(*  Title:      HOLCF/Up.thy

    Author:     Franz Regensburger

    Author:     Brian Huffman

*)

header {* The type of lifted values *}

theory Up

imports Cfun

begin

default_sort cpo

subsection {* Definition of new type for lifting *}

datatype 'a u = Ibottom | Iup 'a

type_notation (xsymbols)

  u  ("(_\<^sub>\<bottom>)" [1000] 999)

primrec Ifup :: "('a \<rightarrow> 'b::pcpo) \<Rightarrow> 'a u \<Rightarrow> 'b" where

    "Ifup f Ibottom = \<bottom>"

 |  "Ifup f (Iup x) = f\<cdot>x"

subsection {* Ordering on lifted cpo *}

instantiation u :: (cpo) below

begin

definition

  below_up_def:

    "(op \<sqsubseteq>) \<equiv> (\<lambda>x y. case x of Ibottom \<Rightarrow> True | Iup a \<Rightarrow>

      (case y of Ibottom \<Rightarrow> False | Iup b \<Rightarrow> a \<sqsubseteq> b))"

instance ..

end

lemma minimal_up [iff]: "Ibottom \<sqsubseteq> z"

by (simp add: below_up_def)

lemma not_Iup_below [iff]: "\<not> Iup x \<sqsubseteq> Ibottom"

by (simp add: below_up_def)

lemma Iup_below [iff]: "(Iup x \<sqsubseteq> Iup y) = (x \<sqsubseteq> y)"

by (simp add: below_up_def)

subsection {* Lifted cpo is a partial order *}

instance u :: (cpo) po

proof

  fix x :: "'a u"

  show "x \<sqsubseteq> x"

    unfolding below_up_def by (simp split: u.split)

next

  fix x y :: "'a u"

  assume "x \<sqsubseteq> y" "y \<sqsubseteq> x" thus "x = y"

    unfolding below_up_def

    by (auto split: u.split_asm intro: below_antisym)

next

  fix x y z :: "'a u"

  assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"

    unfolding below_up_def

    by (auto split: u.split_asm intro: below_trans)

qed

subsection {* Lifted cpo is a cpo *}

lemma is_lub_Iup:

  "range S <<| x \<Longrightarrow> range (\<lambda>i. Iup (S i)) <<| Iup x"

unfolding is_lub_def is_ub_def ball_simps

by (auto simp add: below_up_def split: u.split)

lemma up_chain_lemma:

  assumes Y: "chain Y" obtains "\<forall>i. Y i = Ibottom"

  | A k where "\<forall>i. Iup (A i) = Y (i + k)" and "chain A" and "range Y <<| Iup (\<Squnion>i. A i)"

proof (cases "\<exists>k. Y k \<noteq> Ibottom")

  case True

  then obtain k where k: "Y k \<noteq> Ibottom" ..

  def A \<equiv> "\<lambda>i. THE a. Iup a = Y (i + k)"

  have Iup_A: "\<forall>i. Iup (A i) = Y (i + k)"

  proof

    fix i :: nat

    from Y le_add2 have "Y k \<sqsubseteq> Y (i + k)" by (rule chain_mono)

    with k have "Y (i + k) \<noteq> Ibottom" by (cases "Y k", auto)

    thus "Iup (A i) = Y (i + k)"

      by (cases "Y (i + k)", simp_all add: A_def)

  qed

  from Y have chain_A: "chain A"

    unfolding chain_def Iup_below [symmetric]

    by (simp add: Iup_A)

  hence "range A <<| (\<Squnion>i. A i)"

    by (rule cpo_lubI)

  hence "range (\<lambda>i. Iup (A i)) <<| Iup (\<Squnion>i. A i)"

    by (rule is_lub_Iup)

  hence "range (\<lambda>i. Y (i + k)) <<| Iup (\<Squnion>i. A i)"

    by (simp only: Iup_A)

  hence "range (\<lambda>i. Y i) <<| Iup (\<Squnion>i. A i)"

    by (simp only: is_lub_range_shift [OF Y])

  with Iup_A chain_A show ?thesis ..

next

  case False

  then have "\<forall>i. Y i = Ibottom" by simp

  then show ?thesis ..

qed

instance u :: (cpo) cpo

proof

  fix S :: "nat \<Rightarrow> 'a u"

  assume S: "chain S"

  thus "\<exists>x. range (\<lambda>i. S i) <<| x"

  proof (rule up_chain_lemma)

    assume "\<forall>i. S i = Ibottom"

    hence "range (\<lambda>i. S i) <<| Ibottom"

      by (simp add: is_lub_const)

    thus ?thesis ..

  next

    fix A :: "nat \<Rightarrow> 'a"

    assume "range S <<| Iup (\<Squnion>i. A i)"

    thus ?thesis ..

  qed

qed

subsection {* Lifted cpo is pointed *}

instance u :: (cpo) pcpo

by intro_classes fast

text {* for compatibility with old HOLCF-Version *}

lemma inst_up_pcpo: "\<bottom> = Ibottom"

by (rule minimal_up [THEN UU_I, symmetric])

subsection {* Continuity of \emph{Iup} and \emph{Ifup} *}

text {* continuity for @{term Iup} *}

lemma cont_Iup: "cont Iup"

apply (rule contI)

apply (rule is_lub_Iup)

apply (erule cpo_lubI)

done

text {* continuity for @{term Ifup} *}

lemma cont_Ifup1: "cont (\<lambda>f. Ifup f x)"

by (induct x, simp_all)

lemma monofun_Ifup2: "monofun (\<lambda>x. Ifup f x)"

apply (rule monofunI)

apply (case_tac x, simp)

apply (case_tac y, simp)

apply (simp add: monofun_cfun_arg)

done

lemma cont_Ifup2: "cont (\<lambda>x. Ifup f x)"

proof (rule contI2)

  fix Y assume Y: "chain Y" and Y': "chain (\<lambda>i. Ifup f (Y i))"

  from Y show "Ifup f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. Ifup f (Y i))"

  proof (rule up_chain_lemma)

    fix A and k

    assume A: "\<forall>i. Iup (A i) = Y (i + k)"

    assume "chain A" and "range Y <<| Iup (\<Squnion>i. A i)"

    hence "Ifup f (\<Squnion>i. Y i) = (\<Squnion>i. Ifup f (Iup (A i)))"

      by (simp add: lub_eqI contlub_cfun_arg)

    also have "\<dots> = (\<Squnion>i. Ifup f (Y (i + k)))"

      by (simp add: A)

    also have "\<dots> = (\<Squnion>i. Ifup f (Y i))"

      using Y' by (rule lub_range_shift)

    finally show ?thesis by simp

  qed simp

qed (rule monofun_Ifup2)

subsection {* Continuous versions of constants *}

definition

  up  :: "'a \<rightarrow> 'a u" where

  "up = (\<Lambda> x. Iup x)"

definition

  fup :: "('a \<rightarrow> 'b::pcpo) \<rightarrow> 'a u \<rightarrow> 'b" where

  "fup = (\<Lambda> f p. Ifup f p)"

translations

  "case l of XCONST up\<cdot>x \<Rightarrow> t" == "CONST fup\<cdot>(\<Lambda> x. t)\<cdot>l"

  "\<Lambda>(XCONST up\<cdot>x). t" == "CONST fup\<cdot>(\<Lambda> x. t)"

text {* continuous versions of lemmas for @{typ "('a)u"} *}

lemma Exh_Up: "z = \<bottom> \<or> (\<exists>x. z = up\<cdot>x)"

apply (induct z)

apply (simp add: inst_up_pcpo)

apply (simp add: up_def cont_Iup)

done

lemma up_eq [simp]: "(up\<cdot>x = up\<cdot>y) = (x = y)"

by (simp add: up_def cont_Iup)

lemma up_inject: "up\<cdot>x = up\<cdot>y \<Longrightarrow> x = y"

by simp

lemma up_defined [simp]: "up\<cdot>x \<noteq> \<bottom>"

by (simp add: up_def cont_Iup inst_up_pcpo)

lemma not_up_less_UU: "\<not> up\<cdot>x \<sqsubseteq> \<bottom>"

by simp (* FIXME: remove? *)

lemma up_below [simp]: "up\<cdot>x \<sqsubseteq> up\<cdot>y \<longleftrightarrow> x \<sqsubseteq> y"

by (simp add: up_def cont_Iup)

lemma upE [case_names bottom up, cases type: u]:

  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x. p = up\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

apply (cases p)

apply (simp add: inst_up_pcpo)

apply (simp add: up_def cont_Iup)

done

lemma up_induct [case_names bottom up, induct type: u]:

  "\<lbrakk>P \<bottom>; \<And>x. P (up\<cdot>x)\<rbrakk> \<Longrightarrow> P x"

by (cases x, simp_all)

text {* lifting preserves chain-finiteness *}

lemma up_chain_cases:

  assumes Y: "chain Y" obtains "\<forall>i. Y i = \<bottom>"

  | A k where "\<forall>i. up\<cdot>(A i) = Y (i + k)" and "chain A" and "(\<Squnion>i. Y i) = up\<cdot>(\<Squnion>i. A i)"

apply (rule up_chain_lemma [OF Y])

apply (simp_all add: inst_up_pcpo up_def cont_Iup lub_eqI)

done

lemma compact_up: "compact x \<Longrightarrow> compact (up\<cdot>x)"

apply (rule compactI2)

apply (erule up_chain_cases)

apply simp

apply (drule (1) compactD2, simp)

apply (erule exE)

apply (drule_tac f="up" and x="x" in monofun_cfun_arg)

apply (simp, erule exI)

done

lemma compact_upD: "compact (up\<cdot>x) \<Longrightarrow> compact x"

unfolding compact_def

by (drule adm_subst [OF cont_Rep_cfun2 [where f=up]], simp)

lemma compact_up_iff [simp]: "compact (up\<cdot>x) = compact x"

by (safe elim!: compact_up compact_upD)

instance u :: (chfin) chfin

apply intro_classes

apply (erule compact_imp_max_in_chain)

apply (rule_tac p="\<Squnion>i. Y i" in upE, simp_all)

done

text {* properties of fup *}

lemma fup1 [simp]: "fup\<cdot>f\<cdot>\<bottom> = \<bottom>"

by (simp add: fup_def cont_Ifup1 cont_Ifup2 inst_up_pcpo cont2cont_LAM)

lemma fup2 [simp]: "fup\<cdot>f\<cdot>(up\<cdot>x) = f\<cdot>x"

by (simp add: up_def fup_def cont_Iup cont_Ifup1 cont_Ifup2 cont2cont_LAM)

lemma fup3 [simp]: "fup\<cdot>up\<cdot>x = x"

by (cases x, simp_all)

end