Theory Product_Cpo

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

section ‹The cpo of cartesian products›

theory Product_Cpo
  imports Adm
begin

default_sort cpo


subsection ‹Unit type is a pcpo›

instantiation unit :: discrete_cpo
begin

definition below_unit_def [simp]: "x ⊑ (y::unit) ⟷ True"

instance
  by standard simp

end

instance unit :: pcpo
  by standard simp


subsection ‹Product type is a partial order›

instantiation prod :: (below, below) below
begin

definition below_prod_def: "(⊑) ≡ λp1 p2. (fst p1 ⊑ fst p2 ∧ snd p1 ⊑ snd p2)"

instance ..

end

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


subsection ‹Monotonicity of \emph{Pair}, \emph{fst}, \emph{snd}›

lemma prod_belowI: "fst p ⊑ fst q ⟹ snd p ⊑ snd q ⟹ p ⊑ q"
  by (simp add: below_prod_def)

lemma Pair_below_iff [simp]: "(a, b) ⊑ (c, d) ⟷ a ⊑ c ∧ b ⊑ d"
  by (simp add: below_prod_def)

text ‹Pair ‹(_,_)›  is monotone in both arguments›

lemma monofun_pair1: "monofun (λx. (x, y))"
  by (simp add: monofun_def)

lemma monofun_pair2: "monofun (λy. (x, y))"
  by (simp add: monofun_def)

lemma monofun_pair: "x1 ⊑ x2 ⟹ y1 ⊑ y2 ⟹ (x1, y1) ⊑ (x2, y2)"
  by simp

lemma ch2ch_Pair [simp]: "chain X ⟹ chain Y ⟹ chain (λi. (X i, Y i))"
  by (rule chainI, simp add: chainE)

text ‹\<^term>‹fst› and \<^term>‹snd› are monotone›

lemma fst_monofun: "x ⊑ y ⟹ fst x ⊑ fst y"
  by (simp add: below_prod_def)

lemma snd_monofun: "x ⊑ y ⟹ snd x ⊑ 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 = (λi. (A i, B i))"
proof
  from chain show "chain (λi. fst (Y i))"
    by (rule ch2ch_fst)
  from chain show "chain (λi. snd (Y i))"
    by (rule ch2ch_snd)
  show "Y = (λi. (fst (Y i), snd (Y i)))"
    by simp
qed


subsection ‹Product type is a cpo›

lemma is_lub_Pair: "range A <<| x ⟹ range B <<| y ⟹ range (λi. (A i, B i)) <<| (x, y)"
  by (simp add: is_lub_def is_ub_def below_prod_def)

lemma lub_Pair: "chain A ⟹ chain B ⟹ (⨆i. (A i, B i)) = (⨆i. A i, ⨆i. B i)"
  for A :: "nat ⇒ 'a::cpo" and B :: "nat ⇒ 'b::cpo"
  by (fast intro: lub_eqI is_lub_Pair elim: thelubE)

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

lemma lub_prod: "chain S ⟹ (⨆i. S i) = (⨆i. fst (S i), ⨆i. snd (S i))"
  for S :: "nat ⇒ 'a::cpo × 'b::cpo"
  by (rule is_lub_prod [THEN lub_eqI])

instance prod :: (cpo, cpo) cpo
proof
  fix S :: "nat ⇒ ('a × 'b)"
  assume "chain S"
  then have "range S <<| (⨆i. fst (S i), ⨆i. snd (S i))"
    by (rule is_lub_prod)
  then show "∃x. range S <<| x" ..
qed

instance prod :: (discrete_cpo, discrete_cpo) discrete_cpo
proof
  fix x y :: "'a × 'b"
  show "x ⊑ y ⟷ x = y"
    by (simp add: below_prod_def prod_eq_iff)
qed


subsection ‹Product type is pointed›

lemma minimal_prod: "(⊥, ⊥) ⊑ p"
  by (simp add: below_prod_def)

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

lemma inst_prod_pcpo: "⊥ = (⊥, ⊥)"
  by (rule minimal_prod [THEN bottomI, symmetric])

lemma Pair_bottom_iff [simp]: "(x, y) = ⊥ ⟷ x = ⊥ ∧ y = ⊥"
  by (simp add: inst_prod_pcpo)

lemma fst_strict [simp]: "fst ⊥ = ⊥"
  unfolding inst_prod_pcpo by (rule fst_conv)

lemma snd_strict [simp]: "snd ⊥ = ⊥"
  unfolding inst_prod_pcpo by (rule snd_conv)

lemma Pair_strict [simp]: "(⊥, ⊥) = ⊥"
  by simp

lemma split_strict [simp]: "case_prod f ⊥ = f ⊥ ⊥"
  by (simp add: split_def)


subsection ‹Continuity of \emph{Pair}, \emph{fst}, \emph{snd}›

lemma cont_pair1: "cont (λx. (x, y))"
  apply (rule contI)
  apply (rule is_lub_Pair)
   apply (erule cpo_lubI)
  apply (rule is_lub_const)
  done

lemma cont_pair2: "cont (λy. (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 (λx. f x)"
  assumes g: "cont (λx. g x)"
  shows "cont (λx. (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: "⋀a b. cont (λx. f x a b)"
  assumes f2: "⋀x b. cont (λa. f x a b)"
  assumes f3: "⋀x a. cont (λb. f x a b)"
  assumes g: "cont (λx. g x)"
  shows "cont (λx. case g x of (a, b) ⇒ 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: "⋀y. cont (λx. f (x, y))"
  assumes f2: "⋀x. cont (λy. f (x, y))"
  shows "cont f"
proof -
  have "cont (λ(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 ⟷ (∀y. cont (λx. f (x, y))) ∧ (∀x. cont (λy. 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 (λp. f (fst p) (fst (snd p)) (snd (snd p)))"
  assumes g: "cont (λx. g x)"
  shows "cont (λx. case_prod (f x) (g x))"
  using assms by (simp add: cont2cont_case_prod prod_cont_iff)

text ‹The simple version (due to Joachim Breitner) is needed if
  either element type of the pair is not a cpo.›

lemma cont2cont_split_simple [simp, cont2cont]:
  assumes "⋀a b. cont (λx. f x a b)"
  shows "cont (λx. case p of (a, b) ⇒ f x a b)"
  using assms by (cases p) auto

text ‹Admissibility of predicates on product types.›

lemma adm_case_prod [simp]:
  assumes "adm (λx. P x (fst (f x)) (snd (f x)))"
  shows "adm (λx. case f x of (a, b) ⇒ P x a b)"
  unfolding case_prod_beta using assms .


subsection ‹Compactness and chain-finiteness›

lemma fst_below_iff: "fst x ⊑ y ⟷ x ⊑ (y, snd x)"
  for x :: "'a × 'b"
  by (simp add: below_prod_def)

lemma snd_below_iff: "snd x ⊑ y ⟷ x ⊑ (fst x, y)"
  for x :: "'a × 'b"
  by (simp add: below_prod_def)

lemma compact_fst: "compact x ⟹ compact (fst x)"
  by (rule compactI) (simp add: fst_below_iff)

lemma compact_snd: "compact x ⟹ compact (snd x)"
  by (rule compactI) (simp add: snd_below_iff)

lemma compact_Pair: "compact x ⟹ compact y ⟹ compact (x, y)"
  by (rule compactI) (simp add: below_prod_def)

lemma compact_Pair_iff [simp]: "compact (x, y) ⟷ compact x ∧ 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 "⨆i. Y i", simp)
  done

end