# Theory Sum_Cpo

theory Sum_Cpo
imports HOLCF
(*  Title:      HOL/HOLCF/Library/Sum_Cpo.thy
Author:     Brian Huffman
*)

section ‹The cpo of disjoint sums›

theory Sum_Cpo
imports HOLCF
begin

subsection ‹Ordering on sum type›

instantiation sum :: (below, below) below
begin

definition below_sum_def:
"x ⊑ y ≡ case x of
Inl a ⇒ (case y of Inl b ⇒ a ⊑ b | Inr b ⇒ False) |
Inr a ⇒ (case y of Inl b ⇒ False | Inr b ⇒ a ⊑ b)"

instance ..
end

lemma Inl_below_Inl [simp]: "Inl x ⊑ Inl y ⟷ x ⊑ y"
unfolding below_sum_def by simp

lemma Inr_below_Inr [simp]: "Inr x ⊑ Inr y ⟷ x ⊑ y"
unfolding below_sum_def by simp

lemma Inl_below_Inr [simp]: "¬ Inl x ⊑ Inr y"
unfolding below_sum_def by simp

lemma Inr_below_Inl [simp]: "¬ Inr x ⊑ Inl y"
unfolding below_sum_def by simp

lemma Inl_mono: "x ⊑ y ⟹ Inl x ⊑ Inl y"
by simp

lemma Inr_mono: "x ⊑ y ⟹ Inr x ⊑ Inr y"
by simp

lemma Inl_belowE: "⟦Inl a ⊑ x; ⋀b. ⟦x = Inl b; a ⊑ b⟧ ⟹ R⟧ ⟹ R"
by (cases x, simp_all)

lemma Inr_belowE: "⟦Inr a ⊑ x; ⋀b. ⟦x = Inr b; a ⊑ b⟧ ⟹ R⟧ ⟹ R"
by (cases x, simp_all)

lemmas sum_below_elims = Inl_belowE Inr_belowE

lemma sum_below_cases:
"⟦x ⊑ y;
⋀a b. ⟦x = Inl a; y = Inl b; a ⊑ b⟧ ⟹ R;
⋀a b. ⟦x = Inr a; y = Inr b; a ⊑ b⟧ ⟹ R⟧
⟹ R"
by (cases x, safe elim!: sum_below_elims, auto)

subsection ‹Sum type is a complete partial order›

instance sum :: (po, po) po
proof
fix x :: "'a + 'b"
show "x ⊑ x"
by (induct x, simp_all)
next
fix x y :: "'a + 'b"
assume "x ⊑ y" and "y ⊑ x" thus "x = y"
by (induct x, auto elim!: sum_below_elims intro: below_antisym)
next
fix x y z :: "'a + 'b"
assume "x ⊑ y" and "y ⊑ z" thus "x ⊑ z"
by (induct x, auto elim!: sum_below_elims intro: below_trans)
qed

lemma monofun_inv_Inl: "monofun (λp. THE a. p = Inl a)"
by (rule monofunI, erule sum_below_cases, simp_all)

lemma monofun_inv_Inr: "monofun (λp. THE b. p = Inr b)"
by (rule monofunI, erule sum_below_cases, simp_all)

lemma sum_chain_cases:
assumes Y: "chain Y"
assumes A: "⋀A. ⟦chain A; Y = (λi. Inl (A i))⟧ ⟹ R"
assumes B: "⋀B. ⟦chain B; Y = (λi. Inr (B i))⟧ ⟹ R"
shows "R"
apply (cases "Y 0")
apply (rule A)
apply (rule ch2ch_monofun [OF monofun_inv_Inl Y])
apply (rule ext)
apply (cut_tac j=i in chain_mono [OF Y le0], simp)
apply (erule Inl_belowE, simp)
apply (rule B)
apply (rule ch2ch_monofun [OF monofun_inv_Inr Y])
apply (rule ext)
apply (cut_tac j=i in chain_mono [OF Y le0], simp)
apply (erule Inr_belowE, simp)
done

lemma is_lub_Inl: "range S <<| x ⟹ range (λi. Inl (S i)) <<| Inl x"
apply (rule is_lubI)
apply (rule ub_rangeI)
apply (frule ub_rangeD [where i=arbitrary])
apply (erule Inl_belowE, simp)
apply (erule is_lubD2)
apply (rule ub_rangeI)
apply (drule ub_rangeD, simp)
done

lemma is_lub_Inr: "range S <<| x ⟹ range (λi. Inr (S i)) <<| Inr x"
apply (rule is_lubI)
apply (rule ub_rangeI)
apply (frule ub_rangeD [where i=arbitrary])
apply (erule Inr_belowE, simp)
apply (erule is_lubD2)
apply (rule ub_rangeI)
apply (drule ub_rangeD, simp)
done

instance sum :: (cpo, cpo) cpo
apply intro_classes
apply (erule sum_chain_cases, safe)
apply (rule exI)
apply (rule is_lub_Inl)
apply (erule cpo_lubI)
apply (rule exI)
apply (rule is_lub_Inr)
apply (erule cpo_lubI)
done

subsection ‹Continuity of \emph{Inl}, \emph{Inr}, and case function›

lemma cont_Inl: "cont Inl"
by (intro contI is_lub_Inl cpo_lubI)

lemma cont_Inr: "cont Inr"
by (intro contI is_lub_Inr cpo_lubI)

lemmas cont2cont_Inl [simp, cont2cont] = cont_compose [OF cont_Inl]
lemmas cont2cont_Inr [simp, cont2cont] = cont_compose [OF cont_Inr]

lemmas ch2ch_Inl [simp] = ch2ch_cont [OF cont_Inl]
lemmas ch2ch_Inr [simp] = ch2ch_cont [OF cont_Inr]

lemmas lub_Inl = cont2contlubE [OF cont_Inl, symmetric]
lemmas lub_Inr = cont2contlubE [OF cont_Inr, symmetric]

lemma cont_case_sum1:
assumes f: "⋀a. cont (λx. f x a)"
assumes g: "⋀b. cont (λx. g x b)"
shows "cont (λx. case y of Inl a ⇒ f x a | Inr b ⇒ g x b)"

lemma cont_case_sum2: "⟦cont f; cont g⟧ ⟹ cont (case_sum f g)"
apply (rule contI)
apply (erule sum_chain_cases)
apply (simp add: cont2contlubE [OF cont_Inl, symmetric] contE)
apply (simp add: cont2contlubE [OF cont_Inr, symmetric] contE)
done

lemma cont2cont_case_sum:
assumes f1: "⋀a. cont (λx. f x a)" and f2: "⋀x. cont (λa. f x a)"
assumes g1: "⋀b. cont (λx. g x b)" and g2: "⋀x. cont (λb. g x b)"
assumes h: "cont (λx. h x)"
shows "cont (λx. case h x of Inl a ⇒ f x a | Inr b ⇒ g x b)"
apply (rule cont_apply [OF h])
apply (rule cont_case_sum2 [OF f2 g2])
apply (rule cont_case_sum1 [OF f1 g1])
done

lemma cont2cont_case_sum' [simp, cont2cont]:
assumes f: "cont (λp. f (fst p) (snd p))"
assumes g: "cont (λp. g (fst p) (snd p))"
assumes h: "cont (λx. h x)"
shows "cont (λx. case h x of Inl a ⇒ f x a | Inr b ⇒ g x b)"
using assms by (simp add: cont2cont_case_sum prod_cont_iff)

text ‹Continuity of map function.›

lemma map_sum_eq: "map_sum f g = case_sum (λa. Inl (f a)) (λb. Inr (g b))"
by (rule ext, case_tac x, simp_all)

lemma cont2cont_map_sum [simp, cont2cont]:
assumes f: "cont (λ(x, y). f x y)"
assumes g: "cont (λ(x, y). g x y)"
assumes h: "cont (λx. h x)"
shows "cont (λx. map_sum (λy. f x y) (λy. g x y) (h x))"
using assms by (simp add: map_sum_eq prod_cont_iff)

subsection ‹Compactness and chain-finiteness›

lemma compact_Inl: "compact a ⟹ compact (Inl a)"
apply (rule compactI2)
apply (erule sum_chain_cases, safe)
apply (erule (2) compactD2)
done

lemma compact_Inr: "compact a ⟹ compact (Inr a)"
apply (rule compactI2)
apply (erule sum_chain_cases, safe)
apply (erule (2) compactD2)
done

lemma compact_Inl_rev: "compact (Inl a) ⟹ compact a"
unfolding compact_def
by (drule adm_subst [OF cont_Inl], simp)

lemma compact_Inr_rev: "compact (Inr a) ⟹ compact a"
unfolding compact_def
by (drule adm_subst [OF cont_Inr], simp)

lemma compact_Inl_iff [simp]: "compact (Inl a) = compact a"
by (safe elim!: compact_Inl compact_Inl_rev)

lemma compact_Inr_iff [simp]: "compact (Inr a) = compact a"
by (safe elim!: compact_Inr compact_Inr_rev)

instance sum :: (chfin, chfin) chfin
apply intro_classes
apply (erule compact_imp_max_in_chain)
apply (case_tac "⨆i. Y i", simp_all)
done

instance sum :: (discrete_cpo, discrete_cpo) discrete_cpo
by intro_classes (simp add: below_sum_def split: sum.split)

subsection ‹Using sum types with fixrec›

definition
"match_Inl = (Λ x k. case x of Inl a ⇒ k⋅a | Inr b ⇒ Fixrec.fail)"

definition
"match_Inr = (Λ x k. case x of Inl a ⇒ Fixrec.fail | Inr b ⇒ k⋅b)"

lemma match_Inl_simps [simp]:
"match_Inl⋅(Inl a)⋅k = k⋅a"
"match_Inl⋅(Inr b)⋅k = Fixrec.fail"
unfolding match_Inl_def by simp_all

lemma match_Inr_simps [simp]:
"match_Inr⋅(Inl a)⋅k = Fixrec.fail"
"match_Inr⋅(Inr b)⋅k = k⋅b"
unfolding match_Inr_def by simp_all

setup ‹
[ (@{const_name Inl}, @{const_name match_Inl}),
(@{const_name Inr}, @{const_name match_Inr}) ]
›

subsection ‹Disjoint sum is a predomain›

definition
"encode_sum_u =
(Λ(up⋅x). case x of Inl a ⇒ sinl⋅(up⋅a) | Inr b ⇒ sinr⋅(up⋅b))"

definition
"decode_sum_u = sscase⋅(Λ(up⋅a). up⋅(Inl a))⋅(Λ(up⋅b). up⋅(Inr b))"

lemma decode_encode_sum_u [simp]: "decode_sum_u⋅(encode_sum_u⋅x) = x"
unfolding decode_sum_u_def encode_sum_u_def
by (case_tac x, simp, rename_tac y, case_tac y, simp_all)

lemma encode_decode_sum_u [simp]: "encode_sum_u⋅(decode_sum_u⋅x) = x"
unfolding decode_sum_u_def encode_sum_u_def
apply (case_tac x, simp)
apply (rename_tac a, case_tac a, simp, simp)
apply (rename_tac b, case_tac b, simp, simp)
done

text ‹A deflation combinator for making unpointed types›

definition udefl :: "udom defl → udom u defl"
where "udefl = defl_fun1 (strictify⋅up) (fup⋅ID) ID"

lemma ep_pair_strictify_up:
"ep_pair (strictify⋅up) (fup⋅ID)"
apply (rule ep_pair.intro)
apply (case_tac y, simp, simp add: strictify_conv_if)
done

lemma cast_udefl:
"cast⋅(udefl⋅t) = strictify⋅up oo cast⋅t oo fup⋅ID"
unfolding udefl_def by (simp add: cast_defl_fun1 ep_pair_strictify_up)

definition sum_liftdefl :: "udom u defl → udom u defl → udom u defl"
where "sum_liftdefl = (Λ a b. udefl⋅(ssum_defl⋅(u_liftdefl⋅a)⋅(u_liftdefl⋅b)))"

lemma u_emb_bottom: "u_emb⋅⊥ = ⊥"
by (rule pcpo_ep_pair.e_strict [unfolded pcpo_ep_pair_def, OF ep_pair_u])

(*
definition sum_liftdefl :: "udom u defl → udom u defl → udom u defl"
where "sum_liftdefl = defl_fun2 (u_map⋅emb oo strictify⋅up)
(fup⋅ID oo u_map⋅prj) ssum_map"
*)

instantiation sum :: (predomain, predomain) predomain
begin

definition
"liftemb = (strictify⋅up oo ssum_emb) oo
(ssum_map⋅(u_emb oo liftemb)⋅(u_emb oo liftemb) oo encode_sum_u)"

definition
"liftprj = (decode_sum_u oo ssum_map⋅(liftprj oo u_prj)⋅(liftprj oo u_prj))
oo (ssum_prj oo fup⋅ID)"

definition
"liftdefl (t::('a + 'b) itself) = sum_liftdefl⋅LIFTDEFL('a)⋅LIFTDEFL('b)"

instance proof
show "ep_pair liftemb (liftprj :: udom u → ('a + 'b) u)"
unfolding liftemb_sum_def liftprj_sum_def
by (intro ep_pair_comp ep_pair_ssum_map ep_pair_u predomain_ep
show "cast⋅LIFTDEFL('a + 'b) = liftemb oo (liftprj :: udom u → ('a + 'b) u)"
unfolding liftemb_sum_def liftprj_sum_def liftdefl_sum_def
by (simp add: sum_liftdefl_def cast_udefl cast_ssum_defl cast_u_liftdefl
cast_liftdefl cfcomp1 ssum_map_map u_emb_bottom)
qed

end

subsection ‹Configuring domain package to work with sum type›

lemma liftdefl_sum [domain_defl_simps]:
"LIFTDEFL('a::predomain + 'b::predomain) =
sum_liftdefl⋅LIFTDEFL('a)⋅LIFTDEFL('b)"
by (rule liftdefl_sum_def)

abbreviation map_sum'
where "map_sum' f g ≡ Abs_cfun (map_sum (Rep_cfun f) (Rep_cfun g))"

lemma map_sum_ID [domain_map_ID]: "map_sum' ID ID = ID"
by (simp add: ID_def cfun_eq_iff map_sum.identity id_def)

lemma deflation_map_sum [domain_deflation]:
"⟦deflation d1; deflation d2⟧ ⟹ deflation (map_sum' d1 d2)"
apply standard
apply (induct_tac x, simp_all add: deflation.idem)
apply (induct_tac x, simp_all add: deflation.below)
done

lemma encode_sum_u_map_sum:
"encode_sum_u⋅(u_map⋅(map_sum' f g)⋅(decode_sum_u⋅x))
= ssum_map⋅(u_map⋅f)⋅(u_map⋅g)⋅x"
apply (induct x, simp add: decode_sum_u_def encode_sum_u_def)
apply (case_tac x, simp, simp add: decode_sum_u_def encode_sum_u_def)
apply (case_tac y, simp, simp add: decode_sum_u_def encode_sum_u_def)
done

lemma isodefl_sum [domain_isodefl]:
fixes d :: "'a::predomain → 'a"
assumes "isodefl' d1 t1" and "isodefl' d2 t2"
shows "isodefl' (map_sum' d1 d2) (sum_liftdefl⋅t1⋅t2)"
using assms unfolding isodefl'_def liftemb_sum_def liftprj_sum_def
apply (simp add: sum_liftdefl_def cast_udefl cast_ssum_defl cast_u_liftdefl)