(*  Title:      HOLCF/Ssum.thy

     Author:     Franz Regensburger and Brian Huffman

 *)

 header {* The type of strict sums *}

 theory Ssum

 imports Tr

 begin

 defaultsort pcpo

 subsection {* Definition of strict sum type *}

 pcpodef (Ssum)  ('a, 'b) ssum (infixr "++" 10) = 

   "{p :: tr \<times> ('a \<times> 'b).

     (fst p \<sqsubseteq> TT \<longleftrightarrow> snd (snd p) = \<bottom>) \<and>

     (fst p \<sqsubseteq> FF \<longleftrightarrow> fst (snd p) = \<bottom>)}"

 by simp_all

 instance ssum :: ("{finite_po,pcpo}", "{finite_po,pcpo}") finite_po

 by (rule typedef_finite_po [OF type_definition_Ssum])

 instance ssum :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin

 by (rule typedef_chfin [OF type_definition_Ssum below_Ssum_def])

 type_notation (xsymbols)

   ssum  ("(_ \<oplus>/ _)" [21, 20] 20)

 type_notation (HTML output)

   ssum  ("(_ \<oplus>/ _)" [21, 20] 20)

 subsection {* Definitions of constructors *}

 definition

   sinl :: "'a \<rightarrow> ('a ++ 'b)" where

   "sinl = (\<Lambda> a. Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>))"

 definition

   sinr :: "'b \<rightarrow> ('a ++ 'b)" where

   "sinr = (\<Lambda> b. Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b))"

 lemma sinl_Ssum: "(strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>) \<in> Ssum"

 by (simp add: Ssum_def strictify_conv_if)

 lemma sinr_Ssum: "(strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b) \<in> Ssum"

 by (simp add: Ssum_def strictify_conv_if)

 lemma sinl_Abs_Ssum: "sinl\<cdot>a = Abs_Ssum (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)"

 by (unfold sinl_def, simp add: cont_Abs_Ssum sinl_Ssum)

 lemma sinr_Abs_Ssum: "sinr\<cdot>b = Abs_Ssum (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)"

 by (unfold sinr_def, simp add: cont_Abs_Ssum sinr_Ssum)

 lemma Rep_Ssum_sinl: "Rep_Ssum (sinl\<cdot>a) = (strictify\<cdot>(\<Lambda> _. TT)\<cdot>a, a, \<bottom>)"

 by (simp add: sinl_Abs_Ssum Abs_Ssum_inverse sinl_Ssum)

 lemma Rep_Ssum_sinr: "Rep_Ssum (sinr\<cdot>b) = (strictify\<cdot>(\<Lambda> _. FF)\<cdot>b, \<bottom>, b)"

 by (simp add: sinr_Abs_Ssum Abs_Ssum_inverse sinr_Ssum)

 subsection {* Properties of \emph{sinl} and \emph{sinr} *}

 text {* Ordering *}

 lemma sinl_below [simp]: "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"

 by (simp add: below_Ssum_def Rep_Ssum_sinl strictify_conv_if)

 lemma sinr_below [simp]: "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"

 by (simp add: below_Ssum_def Rep_Ssum_sinr strictify_conv_if)

 lemma sinl_below_sinr [simp]: "(sinl\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x = \<bottom>)"

 by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)

 lemma sinr_below_sinl [simp]: "(sinr\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x = \<bottom>)"

 by (simp add: below_Ssum_def Rep_Ssum_sinl Rep_Ssum_sinr strictify_conv_if)

 text {* Equality *}

 lemma sinl_eq [simp]: "(sinl\<cdot>x = sinl\<cdot>y) = (x = y)"

 by (simp add: po_eq_conv)

 lemma sinr_eq [simp]: "(sinr\<cdot>x = sinr\<cdot>y) = (x = y)"

 by (simp add: po_eq_conv)

 lemma sinl_eq_sinr [simp]: "(sinl\<cdot>x = sinr\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"

 by (subst po_eq_conv, simp)

 lemma sinr_eq_sinl [simp]: "(sinr\<cdot>x = sinl\<cdot>y) = (x = \<bottom> \<and> y = \<bottom>)"

 by (subst po_eq_conv, simp)

 lemma sinl_inject: "sinl\<cdot>x = sinl\<cdot>y \<Longrightarrow> x = y"

 by (rule sinl_eq [THEN iffD1])

 lemma sinr_inject: "sinr\<cdot>x = sinr\<cdot>y \<Longrightarrow> x = y"

 by (rule sinr_eq [THEN iffD1])

 text {* Strictness *}

 lemma sinl_strict [simp]: "sinl\<cdot>\<bottom> = \<bottom>"

 by (simp add: sinl_Abs_Ssum Abs_Ssum_strict)

 lemma sinr_strict [simp]: "sinr\<cdot>\<bottom> = \<bottom>"

 by (simp add: sinr_Abs_Ssum Abs_Ssum_strict)

 lemma sinl_defined_iff [simp]: "(sinl\<cdot>x = \<bottom>) = (x = \<bottom>)"

 by (cut_tac sinl_eq [of "x" "\<bottom>"], simp)

 lemma sinr_defined_iff [simp]: "(sinr\<cdot>x = \<bottom>) = (x = \<bottom>)"

 by (cut_tac sinr_eq [of "x" "\<bottom>"], simp)

 lemma sinl_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinl\<cdot>x \<noteq> \<bottom>"

 by simp

 lemma sinr_defined [intro!]: "x \<noteq> \<bottom> \<Longrightarrow> sinr\<cdot>x \<noteq> \<bottom>"

 by simp

 text {* Compactness *}

 lemma compact_sinl: "compact x \<Longrightarrow> compact (sinl\<cdot>x)"

 by (rule compact_Ssum, simp add: Rep_Ssum_sinl strictify_conv_if)

 lemma compact_sinr: "compact x \<Longrightarrow> compact (sinr\<cdot>x)"

 by (rule compact_Ssum, simp add: Rep_Ssum_sinr strictify_conv_if)

 lemma compact_sinlD: "compact (sinl\<cdot>x) \<Longrightarrow> compact x"

 unfolding compact_def

 by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinl]], simp)

 lemma compact_sinrD: "compact (sinr\<cdot>x) \<Longrightarrow> compact x"

 unfolding compact_def

 by (drule adm_subst [OF cont_Rep_CFun2 [where f=sinr]], simp)

 lemma compact_sinl_iff [simp]: "compact (sinl\<cdot>x) = compact x"

 by (safe elim!: compact_sinl compact_sinlD)

 lemma compact_sinr_iff [simp]: "compact (sinr\<cdot>x) = compact x"

 by (safe elim!: compact_sinr compact_sinrD)

 subsection {* Case analysis *}

 lemma Exh_Ssum: 

   "z = \<bottom> \<or> (\<exists>a. z = sinl\<cdot>a \<and> a \<noteq> \<bottom>) \<or> (\<exists>b. z = sinr\<cdot>b \<and> b \<noteq> \<bottom>)"

 apply (induct z rule: Abs_Ssum_induct)

 apply (case_tac y, rename_tac t a b)

 apply (case_tac t rule: trE)

 apply (rule disjI1)

 apply (simp add: Ssum_def Abs_Ssum_strict)

 apply (rule disjI2, rule disjI1, rule_tac x=a in exI)

 apply (simp add: sinl_Abs_Ssum Ssum_def)

 apply (rule disjI2, rule disjI2, rule_tac x=b in exI)

 apply (simp add: sinr_Abs_Ssum Ssum_def)

 done

 lemma ssumE [case_names bottom sinl sinr, cases type: ssum]:

   "\<lbrakk>p = \<bottom> \<Longrightarrow> Q;

    \<And>x. \<lbrakk>p = sinl\<cdot>x; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q;

    \<And>y. \<lbrakk>p = sinr\<cdot>y; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

 using Exh_Ssum [of p] by auto

 lemma ssum_induct [case_names bottom sinl sinr, induct type: ssum]:

   "\<lbrakk>P \<bottom>;

    \<And>x. x \<noteq> \<bottom> \<Longrightarrow> P (sinl\<cdot>x);

    \<And>y. y \<noteq> \<bottom> \<Longrightarrow> P (sinr\<cdot>y)\<rbrakk> \<Longrightarrow> P x"

 by (cases x, simp_all)

 lemma ssumE2 [case_names sinl sinr]:

   "\<lbrakk>\<And>x. p = sinl\<cdot>x \<Longrightarrow> Q; \<And>y. p = sinr\<cdot>y \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

 by (cases p, simp only: sinl_strict [symmetric], simp, simp)

 lemma below_sinlD: "p \<sqsubseteq> sinl\<cdot>x \<Longrightarrow> \<exists>y. p = sinl\<cdot>y \<and> y \<sqsubseteq> x"

 by (cases p, rule_tac x="\<bottom>" in exI, simp_all)

 lemma below_sinrD: "p \<sqsubseteq> sinr\<cdot>x \<Longrightarrow> \<exists>y. p = sinr\<cdot>y \<and> y \<sqsubseteq> x"

 by (cases p, rule_tac x="\<bottom>" in exI, simp_all)

 subsection {* Case analysis combinator *}

 definition

   sscase :: "('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a ++ 'b) \<rightarrow> 'c" where

   "sscase = (\<Lambda> f g s. (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y fi) (Rep_Ssum s))"

 translations

   "case s of XCONST sinl\<cdot>x \<Rightarrow> t1 | XCONST sinr\<cdot>y \<Rightarrow> t2" == "CONST sscase\<cdot>(\<Lambda> x. t1)\<cdot>(\<Lambda> y. t2)\<cdot>s"

 translations

   "\<Lambda>(XCONST sinl\<cdot>x). t" == "CONST sscase\<cdot>(\<Lambda> x. t)\<cdot>\<bottom>"

   "\<Lambda>(XCONST sinr\<cdot>y). t" == "CONST sscase\<cdot>\<bottom>\<cdot>(\<Lambda> y. t)"

 lemma beta_sscase:

   "sscase\<cdot>f\<cdot>g\<cdot>s = (\<lambda>(t, x, y). If t then f\<cdot>x else g\<cdot>y fi) (Rep_Ssum s)"

 unfolding sscase_def by (simp add: cont_Rep_Ssum [THEN cont_compose])

 lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"

 unfolding beta_sscase by (simp add: Rep_Ssum_strict)

 lemma sscase2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = f\<cdot>x"

 unfolding beta_sscase by (simp add: Rep_Ssum_sinl)

 lemma sscase3 [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sscase\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>y) = g\<cdot>y"

 unfolding beta_sscase by (simp add: Rep_Ssum_sinr)

 lemma sscase4 [simp]: "sscase\<cdot>sinl\<cdot>sinr\<cdot>z = z"

 by (cases z, simp_all)

 subsection {* Strict sum preserves flatness *}

 instance ssum :: (flat, flat) flat

 apply (intro_classes, clarify)

 apply (case_tac x, simp)

 apply (case_tac y, simp_all add: flat_below_iff)

 apply (case_tac y, simp_all add: flat_below_iff)

 done

 subsection {* Map function for strict sums *}

 definition

   ssum_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<oplus> 'c \<rightarrow> 'b \<oplus> 'd"

 where

   "ssum_map = (\<Lambda> f g. sscase\<cdot>(sinl oo f)\<cdot>(sinr oo g))"

 lemma ssum_map_strict [simp]: "ssum_map\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"

 unfolding ssum_map_def by simp

 lemma ssum_map_sinl [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)"

 unfolding ssum_map_def by simp

 lemma ssum_map_sinr [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"

 unfolding ssum_map_def by simp

 lemma ssum_map_sinl': "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)"

 by (cases "x = \<bottom>") simp_all

 lemma ssum_map_sinr': "g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"

 by (cases "x = \<bottom>") simp_all

 lemma ssum_map_ID: "ssum_map\<cdot>ID\<cdot>ID = ID"

 unfolding ssum_map_def by (simp add: expand_cfun_eq eta_cfun)

 lemma ssum_map_map:

   "\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow>

     ssum_map\<cdot>f1\<cdot>g1\<cdot>(ssum_map\<cdot>f2\<cdot>g2\<cdot>p) =

      ssum_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"

 apply (induct p, simp)

 apply (case_tac "f2\<cdot>x = \<bottom>", simp, simp)

 apply (case_tac "g2\<cdot>y = \<bottom>", simp, simp)

 done

 lemma ep_pair_ssum_map:

   assumes "ep_pair e1 p1" and "ep_pair e2 p2"

   shows "ep_pair (ssum_map\<cdot>e1\<cdot>e2) (ssum_map\<cdot>p1\<cdot>p2)"

 proof

   interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact

   interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact

   fix x show "ssum_map\<cdot>p1\<cdot>p2\<cdot>(ssum_map\<cdot>e1\<cdot>e2\<cdot>x) = x"

     by (induct x) simp_all

   fix y show "ssum_map\<cdot>e1\<cdot>e2\<cdot>(ssum_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"

     apply (induct y, simp)

     apply (case_tac "p1\<cdot>x = \<bottom>", simp, simp add: e1p1.e_p_below)

     apply (case_tac "p2\<cdot>y = \<bottom>", simp, simp add: e2p2.e_p_below)

     done

 qed

 lemma deflation_ssum_map:

   assumes "deflation d1" and "deflation d2"

   shows "deflation (ssum_map\<cdot>d1\<cdot>d2)"

 proof

   interpret d1: deflation d1 by fact

   interpret d2: deflation d2 by fact

   fix x

   show "ssum_map\<cdot>d1\<cdot>d2\<cdot>(ssum_map\<cdot>d1\<cdot>d2\<cdot>x) = ssum_map\<cdot>d1\<cdot>d2\<cdot>x"

     apply (induct x, simp)

     apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.idem)

     apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.idem)

     done

   show "ssum_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"

     apply (induct x, simp)

     apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.below)

     apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.below)

     done

 qed

 lemma finite_deflation_ssum_map:

   assumes "finite_deflation d1" and "finite_deflation d2"

   shows "finite_deflation (ssum_map\<cdot>d1\<cdot>d2)"

 proof (intro finite_deflation.intro finite_deflation_axioms.intro)

   interpret d1: finite_deflation d1 by fact

   interpret d2: finite_deflation d2 by fact

   have "deflation d1" and "deflation d2" by fact+

   thus "deflation (ssum_map\<cdot>d1\<cdot>d2)" by (rule deflation_ssum_map)

   have "{x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq>

         (\<lambda>x. sinl\<cdot>x) ` {x. d1\<cdot>x = x} \<union>

         (\<lambda>x. sinr\<cdot>x) ` {x. d2\<cdot>x = x} \<union> {\<bottom>}"

     by (rule subsetI, case_tac x, simp_all)

   thus "finite {x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x}"

     by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)

 qed

 subsection {* Strict sum is a bifinite domain *}

 instantiation ssum :: (bifinite, bifinite) bifinite

 begin

 definition

   approx_ssum_def:

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

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

 unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all

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

 unfolding approx_ssum_def by (cases "x = \<bottom>") simp_all

 instance proof

   fix i :: nat and x :: "'a \<oplus> 'b"

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

     unfolding approx_ssum_def by simp

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

     unfolding approx_ssum_def

     by (cases x, simp_all add: lub_distribs)

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

     by (cases x, simp add: approx_ssum_def, simp, simp)

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

     unfolding approx_ssum_def

     by (intro finite_deflation.finite_fixes

               finite_deflation_ssum_map

               finite_deflation_approx)

 qed

 end

 end