--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Complete_Partial_Order2.thy Fri Mar 18 08:01:49 2016 +0100
@@ -0,0 +1,1708 @@
+(* Title: src/HOL/Library/Complete_Partial_Order2
+ Author: Andreas Lochbihler, ETH Zurich
+*)
+
+section {* Formalisation of chain-complete partial orders, continuity and admissibility *}
+
+theory Complete_Partial_Order2 imports
+ Main
+ "~~/src/HOL/Library/Lattice_Syntax"
+begin
+
+context begin interpretation lifting_syntax .
+
+lemma chain_transfer [transfer_rule]:
+ "((A ===> A ===> op =) ===> rel_set A ===> op =) Complete_Partial_Order.chain Complete_Partial_Order.chain"
+unfolding chain_def[abs_def] by transfer_prover
+
+end
+
+lemma linorder_chain [simp, intro!]:
+ fixes Y :: "_ :: linorder set"
+ shows "Complete_Partial_Order.chain op \<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 op \<le>) (mk_less (fun_ord op \<le>))"
+ by standard (auto 4 3 simp add: mk_less_def fun_ord_def fun_lub_apply
+ intro: order.trans antisym chain_imageI ccpo_Sup_upper ccpo_Sup_least)
+
+lemma ccpo_Sup_below_iff: "Complete_Partial_Order.chain op \<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 op \<le> A"
+ shows "\<Squnion>(A - {\<Squnion>{}}) = \<Squnion>A"
+apply(rule antisym)
+ apply(blast intro: ccpo_Sup_least chain_Diff[OF chain] ccpo_Sup_upper[OF chain])
+apply(rule ccpo_Sup_least[OF chain])
+apply(case_tac "x = \<Squnion>{}")
+by(blast intro: ccpo_Sup_least chain_empty ccpo_Sup_upper[OF chain_Diff[OF chain]])+
+
+lemma mono_lub:
+ fixes le_b (infix "\<sqsubseteq>" 60)
+ assumes chain: "Complete_Partial_Order.chain (fun_ord op \<le>) Y"
+ and mono: "\<And>f. f \<in> Y \<Longrightarrow> monotone le_b op \<le> f"
+ shows "monotone op \<sqsubseteq> op \<le> (fun_lub Sup Y)"
+proof(rule monotoneI)
+ fix x y
+ assume "x \<sqsubseteq> y"
+
+ have chain'': "\<And>x. Complete_Partial_Order.chain op \<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 `x' = f x` also
+ from `f \<in> Y` `x \<sqsubseteq> y` 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: `f \<in> Y`)
+ 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 op \<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 op \<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 `y' \<in> Y`] 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: `y' \<in> Y`)
+ 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 antisym)
+ have chain1: "Complete_Partial_Order.chain op \<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 op \<le> (f y' ` Y)" using chain
+ by(rule chain_imageI)(auto dest: mono2)
+ have chain3: "Complete_Partial_Order.chain op \<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 `y' \<in> Y`]
+ 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
+ def y'' \<equiv> "if y \<sqsubseteq> y' then y' else y"
+ from chain `y \<in> Y` `y' \<in> Y` have "y \<sqsubseteq> y' \<or> y' \<sqsubseteq> y" by(rule chainD)
+ hence "f y' y \<le> f y'' y''" using `y \<in> Y` `y' \<in> Y`
+ by(auto simp add: y''_def intro: mono2 monotoneD[OF mono1])
+ also from `y \<in> Y` `y' \<in> Y` have "y'' \<in> Y" by(simp add: y''_def)
+ from chain3 have "f y'' y'' \<le> ?rhs" by(rule ccpo_Sup_upper)(simp add: `y'' \<in> Y`)
+ 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 `x \<in> Y`] have "\<dots> \<le> \<Squnion>(f x ` Y)"
+ by(rule ccpo_Sup_upper)(simp add: `x \<in> Y`)
+ also have "\<dots> \<le> ?lhs" by(rule ccpo_Sup_upper[OF chain1])(simp add: `x \<in> Y`)
+ finally show "y \<le> ?lhs" .
+ qed
+qed
+
+end
+
+lemma Sup_image_mono_le:
+ fixes le_b (infix "\<sqsubseteq>" 60) and Sup_b ("\<Or>_" [900] 900)
+ assumes ccpo: "class.ccpo Sup_b op \<sqsubseteq> lt_b"
+ assumes chain: "Complete_Partial_Order.chain op \<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 op \<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 `y \<in> Y` by(rule ccpo.ccpo_Sup_upper)
+ hence "f y \<le> f (\<Or>Y)" using `y \<in> Y` by(rule mono)
+ finally show "x \<le> \<dots>" .
+qed
+
+lemma swap_Sup:
+ fixes le_b (infix "\<sqsubseteq>" 60)
+ assumes Y: "Complete_Partial_Order.chain op \<sqsubseteq> Y"
+ and Z: "Complete_Partial_Order.chain (fun_ord op \<le>) Z"
+ and mono: "\<And>f. f \<in> Z \<Longrightarrow> monotone op \<sqsubseteq> op \<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: strong_SUP_cong)
+next
+ case False
+ have chain1: "\<And>f. f \<in> Z \<Longrightarrow> Complete_Partial_Order.chain op \<le> (f ` Y)"
+ by(rule chain_imageI[OF Y])(rule monotoneD[OF mono])
+ have chain2: "Complete_Partial_Order.chain op \<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 op \<le> f g"
+ from chain1[OF `f \<in> Z`] 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 `fun_ord op \<le> f g` by(simp add: fun_ord_def)
+ also have "\<dots> \<le> \<Squnion>(g ` Y)" using chain1[OF `g \<in> Z`]
+ by(rule ccpo_Sup_upper)(simp add: `y \<in> Y`)
+ finally show "x \<le> \<Squnion>(g ` Y)" .
+ qed
+ qed
+ have chain3: "\<And>x. Complete_Partial_Order.chain op \<le> ((\<lambda>f. f x) ` Z)"
+ using Z by(rule chain_imageI)(simp add: fun_ord_def)
+ have chain4: "Complete_Partial_Order.chain op \<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 `f \<in> Z` `x \<sqsubseteq> y` by(rule monotoneD[OF mono])
+ also have "f y \<le> ?rhs" using chain3
+ by(rule ccpo_Sup_upper)(simp add: `f \<in> Z`)
+ 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 `f \<in> Z`]
+ 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: `f \<in> Z`)
+ also have "\<dots> \<le> ?rhs" using chain4 by(rule ccpo_Sup_upper)(simp add: `y \<in> Y`)
+ 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 `f \<in> Z`]
+ by(rule ccpo_Sup_upper)(simp add: `y \<in> Y`)
+ also have "\<dots> \<le> ?lhs" using chain2
+ by(rule ccpo_Sup_upper)(simp add: `f \<in> Z`)
+ finally show "x' \<le> ?lhs" .
+ qed
+ finally show "x \<le> ?lhs" .
+ qed
+ ultimately show "?lhs = ?rhs" by(rule antisym)
+qed
+
+lemma fixp_mono:
+ assumes fg: "fun_ord op \<le> f g"
+ and f: "monotone op \<le> op \<le> f"
+ and g: "monotone op \<le> op \<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 op \<le>) F"
+ and mono: "\<And>f. monotone op \<sqsubseteq> op \<le> f \<Longrightarrow> monotone op \<sqsubseteq> op \<le> (F f)"
+ shows "monotone op \<sqsubseteq> op \<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 op \<le>) op \<le> (\<lambda>f. F f x)"
+ and mono2: "\<And>f. monotone op \<sqsubseteq> op \<le> f \<Longrightarrow> monotone op \<sqsubseteq> op \<le> (F f)"
+ shows "monotone op \<sqsubseteq> op \<le> (ccpo.fixp (fun_lub Sup) (fun_ord op \<le>) F)"
+ (is "monotone _ _ ?fixp")
+proof(rule monotoneI)
+ have mono: "monotone (fun_ord op \<le>) (fun_ord op \<le>) F"
+ by(rule monotoneI)(auto simp add: fun_ord_def intro: monotoneD[OF mono])
+ let ?iter = "ccpo.iterates (fun_lub Sup) (fun_ord op \<le>) F"
+ have chain: "\<And>x. Complete_Partial_Order.chain op \<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"
+ unfolding 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 `f \<in> ?iter` mono2]])(blast intro: `x \<sqsubseteq> y`)+
+ also have "f y \<le> \<Squnion>((\<lambda>f. f y) ` ?iter)" using chain
+ by(rule ccpo_Sup_upper)(simp add: `f \<in> ?iter`)
+ 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 {* Continuity *}
+
+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 {* Theorem collection @{text cont_intro} *}
+
+named_theorems cont_intro "continuity and admissibility intro rules"
+ML {*
+(* 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 cont_intro})))
+ 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
+ in
+ case Thm.term_of ct of
+ t as Const (@{const_name ccpo.admissible}, _) $ _ $ _ $ _ => mk_thm t
+ | t as Const (@{const_name mcont}, _) $ _ $ _ $ _ $ _ $ _ => mk_thm t
+ | t as Const (@{const_name monotone}, _) $ _ $ _ $ _ => mk_thm t
+ | _ => NONE
+ end
+ handle THM _ => NONE
+ | TYPE _ => NONE
+*}
+
+simproc_setup "cont_intro"
+ ( "ccpo.admissible lub ord P"
+ | "mcont lub ord lub' ord' f"
+ | "monotone ord ord' f"
+ ) = {* K cont_intro_simproc *}
+
+lemmas [cont_intro] =
+ call_mono
+ let_mono
+ if_mono
+ option.const_mono
+ tailrec.const_mono
+ bind_mono
+
+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
+
+lemma transp_le [simp, cont_intro]: "transp op \<le>"
+by(rule transpI)(rule order_trans)
+
+lemma monotone_const [simp, cont_intro]: "monotone ord op \<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 op \<le> (\<lambda>x. c)"
+by (rule contI) (simp add: image_constant_conv cong del: strong_SUP_cong)
+
+lemma mcont_const [cont_intro, simp]:
+ "mcont luba orda Sup op \<le> (\<lambda>x. c)"
+by(simp add: mcont_def)
+
+lemma cont_apply:
+ assumes 2: "\<And>x. cont lubb ordb Sup op \<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 op \<le> (\<lambda>x. f x y)"
+ and mono: "monotone orda ordb (\<lambda>x. t x)"
+ and mono2: "\<And>x. monotone ordb op \<le> (\<lambda>y. f x y)"
+ and mono1: "\<And>y. monotone orda op \<le> (\<lambda>x. f x y)"
+ shows "cont luba orda Sup op \<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 op \<le> (\<lambda>y. f x y);
+ \<And>y. mcont lub ord Sup op \<le> (\<lambda>x. f x y);
+ mcont lub ord lub' ord' (\<lambda>y. t y) \<rbrakk>
+ \<Longrightarrow> mcont lub ord Sup op \<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 op \<le> (\<lambda>x. f x); mcont lub ord lub' ord' (\<lambda>x. t x)\<rbrakk>
+ \<Longrightarrow> mcont lub ord Sup op \<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>_" [900] 900)
+begin
+
+lemma cont_fun_lub_Sup:
+ assumes chainM: "Complete_Partial_Order.chain (fun_ord op \<le>) M"
+ and mcont [rule_format]: "\<forall>f\<in>M. mcont lub op \<sqsubseteq> Sup op \<le> f"
+ shows "cont lub op \<sqsubseteq> Sup op \<le> (fun_lub Sup M)"
+proof(rule contI)
+ fix Y
+ assume chain: "Complete_Partial_Order.chain op \<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 op \<le>) M;
+ \<forall>f\<in>M. mcont lub ord Sup op \<le> f \<rbrakk>
+ \<Longrightarrow> mcont lub op \<sqsubseteq> Sup op \<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 op \<le>) F"
+ and mono: "\<And>f. mcont lub op \<sqsubseteq> Sup op \<le> f \<Longrightarrow> mcont lub op \<sqsubseteq> Sup op \<le> (F f)"
+ shows "mcont lub op \<sqsubseteq> Sup op \<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 op \<le>) op \<le> (\<lambda>f. F f x)"
+ and mcont: "\<And>f. mcont lub op \<sqsubseteq> Sup op \<le> f \<Longrightarrow> mcont lub op \<sqsubseteq> Sup op \<le> (F f)"
+ shows "mcont lub op \<sqsubseteq> Sup op \<le> (ccpo.fixp (fun_lub Sup) (fun_ord op \<le>) F)"
+ (is "mcont _ _ _ _ ?fixp")
+unfolding mcont_def
+proof(intro conjI monotoneI contI)
+ have mono: "monotone (fun_ord op \<le>) (fun_ord op \<le>) F"
+ by(rule monotoneI)(auto simp add: fun_ord_def intro: monotoneD[OF mono])
+ let ?iter = "ccpo.iterates (fun_lub Sup) (fun_ord op \<le>) F"
+ have chain: "\<And>x. Complete_Partial_Order.chain op \<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"
+ unfolding 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 _ `x \<sqsubseteq> y` have "f x \<le> f y"
+ by(rule mcont_monoD[OF iterates_mcont[OF `f \<in> ?iter` mcont]])
+ also have "f y \<le> \<Squnion>((\<lambda>f. f y) ` ?iter)" using chain
+ by(rule ccpo_Sup_upper)(simp add: `f \<in> ?iter`)
+ finally show "x' \<le> \<dots>" .
+ qed
+ next
+ fix Y
+ assume chain: "Complete_Partial_Order.chain op \<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 op \<le>) op \<le> (\<lambda>f. U (F (C f)) x)"
+ and eq: "f \<equiv> C (ccpo.fixp (fun_lub Sup) (fun_ord op \<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 op \<le> (U f) \<Longrightarrow> monotone ord op \<le> (U (F f))"
+ shows "monotone ord op \<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 op \<le> (U f) \<Longrightarrow> mcont lubb ordb Sup op \<le> (U (F f))"
+ shows "mcont lubb ordb Sup op \<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 op \<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>_" [900] 900) and ord (infix "\<sqsubseteq>" 60)
+ assumes ccpo: "class.ccpo lub op \<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 op \<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 op \<le> lub op \<sqsubseteq> (\<lambda>x. if x \<le> bound then bot else f x)" (is "mcont _ _ _ _ ?g")
+proof(intro mcontI contI)
+ interpret c: ccpo lub "op \<sqsubseteq>" lt by(fact ccpo)
+ show "monotone op \<le> op \<sqsubseteq> ?g" by(rule monotone_if_bot)(simp_all add: mono bot)
+
+ fix Y
+ assume chain: "Complete_Partial_Order.chain op \<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.strong_SUP_cong)
+ next
+ case False
+ let ?Y = "Y \<inter> {x. \<not> x \<le> bound}"
+ have chain': "Complete_Partial_Order.chain op \<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 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 op \<sqsubseteq> (insert bot (f ` ?Y))"
+ using chain by(auto intro!: chainI bot dest: chainD intro: mono)
+ hence chain''': "Complete_Partial_Order.chain op \<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.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 {* Admissibility *}
+
+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 op \<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 op \<le> (\<lambda>x. f x)"
+ and g: "mcont luba orda Sup op \<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 op \<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 `y \<in> A` by simp
+ also have "Complete_Partial_Order.chain op \<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: `y \<in> A`)
+ 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>_" [900] 900)
+ assumes "class.ccpo lub op \<sqsubseteq> (mk_less op \<sqsubseteq>)"
+ and "mcont luba orda lub op \<sqsubseteq> (\<lambda>x. f x)"
+ and "mcont luba orda lub op \<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 op \<le> (\<lambda>x. \<not> op \<le> x y)"
+by(rule ccpo.admissibleI)(simp add: ccpo_Sup_below_iff)
+
+end
+
+lemma (in preorder) preorder [cont_intro, simp]: "class.preorder op \<le> (mk_less op \<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
+
+
+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"
+
+hide_fact (open) compact
+
+context ccpo begin
+
+lemma compactI:
+ assumes "ccpo.admissible Sup op \<le> (\<lambda>y. \<not> x \<le> y)"
+ shows "compact Sup op \<le> x"
+using assms
+proof(rule 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 op \<le> (\<lambda>y. x \<noteq> y)"
+ by(subst neq)(rule admissible_disj admissible_not_below assms)+
+qed
+
+lemma compact_bot:
+ assumes "x = Sup {}"
+ shows "compact Sup op \<le> x"
+proof(rule compactI)
+ show "ccpo.admissible Sup op \<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: "compact lub ord k \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. k \<noteq> x)"
+by(simp add: compact.simps)
+
+lemma admissible_neq_compact' [THEN admissible_subst, cont_intro, simp]:
+ shows admissible_neq_compact: "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 op \<le> P"
+ and mono: "monotone op \<le> op \<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 op \<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 {* @{term "op ="} as order *}
+
+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 op = (mk_less op =)"
+by(unfold_locales)(simp_all add: mk_less_def)
+
+lemma monotone_eqI [cont_intro]:
+ assumes "class.preorder ord (mk_less ord)"
+ shows "monotone op = 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 op = lub ord f"
+proof(rule contI)
+ fix Y :: "'a set"
+ assume "Complete_Partial_Order.chain op = 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 op = lub ord f"
+by(simp add: mcont_def cont_eqI monotone_eqI)
+
+subsection {* ccpo for products *}
+
+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 `Y \<noteq> {}` 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) op \<le> f"
+ and cont1: "\<And>x. cont lubb ordb Sup op \<le> (\<lambda>y. f (x, y))"
+ and cont2: "\<And>y. cont luba orda Sup op \<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 op \<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] `Y \<noteq> {}`)
+ 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] `Y \<noteq> {}`)
+ 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) op \<le> (case_prod f)"
+ and "\<And>x. cont lubb ordb Sup op \<le> (\<lambda>y. f x y)"
+ and "\<And>y. cont luba orda Sup op \<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 op \<le> (case_prod f)"
+by(rule cont_prodI)(simp_all add: assms)
+
+lemma cont_case_prod_iff:
+ "\<lbrakk> monotone (rel_prod orda ordb) op \<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 op \<le> (case_prod f) \<longleftrightarrow>
+ (\<forall>x. cont lubb ordb Sup op \<le> (\<lambda>y. f x y)) \<and> (\<forall>y. cont luba orda Sup op \<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 {* Complete lattices as ccpo *}
+
+context complete_lattice begin
+
+lemma complete_lattice_ccpo: "class.ccpo Sup op \<le> op <"
+by(unfold_locales)(fast intro: Sup_upper Sup_least)+
+
+lemma complete_lattice_ccpo': "class.ccpo Sup op \<le> (mk_less op \<le>)"
+by(unfold_locales)(auto simp add: mk_less_def intro: Sup_upper Sup_least)
+
+lemma complete_lattice_partial_function_definitions:
+ "partial_function_definitions op \<le> Sup"
+by(unfold_locales)(auto intro: Sup_least Sup_upper)
+
+lemma complete_lattice_partial_function_definitions_dual:
+ "partial_function_definitions op \<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 op \<le> (\<lambda>x. f x)"
+ and g: "monotone ord op \<le> (\<lambda>x. g x)"
+ shows "monotone ord op \<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 op \<le> (\<lambda>_. c)"
+by(rule ccpo.mcont_const[OF complete_lattice_ccpo])
+
+lemma mono2mono_sup:
+ assumes f: "monotone ord op \<le> (\<lambda>x. f x)"
+ and g: "monotone ord op \<le> (\<lambda>x. g x)"
+ shows "monotone ord op \<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>(op \<squnion> x ` Y) = x \<squnion> \<Squnion>Y"
+proof(rule Sup_eqI)
+ fix y
+ assume "y \<in> op \<squnion> x ` Y"
+ then obtain z where "y = x \<squnion> z" and "z \<in> Y" by blast
+ from `z \<in> Y` have "z \<le> \<Squnion>Y" by(rule Sup_upper)
+ with _ show "y \<le> x \<squnion> \<Squnion>Y" unfolding `y = x \<squnion> z` by(rule sup_mono) simp
+next
+ fix y
+ assume upper: "\<And>z. z \<in> op \<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 `z' \<in> Y` `z \<in> insert x Y` by auto
+ also have "\<dots> \<le> y" by(rule upper)(auto split: if_split_asm intro: `z' \<in> Y`)
+ finally show "z \<le> y" .
+ qed
+qed
+
+lemma mcont_sup1: "mcont Sup op \<le> Sup op \<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 op \<le> Sup op \<le> (\<lambda>x. x \<squnion> y)"
+by(subst sup_commute)(rule mcont_sup1)
+
+lemma mcont2mcont_sup [cont_intro, simp]:
+ "\<lbrakk> mcont lub ord Sup op \<le> (\<lambda>x. f x);
+ mcont lub ord Sup op \<le> (\<lambda>x. g x) \<rbrakk>
+ \<Longrightarrow> mcont lub ord Sup op \<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 op \<le> Sup op \<le> (\<lambda>y. x \<sqinter> y)"
+by(auto intro: monotoneI contI simp add: le_infI2 inf_Sup mcont_def)
+
+lemma mcont_inf2: "mcont Sup op \<le> Sup op \<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 op \<le> (\<lambda>x. f x);
+ mcont lub ord Sup op \<le> (\<lambda>x. g x) \<rbrakk>
+ \<Longrightarrow> mcont lub ord Sup op \<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 "op \<le> :: _ :: complete_lattice \<Rightarrow> _" Sup
+by(rule complete_lattice_partial_function_definitions)
+
+declaration {* Partial_Function.init "lfp" @{term lfp.fixp_fun} @{term lfp.mono_body}
+ @{thm lfp.fixp_rule_uc} @{thm lfp.fixp_induct_uc} NONE *}
+
+interpretation gfp: partial_function_definitions "op \<ge> :: _ :: complete_lattice \<Rightarrow> _" Inf
+by(rule complete_lattice_partial_function_definitions_dual)
+
+declaration {* Partial_Function.init "gfp" @{term gfp.fixp_fun} @{term gfp.mono_body}
+ @{thm gfp.fixp_rule_uc} @{thm gfp.fixp_induct_uc} NONE *}
+
+lemma insert_mono [partial_function_mono]:
+ "monotone (fun_ord op \<subseteq>) op \<subseteq> A \<Longrightarrow> monotone (fun_ord op \<subseteq>) op \<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 op \<subseteq> op \<subseteq> (insert x)"
+by(rule monotoneI) blast
+
+lemma mcont2mcont_insert[THEN lfp.mcont2mcont, cont_intro, simp]:
+ shows mcont_insert: "mcont Union op \<subseteq> Union op \<subseteq> (insert x)"
+by(blast intro: mcontI contI monotone_insert)
+
+lemma mono2mono_image [THEN lfp.mono2mono, cont_intro, simp]:
+ shows monotone_image: "monotone op \<subseteq> op \<subseteq> (op ` f)"
+by(rule monotoneI) blast
+
+lemma cont_image: "cont Union op \<subseteq> Union op \<subseteq> (op ` f)"
+by(rule contI)(auto)
+
+lemma mcont2mcont_image [THEN lfp.mcont2mcont, cont_intro, simp]:
+ shows mcont_image: "mcont Union op \<subseteq> Union op \<subseteq> (op ` f)"
+by(blast intro: mcontI monotone_image cont_image)
+
+context complete_lattice begin
+
+lemma monotone_Sup [cont_intro, simp]:
+ "monotone ord op \<subseteq> f \<Longrightarrow> monotone ord op \<le> (\<lambda>x. \<Squnion>f x)"
+by(blast intro: monotoneI Sup_least Sup_upper dest: monotoneD)
+
+lemma cont_Sup:
+ assumes "cont lub ord Union op \<subseteq> f"
+ shows "cont lub ord Sup op \<le> (\<lambda>x. \<Squnion>f x)"
+apply(rule contI)
+apply(simp add: contD[OF assms])
+apply(blast intro: Sup_least Sup_upper order_trans antisym)
+done
+
+lemma mcont_Sup: "mcont lub ord Union op \<subseteq> f \<Longrightarrow> mcont lub ord Sup op \<le> (\<lambda>x. \<Squnion>f x)"
+unfolding mcont_def by(blast intro: monotone_Sup cont_Sup)
+
+lemma monotone_SUP:
+ "\<lbrakk> monotone ord op \<subseteq> f; \<And>y. monotone ord op \<le> (\<lambda>x. g x y) \<rbrakk> \<Longrightarrow> monotone ord op \<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 op \<le> (\<lambda>x. g x y)) \<Longrightarrow> monotone ord op \<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 op \<subseteq> f"
+ and g: "\<And>y. mcont lub ord Sup op \<le> (\<lambda>x. g x y)"
+ shows "cont lub ord Sup op \<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 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 `y \<in> Y` `y' \<in> Y` have "ord y y' \<or> ord y' y" by(rule chainD)
+ thus "u \<le> ?rhs"
+ proof
+ note `u = g y' z` also
+ assume "ord y y'"
+ with f have "f y \<subseteq> f y'" by(rule mcont_monoD)
+ with `z \<in> f y`
+ have "g y' z \<le> \<Squnion>(g y' ` f y')" by(auto intro: Sup_upper)
+ also have "\<dots> \<le> ?rhs" using `y' \<in> Y` by(auto intro: Sup_upper)
+ finally show ?thesis .
+ next
+ note `u = g y' z` 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 `z \<in> f y`
+ by(auto intro: Sup_upper)
+ also have "\<dots> \<le> ?rhs" using `y \<in> Y` 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 `u = g y z`
+ also have "g y z \<le> \<Squnion>((\<lambda>x. g x z) ` Y)"
+ using `y \<in> Y` 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 `z \<in> f y` `y \<in> Y`
+ 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 op \<subseteq> f; \<And>y. mcont lub ord Sup op \<le> (\<lambda>x. g x y) \<rbrakk>
+ \<Longrightarrow> mcont lub ord Sup op \<le> (\<lambda>x. \<Squnion>y\<in>f x. g x y)"
+by(blast intro: mcontI cont_SUP[OF assms] 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 op \<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 op \<subseteq> (\<lambda>A. \<exists>x\<in>A. P x)"
+by(rule ccpo.admissibleI)(auto)
+
+subsection {* Parallel fixpoint induction *}
+
+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.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)
+
+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
+
+end