(*  Title:      HOLCF/Sprod.thy

     Author:     Franz Regensburger and Brian Huffman

 *)

 header {* The type of strict products *}

 theory Sprod

 imports Cprod

 begin

 defaultsort pcpo

 subsection {* Definition of strict product type *}

 pcpodef (Sprod)  ('a, 'b) "**" (infixr "**" 20) =

         "{p::'a \<times> 'b. p = \<bottom> \<or> (cfst\<cdot>p \<noteq> \<bottom> \<and> csnd\<cdot>p \<noteq> \<bottom>)}"

 by simp_all

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

 by (rule typedef_finite_po [OF type_definition_Sprod])

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

 by (rule typedef_chfin [OF type_definition_Sprod below_Sprod_def])

 syntax (xsymbols)

   "**"		:: "[type, type] => type"	 ("(_ \<otimes>/ _)" [21,20] 20)

 syntax (HTML output)

   "**"		:: "[type, type] => type"	 ("(_ \<otimes>/ _)" [21,20] 20)

 lemma spair_lemma:

   "<strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a> \<in> Sprod"

 by (simp add: Sprod_def strictify_conv_if)

 subsection {* Definitions of constants *}

 definition

   sfst :: "('a ** 'b) \<rightarrow> 'a" where

   "sfst = (\<Lambda> p. cfst\<cdot>(Rep_Sprod p))"

 definition

   ssnd :: "('a ** 'b) \<rightarrow> 'b" where

   "ssnd = (\<Lambda> p. csnd\<cdot>(Rep_Sprod p))"

 definition

   spair :: "'a \<rightarrow> 'b \<rightarrow> ('a ** 'b)" where

   "spair = (\<Lambda> a b. Abs_Sprod

              <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>)"

 definition

   ssplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a ** 'b) \<rightarrow> 'c" where

   "ssplit = (\<Lambda> f. strictify\<cdot>(\<Lambda> p. f\<cdot>(sfst\<cdot>p)\<cdot>(ssnd\<cdot>p)))"

 syntax

   "@stuple" :: "['a, args] => 'a ** 'b"  ("(1'(:_,/ _:'))")

 translations

   "(:x, y, z:)" == "(:x, (:y, z:):)"

   "(:x, y:)"    == "CONST spair\<cdot>x\<cdot>y"

 translations

   "\<Lambda>(CONST spair\<cdot>x\<cdot>y). t" == "CONST ssplit\<cdot>(\<Lambda> x y. t)"

 subsection {* Case analysis *}

 lemma Rep_Sprod_spair:

   "Rep_Sprod (:a, b:) = <strictify\<cdot>(\<Lambda> b. a)\<cdot>b, strictify\<cdot>(\<Lambda> a. b)\<cdot>a>"

 unfolding spair_def

 by (simp add: cont_Abs_Sprod Abs_Sprod_inverse spair_lemma)

 lemmas Rep_Sprod_simps =

   Rep_Sprod_inject [symmetric] below_Sprod_def

   Rep_Sprod_strict Rep_Sprod_spair

 lemma Exh_Sprod:

   "z = \<bottom> \<or> (\<exists>a b. z = (:a, b:) \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"

 apply (insert Rep_Sprod [of z])

 apply (simp add: Rep_Sprod_simps eq_cprod)

 apply (simp add: Sprod_def)

 apply (erule disjE, simp)

 apply (simp add: strictify_conv_if)

 apply fast

 done

 lemma sprodE [cases type: **]:

   "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x y. \<lbrakk>p = (:x, y:); x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

 by (cut_tac z=p in Exh_Sprod, auto)

 lemma sprod_induct [induct type: **]:

   "\<lbrakk>P \<bottom>; \<And>x y. \<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> P (:x, y:)\<rbrakk> \<Longrightarrow> P x"

 by (cases x, simp_all)

 subsection {* Properties of @{term spair} *}

 lemma spair_strict1 [simp]: "(:\<bottom>, y:) = \<bottom>"

 by (simp add: Rep_Sprod_simps strictify_conv_if)

 lemma spair_strict2 [simp]: "(:x, \<bottom>:) = \<bottom>"

 by (simp add: Rep_Sprod_simps strictify_conv_if)

 lemma spair_strict_iff [simp]: "((:x, y:) = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"

 by (simp add: Rep_Sprod_simps strictify_conv_if)

 lemma spair_below_iff:

   "((:a, b:) \<sqsubseteq> (:c, d:)) = (a = \<bottom> \<or> b = \<bottom> \<or> (a \<sqsubseteq> c \<and> b \<sqsubseteq> d))"

 by (simp add: Rep_Sprod_simps strictify_conv_if)

 lemma spair_eq_iff:

   "((:a, b:) = (:c, d:)) =

     (a = c \<and> b = d \<or> (a = \<bottom> \<or> b = \<bottom>) \<and> (c = \<bottom> \<or> d = \<bottom>))"

 by (simp add: Rep_Sprod_simps strictify_conv_if)

 lemma spair_strict: "x = \<bottom> \<or> y = \<bottom> \<Longrightarrow> (:x, y:) = \<bottom>"

 by simp

 lemma spair_strict_rev: "(:x, y:) \<noteq> \<bottom> \<Longrightarrow> x \<noteq> \<bottom> \<and> y \<noteq> \<bottom>"

 by simp

 lemma spair_defined: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<noteq> \<bottom>"

 by simp

 lemma spair_defined_rev: "(:x, y:) = \<bottom> \<Longrightarrow> x = \<bottom> \<or> y = \<bottom>"

 by simp

 lemma spair_eq:

   "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ((:x, y:) = (:a, b:)) = (x = a \<and> y = b)"

 by (simp add: spair_eq_iff)

 lemma spair_inject:

   "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>; (:x, y:) = (:a, b:)\<rbrakk> \<Longrightarrow> x = a \<and> y = b"

 by (rule spair_eq [THEN iffD1])

 lemma inst_sprod_pcpo2: "UU = (:UU,UU:)"

 by simp

 subsection {* Properties of @{term sfst} and @{term ssnd} *}

 lemma sfst_strict [simp]: "sfst\<cdot>\<bottom> = \<bottom>"

 by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_strict)

 lemma ssnd_strict [simp]: "ssnd\<cdot>\<bottom> = \<bottom>"

 by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_strict)

 lemma sfst_spair [simp]: "y \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>(:x, y:) = x"

 by (simp add: sfst_def cont_Rep_Sprod Rep_Sprod_spair)

 lemma ssnd_spair [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>(:x, y:) = y"

 by (simp add: ssnd_def cont_Rep_Sprod Rep_Sprod_spair)

 lemma sfst_defined_iff [simp]: "(sfst\<cdot>p = \<bottom>) = (p = \<bottom>)"

 by (cases p, simp_all)

 lemma ssnd_defined_iff [simp]: "(ssnd\<cdot>p = \<bottom>) = (p = \<bottom>)"

 by (cases p, simp_all)

 lemma sfst_defined: "p \<noteq> \<bottom> \<Longrightarrow> sfst\<cdot>p \<noteq> \<bottom>"

 by simp

 lemma ssnd_defined: "p \<noteq> \<bottom> \<Longrightarrow> ssnd\<cdot>p \<noteq> \<bottom>"

 by simp

 lemma surjective_pairing_Sprod2: "(:sfst\<cdot>p, ssnd\<cdot>p:) = p"

 by (cases p, simp_all)

 lemma below_sprod: "x \<sqsubseteq> y = (sfst\<cdot>x \<sqsubseteq> sfst\<cdot>y \<and> ssnd\<cdot>x \<sqsubseteq> ssnd\<cdot>y)"

 apply (simp add: below_Sprod_def sfst_def ssnd_def cont_Rep_Sprod)

 apply (rule below_cprod)

 done

 lemma eq_sprod: "(x = y) = (sfst\<cdot>x = sfst\<cdot>y \<and> ssnd\<cdot>x = ssnd\<cdot>y)"

 by (auto simp add: po_eq_conv below_sprod)

 lemma spair_below:

   "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> (:x, y:) \<sqsubseteq> (:a, b:) = (x \<sqsubseteq> a \<and> y \<sqsubseteq> b)"

 apply (cases "a = \<bottom>", simp)

 apply (cases "b = \<bottom>", simp)

 apply (simp add: below_sprod)

 done

 lemma sfst_below_iff: "sfst\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:y, ssnd\<cdot>x:)"

 apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)

 apply (simp add: below_sprod)

 done

 lemma ssnd_below_iff: "ssnd\<cdot>x \<sqsubseteq> y = x \<sqsubseteq> (:sfst\<cdot>x, y:)"

 apply (cases "x = \<bottom>", simp, cases "y = \<bottom>", simp)

 apply (simp add: below_sprod)

 done

 subsection {* Compactness *}

 lemma compact_sfst: "compact x \<Longrightarrow> compact (sfst\<cdot>x)"

 by (rule compactI, simp add: sfst_below_iff)

 lemma compact_ssnd: "compact x \<Longrightarrow> compact (ssnd\<cdot>x)"

 by (rule compactI, simp add: ssnd_below_iff)

 lemma compact_spair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (:x, y:)"

 by (rule compact_Sprod, simp add: Rep_Sprod_spair strictify_conv_if)

 lemma compact_spair_iff:

   "compact (:x, y:) = (x = \<bottom> \<or> y = \<bottom> \<or> (compact x \<and> compact y))"

 apply (safe elim!: compact_spair)

 apply (drule compact_sfst, simp)

 apply (drule compact_ssnd, simp)

 apply simp

 apply simp

 done

 subsection {* Properties of @{term ssplit} *}

 lemma ssplit1 [simp]: "ssplit\<cdot>f\<cdot>\<bottom> = \<bottom>"

 by (simp add: ssplit_def)

 lemma ssplit2 [simp]: "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> ssplit\<cdot>f\<cdot>(:x, y:) = f\<cdot>x\<cdot>y"

 by (simp add: ssplit_def)

 lemma ssplit3 [simp]: "ssplit\<cdot>spair\<cdot>z = z"

 by (cases z, simp_all)

 subsection {* Strict product preserves flatness *}

 instance "**" :: (flat, flat) flat

 proof

   fix x y :: "'a \<otimes> 'b"

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

     apply (induct x, simp)

     apply (induct y, simp)

     apply (simp add: spair_below_iff flat_below_iff)

     done

 qed

 subsection {* Strict product is a bifinite domain *}

 instantiation "**" :: (bifinite, bifinite) bifinite

 begin

 definition

   approx_sprod_def:

     "approx = (\<lambda>n. \<Lambda>(:x, y:). (:approx n\<cdot>x, approx n\<cdot>y:))"

 instance proof

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

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

     unfolding approx_sprod_def by simp

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

     unfolding approx_sprod_def

     by (simp add: lub_distribs eta_cfun)

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

     unfolding approx_sprod_def

     by (simp add: ssplit_def strictify_conv_if)

   have "Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x} \<subseteq> {x. approx i\<cdot>x = x}"

     unfolding approx_sprod_def

     apply (clarify, case_tac x)

      apply (simp add: Rep_Sprod_strict)

     apply (simp add: Rep_Sprod_spair spair_eq_iff)

     done

   hence "finite (Rep_Sprod ` {x::'a \<otimes> 'b. approx i\<cdot>x = x})"

     using finite_fixes_approx by (rule finite_subset)

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

     by (rule finite_imageD, simp add: inj_on_def Rep_Sprod_inject)

 qed

 end

 lemma approx_spair [simp]:

   "approx i\<cdot>(:x, y:) = (:approx i\<cdot>x, approx i\<cdot>y:)"

 unfolding approx_sprod_def

 by (simp add: ssplit_def strictify_conv_if)

 end