Theory Product_Cpo

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theory Product_Cpo
imports Adm

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


header {* The cpo of cartesian products *}

theory Product_Cpo
imports Adm
begin


default_sort cpo

subsection {* Unit type is a pcpo *}

instantiation unit :: below
begin


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


instance ..
end

instance unit :: discrete_cpo
by intro_classes simp

instance unit :: finite_po ..

instance unit :: pcpo
by intro_classes simp


subsection {* Product type is a partial order *}

instantiation "*" :: (below, below) below
begin


definition
below_prod_def: "(op \<sqsubseteq>) ≡ λp1 p2. (fst p1 \<sqsubseteq> fst p2 ∧ snd p1 \<sqsubseteq> snd p2)"


instance ..
end

instance "*" :: (po, po) po
proof
fix x :: "'a × 'b"
show "x \<sqsubseteq> x"
unfolding below_prod_def by simp
next
fix x y :: "'a × 'b"
assume "x \<sqsubseteq> y" "y \<sqsubseteq> x" thus "x = y"
unfolding below_prod_def Pair_fst_snd_eq
by (fast intro: below_antisym)
next
fix x y z :: "'a × 'b"
assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> 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 \<sqsubseteq> fst q; snd p \<sqsubseteq> snd q|] ==> p \<sqsubseteq> q"
unfolding below_prod_def by simp

lemma Pair_below_iff [simp]: "(a, b) \<sqsubseteq> (c, d) <-> a \<sqsubseteq> c ∧ b \<sqsubseteq> d"
unfolding below_prod_def by simp

text {* Pair @{text "(_,_)"} 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 \<sqsubseteq> x2; y1 \<sqsubseteq> y2|] ==> (x1, y1) \<sqsubseteq> (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 \<sqsubseteq> y ==> fst x \<sqsubseteq> fst y"
unfolding below_prod_def by simp

lemma snd_monofun: "x \<sqsubseteq> y ==> snd x \<sqsubseteq> 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)"

apply (rule is_lubI [OF ub_rangeI])
apply (simp add: is_ub_lub)
apply (frule ub2ub_monofun [OF monofun_fst])
apply (drule ub2ub_monofun [OF monofun_snd])
apply (simp add: below_prod_def is_lub_lub)
done

lemma thelub_Pair:
"[|chain (A::nat => 'a::cpo); chain (B::nat => 'b::cpo)|]
==> (\<Squnion>i. (A i, B i)) = (\<Squnion>i. A i, \<Squnion>i. B i)"

by (fast intro: thelubI is_lub_Pair elim: thelubE)

lemma lub_cprod:
fixes S :: "nat => ('a::cpo × 'b::cpo)"
assumes S: "chain S"
shows "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"

proof -
from `chain S` have "chain (λi. fst (S i))"
by (rule ch2ch_fst)
hence 1: "range (λi. fst (S i)) <<| (\<Squnion>i. fst (S i))"
by (rule cpo_lubI)
from `chain S` have "chain (λi. snd (S i))"
by (rule ch2ch_snd)
hence 2: "range (λi. snd (S i)) <<| (\<Squnion>i. snd (S i))"
by (rule cpo_lubI)
show "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
using is_lub_Pair [OF 1 2] by simp
qed

lemma thelub_cprod:
"chain (S::nat => 'a::cpo × 'b::cpo)
==> (\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"

by (rule lub_cprod [THEN thelubI])

instance "*" :: (cpo, cpo) cpo
proof
fix S :: "nat => ('a × 'b)"
assume "chain S"
hence "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
by (rule lub_cprod)
thus "∃x. range S <<| x" ..
qed

instance "*" :: (finite_po, finite_po) finite_po ..

instance "*" :: (discrete_cpo, discrete_cpo) discrete_cpo
proof
fix x y :: "'a × 'b"
show "x \<sqsubseteq> y <-> x = y"
unfolding below_prod_def Pair_fst_snd_eq
by simp
qed

subsection {* Product type is pointed *}

lemma minimal_cprod: "(⊥, ⊥) \<sqsubseteq> p"
by (simp add: below_prod_def)

instance "*" :: (pcpo, pcpo) pcpo
by intro_classes (fast intro: minimal_cprod)

lemma inst_cprod_pcpo: "⊥ = (⊥, ⊥)"
by (rule minimal_cprod [THEN UU_I, symmetric])

lemma Pair_defined_iff [simp]: "(x, y) = ⊥ <-> x = ⊥ ∧ y = ⊥"
unfolding inst_cprod_pcpo by simp

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

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

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

lemma split_strict [simp]: "split 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 lub_const)
done

lemma cont_pair2: "cont (λy. (x, y))"
apply (rule contI)
apply (rule is_lub_Pair)
apply (rule lub_const)
apply (erule cpo_lubI)
done

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

lemma cont_snd: "cont snd"
apply (rule contI)
apply (simp add: thelub_cprod)
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_split:
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. split (λa b. f x a b) (g x))"

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 cont_fst_snd_D1:
"cont (λp. f (fst p) (snd p)) ==> cont (λx. f x y)"

by (drule cont_compose [OF _ cont_pair1], simp)

lemma cont_fst_snd_D2:
"cont (λp. f (fst p) (snd p)) ==> cont (λy. f x y)"

by (drule cont_compose [OF _ cont_pair2], simp)

lemma cont2cont_split' [simp, cont2cont]:
assumes f: "cont (λp. f (fst p) (fst (snd p)) (snd (snd p)))"
assumes g: "cont (λx. g x)"
shows "cont (λx. split (f x) (g x))"

proof -
note f1 = f [THEN cont_fst_snd_D1]
note f2 = f [THEN cont_fst_snd_D2, THEN cont_fst_snd_D1]
note f3 = f [THEN cont_fst_snd_D2, THEN cont_fst_snd_D2]
show ?thesis
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
qed

subsection {* Compactness and chain-finiteness *}

lemma fst_below_iff: "fst (x::'a × 'b) \<sqsubseteq> y <-> x \<sqsubseteq> (y, snd x)"
unfolding below_prod_def by simp

lemma snd_below_iff: "snd (x::'a × 'b) \<sqsubseteq> y <-> x \<sqsubseteq> (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 "*" :: (chfin, chfin) chfin
apply intro_classes
apply (erule compact_imp_max_in_chain)
apply (case_tac "\<Squnion>i. Y i", simp)
done

end