(blast intro: mcontI contI monotone_insert)
 
 lemma mono2mono_image [THEN lfp.mono2mono, cont_intro, simp]:
   shows monotone_image: "monotone (\<subseteq>) (\<subseteq>) ((`) f)"
 by(rule monotoneI) blast
 
 lemma cont_image: "cont Union (\<subseteq>) Union (\<subseteq>) ((`) f)"
 by(rule contI)(auto)
 
 lemma mcont2mcont_image [THEN lfp.mcont2mcont, cont_intro, simp]:
   shows mcont_image: "mcont Union (\<subseteq>) Union (\<subseteq>) ((`) f)"
 by(blast intro: mcontI monotone_image cont_image)
 
 context complete_lattice begin
 
 lemma monotone_Sup [cont_intro, simp]:
   "monotone ord (\<subseteq>) f \<Longrightarrow> monotone ord (\<le>) (\<lambda>x. \<Squnion>f x)"
 by(blast intro: monotoneI Sup_least Sup_upper dest: monotoneD)
 
 lemma cont_Sup:
   assumes "cont lub ord Union (\<subseteq>) f"
   shows "cont lub ord Sup (\<le>) (\<lambda>x. \<Squnion>f x)"
 apply(rule contI)
 apply(simp add: contD[OF assms])
 apply(blast intro: Sup_least Sup_upper order_trans order.antisym)
 done
 
 lemma mcont_Sup: "mcont lub ord Union (\<subseteq>) f \<Longrightarrow> mcont lub ord Sup (\<le>) (\<lambda>x. \<Squnion>f x)"
 unfolding mcont_def by(blast intro: monotone_Sup cont_Sup)
 
 lemma monotone_SUP:
   "\<lbrakk> monotone ord (\<subseteq>) f; \<And>y. monotone ord (\<le>) (\<lambda>x. g x y) \<rbrakk> \<Longrightarrow> monotone ord (\<le>) (\<lambda>x. \<Squnion>y\<in>f x. g x y)"
 by(rule monotoneI)(blast dest: monotoneD intro: Sup_upper order_trans intro!: Sup_least)
 
 lemma monotone_SUP2:
   "(\<And>y. y \<in> A \<Longrightarrow> monotone ord (\<le>) (\<lambda>x. g x y)) \<Longrightarrow> monotone ord (\<le>) (\<lambda>x. \<Squnion>y\<in>A. g x y)"
 by(rule monotoneI)(blast intro: Sup_upper order_trans dest: monotoneD intro!: Sup_least)
 
 lemma cont_SUP:
   assumes f: "mcont lub ord Union (\<subseteq>) f"
   and g: "\<And>y. mcont lub ord Sup (\<le>) (\<lambda>x. g x y)"
   shows "cont lub ord Sup (\<le>) (\<lambda>x. \<Squnion>y\<in>f x. g x y)"
 proof(rule contI)
   fix Y
   assume chain: "Complete_Partial_Order.chain ord Y"
     and Y: "Y \<noteq> {}"
   show "\<Squnion>(g (lub Y) ` f (lub Y)) = \<Squnion>((\<lambda>x. \<Squnion>(g x ` f x)) ` Y)" (is "?lhs = ?rhs")
   proof(rule order.antisym)
     show "?lhs \<le> ?rhs"
     proof(rule Sup_least)
       fix x
       assume "x \<in> g (lub Y) ` f (lub Y)"
       with mcont_contD[OF f chain Y] mcont_contD[OF g chain Y]
       obtain y z where "y \<in> Y" "z \<in> f y"
         and x: "x = \<Squnion>((\<lambda>x. g x z) ` Y)" by auto
       show "x \<le> ?rhs" unfolding x
       proof(rule Sup_least)
         fix u
         assume "u \<in> (\<lambda>x. g x z) ` Y"
         then obtain y' where "u = g y' z" "y' \<in> Y" by auto
         from chain \<open>y \<in> Y\<close> \<open>y' \<in> Y\<close> have "ord y y' \<or> ord y' y" by(rule chainD)
         thus "u \<le> ?rhs"
         proof
           note \<open>u = g y' z\<close> also
           assume "ord y y'"
           with f have "f y \<subseteq> f y'" by(rule mcont_monoD)
           with \<open>z \<in> f y\<close>
           have "g y' z \<le> \<Squnion>(g y' ` f y')" by(auto intro: Sup_upper)
           also have "\<dots> \<le> ?rhs" using \<open>y' \<in> Y\<close> by(auto intro: Sup_upper)
           finally show ?thesis .
         next
           note \<open>u = g y' z\<close> also
           assume "ord y' y"
           with g have "g y' z \<le> g y z" by(rule mcont_monoD)
           also have "\<dots> \<le> \<Squnion>(g y ` f y)" using \<open>z \<in> f y\<close>
             by(auto intro: Sup_upper)
           also have "\<dots> \<le> ?rhs" using \<open>y \<in> Y\<close> by(auto intro: Sup_upper)
           finally show ?thesis .
         qed
       qed
     qed
   next
     show "?rhs \<le> ?lhs"
     proof(rule Sup_least)
       fix x
       assume "x \<in> (\<lambda>x. \<Squnion>(g x ` f x)) ` Y"
       then obtain y where x: "x = \<Squnion>(g y ` f y)" and "y \<in> Y" by auto
       show "x \<le> ?lhs" unfolding x
       proof(rule Sup_least)
         fix u
         assume "u \<in> g y ` f y"
         then obtain z where "u = g y z" "z \<in> f y" by auto
         note \<open>u = g y z\<close>
         also have "g y z \<le> \<Squnion>((\<lambda>x. g x z) ` Y)"
           using \<open>y \<in> Y\<close> by(auto intro: Sup_upper)
         also have "\<dots> = g (lub Y) z" by(simp add: mcont_contD[OF g chain Y])
         also have "\<dots> \<le> ?lhs" using \<open>z \<in> f y\<close> \<open>y \<in> Y\<close>
           by(auto intro: Sup_upper simp add: mcont_contD[OF f chain Y])
         finally show "u \<le> ?lhs" .
       qed
     qed
   qed
 qed
 
 lemma mcont_SUP [cont_intro, simp]:
   "\<lbrakk> mcont lub ord Union (\<subseteq>) f; \<And>y. mcont lub ord Sup (\<le>) (\<lambda>x. g x y) \<rbrakk>
   \<Longrightarrow> mcont lub ord Sup (\<le>) (\<lambda>x. \<Squnion>y\<in>f x. g x y)"
 by(blast intro: mcontI cont_SUP monotone_SUP mcont_mono)
 
 end
 
 lemma admissible_Ball [cont_intro, simp]:
   "\<lbrakk> \<And>x. ccpo.admissible lub ord (\<lambda>A. P A x);
      mcont lub ord Union (\<subseteq>) f;
      class.ccpo lub ord (mk_less ord) \<rbrakk>
   \<Longrightarrow> ccpo.admissible lub ord (\<lambda>A. \<forall>x\<in>f A. P A x)"
 unfolding Ball_def by simp
 
 lemma admissible_Bex'[THEN admissible_subst, cont_intro, simp]:
   shows admissible_Bex: "ccpo.admissible Union (\<subseteq>) (\<lambda>A. \<exists>x\<in>A. P x)"
 by(rule ccpo.admissibleI)(auto)
 
 subsection \<open>Parallel fixpoint induction\<close>
 
 context
   fixes luba :: "'a set \<Rightarrow> 'a"
   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
   and lubb :: "'b set \<Rightarrow> 'b"
   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
   assumes a: "class.ccpo luba orda (mk_less orda)"
   and b: "class.ccpo lubb ordb (mk_less ordb)"
 begin
 
 interpretation a: ccpo luba orda "mk_less orda" by(rule a)
 interpretation b: ccpo lubb ordb "mk_less ordb" by(rule b)
 
 lemma ccpo_rel_prodI:
   "class.ccpo (prod_lub luba lubb) (rel_prod orda ordb) (mk_less (rel_prod orda ordb))"
   (is "class.ccpo ?lub ?ord ?ord'")
 proof(intro class.ccpo.intro class.ccpo_axioms.intro)
   show "class.order ?ord ?ord'" by(rule order_rel_prodI) intro_locales
 qed(auto 4 4 simp add: prod_lub_def intro: a.ccpo_Sup_upper b.ccpo_Sup_upper a.ccpo_Sup_least b.ccpo_Sup_least rev_image_eqI dest: chain_rel_prodD1 chain_rel_prodD2)
 
 interpretation ab: ccpo "prod_lub luba lubb" "rel_prod orda ordb" "mk_less (rel_prod orda ordb)"
 by(rule ccpo_rel_prodI)
 
 lemma monotone_map_prod [simp]:
   "monotone (rel_prod orda ordb) (rel_prod ordc ordd) (map_prod f g) \<longleftrightarrow>
    monotone orda ordc f \<and> monotone ordb ordd g"
 by(auto simp add: monotone_def)
 
 lemma parallel_fixp_induct:
   assumes adm: "ccpo.admissible (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. P (fst x) (snd x))"
   and f: "monotone orda orda f"
   and g: "monotone ordb ordb g"
   and bot: "P (luba {}) (lubb {})"
   and step: "\<And>x y. P x y \<Longrightarrow> P (f x) (g y)"
   shows "P (ccpo.fixp luba orda f) (ccpo.fixp lubb ordb g)"
 proof -
   let ?lub = "prod_lub luba lubb"
     and ?ord = "rel_prod orda ordb"
     and ?P = "\<lambda>(x, y). P x y"
   from adm have adm': "ccpo.admissible ?lub ?ord ?P" by(simp add: split_def)
   hence "?P (ccpo.fixp (prod_lub luba lubb) (rel_prod orda ordb) (map_prod f g))"
     by(rule ab.fixp_induct)(auto simp add: f g step bot)
   also have "ccpo.fixp (prod_lub luba lubb) (rel_prod orda ordb) (map_prod f g) = 
             (ccpo.fixp luba orda f, ccpo.fixp lubb ordb g)" (is "?lhs = (?rhs1, ?rhs2)")
   proof(rule ab.order.antisym)
     have "ccpo.admissible ?lub ?ord (\<lambda>xy. ?ord xy (?rhs1, ?rhs2))"
       by(rule admissible_leI[OF ccpo_rel_prodI])(auto simp add: prod_lub_def chain_empty intro: a.ccpo_Sup_least b.ccpo_Sup_least)
     thus "?ord ?lhs (?rhs1, ?rhs2)"
       by(rule ab.fixp_induct)(auto 4 3 dest: monotoneD[OF f] monotoneD[OF g] simp add: b.fixp_unfold[OF g, symmetric] a.fixp_unfold[OF f, symmetric] f g intro: a.ccpo_Sup_least b.ccpo_Sup_least chain_empty)
   next
     have "ccpo.admissible luba orda (\<lambda>x. orda x (fst ?lhs))"
       by(rule admissible_leI[OF a])(auto intro: a.ccpo_Sup_least simp add: chain_empty)
     hence "orda ?rhs1 (fst ?lhs)" using f
     proof(rule a.fixp_induct)
       fix x
       assume "orda x (fst ?lhs)"
       thus "orda (f x) (fst ?lhs)"
         by(subst ab.fixp_unfold)(auto simp add: f g dest: monotoneD[OF f])
     qed(auto intro: a.ccpo_Sup_least chain_empty)
     moreover
     have "ccpo.admissible lubb ordb (\<lambda>y. ordb y (snd ?lhs))"
       by(rule admissible_leI[OF b])(auto intro: b.ccpo_Sup_least simp add: chain_empty)
     hence "ordb ?rhs2 (snd ?lhs)" using g
     proof(rule b.fixp_induct)
       fix y
       assume "ordb y (snd ?lhs)"
       thus "ordb (g y) (snd ?lhs)"
         by(subst ab.fixp_unfold)(auto simp add: f g dest: monotoneD[OF g])
     qed(auto intro: b.ccpo_Sup_least chain_empty)
     ultimately show "?ord (?rhs1, ?rhs2) ?lhs"
       by(simp add: rel_prod_conv split_beta)
   qed
   finally show ?thesis by simp
 qed
 
 end
 
 lemma parallel_fixp_induct_uc:
   assumes a: "partial_function_definitions orda luba"
   and b: "partial_function_definitions ordb lubb"
   and F: "\<And>x. monotone (fun_ord orda) orda (\<lambda>f. U1 (F (C1 f)) x)"
   and G: "\<And>y. monotone (fun_ord ordb) ordb (\<lambda>g. U2 (G (C2 g)) y)"
   and eq1: "f \<equiv> C1 (ccpo.fixp (fun_lub luba) (fun_ord orda) (\<lambda>f. U1 (F (C1 f))))"
   and eq2: "g \<equiv> C2 (ccpo.fixp (fun_lub lubb) (fun_ord ordb) (\<lambda>g. U2 (G (C2 g))))"
   and inverse: "\<And>f. U1 (C1 f) = f"
   and inverse2: "\<And>g. U2 (C2 g) = g"
   and adm: "ccpo.admissible (prod_lub (fun_lub luba) (fun_lub lubb)) (rel_prod (fun_ord orda) (fun_ord ordb)) (\<lambda>x. P (fst x) (snd x))"
   and bot: "P (\<lambda>_. luba {}) (\<lambda>_. lubb {})"
   and step: "\<And>f g. P (U1 f) (U2 g) \<Longrightarrow> P (U1 (F f)) (U2 (G g))"
   shows "P (U1 f) (U2 g)"
 apply(unfold eq1 eq2 inverse inverse2)
 apply(rule parallel_fixp_induct[OF partial_function_definitions.ccpo[OF a] partial_function_definitions.ccpo[OF b] adm])
 using F apply(simp add: monotone_def fun_ord_def)
 using G apply(simp add: monotone_def fun_ord_def)
 apply(simp add: fun_lub_def bot)
 apply(rule step, simp add: inverse inverse2)
 done
 
 lemmas parallel_fixp_induct_1_1 = parallel_fixp_induct_uc[
   of _ _ _ _ "\<lambda>x. x" _ "\<lambda>x. x" "\<lambda>x. x" _ "\<lambda>x. x",
   OF _ _ _ _ _ _ refl refl]
 
 lemmas parallel_fixp_induct_2_2 = parallel_fixp_induct_uc[
   of _ _ _ _ "case_prod" _ "curry" "case_prod" _ "curry",
   where P="\<lambda>f g. P (curry f) (curry g)",
   unfolded case_prod_curry curry_case_prod curry_K,
   OF _ _ _ _ _ _ refl refl]
   for P
 
 lemma monotone_fst: "monotone (rel_prod orda ordb) orda fst"
 by(auto intro: monotoneI)
 
 lemma mcont_fst: "mcont (prod_lub luba lubb) (rel_prod orda ordb) luba orda fst"
 by(auto intro!: mcontI monotoneI contI simp add: prod_lub_def)
 
 lemma mcont2mcont_fst [cont_intro, simp]:
   "mcont lub ord (prod_lub luba lubb) (rel_prod orda ordb) t
   \<Longrightarrow> mcont lub ord luba orda (\<lambda>x. fst (t x))"
 by(auto intro!: mcontI monotoneI contI dest: mcont_monoD mcont_contD simp add: rel_prod_sel split_beta prod_lub_def image_image)
 
 lemma monotone_snd: "monotone (rel_prod orda ordb) ordb snd"
 by(auto intro: monotoneI)
 
 lemma mcont_snd: "mcont (prod_lub luba lubb) (rel_prod orda ordb) lubb ordb snd"
 by(auto intro!: mcontI monotoneI contI simp add: prod_lub_def)
 
 lemma mcont2mcont_snd [cont_intro, simp]:
   "mcont lub ord (prod_lub luba lubb) (rel_prod orda ordb) t
   \<Longrightarrow> mcont lub ord lubb ordb (\<lambda>x. snd (t x))"
 by(auto intro!: mcontI monotoneI contI dest: mcont_monoD mcont_contD simp add: rel_prod_sel split_beta prod_lub_def image_image)
 
 lemma monotone_Pair:
   "\<lbrakk> monotone ord orda f; monotone ord ordb g \<rbrakk>
   \<Longrightarrow> monotone ord (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
 by(simp add: monotone_def)
 
 lemma cont_Pair:
   "\<lbrakk> cont lub ord luba orda f; cont lub ord lubb ordb g \<rbrakk>
   \<Longrightarrow> cont lub ord (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
 by(rule contI)(auto simp add: prod_lub_def image_image dest!: contD)
 
 lemma mcont_Pair:
   "\<lbrakk> mcont lub ord luba orda f; mcont lub ord lubb ordb g \<rbrakk>
   \<Longrightarrow> mcont lub ord (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
 by(rule mcontI)(simp_all add: monotone_Pair mcont_mono cont_Pair)
 
 context partial_function_definitions begin
 text \<open>Specialised versions of @{thm [source] mcont_call} for admissibility proofs for parallel fixpoint inductions\<close>
 lemmas mcont_call_fst [cont_intro] = mcont_call[THEN mcont2mcont, OF mcont_fst]
 lemmas mcont_call_snd [cont_intro] = mcont_call[THEN mcont2mcont, OF mcont_snd]
 end
 
 lemma map_option_mono [partial_function_mono]:
   "mono_option B \<Longrightarrow> mono_option (\<lambda>f. map_option g (B f))"
 unfolding map_conv_bind_option by(rule bind_mono) simp_all
 
 lemma compact_flat_lub [cont_intro]: "ccpo.compact (flat_lub x) (flat_ord x) y"
 using flat_interpretation[THEN ccpo]
 proof(rule ccpo.compactI[OF _ ccpo.admissibleI])
   fix A
   assume chain: "Complete_Partial_Order.chain (flat_ord x) A"
     and A: "A \<noteq> {}"
     and *: "\<forall>z\<in>A. \<not> flat_ord x y z"
   from A obtain z where "z \<in> A" by blast
   with * have z: "\<not> flat_ord x y z" ..
   hence y: "x \<noteq> y" "y \<noteq> z" by(auto simp add: flat_ord_def)
   { assume "\<not> A \<subseteq> {x}"
     then obtain z' where "z' \<in> A" "z' \<noteq> x" by auto
     then have "(THE z. z \<in> A - {x}) = z'"
       by(intro the_equality)(auto dest: chainD[OF chain] simp add: flat_ord_def)
     moreover have "z' \<noteq> y" using \<open>z' \<in> A\<close> * by(auto simp add: flat_ord_def)
     ultimately have "y \<noteq> (THE z. z \<in> A - {x})" by simp }
   with z show "\<not> flat_ord x y (flat_lub x A)" by(simp add: flat_ord_def flat_lub_def)
 qed
 
 end