--- a/src/HOLCF/Domain.thy Wed Nov 10 18:15:21 2010 -0800
+++ b/src/HOLCF/Domain.thy Wed Nov 10 18:30:17 2010 -0800
@@ -5,10 +5,330 @@
header {* Domain package *}
theory Domain
-imports Representable
+imports Bifinite Domain_Aux
uses
- "Tools/Domain/domain_axioms.ML"
- "Tools/Domain/domain.ML"
+ ("Tools/repdef.ML")
+ ("Tools/Domain/domain_isomorphism.ML")
+ ("Tools/Domain/domain_axioms.ML")
+ ("Tools/Domain/domain.ML")
begin
+default_sort "domain"
+
+subsection {* Representations of types *}
+
+lemma emb_prj: "emb\<cdot>((prj\<cdot>x)::'a) = cast\<cdot>DEFL('a)\<cdot>x"
+by (simp add: cast_DEFL)
+
+lemma emb_prj_emb:
+ fixes x :: "'a"
+ assumes "DEFL('a) \<sqsubseteq> DEFL('b)"
+ shows "emb\<cdot>(prj\<cdot>(emb\<cdot>x) :: 'b) = emb\<cdot>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) \<sqsubseteq> DEFL('b)"
+ shows "prj\<cdot>(emb\<cdot>(prj\<cdot>x :: 'b)) = (prj\<cdot>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 \<rightarrow> 'b) \<equiv> prj oo emb"
+ assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
+ shows "rep\<cdot>(abs\<cdot>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 \<rightarrow> 'b) \<equiv> prj oo emb"
+ assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
+ shows "abs\<cdot>(rep\<cdot>x) = x"
+unfolding abs_def rep_def
+by (simp add: emb_prj_emb DEFL)
+
+subsection {* Deflations as sets *}
+
+definition defl_set :: "defl \<Rightarrow> udom set"
+where "defl_set A = {x. cast\<cdot>A\<cdot>x = x}"
+
+lemma adm_defl_set: "adm (\<lambda>x. x \<in> defl_set A)"
+unfolding defl_set_def by simp
+
+lemma defl_set_bottom: "\<bottom> \<in> defl_set A"
+unfolding defl_set_def by simp
+
+lemma defl_set_cast [simp]: "cast\<cdot>A\<cdot>x \<in> defl_set A"
+unfolding defl_set_def by simp
+
+lemma defl_set_subset_iff: "defl_set A \<subseteq> defl_set B \<longleftrightarrow> A \<sqsubseteq> 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 \<Rightarrow> defl"})
+ , (@{const_name emb}, SOME @{typ "'a::pcpo \<rightarrow> udom"})
+ , (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"})
+ , (@{const_name liftdefl}, SOME @{typ "'a::pcpo itself \<Rightarrow> defl"})
+ , (@{const_name liftemb}, SOME @{typ "'a::pcpo u \<rightarrow> udom"})
+ , (@{const_name liftprj}, SOME @{typ "udom \<rightarrow> 'a::pcpo u"}) ]
+*}
+
+lemma typedef_rep_class:
+ fixes Rep :: "'a::pcpo \<Rightarrow> udom"
+ fixes Abs :: "udom \<Rightarrow> 'a::pcpo"
+ fixes t :: defl
+ assumes type: "type_definition Rep Abs (defl_set t)"
+ assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y"
+ assumes emb: "emb \<equiv> (\<Lambda> x. Rep x)"
+ assumes prj: "prj \<equiv> (\<Lambda> x. Abs (cast\<cdot>t\<cdot>x))"
+ assumes defl: "defl \<equiv> (\<lambda> a::'a itself. t)"
+ assumes liftemb: "(liftemb :: 'a u \<rightarrow> udom) \<equiv> udom_emb u_approx oo u_map\<cdot>emb"
+ assumes liftprj: "(liftprj :: udom \<rightarrow> 'a u) \<equiv> u_map\<cdot>prj oo udom_prj u_approx"
+ assumes liftdefl: "(liftdefl :: 'a itself \<Rightarrow> defl) \<equiv> (\<lambda>t. u_defl\<cdot>DEFL('a))"
+ shows "OFCLASS('a, liftdomain_class)"
+using liftemb [THEN meta_eq_to_obj_eq]
+using liftprj [THEN meta_eq_to_obj_eq]
+proof (rule liftdomain_class_intro)
+ have emb_beta: "\<And>x. emb\<cdot>x = Rep x"
+ unfolding emb
+ apply (rule beta_cfun)
+ apply (rule typedef_cont_Rep [OF type below adm_defl_set])
+ done
+ have prj_beta: "\<And>y. prj\<cdot>y = Abs (cast\<cdot>t\<cdot>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: "\<And>x::'a. prj\<cdot>(emb\<cdot>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: "\<And>y. emb\<cdot>(prj\<cdot>y :: 'a) = cast\<cdot>t\<cdot>y"
+ unfolding prj_beta emb_beta
+ by (simp add: type_definition.Abs_inverse [OF type])
+ show "ep_pair (emb :: 'a \<rightarrow> udom) prj"
+ apply default
+ apply (simp add: prj_emb)
+ apply (simp add: emb_prj cast.below)
+ done
+ show "cast\<cdot>DEFL('a) = emb oo (prj :: udom \<rightarrow> 'a)"
+ by (rule cfun_eqI, simp add: defl emb_prj)
+ show "LIFTDEFL('a) = u_defl\<cdot>DEFL('a)"
+ unfolding liftdefl ..
+qed
+
+lemma typedef_DEFL:
+ assumes "defl \<equiv> (\<lambda>a::'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 \<Rightarrow> defl"})
+ , (@{const_name emb}, SOME @{typ "'a::domain \<rightarrow> udom"})
+ , (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::domain"})
+ , (@{const_name liftdefl}, SOME @{typ "'a::predomain itself \<Rightarrow> defl"})
+ , (@{const_name liftemb}, SOME @{typ "'a::predomain u \<rightarrow> udom"})
+ , (@{const_name liftprj}, SOME @{typ "udom \<rightarrow> 'a::predomain u"}) ]
+*}
+
+use "Tools/repdef.ML"
+
+subsection {* Isomorphic deflations *}
+
+definition
+ isodefl :: "('a \<rightarrow> 'a) \<Rightarrow> defl \<Rightarrow> bool"
+where
+ "isodefl d t \<longleftrightarrow> cast\<cdot>t = emb oo d oo prj"
+
+lemma isodeflI: "(\<And>x. cast\<cdot>t\<cdot>x = emb\<cdot>(d\<cdot>(prj\<cdot>x))) \<Longrightarrow> isodefl d t"
+unfolding isodefl_def by (simp add: cfun_eqI)
+
+lemma cast_isodefl: "isodefl d t \<Longrightarrow> cast\<cdot>t = (\<Lambda> x. emb\<cdot>(d\<cdot>(prj\<cdot>x)))"
+unfolding isodefl_def by (simp add: cfun_eqI)
+
+lemma isodefl_strict: "isodefl d t \<Longrightarrow> d\<cdot>\<bottom> = \<bottom>"
+unfolding isodefl_def
+by (drule cfun_fun_cong [where x="\<bottom>"], simp)
+
+lemma isodefl_imp_deflation:
+ fixes d :: "'a \<rightarrow> 'a"
+ assumes "isodefl d t" shows "deflation d"
+proof
+ note assms [unfolded isodefl_def, simp]
+ fix x :: 'a
+ show "d\<cdot>(d\<cdot>x) = d\<cdot>x"
+ using cast.idem [of t "emb\<cdot>x"] by simp
+ show "d\<cdot>x \<sqsubseteq> x"
+ using cast.below [of t "emb\<cdot>x"] by simp
+qed
+
+lemma isodefl_ID_DEFL: "isodefl (ID :: 'a \<rightarrow> 'a) DEFL('a)"
+unfolding isodefl_def by (simp add: cast_DEFL)
+
+lemma isodefl_LIFTDEFL:
+ "isodefl (u_map\<cdot>(ID :: 'a \<rightarrow> 'a)) LIFTDEFL('a::predomain)"
+unfolding u_map_ID DEFL_u [symmetric]
+by (rule isodefl_ID_DEFL)
+
+lemma isodefl_DEFL_imp_ID: "isodefl (d :: 'a \<rightarrow> 'a) DEFL('a) \<Longrightarrow> d = ID"
+unfolding isodefl_def
+apply (simp add: cast_DEFL)
+apply (simp add: cfun_eq_iff)
+apply (rule allI)
+apply (drule_tac x="emb\<cdot>x" in spec)
+apply simp
+done
+
+lemma isodefl_bottom: "isodefl \<bottom> \<bottom>"
+unfolding isodefl_def by (simp add: cfun_eq_iff)
+
+lemma adm_isodefl:
+ "cont f \<Longrightarrow> cont g \<Longrightarrow> adm (\<lambda>x. isodefl (f x) (g x))"
+unfolding isodefl_def by simp
+
+lemma isodefl_lub:
+ assumes "chain d" and "chain t"
+ assumes "\<And>i. isodefl (d i) (t i)"
+ shows "isodefl (\<Squnion>i. d i) (\<Squnion>i. t i)"
+using prems unfolding isodefl_def
+by (simp add: contlub_cfun_arg contlub_cfun_fun)
+
+lemma isodefl_fix:
+ assumes "\<And>d t. isodefl d t \<Longrightarrow> isodefl (f\<cdot>d) (g\<cdot>t)"
+ shows "isodefl (fix\<cdot>f) (fix\<cdot>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 \<rightarrow> 'b) \<equiv> prj oo emb"
+ assumes rep_def: "(rep :: 'b \<rightarrow> 'a) \<equiv> prj oo emb"
+ shows "isodefl d t \<Longrightarrow> 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_cfun:
+ "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
+ isodefl (cfun_map\<cdot>d1\<cdot>d2) (cfun_defl\<cdot>t1\<cdot>t2)"
+apply (rule isodeflI)
+apply (simp add: cast_cfun_defl cast_isodefl)
+apply (simp add: emb_cfun_def prj_cfun_def)
+apply (simp add: cfun_map_map cfcomp1)
+done
+
+lemma isodefl_ssum:
+ "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
+ isodefl (ssum_map\<cdot>d1\<cdot>d2) (ssum_defl\<cdot>t1\<cdot>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 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
+ isodefl (sprod_map\<cdot>d1\<cdot>d2) (sprod_defl\<cdot>t1\<cdot>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_cprod:
+ "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
+ isodefl (cprod_map\<cdot>d1\<cdot>d2) (prod_defl\<cdot>t1\<cdot>t2)"
+apply (rule isodeflI)
+apply (simp add: cast_prod_defl cast_isodefl)
+apply (simp add: emb_prod_def prj_prod_def)
+apply (simp add: cprod_map_map cfcomp1)
+done
+
+lemma isodefl_u:
+ fixes d :: "'a::liftdomain \<rightarrow> 'a"
+ shows "isodefl (d :: 'a \<rightarrow> 'a) t \<Longrightarrow> isodefl (u_map\<cdot>d) (u_defl\<cdot>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)
+apply (simp add: u_map_map)
+done
+
+lemma encode_prod_u_map:
+ "encode_prod_u\<cdot>(u_map\<cdot>(cprod_map\<cdot>f\<cdot>g)\<cdot>(decode_prod_u\<cdot>x))
+ = sprod_map\<cdot>(u_map\<cdot>f)\<cdot>(u_map\<cdot>g)\<cdot>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_cprod_u:
+ assumes "isodefl (u_map\<cdot>d1) t1" and "isodefl (u_map\<cdot>d2) t2"
+ shows "isodefl (u_map\<cdot>(cprod_map\<cdot>d1\<cdot>d2)) (sprod_defl\<cdot>t1\<cdot>t2)"
+using assms unfolding isodefl_def
+apply (simp add: emb_u_def prj_u_def liftemb_prod_def liftprj_prod_def)
+apply (simp add: emb_u_def [symmetric] prj_u_def [symmetric])
+apply (simp add: cfcomp1 encode_prod_u_map cast_sprod_defl sprod_map_map)
+done
+
+subsection {* Setting up the domain package *}
+
+use "Tools/Domain/domain_isomorphism.ML"
+use "Tools/Domain/domain_axioms.ML"
+use "Tools/Domain/domain.ML"
+
+setup Domain_Isomorphism.setup
+
+lemmas [domain_defl_simps] =
+ DEFL_cfun DEFL_ssum DEFL_sprod DEFL_prod DEFL_u
+ liftdefl_eq LIFTDEFL_prod
+
+lemmas [domain_map_ID] =
+ cfun_map_ID ssum_map_ID sprod_map_ID cprod_map_ID u_map_ID
+
+lemmas [domain_isodefl] =
+ isodefl_u isodefl_cfun isodefl_ssum isodefl_sprod
+ isodefl_cprod isodefl_cprod_u
+
+lemmas [domain_deflation] =
+ deflation_cfun_map deflation_ssum_map deflation_sprod_map
+ deflation_cprod_map deflation_u_map
+
+setup {*
+ fold Domain_Take_Proofs.add_map_function
+ [(@{type_name cfun}, @{const_name cfun_map}, [true, true]),
+ (@{type_name ssum}, @{const_name ssum_map}, [true, true]),
+ (@{type_name sprod}, @{const_name sprod_map}, [true, true]),
+ (@{type_name prod}, @{const_name cprod_map}, [true, true]),
+ (@{type_name "u"}, @{const_name u_map}, [true])]
+*}
+
end