(* Title: HOLCF/Representable.thy
Author: Brian Huffman
*)
header {* Representable Types *}
theory Representable
imports Algebraic Bifinite Universal Ssum One Fixrec Domain_Aux
uses
("Tools/repdef.ML")
("Tools/Domain/domain_isomorphism.ML")
begin
default_sort bifinite
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 for @{term emb} and @{term prj}.
*}
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"}) ]
*}
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)"
shows "OFCLASS('a, bifinite_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])
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
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::bifinite itself \<Rightarrow> defl"})
, (@{const_name emb}, SOME @{typ "'a::bifinite \<rightarrow> udom"})
, (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::bifinite"}) ]
*}
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_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:
"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)
apply (simp add: u_map_map)
done
subsection {* Constructing Domain Isomorphisms *}
use "Tools/Domain/domain_isomorphism.ML"
setup Domain_Isomorphism.setup
lemmas [domain_defl_simps] =
DEFL_cfun DEFL_ssum DEFL_sprod DEFL_prod DEFL_u
lemmas [domain_map_ID] =
cfun_map_ID ssum_map_ID sprod_map_ID cprod_map_ID u_map_ID
lemmas [domain_isodefl] =
isodefl_cfun isodefl_ssum isodefl_sprod isodefl_cprod isodefl_u
lemmas [domain_deflation] =
deflation_cfun_map deflation_ssum_map deflation_sprod_map
deflation_cprod_map deflation_u_map
setup {*
fold Domain_Isomorphism.add_type_constructor
[(@{type_name cfun}, @{const_name cfun_defl}, @{const_name cfun_map}, [true, true]),
(@{type_name ssum}, @{const_name ssum_defl}, @{const_name ssum_map}, [true, true]),
(@{type_name sprod}, @{const_name sprod_defl}, @{const_name sprod_map}, [true, true]),
(@{type_name prod}, @{const_name prod_defl}, @{const_name cprod_map}, [true, true]),
(@{type_name "u"}, @{const_name u_defl}, @{const_name u_map}, [true])]
*}
end