# Theory Product_Cpo

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

section ‹The cpo of cartesian products›

theory Product_Cpo
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 proof
qed simp

end

instance unit :: pcpo
by intro_classes simp

subsection ‹Product type is a partial order›

instantiation prod :: (below, below) below
begin

definition
below_prod_def: "(op ⊑) ≡ 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"
unfolding below_prod_def by simp
next
fix x y :: "'a × 'b"
assume "x ⊑ y" "y ⊑ x" thus "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" thus "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"
unfolding below_prod_def by simp

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

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"
unfolding below_prod_def by simp

lemma snd_monofun: "x ⊑ y ⟹ snd x ⊑ snd y"
unfolding below_prod_def by simp

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 Y"
obtains A B
where "chain A" and "chain B" and "Y = (λi. (A i, B i))"
proof
from ‹chain Y› show "chain (λi. fst (Y i))" by (rule ch2ch_fst)
from ‹chain Y› 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)"
unfolding is_lub_def is_ub_def ball_simps below_prod_def by simp

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

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

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

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

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

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 ⊥ ⊥"
unfolding split_def by simp

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)
thus "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.›

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::'a × 'b) ⊑ y ⟷ x ⊑ (y, snd x)"
unfolding below_prod_def by simp

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

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