src/HOL/HOLCF/Domain.thy

simpler proof

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

header {* Domain package *}

theory Domain
imports Representable Domain_Aux
keywords
  "domaindef" :: thy_decl and "lazy" "unsafe" and
  "domain_isomorphism" "domain" :: thy_decl
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 :: "'a::bifinite defl \<Rightarrow> 'a 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> udom 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> udom u defl"})
  , (@{const_name liftemb}, SOME @{typ "'a::pcpo u \<rightarrow> udom u"})
  , (@{const_name liftprj}, SOME @{typ "udom u \<rightarrow> 'a::pcpo u"}) ]
*}

lemma typedef_domain_class:
  fixes Rep :: "'a::pcpo \<Rightarrow> udom"
  fixes Abs :: "udom \<Rightarrow> 'a::pcpo"
  fixes t :: "udom 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 u) \<equiv> u_map\<cdot>emb"
  assumes liftprj: "(liftprj :: udom u \<rightarrow> 'a u) \<equiv> u_map\<cdot>prj"
  assumes liftdefl: "(liftdefl :: 'a itself \<Rightarrow> _) \<equiv> (\<lambda>t. liftdefl_of\<cdot>DEFL('a))"
  shows "OFCLASS('a, domain_class)"
proof
  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 cont_id])
    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)
qed (simp_all only: liftemb liftprj liftdefl)

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> udom 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> udom u defl"})
  , (@{const_name liftemb}, SOME @{typ "'a::predomain u \<rightarrow> udom u"})
  , (@{const_name liftprj}, SOME @{typ "udom u \<rightarrow> 'a::predomain u"}) ]
*}

ML_file "Tools/domaindef.ML"

subsection {* Isomorphic deflations *}

definition isodefl :: "('a \<rightarrow> 'a) \<Rightarrow> udom defl \<Rightarrow> bool"
  where "isodefl d t \<longleftrightarrow> cast\<cdot>t = emb oo d oo prj"

definition isodefl' :: "('a::predomain \<rightarrow> 'a) \<Rightarrow> udom u defl \<Rightarrow> bool"
  where "isodefl' d t \<longleftrightarrow> cast\<cdot>t = liftemb oo u_map\<cdot>d oo liftprj"

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' (ID :: 'a \<rightarrow> 'a) LIFTDEFL('a::predomain)"
unfolding isodefl'_def by (simp add: cast_liftdefl u_map_ID)

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 assms 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'_liftdefl_of: "isodefl d t \<Longrightarrow> isodefl' d (liftdefl_of\<cdot>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 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
    isodefl (sfun_map\<cdot>d1\<cdot>d2) (sfun_defl\<cdot>t1\<cdot>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 \<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_prod:
  "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
    isodefl (prod_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: prod_map_map cfcomp1)
done

lemma isodefl_u:
  "isodefl d 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 u_map_map)
done

lemma isodefl_u_liftdefl:
  "isodefl' d t \<Longrightarrow> isodefl (u_map\<cdot>d) (u_liftdefl\<cdot>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\<cdot>(u_map\<cdot>(prod_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_prod_u:
  assumes "isodefl' d1 t1" and "isodefl' d2 t2"
  shows "isodefl' (prod_map\<cdot>d1\<cdot>d2) (prod_liftdefl\<cdot>t1\<cdot>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\<cdot>(cfun_map\<cdot>f\<cdot>g\<cdot>(decode_cfun\<cdot>x))
    = sfun_map\<cdot>(u_map\<cdot>f)\<cdot>g\<cdot>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\<cdot>d1) t1" and "isodefl d2 t2"
  shows "isodefl (cfun_map\<cdot>d1\<cdot>d2) (sfun_defl\<cdot>t1\<cdot>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$DEFL('a)"
named_theorems domain_isodefl "theorems like isodefl d t ==> isodefl (foo_map$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