(*  Title:      HOLCF/Up.thy

     Author:     Franz Regensburger and Brian Huffman

 *)

 header {* The type of lifted values *}

 theory Up

 imports Bifinite

 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

 lemma u_UNIV: "UNIV = insert Ibottom (range Iup)"

 by (auto, case_tac x, auto)

 instance u :: (finite_po) finite_po

 by (intro_classes, simp add: u_UNIV)

 subsection {* Lifted cpo is a cpo *}

 lemma is_lub_Iup:

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

 apply (rule is_lubI)

 apply (rule ub_rangeI)

 apply (subst Iup_below)

 apply (erule is_ub_lub)

 apply (case_tac u)

 apply (drule ub_rangeD)

 apply simp

 apply simp

 apply (erule is_lub_lub)

 apply (rule ub_rangeI)

 apply (drule_tac i=i in ub_rangeD)

 apply simp

 done

 text {* Now some lemmas about chains of @{typ "'a u"} elements *}

 lemma up_lemma1: "z \<noteq> Ibottom \<Longrightarrow> Iup (THE a. Iup a = z) = z"

 by (case_tac z, simp_all)

 lemma up_lemma2:

   "\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow> Y (i + j) \<noteq> Ibottom"

 apply (erule contrapos_nn)

 apply (drule_tac i="j" and j="i + j" in chain_mono)

 apply (rule le_add2)

 apply (case_tac "Y j")

 apply assumption

 apply simp

 done

 lemma up_lemma3:

   "\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow> Iup (THE a. Iup a = Y (i + j)) = Y (i + j)"

 by (rule up_lemma1 [OF up_lemma2])

 lemma up_lemma4:

   "\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow> chain (\<lambda>i. THE a. Iup a = Y (i + j))"

 apply (rule chainI)

 apply (rule Iup_below [THEN iffD1])

 apply (subst up_lemma3, assumption+)+

 apply (simp add: chainE)

 done

 lemma up_lemma5:

   "\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow>

     (\<lambda>i. Y (i + j)) = (\<lambda>i. Iup (THE a. Iup a = Y (i + j)))"

 by (rule ext, rule up_lemma3 [symmetric])

 lemma up_lemma6:

   "\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk>

       \<Longrightarrow> range Y <<| Iup (\<Squnion>i. THE a. Iup a = Y(i + j))"

 apply (rule_tac j1 = j in is_lub_range_shift [THEN iffD1])

 apply assumption

 apply (subst up_lemma5, assumption+)

 apply (rule is_lub_Iup)

 apply (rule cpo_lubI)

 apply (erule (1) up_lemma4)

 done

 lemma up_chain_lemma:

   "chain Y \<Longrightarrow>

    (\<exists>A. chain A \<and> (\<Squnion>i. Y i) = Iup (\<Squnion>i. A i) \<and>

    (\<exists>j. \<forall>i. Y (i + j) = Iup (A i))) \<or> (Y = (\<lambda>i. Ibottom))"

 apply (rule disjCI)

 apply (simp add: expand_fun_eq)

 apply (erule exE, rename_tac j)

 apply (rule_tac x="\<lambda>i. THE a. Iup a = Y (i + j)" in exI)

 apply (simp add: up_lemma4)

 apply (simp add: up_lemma6 [THEN thelubI])

 apply (rule_tac x=j in exI)

 apply (simp add: up_lemma3)

 done

 lemma cpo_up: "chain (Y::nat \<Rightarrow> 'a u) \<Longrightarrow> \<exists>x. range Y <<| x"

 apply (frule up_chain_lemma, safe)

 apply (rule_tac x="Iup (\<Squnion>i. A i)" in exI)

 apply (erule_tac j="j" in is_lub_range_shift [THEN iffD1, standard])

 apply (simp add: is_lub_Iup cpo_lubI)

 apply (rule exI, rule lub_const)

 done

 instance u :: (cpo) cpo

 by intro_classes (rule cpo_up)

 subsection {* Lifted cpo is pointed *}

 lemma least_up: "\<exists>x::'a u. \<forall>y. x \<sqsubseteq> y"

 apply (rule_tac x = "Ibottom" in exI)

 apply (rule minimal_up [THEN allI])

 done

 instance u :: (cpo) pcpo

 by intro_classes (rule least_up)

 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)"

 apply (rule contI)

 apply (frule up_chain_lemma, safe)

 apply (rule_tac j="j" in is_lub_range_shift [THEN iffD1, standard])

 apply (erule monofun_Ifup2 [THEN ch2ch_monofun])

 apply (simp add: cont_cfun_arg)

 apply (simp add: lub_const)

 done

 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:

   "chain Y \<Longrightarrow>

   (\<exists>A. chain A \<and> (\<Squnion>i. Y i) = up\<cdot>(\<Squnion>i. A i) \<and>

   (\<exists>j. \<forall>i. Y (i + j) = up\<cdot>(A i))) \<or> Y = (\<lambda>i. \<bottom>)"

 by (simp add: inst_up_pcpo up_def cont_Iup up_chain_lemma)

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

 apply (rule compactI2)

 apply (drule up_chain_cases, safe)

 apply (drule (1) compactD2, simp)

 apply (erule exE, rule_tac x="i + j" in exI)

 apply simp

 apply simp

 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)

 subsection {* Map function for lifted cpo *}

 definition

   u_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a u \<rightarrow> 'b u"

 where

   "u_map = (\<Lambda> f. fup\<cdot>(up oo f))"

 lemma u_map_strict [simp]: "u_map\<cdot>f\<cdot>\<bottom> = \<bottom>"

 unfolding u_map_def by simp

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

 unfolding u_map_def by simp

 lemma u_map_ID: "u_map\<cdot>ID = ID"

 unfolding u_map_def by (simp add: expand_cfun_eq eta_cfun)

 lemma u_map_map: "u_map\<cdot>f\<cdot>(u_map\<cdot>g\<cdot>p) = u_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>p"

 by (induct p) simp_all

 lemma ep_pair_u_map: "ep_pair e p \<Longrightarrow> ep_pair (u_map\<cdot>e) (u_map\<cdot>p)"

 apply default

 apply (case_tac x, simp, simp add: ep_pair.e_inverse)

 apply (case_tac y, simp, simp add: ep_pair.e_p_below)

 done

 lemma deflation_u_map: "deflation d \<Longrightarrow> deflation (u_map\<cdot>d)"

 apply default

 apply (case_tac x, simp, simp add: deflation.idem)

 apply (case_tac x, simp, simp add: deflation.below)

 done

 lemma finite_deflation_u_map:

   assumes "finite_deflation d" shows "finite_deflation (u_map\<cdot>d)"

 proof (intro finite_deflation.intro finite_deflation_axioms.intro)

   interpret d: finite_deflation d by fact

   have "deflation d" by fact

   thus "deflation (u_map\<cdot>d)" by (rule deflation_u_map)

   have "{x. u_map\<cdot>d\<cdot>x = x} \<subseteq> insert \<bottom> ((\<lambda>x. up\<cdot>x) ` {x. d\<cdot>x = x})"

     by (rule subsetI, case_tac x, simp_all)

   thus "finite {x. u_map\<cdot>d\<cdot>x = x}"

     by (rule finite_subset, simp add: d.finite_fixes)

 qed

 subsection {* Lifted cpo is a bifinite domain *}

 instantiation u :: (profinite) bifinite

 begin

 definition

   approx_up_def:

     "approx = (\<lambda>n. u_map\<cdot>(approx n))"

 instance proof

   fix i :: nat and x :: "'a u"

   show "chain (approx :: nat \<Rightarrow> 'a u \<rightarrow> 'a u)"

     unfolding approx_up_def by simp

   show "(\<Squnion>i. approx i\<cdot>x) = x"

     unfolding approx_up_def

     by (induct x, simp, simp add: lub_distribs)

   show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"

     unfolding approx_up_def

     by (induct x) simp_all

   show "finite {x::'a u. approx i\<cdot>x = x}"

     unfolding approx_up_def

     by (intro finite_deflation.finite_fixes

               finite_deflation_u_map

               finite_deflation_approx)

 qed

 end

 lemma approx_up [simp]: "approx i\<cdot>(up\<cdot>x) = up\<cdot>(approx i\<cdot>x)"

 unfolding approx_up_def by simp

 end