# Theory Domain

theory Domain
imports Representable Domain_Aux
```(*  Title:      HOL/HOLCF/Domain.thy
Author:     Brian Huffman
*)

section ‹Domain package›

theory Domain
imports Representable Domain_Aux
keywords
"lazy" "unsafe" and
"domaindef" "domain_isomorphism" "domain" :: thy_decl
begin

default_sort "domain"

subsection ‹Representations of types›

lemma emb_prj: "emb⋅((prj⋅x)::'a) = cast⋅DEFL('a)⋅x"

lemma emb_prj_emb:
fixes x :: "'a"
assumes "DEFL('a) ⊑ DEFL('b)"
shows "emb⋅(prj⋅(emb⋅x) :: 'b) = emb⋅x"
unfolding emb_prj
apply (rule cast.belowD)
apply (rule monofun_cfun_arg [OF assms])
done

lemma prj_emb_prj:
assumes "DEFL('a) ⊑ DEFL('b)"
shows "prj⋅(emb⋅(prj⋅x :: 'b)) = (prj⋅x :: 'a)"
apply (rule emb_eq_iff [THEN iffD1])
apply (simp only: emb_prj)
apply (rule deflation_below_comp1)
apply (rule deflation_cast)
apply (rule deflation_cast)
apply (rule monofun_cfun_arg [OF assms])
done

text ‹Isomorphism lemmas used internally by the domain package:›

lemma domain_abs_iso:
fixes abs and rep
assumes DEFL: "DEFL('b) = DEFL('a)"
assumes abs_def: "(abs :: 'a → 'b) ≡ prj oo emb"
assumes rep_def: "(rep :: 'b → 'a) ≡ prj oo emb"
shows "rep⋅(abs⋅x) = x"
unfolding abs_def rep_def

lemma domain_rep_iso:
fixes abs and rep
assumes DEFL: "DEFL('b) = DEFL('a)"
assumes abs_def: "(abs :: 'a → 'b) ≡ prj oo emb"
assumes rep_def: "(rep :: 'b → 'a) ≡ prj oo emb"
shows "abs⋅(rep⋅x) = x"
unfolding abs_def rep_def

subsection ‹Deflations as sets›

definition defl_set :: "'a::bifinite defl ⇒ 'a set"
where "defl_set A = {x. cast⋅A⋅x = x}"

unfolding defl_set_def by simp

lemma defl_set_bottom: "⊥ ∈ defl_set A"
unfolding defl_set_def by simp

lemma defl_set_cast [simp]: "cast⋅A⋅x ∈ defl_set A"
unfolding defl_set_def by simp

lemma defl_set_subset_iff: "defl_set A ⊆ defl_set B ⟷ A ⊑ B"
apply (simp add: defl_set_def subset_eq cast_below_cast [symmetric])
apply (auto simp add: cast.belowI cast.belowD)
done

subsection ‹Proving a subtype is representable›

text ‹Temporarily relax type constraints.›

setup ‹
[ (@{const_name defl}, SOME @{typ "'a::pcpo itself ⇒ udom defl"})
, (@{const_name emb}, SOME @{typ "'a::pcpo → udom"})
, (@{const_name prj}, SOME @{typ "udom → 'a::pcpo"})
, (@{const_name liftdefl}, SOME @{typ "'a::pcpo itself ⇒ udom u defl"})
, (@{const_name liftemb}, SOME @{typ "'a::pcpo u → udom u"})
, (@{const_name liftprj}, SOME @{typ "udom u → 'a::pcpo u"}) ]
›

lemma typedef_domain_class:
fixes Rep :: "'a::pcpo ⇒ udom"
fixes Abs :: "udom ⇒ 'a::pcpo"
fixes t :: "udom defl"
assumes type: "type_definition Rep Abs (defl_set t)"
assumes below: "op ⊑ ≡ λx y. Rep x ⊑ Rep y"
assumes emb: "emb ≡ (Λ x. Rep x)"
assumes prj: "prj ≡ (Λ x. Abs (cast⋅t⋅x))"
assumes defl: "defl ≡ (λ a::'a itself. t)"
assumes liftemb: "(liftemb :: 'a u → udom u) ≡ u_map⋅emb"
assumes liftprj: "(liftprj :: udom u → 'a u) ≡ u_map⋅prj"
assumes liftdefl: "(liftdefl :: 'a itself ⇒ _) ≡ (λt. liftdefl_of⋅DEFL('a))"
shows "OFCLASS('a, domain_class)"
proof
have emb_beta: "⋀x. emb⋅x = Rep x"
unfolding emb
apply (rule beta_cfun)
apply (rule typedef_cont_Rep [OF type below adm_defl_set cont_id])
done
have prj_beta: "⋀y. prj⋅y = Abs (cast⋅t⋅y)"
unfolding prj
apply (rule beta_cfun)
apply (rule typedef_cont_Abs [OF type below adm_defl_set])
apply simp_all
done
have prj_emb: "⋀x::'a. prj⋅(emb⋅x) = x"
using type_definition.Rep [OF type]
unfolding prj_beta emb_beta defl_set_def
by (simp add: type_definition.Rep_inverse [OF type])
have emb_prj: "⋀y. emb⋅(prj⋅y :: 'a) = cast⋅t⋅y"
unfolding prj_beta emb_beta
by (simp add: type_definition.Abs_inverse [OF type])
show "ep_pair (emb :: 'a → udom) prj"
apply standard
done
show "cast⋅DEFL('a) = emb oo (prj :: udom → 'a)"
by (rule cfun_eqI, simp add: defl emb_prj)
qed (simp_all only: liftemb liftprj liftdefl)

lemma typedef_DEFL:
assumes "defl ≡ (λa::'a::pcpo itself. t)"
shows "DEFL('a::pcpo) = t"
unfolding assms ..

text ‹Restore original typing constraints.›

setup ‹
[(@{const_name defl}, SOME @{typ "'a::domain itself ⇒ udom defl"}),
(@{const_name emb}, SOME @{typ "'a::domain → udom"}),
(@{const_name prj}, SOME @{typ "udom → 'a::domain"}),
(@{const_name liftdefl}, SOME @{typ "'a::predomain itself ⇒ udom u defl"}),
(@{const_name liftemb}, SOME @{typ "'a::predomain u → udom u"}),
(@{const_name liftprj}, SOME @{typ "udom u → 'a::predomain u"})]
›

ML_file "Tools/domaindef.ML"

subsection ‹Isomorphic deflations›

definition isodefl :: "('a → 'a) ⇒ udom defl ⇒ bool"
where "isodefl d t ⟷ cast⋅t = emb oo d oo prj"

definition isodefl' :: "('a::predomain → 'a) ⇒ udom u defl ⇒ bool"
where "isodefl' d t ⟷ cast⋅t = liftemb oo u_map⋅d oo liftprj"

lemma isodeflI: "(⋀x. cast⋅t⋅x = emb⋅(d⋅(prj⋅x))) ⟹ isodefl d t"
unfolding isodefl_def by (simp add: cfun_eqI)

lemma cast_isodefl: "isodefl d t ⟹ cast⋅t = (Λ x. emb⋅(d⋅(prj⋅x)))"
unfolding isodefl_def by (simp add: cfun_eqI)

lemma isodefl_strict: "isodefl d t ⟹ d⋅⊥ = ⊥"
unfolding isodefl_def
by (drule cfun_fun_cong [where x="⊥"], simp)

lemma isodefl_imp_deflation:
fixes d :: "'a → 'a"
assumes "isodefl d t" shows "deflation d"
proof
note assms [unfolded isodefl_def, simp]
fix x :: 'a
show "d⋅(d⋅x) = d⋅x"
using cast.idem [of t "emb⋅x"] by simp
show "d⋅x ⊑ x"
using cast.below [of t "emb⋅x"] by simp
qed

lemma isodefl_ID_DEFL: "isodefl (ID :: 'a → 'a) DEFL('a)"
unfolding isodefl_def by (simp add: cast_DEFL)

lemma isodefl_LIFTDEFL:
"isodefl' (ID :: 'a → 'a) LIFTDEFL('a::predomain)"
unfolding isodefl'_def by (simp add: cast_liftdefl u_map_ID)

lemma isodefl_DEFL_imp_ID: "isodefl (d :: 'a → 'a) DEFL('a) ⟹ d = ID"
unfolding isodefl_def
apply (rule allI)
apply (drule_tac x="emb⋅x" in spec)
apply simp
done

lemma isodefl_bottom: "isodefl ⊥ ⊥"
unfolding isodefl_def by (simp add: cfun_eq_iff)

"cont f ⟹ cont g ⟹ adm (λx. isodefl (f x) (g x))"
unfolding isodefl_def by simp

lemma isodefl_lub:
assumes "chain d" and "chain t"
assumes "⋀i. isodefl (d i) (t i)"
shows "isodefl (⨆i. d i) (⨆i. t i)"
using assms unfolding isodefl_def

lemma isodefl_fix:
assumes "⋀d t. isodefl d t ⟹ isodefl (f⋅d) (g⋅t)"
shows "isodefl (fix⋅f) (fix⋅g)"
unfolding fix_def2
apply (rule isodefl_lub, simp, simp)
apply (induct_tac i)
done

lemma isodefl_abs_rep:
fixes abs and rep and d
assumes DEFL: "DEFL('b) = DEFL('a)"
assumes abs_def: "(abs :: 'a → 'b) ≡ prj oo emb"
assumes rep_def: "(rep :: 'b → 'a) ≡ prj oo emb"
shows "isodefl d t ⟹ isodefl (abs oo d oo rep) t"
unfolding isodefl_def
by (simp add: cfun_eq_iff assms prj_emb_prj emb_prj_emb)

lemma isodefl'_liftdefl_of: "isodefl d t ⟹ isodefl' d (liftdefl_of⋅t)"
unfolding isodefl_def isodefl'_def
by (simp add: cast_liftdefl_of u_map_oo liftemb_eq liftprj_eq)

lemma isodefl_sfun:
"isodefl d1 t1 ⟹ isodefl d2 t2 ⟹
isodefl (sfun_map⋅d1⋅d2) (sfun_defl⋅t1⋅t2)"
apply (rule isodeflI)
done

lemma isodefl_ssum:
"isodefl d1 t1 ⟹ isodefl d2 t2 ⟹
isodefl (ssum_map⋅d1⋅d2) (ssum_defl⋅t1⋅t2)"
apply (rule isodeflI)
done

lemma isodefl_sprod:
"isodefl d1 t1 ⟹ isodefl d2 t2 ⟹
isodefl (sprod_map⋅d1⋅d2) (sprod_defl⋅t1⋅t2)"
apply (rule isodeflI)
done

lemma isodefl_prod:
"isodefl d1 t1 ⟹ isodefl d2 t2 ⟹
isodefl (prod_map⋅d1⋅d2) (prod_defl⋅t1⋅t2)"
apply (rule isodeflI)
done

lemma isodefl_u:
"isodefl d t ⟹ isodefl (u_map⋅d) (u_defl⋅t)"
apply (rule isodeflI)
apply (simp add: emb_u_def prj_u_def liftemb_eq liftprj_eq u_map_map)
done

lemma isodefl_u_liftdefl:
"isodefl' d t ⟹ isodefl (u_map⋅d) (u_liftdefl⋅t)"
apply (rule isodeflI)
apply (simp add: emb_u_def prj_u_def liftemb_eq liftprj_eq)
done

lemma encode_prod_u_map:
"encode_prod_u⋅(u_map⋅(prod_map⋅f⋅g)⋅(decode_prod_u⋅x))
= sprod_map⋅(u_map⋅f)⋅(u_map⋅g)⋅x"
unfolding encode_prod_u_def decode_prod_u_def
apply (case_tac x, simp, rename_tac a b)
apply (case_tac a, simp, case_tac b, simp, simp)
done

lemma isodefl_prod_u:
assumes "isodefl' d1 t1" and "isodefl' d2 t2"
shows "isodefl' (prod_map⋅d1⋅d2) (prod_liftdefl⋅t1⋅t2)"
using assms unfolding isodefl'_def
unfolding liftemb_prod_def liftprj_prod_def
by (simp add: cast_prod_liftdefl cfcomp1 encode_prod_u_map sprod_map_map)

lemma encode_cfun_map:
"encode_cfun⋅(cfun_map⋅f⋅g⋅(decode_cfun⋅x))
= sfun_map⋅(u_map⋅f)⋅g⋅x"
unfolding encode_cfun_def decode_cfun_def
apply (simp add: sfun_eq_iff cfun_map_def sfun_map_def)
apply (rule cfun_eqI, rename_tac y, case_tac y, simp_all)
done

lemma isodefl_cfun:
assumes "isodefl (u_map⋅d1) t1" and "isodefl d2 t2"
shows "isodefl (cfun_map⋅d1⋅d2) (sfun_defl⋅t1⋅t2)"
using isodefl_sfun [OF assms] unfolding isodefl_def
by (simp add: emb_cfun_def prj_cfun_def cfcomp1 encode_cfun_map)

subsection ‹Setting up the domain package›

named_theorems domain_defl_simps "theorems like DEFL('a t) = t_defl\$DEFL('a)"
and domain_isodefl "theorems like isodefl d t ==> isodefl (foo_map\$d) (foo_defl\$t)"

ML_file "Tools/Domain/domain_isomorphism.ML"
ML_file "Tools/Domain/domain_axioms.ML"
ML_file "Tools/Domain/domain.ML"

lemmas [domain_defl_simps] =
DEFL_cfun DEFL_sfun DEFL_ssum DEFL_sprod DEFL_prod DEFL_u
liftdefl_eq LIFTDEFL_prod u_liftdefl_liftdefl_of

lemmas [domain_map_ID] =
cfun_map_ID sfun_map_ID ssum_map_ID sprod_map_ID prod_map_ID u_map_ID

lemmas [domain_isodefl] =
isodefl_u isodefl_sfun isodefl_ssum isodefl_sprod
isodefl_cfun isodefl_prod isodefl_prod_u isodefl'_liftdefl_of
isodefl_u_liftdefl

lemmas [domain_deflation] =
deflation_cfun_map deflation_sfun_map deflation_ssum_map
deflation_sprod_map deflation_prod_map deflation_u_map

setup ‹