(* Title:    HOL/Complete_Partial_Order.thy

    Author:   Brian Huffman, Portland State University

    Author:   Alexander Krauss, TU Muenchen

 *)

 header {* Chain-complete partial orders and their fixpoints *}

 theory Complete_Partial_Order

 imports Product_Type

 begin

 subsection {* Monotone functions *}

 text {* Dictionary-passing version of @{const Orderings.mono}. *}

 definition monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"

 where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"

 lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y))

  \<Longrightarrow> monotone orda ordb f"

 unfolding monotone_def by iprover

 lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"

 unfolding monotone_def by iprover

 subsection {* Chains *}

 text {* A chain is a totally-ordered set. Chains are parameterized over

   the order for maximal flexibility, since type classes are not enough.

 *}

 definition

   chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"

 where

   "chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"

 lemma chainI:

   assumes "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> ord x y \<or> ord y x"

   shows "chain ord S"

 using assms unfolding chain_def by fast

 lemma chainD:

   assumes "chain ord S" and "x \<in> S" and "y \<in> S"

   shows "ord x y \<or> ord y x"

 using assms unfolding chain_def by fast

 lemma chainE:

   assumes "chain ord S" and "x \<in> S" and "y \<in> S"

   obtains "ord x y" | "ord y x"

 using assms unfolding chain_def by fast

 subsection {* Chain-complete partial orders *}

 text {*

   A ccpo has a least upper bound for any chain.  In particular, the

   empty set is a chain, so every ccpo must have a bottom element.

 *}

 class ccpo = order +

   fixes lub :: "'a set \<Rightarrow> 'a"

   assumes lub_upper: "chain (op \<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> lub A"

   assumes lub_least: "chain (op \<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> lub A \<le> z"

 begin

 subsection {* Transfinite iteration of a function *}

 inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set"

 for f :: "'a \<Rightarrow> 'a"

 where

   step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"

 | lub: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> lub M \<in> iterates f"

 lemma iterates_le_f:

   "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"

 by (induct x rule: iterates.induct)

   (force dest: monotoneD intro!: lub_upper lub_least)+

 lemma chain_iterates:

   assumes f: "monotone (op \<le>) (op \<le>) f"

   shows "chain (op \<le>) (iterates f)" (is "chain _ ?C")

 proof (rule chainI)

   fix x y assume "x \<in> ?C" "y \<in> ?C"

   then show "x \<le> y \<or> y \<le> x"

   proof (induct x arbitrary: y rule: iterates.induct)

     fix x y assume y: "y \<in> ?C"

     and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"

     from y show "f x \<le> y \<or> y \<le> f x"

     proof (induct y rule: iterates.induct)

       case (step y) with IH f show ?case by (auto dest: monotoneD)

     next

       case (lub M)

       then have chM: "chain (op \<le>) M"

         and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto

       show "f x \<le> lub M \<or> lub M \<le> f x"

       proof (cases "\<exists>z\<in>M. f x \<le> z")

         case True then have "f x \<le> lub M"

           apply rule

           apply (erule order_trans)

           by (rule lub_upper[OF chM])

         thus ?thesis ..

       next

         case False with IH'

         show ?thesis by (auto intro: lub_least[OF chM])

       qed

     qed

   next

     case (lub M y)

     show ?case

     proof (cases "\<exists>x\<in>M. y \<le> x")

       case True then have "y \<le> lub M"

         apply rule

         apply (erule order_trans)

         by (rule lub_upper[OF lub(1)])

       thus ?thesis ..

     next

       case False with lub

       show ?thesis by (auto intro: lub_least)

     qed

   qed

 qed

 subsection {* Fixpoint combinator *}

 definition

   fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"

 where

   "fixp f = lub (iterates f)"

 lemma iterates_fixp:

   assumes f: "monotone (op \<le>) (op \<le>) f" shows "fixp f \<in> iterates f"

 unfolding fixp_def

 by (simp add: iterates.lub chain_iterates f)

 lemma fixp_unfold:

   assumes f: "monotone (op \<le>) (op \<le>) f"

   shows "fixp f = f (fixp f)"

 proof (rule antisym)

   show "fixp f \<le> f (fixp f)"

     by (intro iterates_le_f iterates_fixp f)

   have "f (fixp f) \<le> lub (iterates f)"

     by (intro lub_upper chain_iterates f iterates.step iterates_fixp)

   thus "f (fixp f) \<le> fixp f"

     unfolding fixp_def .

 qed

 lemma fixp_lowerbound:

   assumes f: "monotone (op \<le>) (op \<le>) f" and z: "f z \<le> z" shows "fixp f \<le> z"

 unfolding fixp_def

 proof (rule lub_least[OF chain_iterates[OF f]])

   fix x assume "x \<in> iterates f"

   thus "x \<le> z"

   proof (induct x rule: iterates.induct)

     fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD)

     also note z finally show "f x \<le> z" .

   qed (auto intro: lub_least)

 qed

 subsection {* Fixpoint induction *}

 definition

   admissible :: "('a \<Rightarrow> bool) \<Rightarrow> bool"

 where

   "admissible P = (\<forall>A. chain (op \<le>) A \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"

 lemma admissibleI:

   assumes "\<And>A. chain (op \<le>) A \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (lub A)"

   shows "admissible P"

 using assms unfolding admissible_def by fast

 lemma admissibleD:

   assumes "admissible P"

   assumes "chain (op \<le>) A"

   assumes "\<And>x. x \<in> A \<Longrightarrow> P x"

   shows "P (lub A)"

 using assms by (auto simp: admissible_def)

 lemma fixp_induct:

   assumes adm: "admissible P"

   assumes mono: "monotone (op \<le>) (op \<le>) f"

   assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"

   shows "P (fixp f)"

 unfolding fixp_def using adm chain_iterates[OF mono]

 proof (rule admissibleD)

   fix x assume "x \<in> iterates f"

   thus "P x"

     by (induct rule: iterates.induct)

       (auto intro: step admissibleD adm)

 qed

 lemma admissible_True: "admissible (\<lambda>x. True)"

 unfolding admissible_def by simp

 lemma admissible_False: "\<not> admissible (\<lambda>x. False)"

 unfolding admissible_def chain_def by simp

 lemma admissible_const: "admissible (\<lambda>x. t) = t"

 by (cases t, simp_all add: admissible_True admissible_False)

 lemma admissible_conj:

   assumes "admissible (\<lambda>x. P x)"

   assumes "admissible (\<lambda>x. Q x)"

   shows "admissible (\<lambda>x. P x \<and> Q x)"

 using assms unfolding admissible_def by simp

 lemma admissible_all:

   assumes "\<And>y. admissible (\<lambda>x. P x y)"

   shows "admissible (\<lambda>x. \<forall>y. P x y)"

 using assms unfolding admissible_def by fast

 lemma admissible_ball:

   assumes "\<And>y. y \<in> A \<Longrightarrow> admissible (\<lambda>x. P x y)"

   shows "admissible (\<lambda>x. \<forall>y\<in>A. P x y)"

 using assms unfolding admissible_def by fast

 lemma chain_compr: "chain (op \<le>) A \<Longrightarrow> chain (op \<le>) {x \<in> A. P x}"

 unfolding chain_def by fast

 lemma admissible_disj_lemma:

   assumes A: "chain (op \<le>)A"

   assumes P: "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y"

   shows "lub A = lub {x \<in> A. P x}"

 proof (rule antisym)

   have *: "chain (op \<le>) {x \<in> A. P x}"

     by (rule chain_compr [OF A])

   show "lub A \<le> lub {x \<in> A. P x}"

     apply (rule lub_least [OF A])

     apply (drule P [rule_format], clarify)

     apply (erule order_trans)

     apply (simp add: lub_upper [OF *])

     done

   show "lub {x \<in> A. P x} \<le> lub A"

     apply (rule lub_least [OF *])

     apply clarify

     apply (simp add: lub_upper [OF A])

     done

 qed

 lemma admissible_disj:

   fixes P Q :: "'a \<Rightarrow> bool"

   assumes P: "admissible (\<lambda>x. P x)"

   assumes Q: "admissible (\<lambda>x. Q x)"

   shows "admissible (\<lambda>x. P x \<or> Q x)"

 proof (rule admissibleI)

   fix A :: "'a set" assume A: "chain (op \<le>) A"

   assume "\<forall>x\<in>A. P x \<or> Q x"

   hence "(\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)"

     using chainD[OF A] by blast

   hence "lub A = lub {x \<in> A. P x} \<or> lub A = lub {x \<in> A. Q x}"

     using admissible_disj_lemma [OF A] by fast

   thus "P (lub A) \<or> Q (lub A)"

     apply (rule disjE, simp_all)

     apply (rule disjI1, rule admissibleD [OF P chain_compr [OF A]], simp)

     apply (rule disjI2, rule admissibleD [OF Q chain_compr [OF A]], simp)

     done

 qed

 end

 hide_const (open) lub iterates fixp admissible

 end