Theory Domain

(*  Title:      HOL/HOLCF/Domain.thy
    Author:     Brian Huffman
*)

section ‹Domain package›

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

default_sort "domain"

subsection ‹Representations of types›

lemma emb_prj: "emb⋅((prj⋅x)::'a) = cast⋅DEFL('a)⋅x"
by (simp add: cast_DEFL)

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])
apply (simp add: cast_DEFL)
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
by (simp add: emb_prj_emb DEFL)

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
by (simp add: emb_prj_emb DEFL)

subsection ‹Deflations as sets›

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

lemma adm_defl_set: "adm (λx. x ∈ defl_set A)"
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 ‹
  fold Sign.add_const_constraint
  [ (\<^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: "(⊑) ≡ 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
    apply (simp add: prj_emb)
    apply (simp add: emb_prj cast.below)
    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 ‹
  fold Sign.add_const_constraint
   [(\<^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 (simp add: cast_DEFL)
apply (simp add: cfun_eq_iff)
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)

lemma adm_isodefl:
  "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
by (simp add: contlub_cfun_arg contlub_cfun_fun)

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)
apply (simp add: isodefl_bottom)
apply (simp add: assms)
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)
apply (simp add: cast_sfun_defl cast_isodefl)
apply (simp add: emb_sfun_def prj_sfun_def)
apply (simp add: sfun_map_map isodefl_strict)
done

lemma isodefl_ssum:
  "isodefl d1 t1 ⟹ isodefl d2 t2 ⟹
    isodefl (ssum_map⋅d1⋅d2) (ssum_defl⋅t1⋅t2)"
apply (rule isodeflI)
apply (simp add: cast_ssum_defl cast_isodefl)
apply (simp add: emb_ssum_def prj_ssum_def)
apply (simp add: ssum_map_map isodefl_strict)
done

lemma isodefl_sprod:
  "isodefl d1 t1 ⟹ isodefl d2 t2 ⟹
    isodefl (sprod_map⋅d1⋅d2) (sprod_defl⋅t1⋅t2)"
apply (rule isodeflI)
apply (simp add: cast_sprod_defl cast_isodefl)
apply (simp add: emb_sprod_def prj_sprod_def)
apply (simp add: sprod_map_map isodefl_strict)
done

lemma isodefl_prod:
  "isodefl d1 t1 ⟹ isodefl d2 t2 ⟹
    isodefl (prod_map⋅d1⋅d2) (prod_defl⋅t1⋅t2)"
apply (rule isodeflI)
apply (simp add: cast_prod_defl cast_isodefl)
apply (simp add: emb_prod_def prj_prod_def)
apply (simp add: prod_map_map cfcomp1)
done

lemma isodefl_u:
  "isodefl d t ⟹ isodefl (u_map⋅d) (u_defl⋅t)"
apply (rule isodeflI)
apply (simp add: cast_u_defl cast_isodefl)
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: cast_u_liftdefl isodefl'_def)
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 ‹
  fold Domain_Take_Proofs.add_rec_type
    [(\<^type_name>‹cfun›, [true, true]),
     (\<^type_name>‹sfun›, [true, true]),
     (\<^type_name>‹ssum›, [true, true]),
     (\<^type_name>‹sprod›, [true, true]),
     (\<^type_name>‹prod›, [true, true]),
     (\<^type_name>‹u›, [true])]
›

end