(* Title: HOLCF/Bifinite.thy
Author: Brian Huffman
*)
header {* Bifinite domains *}
theory Bifinite
imports Algebraic
begin
subsection {* Class of bifinite domains *}
text {*
We define a bifinite domain as a pcpo that is isomorphic to some
algebraic deflation over the universal domain.
*}
class bifinite = pcpo +
fixes emb :: "'a::pcpo \<rightarrow> udom"
fixes prj :: "udom \<rightarrow> 'a::pcpo"
fixes sfp :: "'a itself \<Rightarrow> sfp"
assumes ep_pair_emb_prj: "ep_pair emb prj"
assumes cast_SFP: "cast\<cdot>(sfp TYPE('a)) = emb oo prj"
syntax "_SFP" :: "type \<Rightarrow> sfp" ("(1SFP/(1'(_')))")
translations "SFP('t)" \<rightleftharpoons> "CONST sfp TYPE('t)"
interpretation bifinite:
pcpo_ep_pair "emb :: 'a::bifinite \<rightarrow> udom" "prj :: udom \<rightarrow> 'a::bifinite"
unfolding pcpo_ep_pair_def
by (rule ep_pair_emb_prj)
lemmas emb_inverse = bifinite.e_inverse
lemmas emb_prj_below = bifinite.e_p_below
lemmas emb_eq_iff = bifinite.e_eq_iff
lemmas emb_strict = bifinite.e_strict
lemmas prj_strict = bifinite.p_strict
subsection {* Bifinite domains have a countable compact basis *}
text {*
Eventually it should be possible to generalize this to an unpointed
variant of the bifinite class.
*}
interpretation compact_basis:
ideal_completion below Rep_compact_basis "approximants::'a::bifinite \<Rightarrow> _"
proof -
obtain Y where Y: "\<forall>i. Y i \<sqsubseteq> Y (Suc i)"
and SFP: "SFP('a) = (\<Squnion>i. sfp_principal (Y i))"
by (rule sfp.obtain_principal_chain)
def approx \<equiv> "\<lambda>i. (prj oo cast\<cdot>(sfp_principal (Y i)) oo emb) :: 'a \<rightarrow> 'a"
interpret sfp_approx: approx_chain approx
proof (rule approx_chain.intro)
show "chain (\<lambda>i. approx i)"
unfolding approx_def by (simp add: Y)
show "(\<Squnion>i. approx i) = ID"
unfolding approx_def
by (simp add: lub_distribs Y SFP [symmetric] cast_SFP expand_cfun_eq)
show "\<And>i. finite_deflation (approx i)"
unfolding approx_def
apply (rule bifinite.finite_deflation_p_d_e)
apply (rule finite_deflation_cast)
apply (rule sfp.compact_principal)
apply (rule below_trans [OF monofun_cfun_fun])
apply (rule is_ub_thelub, simp add: Y)
apply (simp add: lub_distribs Y SFP [symmetric] cast_SFP)
done
qed
(* FIXME: why does show ?thesis fail here? *)
show "ideal_completion below Rep_compact_basis (approximants::'a \<Rightarrow> _)" ..
qed
subsection {* Type combinators *}
definition
sfp_fun1 ::
"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) \<Rightarrow> (sfp \<rightarrow> sfp)"
where
"sfp_fun1 approx f =
sfp.basis_fun (\<lambda>a.
sfp_principal (Abs_fin_defl
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)))"
definition
sfp_fun2 ::
"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a))
\<Rightarrow> (sfp \<rightarrow> sfp \<rightarrow> sfp)"
where
"sfp_fun2 approx f =
sfp.basis_fun (\<lambda>a.
sfp.basis_fun (\<lambda>b.
sfp_principal (Abs_fin_defl
(udom_emb approx oo
f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx))))"
lemma cast_sfp_fun1:
assumes approx: "approx_chain approx"
assumes f: "\<And>a. finite_deflation a \<Longrightarrow> finite_deflation (f\<cdot>a)"
shows "cast\<cdot>(sfp_fun1 approx f\<cdot>A) = udom_emb approx oo f\<cdot>(cast\<cdot>A) oo udom_prj approx"
proof -
have 1: "\<And>a. finite_deflation
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)"
apply (rule ep_pair.finite_deflation_e_d_p)
apply (rule approx_chain.ep_pair_udom [OF approx])
apply (rule f, rule finite_deflation_Rep_fin_defl)
done
show ?thesis
by (induct A rule: sfp.principal_induct, simp)
(simp only: sfp_fun1_def
sfp.basis_fun_principal
sfp.basis_fun_mono
sfp.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
cast_sfp_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed
lemma cast_sfp_fun2:
assumes approx: "approx_chain approx"
assumes f: "\<And>a b. finite_deflation a \<Longrightarrow> finite_deflation b \<Longrightarrow>
finite_deflation (f\<cdot>a\<cdot>b)"
shows "cast\<cdot>(sfp_fun2 approx f\<cdot>A\<cdot>B) =
udom_emb approx oo f\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj approx"
proof -
have 1: "\<And>a b. finite_deflation (udom_emb approx oo
f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx)"
apply (rule ep_pair.finite_deflation_e_d_p)
apply (rule ep_pair_udom [OF approx])
apply (rule f, (rule finite_deflation_Rep_fin_defl)+)
done
show ?thesis
by (induct A B rule: sfp.principal_induct2, simp, simp)
(simp only: sfp_fun2_def
sfp.basis_fun_principal
sfp.basis_fun_mono
sfp.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
cast_sfp_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed
subsection {* Instance for universal domain type *}
instantiation udom :: bifinite
begin
definition [simp]:
"emb = (ID :: udom \<rightarrow> udom)"
definition [simp]:
"prj = (ID :: udom \<rightarrow> udom)"
definition
"sfp (t::udom itself) = (\<Squnion>i. sfp_principal (Abs_fin_defl (udom_approx i)))"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> udom)"
by (simp add: ep_pair.intro)
next
show "cast\<cdot>SFP(udom) = emb oo (prj :: udom \<rightarrow> udom)"
unfolding sfp_udom_def
apply (subst contlub_cfun_arg)
apply (rule chainI)
apply (rule sfp.principal_mono)
apply (simp add: below_fin_defl_def)
apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
apply (rule chainE)
apply (rule chain_udom_approx)
apply (subst cast_sfp_principal)
apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
done
qed
end
subsection {* Instance for continuous function space *}
definition
cfun_approx :: "nat \<Rightarrow> (udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom)"
where
"cfun_approx = (\<lambda>i. cfun_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
lemma cfun_approx: "approx_chain cfun_approx"
proof (rule approx_chain.intro)
show "chain (\<lambda>i. cfun_approx i)"
unfolding cfun_approx_def by simp
show "(\<Squnion>i. cfun_approx i) = ID"
unfolding cfun_approx_def
by (simp add: lub_distribs cfun_map_ID)
show "\<And>i. finite_deflation (cfun_approx i)"
unfolding cfun_approx_def
by (intro finite_deflation_cfun_map finite_deflation_udom_approx)
qed
definition cfun_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp"
where "cfun_sfp = sfp_fun2 cfun_approx cfun_map"
lemma cast_cfun_sfp:
"cast\<cdot>(cfun_sfp\<cdot>A\<cdot>B) =
udom_emb cfun_approx oo cfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj cfun_approx"
unfolding cfun_sfp_def
apply (rule cast_sfp_fun2 [OF cfun_approx])
apply (erule (1) finite_deflation_cfun_map)
done
instantiation cfun :: (bifinite, bifinite) bifinite
begin
definition
"emb = udom_emb cfun_approx oo cfun_map\<cdot>prj\<cdot>emb"
definition
"prj = cfun_map\<cdot>emb\<cdot>prj oo udom_prj cfun_approx"
definition
"sfp (t::('a \<rightarrow> 'b) itself) = cfun_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
unfolding emb_cfun_def prj_cfun_def
using ep_pair_udom [OF cfun_approx]
by (intro ep_pair_comp ep_pair_cfun_map ep_pair_emb_prj)
next
show "cast\<cdot>SFP('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
unfolding emb_cfun_def prj_cfun_def sfp_cfun_def cast_cfun_sfp
by (simp add: cast_SFP oo_def expand_cfun_eq cfun_map_map)
qed
end
lemma SFP_cfun:
"SFP('a::bifinite \<rightarrow> 'b::bifinite) = cfun_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
by (rule sfp_cfun_def)
end