(* Title: HOLCF/Bifinite.thy
Author: Brian Huffman
*)
header {* Bifinite domains *}
theory Bifinite
imports Algebraic Map_Functions Countable
begin
subsection {* Class of bifinite domains *}
text {*
We define a ``domain'' as a pcpo that is isomorphic to some
algebraic deflation over the universal domain; this is equivalent
to being omega-bifinite.
A predomain is a cpo that, when lifted, becomes a domain.
*}
class predomain = cpo +
fixes liftdefl :: "('a::cpo) itself \<Rightarrow> defl"
fixes liftemb :: "'a\<^sub>\<bottom> \<rightarrow> udom"
fixes liftprj :: "udom \<rightarrow> 'a\<^sub>\<bottom>"
assumes predomain_ep: "ep_pair liftemb liftprj"
assumes cast_liftdefl: "cast\<cdot>(liftdefl TYPE('a::cpo)) = liftemb oo liftprj"
syntax "_LIFTDEFL" :: "type \<Rightarrow> logic" ("(1LIFTDEFL/(1'(_')))")
translations "LIFTDEFL('t)" \<rightleftharpoons> "CONST liftdefl TYPE('t)"
class "domain" = predomain + pcpo +
fixes emb :: "'a::cpo \<rightarrow> udom"
fixes prj :: "udom \<rightarrow> 'a::cpo"
fixes defl :: "'a itself \<Rightarrow> defl"
assumes ep_pair_emb_prj: "ep_pair emb prj"
assumes cast_DEFL: "cast\<cdot>(defl TYPE('a)) = emb oo prj"
syntax "_DEFL" :: "type \<Rightarrow> defl" ("(1DEFL/(1'(_')))")
translations "DEFL('t)" \<rightleftharpoons> "CONST defl TYPE('t)"
interpretation "domain": pcpo_ep_pair emb prj
unfolding pcpo_ep_pair_def
by (rule ep_pair_emb_prj)
lemmas emb_inverse = domain.e_inverse
lemmas emb_prj_below = domain.e_p_below
lemmas emb_eq_iff = domain.e_eq_iff
lemmas emb_strict = domain.e_strict
lemmas prj_strict = domain.p_strict
subsection {* Domains have a countable compact basis *}
text {*
Eventually it should be possible to generalize this to an unpointed
variant of the domain class.
*}
interpretation compact_basis:
ideal_completion below Rep_compact_basis "approximants::'a::domain \<Rightarrow> _"
proof -
obtain Y where Y: "\<forall>i. Y i \<sqsubseteq> Y (Suc i)"
and DEFL: "DEFL('a) = (\<Squnion>i. defl_principal (Y i))"
by (rule defl.obtain_principal_chain)
def approx \<equiv> "\<lambda>i. (prj oo cast\<cdot>(defl_principal (Y i)) oo emb) :: 'a \<rightarrow> 'a"
interpret defl_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 DEFL [symmetric] cast_DEFL cfun_eq_iff)
show "\<And>i. finite_deflation (approx i)"
unfolding approx_def
apply (rule domain.finite_deflation_p_d_e)
apply (rule finite_deflation_cast)
apply (rule defl.compact_principal)
apply (rule below_trans [OF monofun_cfun_fun])
apply (rule is_ub_thelub, simp add: Y)
apply (simp add: lub_distribs Y DEFL [symmetric] cast_DEFL)
done
qed
(* FIXME: why does show ?thesis fail here? *)
show "ideal_completion below Rep_compact_basis (approximants::'a \<Rightarrow> _)" ..
qed
subsection {* Chains of approx functions *}
definition u_approx :: "nat \<Rightarrow> udom\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"
where "u_approx = (\<lambda>i. u_map\<cdot>(udom_approx i))"
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))"
definition prod_approx :: "nat \<Rightarrow> udom \<times> udom \<rightarrow> udom \<times> udom"
where "prod_approx = (\<lambda>i. cprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
definition sprod_approx :: "nat \<Rightarrow> udom \<otimes> udom \<rightarrow> udom \<otimes> udom"
where "sprod_approx = (\<lambda>i. sprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
definition ssum_approx :: "nat \<Rightarrow> udom \<oplus> udom \<rightarrow> udom \<oplus> udom"
where "ssum_approx = (\<lambda>i. ssum_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
lemma approx_chain_lemma1:
assumes "m\<cdot>ID = ID"
assumes "\<And>d. finite_deflation d \<Longrightarrow> finite_deflation (m\<cdot>d)"
shows "approx_chain (\<lambda>i. m\<cdot>(udom_approx i))"
by (rule approx_chain.intro)
(simp_all add: lub_distribs finite_deflation_udom_approx assms)
lemma approx_chain_lemma2:
assumes "m\<cdot>ID\<cdot>ID = ID"
assumes "\<And>a b. \<lbrakk>finite_deflation a; finite_deflation b\<rbrakk>
\<Longrightarrow> finite_deflation (m\<cdot>a\<cdot>b)"
shows "approx_chain (\<lambda>i. m\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
by (rule approx_chain.intro)
(simp_all add: lub_distribs finite_deflation_udom_approx assms)
lemma u_approx: "approx_chain u_approx"
using u_map_ID finite_deflation_u_map
unfolding u_approx_def by (rule approx_chain_lemma1)
lemma cfun_approx: "approx_chain cfun_approx"
using cfun_map_ID finite_deflation_cfun_map
unfolding cfun_approx_def by (rule approx_chain_lemma2)
lemma prod_approx: "approx_chain prod_approx"
using cprod_map_ID finite_deflation_cprod_map
unfolding prod_approx_def by (rule approx_chain_lemma2)
lemma sprod_approx: "approx_chain sprod_approx"
using sprod_map_ID finite_deflation_sprod_map
unfolding sprod_approx_def by (rule approx_chain_lemma2)
lemma ssum_approx: "approx_chain ssum_approx"
using ssum_map_ID finite_deflation_ssum_map
unfolding ssum_approx_def by (rule approx_chain_lemma2)
subsection {* Type combinators *}
definition
defl_fun1 ::
"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) \<Rightarrow> (defl \<rightarrow> defl)"
where
"defl_fun1 approx f =
defl.basis_fun (\<lambda>a.
defl_principal (Abs_fin_defl
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)))"
definition
defl_fun2 ::
"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a))
\<Rightarrow> (defl \<rightarrow> defl \<rightarrow> defl)"
where
"defl_fun2 approx f =
defl.basis_fun (\<lambda>a.
defl.basis_fun (\<lambda>b.
defl_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_defl_fun1:
assumes approx: "approx_chain approx"
assumes f: "\<And>a. finite_deflation a \<Longrightarrow> finite_deflation (f\<cdot>a)"
shows "cast\<cdot>(defl_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: defl.principal_induct, simp)
(simp only: defl_fun1_def
defl.basis_fun_principal
defl.basis_fun_mono
defl.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
cast_defl_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed
lemma cast_defl_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>(defl_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: defl.principal_induct2, simp, simp)
(simp only: defl_fun2_def
defl.basis_fun_principal
defl.basis_fun_mono
defl.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
cast_defl_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed
definition u_defl :: "defl \<rightarrow> defl"
where "u_defl = defl_fun1 u_approx u_map"
definition cfun_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
where "cfun_defl = defl_fun2 cfun_approx cfun_map"
definition prod_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
where "prod_defl = defl_fun2 prod_approx cprod_map"
definition sprod_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
where "sprod_defl = defl_fun2 sprod_approx sprod_map"
definition ssum_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
where "ssum_defl = defl_fun2 ssum_approx ssum_map"
lemma cast_u_defl:
"cast\<cdot>(u_defl\<cdot>A) =
udom_emb u_approx oo u_map\<cdot>(cast\<cdot>A) oo udom_prj u_approx"
using u_approx finite_deflation_u_map
unfolding u_defl_def by (rule cast_defl_fun1)
lemma cast_cfun_defl:
"cast\<cdot>(cfun_defl\<cdot>A\<cdot>B) =
udom_emb cfun_approx oo cfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj cfun_approx"
using cfun_approx finite_deflation_cfun_map
unfolding cfun_defl_def by (rule cast_defl_fun2)
lemma cast_prod_defl:
"cast\<cdot>(prod_defl\<cdot>A\<cdot>B) = udom_emb prod_approx oo
cprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj prod_approx"
using prod_approx finite_deflation_cprod_map
unfolding prod_defl_def by (rule cast_defl_fun2)
lemma cast_sprod_defl:
"cast\<cdot>(sprod_defl\<cdot>A\<cdot>B) =
udom_emb sprod_approx oo
sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo
udom_prj sprod_approx"
using sprod_approx finite_deflation_sprod_map
unfolding sprod_defl_def by (rule cast_defl_fun2)
lemma cast_ssum_defl:
"cast\<cdot>(ssum_defl\<cdot>A\<cdot>B) =
udom_emb ssum_approx oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj ssum_approx"
using ssum_approx finite_deflation_ssum_map
unfolding ssum_defl_def by (rule cast_defl_fun2)
subsection {* Lemma for proving domain instances *}
text {*
A class of domains where @{const liftemb}, @{const liftprj},
and @{const liftdefl} are all defined in the standard way.
*}
class liftdomain = "domain" +
assumes liftemb_eq: "liftemb = udom_emb u_approx oo u_map\<cdot>emb"
assumes liftprj_eq: "liftprj = u_map\<cdot>prj oo udom_prj u_approx"
assumes liftdefl_eq: "liftdefl TYPE('a::cpo) = u_defl\<cdot>DEFL('a)"
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 liftdomain_class_intro:
assumes liftemb: "(liftemb :: 'a u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
assumes liftprj: "(liftprj :: udom \<rightarrow> 'a u) = u_map\<cdot>prj oo udom_prj u_approx"
assumes liftdefl: "liftdefl TYPE('a) = u_defl\<cdot>DEFL('a)"
assumes ep_pair: "ep_pair emb (prj :: udom \<rightarrow> 'a)"
assumes cast_defl: "cast\<cdot>DEFL('a) = emb oo (prj :: udom \<rightarrow> 'a)"
shows "OFCLASS('a, liftdomain_class)"
proof
show "ep_pair liftemb (liftprj :: udom \<rightarrow> 'a u)"
unfolding liftemb liftprj
by (intro ep_pair_comp ep_pair_u_map ep_pair ep_pair_udom u_approx)
show "cast\<cdot>LIFTDEFL('a) = liftemb oo (liftprj :: udom \<rightarrow> 'a u)"
unfolding liftemb liftprj liftdefl
by (simp add: cfcomp1 cast_u_defl cast_defl u_map_map)
next
qed fact+
text {* Restore original type 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"}) ]
*}
subsection {* The universal domain is a domain *}
instantiation udom :: liftdomain
begin
definition [simp]:
"emb = (ID :: udom \<rightarrow> udom)"
definition [simp]:
"prj = (ID :: udom \<rightarrow> udom)"
definition
"defl (t::udom itself) = (\<Squnion>i. defl_principal (Abs_fin_defl (udom_approx i)))"
definition
"(liftemb :: udom u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
definition
"(liftprj :: udom \<rightarrow> udom u) = u_map\<cdot>prj oo udom_prj u_approx"
definition
"liftdefl (t::udom itself) = u_defl\<cdot>DEFL(udom)"
instance
using liftemb_udom_def liftprj_udom_def liftdefl_udom_def
proof (rule liftdomain_class_intro)
show "ep_pair emb (prj :: udom \<rightarrow> udom)"
by (simp add: ep_pair.intro)
show "cast\<cdot>DEFL(udom) = emb oo (prj :: udom \<rightarrow> udom)"
unfolding defl_udom_def
apply (subst contlub_cfun_arg)
apply (rule chainI)
apply (rule defl.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_defl_principal)
apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
done
qed
end
subsection {* Lifted predomains are domains *}
instantiation u :: (predomain) liftdomain
begin
definition
"emb = liftemb"
definition
"prj = liftprj"
definition
"defl (t::'a u itself) = LIFTDEFL('a)"
definition
"(liftemb :: 'a u u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
definition
"(liftprj :: udom \<rightarrow> 'a u u) = u_map\<cdot>prj oo udom_prj u_approx"
definition
"liftdefl (t::'a u itself) = u_defl\<cdot>DEFL('a u)"
instance
using liftemb_u_def liftprj_u_def liftdefl_u_def
proof (rule liftdomain_class_intro)
show "ep_pair emb (prj :: udom \<rightarrow> 'a u)"
unfolding emb_u_def prj_u_def
by (rule predomain_ep)
show "cast\<cdot>DEFL('a u) = emb oo (prj :: udom \<rightarrow> 'a u)"
unfolding emb_u_def prj_u_def defl_u_def
by (rule cast_liftdefl)
qed
end
lemma DEFL_u: "DEFL('a::predomain u) = LIFTDEFL('a)"
by (rule defl_u_def)
subsection {* Continuous function space is a domain *}
text {* TODO: Allow argument type to be a predomain. *}
instantiation cfun :: ("domain", "domain") liftdomain
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
"defl (t::('a \<rightarrow> 'b) itself) = cfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
definition
"(liftemb :: ('a \<rightarrow> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
definition
"(liftprj :: udom \<rightarrow> ('a \<rightarrow> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx"
definition
"liftdefl (t::('a \<rightarrow> 'b) itself) = u_defl\<cdot>DEFL('a \<rightarrow> 'b)"
instance
using liftemb_cfun_def liftprj_cfun_def liftdefl_cfun_def
proof (rule liftdomain_class_intro)
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)
show "cast\<cdot>DEFL('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
unfolding emb_cfun_def prj_cfun_def defl_cfun_def cast_cfun_defl
by (simp add: cast_DEFL oo_def cfun_eq_iff cfun_map_map)
qed
end
lemma DEFL_cfun:
"DEFL('a::domain \<rightarrow> 'b::domain) = cfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
by (rule defl_cfun_def)
subsection {* Cartesian product is a domain *}
text {*
Types @{typ "('a * 'b) u"} and @{typ "'a u \<otimes> 'b u"} are isomorphic.
*}
definition
"encode_prod_u = (\<Lambda>(up\<cdot>(x, y)). (:up\<cdot>x, up\<cdot>y:))"
definition
"decode_prod_u = (\<Lambda>(:up\<cdot>x, up\<cdot>y:). up\<cdot>(x, y))"
lemma decode_encode_prod_u [simp]: "decode_prod_u\<cdot>(encode_prod_u\<cdot>x) = x"
unfolding encode_prod_u_def decode_prod_u_def
by (case_tac x, simp, rename_tac y, case_tac y, simp)
lemma encode_decode_prod_u [simp]: "encode_prod_u\<cdot>(decode_prod_u\<cdot>y) = y"
unfolding encode_prod_u_def decode_prod_u_def
apply (case_tac y, simp, rename_tac a b)
apply (case_tac a, simp, case_tac b, simp, simp)
done
instantiation prod :: (predomain, predomain) predomain
begin
definition
"liftemb =
(udom_emb sprod_approx oo sprod_map\<cdot>emb\<cdot>emb) oo encode_prod_u"
definition
"liftprj =
decode_prod_u oo (sprod_map\<cdot>prj\<cdot>prj oo udom_prj sprod_approx)"
definition
"liftdefl (t::('a \<times> 'b) itself) = sprod_defl\<cdot>DEFL('a u)\<cdot>DEFL('b u)"
instance proof
have "ep_pair encode_prod_u decode_prod_u"
by (rule ep_pair.intro, simp_all)
thus "ep_pair liftemb (liftprj :: udom \<rightarrow> ('a \<times> 'b) u)"
unfolding liftemb_prod_def liftprj_prod_def
apply (rule ep_pair_comp)
apply (rule ep_pair_comp)
apply (intro ep_pair_sprod_map ep_pair_emb_prj)
apply (rule ep_pair_udom [OF sprod_approx])
done
show "cast\<cdot>LIFTDEFL('a \<times> 'b) = liftemb oo (liftprj :: udom \<rightarrow> ('a \<times> 'b) u)"
unfolding liftemb_prod_def liftprj_prod_def liftdefl_prod_def
by (simp add: cast_sprod_defl cast_DEFL cfcomp1 sprod_map_map)
qed
end
instantiation prod :: ("domain", "domain") "domain"
begin
definition
"emb = udom_emb prod_approx oo cprod_map\<cdot>emb\<cdot>emb"
definition
"prj = cprod_map\<cdot>prj\<cdot>prj oo udom_prj prod_approx"
definition
"defl (t::('a \<times> 'b) itself) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<times> 'b)"
unfolding emb_prod_def prj_prod_def
using ep_pair_udom [OF prod_approx]
by (intro ep_pair_comp ep_pair_cprod_map ep_pair_emb_prj)
next
show "cast\<cdot>DEFL('a \<times> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<times> 'b)"
unfolding emb_prod_def prj_prod_def defl_prod_def cast_prod_defl
by (simp add: cast_DEFL oo_def cfun_eq_iff cprod_map_map)
qed
end
lemma DEFL_prod:
"DEFL('a::domain \<times> 'b::domain) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
by (rule defl_prod_def)
lemma LIFTDEFL_prod:
"LIFTDEFL('a::predomain \<times> 'b::predomain) = sprod_defl\<cdot>DEFL('a u)\<cdot>DEFL('b u)"
by (rule liftdefl_prod_def)
subsection {* Strict product is a domain *}
instantiation sprod :: ("domain", "domain") liftdomain
begin
definition
"emb = udom_emb sprod_approx oo sprod_map\<cdot>emb\<cdot>emb"
definition
"prj = sprod_map\<cdot>prj\<cdot>prj oo udom_prj sprod_approx"
definition
"defl (t::('a \<otimes> 'b) itself) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
definition
"(liftemb :: ('a \<otimes> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
definition
"(liftprj :: udom \<rightarrow> ('a \<otimes> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx"
definition
"liftdefl (t::('a \<otimes> 'b) itself) = u_defl\<cdot>DEFL('a \<otimes> 'b)"
instance
using liftemb_sprod_def liftprj_sprod_def liftdefl_sprod_def
proof (rule liftdomain_class_intro)
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
unfolding emb_sprod_def prj_sprod_def
using ep_pair_udom [OF sprod_approx]
by (intro ep_pair_comp ep_pair_sprod_map ep_pair_emb_prj)
next
show "cast\<cdot>DEFL('a \<otimes> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
unfolding emb_sprod_def prj_sprod_def defl_sprod_def cast_sprod_defl
by (simp add: cast_DEFL oo_def cfun_eq_iff sprod_map_map)
qed
end
lemma DEFL_sprod:
"DEFL('a::domain \<otimes> 'b::domain) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
by (rule defl_sprod_def)
subsection {* Countable discrete cpos are predomains *}
definition discr_approx :: "nat \<Rightarrow> 'a::countable discr u \<rightarrow> 'a discr u"
where "discr_approx = (\<lambda>i. \<Lambda>(up\<cdot>x). if to_nat (undiscr x) < i then up\<cdot>x else \<bottom>)"
lemma chain_discr_approx [simp]: "chain discr_approx"
unfolding discr_approx_def
by (rule chainI, simp add: monofun_cfun monofun_LAM)
lemma lub_discr_approx [simp]: "(\<Squnion>i. discr_approx i) = ID"
apply (rule cfun_eqI)
apply (simp add: contlub_cfun_fun)
apply (simp add: discr_approx_def)
apply (case_tac x, simp)
apply (rule thelubI)
apply (rule is_lubI)
apply (rule ub_rangeI, simp)
apply (drule ub_rangeD)
apply (erule rev_below_trans)
apply simp
apply (rule lessI)
done
lemma inj_on_undiscr [simp]: "inj_on undiscr A"
using Discr_undiscr by (rule inj_on_inverseI)
lemma finite_deflation_discr_approx: "finite_deflation (discr_approx i)"
proof
fix x :: "'a discr u"
show "discr_approx i\<cdot>x \<sqsubseteq> x"
unfolding discr_approx_def
by (cases x, simp, simp)
show "discr_approx i\<cdot>(discr_approx i\<cdot>x) = discr_approx i\<cdot>x"
unfolding discr_approx_def
by (cases x, simp, simp)
show "finite {x::'a discr u. discr_approx i\<cdot>x = x}"
proof (rule finite_subset)
let ?S = "insert (\<bottom>::'a discr u) ((\<lambda>x. up\<cdot>x) ` undiscr -` to_nat -` {..<i})"
show "{x::'a discr u. discr_approx i\<cdot>x = x} \<subseteq> ?S"
unfolding discr_approx_def
by (rule subsetI, case_tac x, simp, simp split: split_if_asm)
show "finite ?S"
by (simp add: finite_vimageI)
qed
qed
lemma discr_approx: "approx_chain discr_approx"
using chain_discr_approx lub_discr_approx finite_deflation_discr_approx
by (rule approx_chain.intro)
instantiation discr :: (countable) predomain
begin
definition
"liftemb = udom_emb discr_approx"
definition
"liftprj = udom_prj discr_approx"
definition
"liftdefl (t::'a discr itself) =
(\<Squnion>i. defl_principal (Abs_fin_defl (liftemb oo discr_approx i oo liftprj)))"
instance proof
show "ep_pair liftemb (liftprj :: udom \<rightarrow> 'a discr u)"
unfolding liftemb_discr_def liftprj_discr_def
by (rule ep_pair_udom [OF discr_approx])
show "cast\<cdot>LIFTDEFL('a discr) = liftemb oo (liftprj :: udom \<rightarrow> 'a discr u)"
unfolding liftemb_discr_def liftprj_discr_def liftdefl_discr_def
apply (subst contlub_cfun_arg)
apply (rule chainI)
apply (rule defl.principal_mono)
apply (simp add: below_fin_defl_def)
apply (simp add: Abs_fin_defl_inverse
ep_pair.finite_deflation_e_d_p [OF ep_pair_udom [OF discr_approx]]
approx_chain.finite_deflation_approx [OF discr_approx])
apply (intro monofun_cfun below_refl)
apply (rule chainE)
apply (rule chain_discr_approx)
apply (subst cast_defl_principal)
apply (simp add: Abs_fin_defl_inverse
ep_pair.finite_deflation_e_d_p [OF ep_pair_udom [OF discr_approx]]
approx_chain.finite_deflation_approx [OF discr_approx])
apply (simp add: lub_distribs)
done
qed
end
subsection {* Strict sum is a domain *}
instantiation ssum :: ("domain", "domain") liftdomain
begin
definition
"emb = udom_emb ssum_approx oo ssum_map\<cdot>emb\<cdot>emb"
definition
"prj = ssum_map\<cdot>prj\<cdot>prj oo udom_prj ssum_approx"
definition
"defl (t::('a \<oplus> 'b) itself) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
definition
"(liftemb :: ('a \<oplus> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
definition
"(liftprj :: udom \<rightarrow> ('a \<oplus> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx"
definition
"liftdefl (t::('a \<oplus> 'b) itself) = u_defl\<cdot>DEFL('a \<oplus> 'b)"
instance
using liftemb_ssum_def liftprj_ssum_def liftdefl_ssum_def
proof (rule liftdomain_class_intro)
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
unfolding emb_ssum_def prj_ssum_def
using ep_pair_udom [OF ssum_approx]
by (intro ep_pair_comp ep_pair_ssum_map ep_pair_emb_prj)
show "cast\<cdot>DEFL('a \<oplus> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
unfolding emb_ssum_def prj_ssum_def defl_ssum_def cast_ssum_defl
by (simp add: cast_DEFL oo_def cfun_eq_iff ssum_map_map)
qed
end
lemma DEFL_ssum:
"DEFL('a::domain \<oplus> 'b::domain) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
by (rule defl_ssum_def)
subsection {* Lifted countable types are bifinite domains *}
instantiation lift :: (countable) liftdomain
begin
definition
"emb = emb oo (\<Lambda> x. Rep_lift x)"
definition
"prj = (\<Lambda> y. Abs_lift y) oo prj"
definition
"defl (t::'a lift itself) = DEFL('a discr u)"
definition
"(liftemb :: 'a lift u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
definition
"(liftprj :: udom \<rightarrow> 'a lift u) = u_map\<cdot>prj oo udom_prj u_approx"
definition
"liftdefl (t::'a lift itself) = u_defl\<cdot>DEFL('a lift)"
instance
using liftemb_lift_def liftprj_lift_def liftdefl_lift_def
proof (rule liftdomain_class_intro)
note [simp] = cont_Rep_lift cont_Abs_lift Rep_lift_inverse Abs_lift_inverse
have "ep_pair (\<Lambda>(x::'a lift). Rep_lift x) (\<Lambda> y. Abs_lift y)"
by (simp add: ep_pair_def)
thus "ep_pair emb (prj :: udom \<rightarrow> 'a lift)"
unfolding emb_lift_def prj_lift_def
using ep_pair_emb_prj by (rule ep_pair_comp)
show "cast\<cdot>DEFL('a lift) = emb oo (prj :: udom \<rightarrow> 'a lift)"
unfolding emb_lift_def prj_lift_def defl_lift_def cast_DEFL
by (simp add: cfcomp1)
qed
end
end