src/HOL/HOLCF/Product_Cpo.thy

Another Nominal example

(*  Title:      HOL/HOLCF/Product_Cpo.thy
    Author:     Franz Regensburger
*)

section \<open>The cpo of cartesian products\<close>

theory Product_Cpo
  imports Adm
begin

default_sort cpo


subsection \<open>Unit type is a pcpo\<close>

instantiation unit :: discrete_cpo
begin

definition below_unit_def [simp]: "x \<sqsubseteq> (y::unit) \<longleftrightarrow> True"

instance
  by standard simp

end

instance unit :: pcpo
  by standard simp


subsection \<open>Product type is a partial order\<close>

instantiation prod :: (below, below) below
begin

definition below_prod_def: "(\<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"

instance ..

end

instance prod :: (po, po) po
proof
  fix x :: "'a \<times> 'b"
  show "x \<sqsubseteq> x"
    by (simp add: below_prod_def)
next
  fix x y :: "'a \<times> 'b"
  assume "x \<sqsubseteq> y" "y \<sqsubseteq> x"
  then show "x = y"
    unfolding below_prod_def prod_eq_iff
    by (fast intro: below_antisym)
next
  fix x y z :: "'a \<times> 'b"
  assume "x \<sqsubseteq> y" "y \<sqsubseteq> z"
  then show "x \<sqsubseteq> z"
    unfolding below_prod_def
    by (fast intro: below_trans)
qed


subsection \<open>Monotonicity of \emph{Pair}, \emph{fst}, \emph{snd}\<close>

lemma prod_belowI: "fst p \<sqsubseteq> fst q \<Longrightarrow> snd p \<sqsubseteq> snd q \<Longrightarrow> p \<sqsubseteq> q"
  by (simp add: below_prod_def)

lemma Pair_below_iff [simp]: "(a, b) \<sqsubseteq> (c, d) \<longleftrightarrow> a \<sqsubseteq> c \<and> b \<sqsubseteq> d"
  by (simp add: below_prod_def)

text \<open>Pair \<open>(_,_)\<close>  is monotone in both arguments\<close>

lemma monofun_pair1: "monofun (\<lambda>x. (x, y))"
  by (simp add: monofun_def)

lemma monofun_pair2: "monofun (\<lambda>y. (x, y))"
  by (simp add: monofun_def)

lemma monofun_pair: "x1 \<sqsubseteq> x2 \<Longrightarrow> y1 \<sqsubseteq> y2 \<Longrightarrow> (x1, y1) \<sqsubseteq> (x2, y2)"
  by simp

lemma ch2ch_Pair [simp]: "chain X \<Longrightarrow> chain Y \<Longrightarrow> chain (\<lambda>i. (X i, Y i))"
  by (rule chainI, simp add: chainE)

text \<open>\<^term>\<open>fst\<close> and \<^term>\<open>snd\<close> are monotone\<close>

lemma fst_monofun: "x \<sqsubseteq> y \<Longrightarrow> fst x \<sqsubseteq> fst y"
  by (simp add: below_prod_def)

lemma snd_monofun: "x \<sqsubseteq> y \<Longrightarrow> snd x \<sqsubseteq> snd y"
  by (simp add: below_prod_def)

lemma monofun_fst: "monofun fst"
  by (simp add: monofun_def below_prod_def)

lemma monofun_snd: "monofun snd"
  by (simp add: monofun_def below_prod_def)

lemmas ch2ch_fst [simp] = ch2ch_monofun [OF monofun_fst]

lemmas ch2ch_snd [simp] = ch2ch_monofun [OF monofun_snd]

lemma prod_chain_cases:
  assumes chain: "chain Y"
  obtains A B
  where "chain A" and "chain B" and "Y = (\<lambda>i. (A i, B i))"
proof
  from chain show "chain (\<lambda>i. fst (Y i))"
    by (rule ch2ch_fst)
  from chain show "chain (\<lambda>i. snd (Y i))"
    by (rule ch2ch_snd)
  show "Y = (\<lambda>i. (fst (Y i), snd (Y i)))"
    by simp
qed


subsection \<open>Product type is a cpo\<close>

lemma is_lub_Pair: "range A <<| x \<Longrightarrow> range B <<| y \<Longrightarrow> range (\<lambda>i. (A i, B i)) <<| (x, y)"
  by (simp add: is_lub_def is_ub_def below_prod_def)

lemma lub_Pair: "chain A \<Longrightarrow> chain B \<Longrightarrow> (\<Squnion>i. (A i, B i)) = (\<Squnion>i. A i, \<Squnion>i. B i)"
  for A :: "nat \<Rightarrow> 'a::cpo" and B :: "nat \<Rightarrow> 'b::cpo"
  by (fast intro: lub_eqI is_lub_Pair elim: thelubE)

lemma is_lub_prod:
  fixes S :: "nat \<Rightarrow> ('a::cpo \<times> 'b::cpo)"
  assumes "chain S"
  shows "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
  using assms by (auto elim: prod_chain_cases simp: is_lub_Pair cpo_lubI)

lemma lub_prod: "chain S \<Longrightarrow> (\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
  for S :: "nat \<Rightarrow> 'a::cpo \<times> 'b::cpo"
  by (rule is_lub_prod [THEN lub_eqI])

instance prod :: (cpo, cpo) cpo
proof
  fix S :: "nat \<Rightarrow> ('a \<times> 'b)"
  assume "chain S"
  then have "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
    by (rule is_lub_prod)
  then show "\<exists>x. range S <<| x" ..
qed

instance prod :: (discrete_cpo, discrete_cpo) discrete_cpo
proof
  fix x y :: "'a \<times> 'b"
  show "x \<sqsubseteq> y \<longleftrightarrow> x = y"
    by (simp add: below_prod_def prod_eq_iff)
qed


subsection \<open>Product type is pointed\<close>

lemma minimal_prod: "(\<bottom>, \<bottom>) \<sqsubseteq> p"
  by (simp add: below_prod_def)

instance prod :: (pcpo, pcpo) pcpo
  by intro_classes (fast intro: minimal_prod)

lemma inst_prod_pcpo: "\<bottom> = (\<bottom>, \<bottom>)"
  by (rule minimal_prod [THEN bottomI, symmetric])

lemma Pair_bottom_iff [simp]: "(x, y) = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
  by (simp add: inst_prod_pcpo)

lemma fst_strict [simp]: "fst \<bottom> = \<bottom>"
  unfolding inst_prod_pcpo by (rule fst_conv)

lemma snd_strict [simp]: "snd \<bottom> = \<bottom>"
  unfolding inst_prod_pcpo by (rule snd_conv)

lemma Pair_strict [simp]: "(\<bottom>, \<bottom>) = \<bottom>"
  by simp

lemma split_strict [simp]: "case_prod f \<bottom> = f \<bottom> \<bottom>"
  by (simp add: split_def)


subsection \<open>Continuity of \emph{Pair}, \emph{fst}, \emph{snd}\<close>

lemma cont_pair1: "cont (\<lambda>x. (x, y))"
  apply (rule contI)
  apply (rule is_lub_Pair)
   apply (erule cpo_lubI)
  apply (rule is_lub_const)
  done

lemma cont_pair2: "cont (\<lambda>y. (x, y))"
  apply (rule contI)
  apply (rule is_lub_Pair)
   apply (rule is_lub_const)
  apply (erule cpo_lubI)
  done

lemma cont_fst: "cont fst"
  apply (rule contI)
  apply (simp add: lub_prod)
  apply (erule cpo_lubI [OF ch2ch_fst])
  done

lemma cont_snd: "cont snd"
  apply (rule contI)
  apply (simp add: lub_prod)
  apply (erule cpo_lubI [OF ch2ch_snd])
  done

lemma cont2cont_Pair [simp, cont2cont]:
  assumes f: "cont (\<lambda>x. f x)"
  assumes g: "cont (\<lambda>x. g x)"
  shows "cont (\<lambda>x. (f x, g x))"
  apply (rule cont_apply [OF f cont_pair1])
  apply (rule cont_apply [OF g cont_pair2])
  apply (rule cont_const)
  done

lemmas cont2cont_fst [simp, cont2cont] = cont_compose [OF cont_fst]

lemmas cont2cont_snd [simp, cont2cont] = cont_compose [OF cont_snd]

lemma cont2cont_case_prod:
  assumes f1: "\<And>a b. cont (\<lambda>x. f x a b)"
  assumes f2: "\<And>x b. cont (\<lambda>a. f x a b)"
  assumes f3: "\<And>x a. cont (\<lambda>b. f x a b)"
  assumes g: "cont (\<lambda>x. g x)"
  shows "cont (\<lambda>x. case g x of (a, b) \<Rightarrow> f x a b)"
  unfolding split_def
  apply (rule cont_apply [OF g])
   apply (rule cont_apply [OF cont_fst f2])
   apply (rule cont_apply [OF cont_snd f3])
   apply (rule cont_const)
  apply (rule f1)
  done

lemma prod_contI:
  assumes f1: "\<And>y. cont (\<lambda>x. f (x, y))"
  assumes f2: "\<And>x. cont (\<lambda>y. f (x, y))"
  shows "cont f"
proof -
  have "cont (\<lambda>(x, y). f (x, y))"
    by (intro cont2cont_case_prod f1 f2 cont2cont)
  then show "cont f"
    by (simp only: case_prod_eta)
qed

lemma prod_cont_iff: "cont f \<longleftrightarrow> (\<forall>y. cont (\<lambda>x. f (x, y))) \<and> (\<forall>x. cont (\<lambda>y. f (x, y)))"
  apply safe
    apply (erule cont_compose [OF _ cont_pair1])
   apply (erule cont_compose [OF _ cont_pair2])
  apply (simp only: prod_contI)
  done

lemma cont2cont_case_prod' [simp, cont2cont]:
  assumes f: "cont (\<lambda>p. f (fst p) (fst (snd p)) (snd (snd p)))"
  assumes g: "cont (\<lambda>x. g x)"
  shows "cont (\<lambda>x. case_prod (f x) (g x))"
  using assms by (simp add: cont2cont_case_prod prod_cont_iff)

text \<open>The simple version (due to Joachim Breitner) is needed if
  either element type of the pair is not a cpo.\<close>

lemma cont2cont_split_simple [simp, cont2cont]:
  assumes "\<And>a b. cont (\<lambda>x. f x a b)"
  shows "cont (\<lambda>x. case p of (a, b) \<Rightarrow> f x a b)"
  using assms by (cases p) auto

text \<open>Admissibility of predicates on product types.\<close>

lemma adm_case_prod [simp]:
  assumes "adm (\<lambda>x. P x (fst (f x)) (snd (f x)))"
  shows "adm (\<lambda>x. case f x of (a, b) \<Rightarrow> P x a b)"
  unfolding case_prod_beta using assms .


subsection \<open>Compactness and chain-finiteness\<close>

lemma fst_below_iff: "fst x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> (y, snd x)"
  for x :: "'a \<times> 'b"
  by (simp add: below_prod_def)

lemma snd_below_iff: "snd x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> (fst x, y)"
  for x :: "'a \<times> 'b"
  by (simp add: below_prod_def)

lemma compact_fst: "compact x \<Longrightarrow> compact (fst x)"
  by (rule compactI) (simp add: fst_below_iff)

lemma compact_snd: "compact x \<Longrightarrow> compact (snd x)"
  by (rule compactI) (simp add: snd_below_iff)

lemma compact_Pair: "compact x \<Longrightarrow> compact y \<Longrightarrow> compact (x, y)"
  by (rule compactI) (simp add: below_prod_def)

lemma compact_Pair_iff [simp]: "compact (x, y) \<longleftrightarrow> compact x \<and> compact y"
  apply (safe intro!: compact_Pair)
   apply (drule compact_fst, simp)
  apply (drule compact_snd, simp)
  done

instance prod :: (chfin, chfin) chfin
  apply intro_classes
  apply (erule compact_imp_max_in_chain)
  apply (case_tac "\<Squnion>i. Y i", simp)
  done

end