--- a/src/HOL/HOLCF/Bifinite.thy Fri Dec 17 22:00:54 2010 +0100
+++ b/src/HOL/HOLCF/Bifinite.thy Mon Dec 20 08:55:36 2010 +0100
@@ -2,659 +2,107 @@
Author: Brian Huffman
*)
-header {* Bifinite domains *}
+header {* Profinite and bifinite cpos *}
theory Bifinite
-imports Algebraic Map_Functions Countable
+imports Map_Functions Countable
+begin
+
+default_sort cpo
+
+subsection {* Chains of finite deflations *}
+
+locale approx_chain =
+ fixes approx :: "nat \<Rightarrow> 'a \<rightarrow> 'a"
+ assumes chain_approx [simp]: "chain (\<lambda>i. approx i)"
+ assumes lub_approx [simp]: "(\<Squnion>i. approx i) = ID"
+ assumes finite_deflation_approx [simp]: "\<And>i. finite_deflation (approx i)"
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
+lemma deflation_approx: "deflation (approx i)"
+using finite_deflation_approx by (rule finite_deflation_imp_deflation)
-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 sfun_approx :: "nat \<Rightarrow> (udom \<rightarrow>! udom) \<rightarrow> (udom \<rightarrow>! udom)"
- where "sfun_approx = (\<lambda>i. sfun_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 approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
+using deflation_approx by (rule deflation.idem)
-lemma sfun_approx: "approx_chain sfun_approx"
-using sfun_map_ID finite_deflation_sfun_map
-unfolding sfun_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 approx_below: "approx i\<cdot>x \<sqsubseteq> x"
+using deflation_approx by (rule deflation.below)
-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 sfun_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
- where "sfun_defl = defl_fun2 sfun_approx sfun_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_sfun_defl:
- "cast\<cdot>(sfun_defl\<cdot>A\<cdot>B) =
- udom_emb sfun_approx oo sfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj sfun_approx"
-using sfun_approx finite_deflation_sfun_map
-unfolding sfun_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)
+lemma finite_range_approx: "finite (range (\<lambda>x. approx i\<cdot>x))"
+apply (rule finite_deflation.finite_range)
+apply (rule finite_deflation_approx)
+done
-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. *}
+lemma compact_approx: "compact (approx n\<cdot>x)"
+apply (rule finite_deflation.compact)
+apply (rule finite_deflation_approx)
+done
-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 {* Class instance proofs *}
-
-subsubsection {* Universal 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
+lemma compact_eq_approx: "compact x \<Longrightarrow> \<exists>i. approx i\<cdot>x = x"
+by (rule admD2, simp_all)
end
-subsubsection {* Lifted cpo *}
-
-instantiation u :: (predomain) liftdomain
-begin
-
-definition
- "emb = liftemb"
-
-definition
- "prj = liftprj"
+subsection {* Omega-profinite and bifinite domains *}
-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)"
+class bifinite = pcpo +
+ assumes bifinite: "\<exists>(a::nat \<Rightarrow> 'a \<rightarrow> 'a). approx_chain a"
-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)
-
-subsubsection {* Strict function space *}
+class profinite = cpo +
+ assumes profinite: "\<exists>(a::nat \<Rightarrow> 'a\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>). approx_chain a"
-instantiation sfun :: ("domain", "domain") liftdomain
-begin
-
-definition
- "emb = udom_emb sfun_approx oo sfun_map\<cdot>prj\<cdot>emb"
-
-definition
- "prj = sfun_map\<cdot>emb\<cdot>prj oo udom_prj sfun_approx"
-
-definition
- "defl (t::('a \<rightarrow>! 'b) itself) = sfun_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)"
+subsection {* Building approx chains *}
-instance
-using liftemb_sfun_def liftprj_sfun_def liftdefl_sfun_def
-proof (rule liftdomain_class_intro)
- show "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow>! 'b)"
- unfolding emb_sfun_def prj_sfun_def
- using ep_pair_udom [OF sfun_approx]
- by (intro ep_pair_comp ep_pair_sfun_map ep_pair_emb_prj)
- show "cast\<cdot>DEFL('a \<rightarrow>! 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow>! 'b)"
- unfolding emb_sfun_def prj_sfun_def defl_sfun_def cast_sfun_defl
- by (simp add: cast_DEFL oo_def sfun_eq_iff sfun_map_map)
-qed
-
-end
-
-lemma DEFL_sfun:
- "DEFL('a::domain \<rightarrow>! 'b::domain) = sfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
-by (rule defl_sfun_def)
-
-subsubsection {* Continuous function space *}
-
-text {*
- Types @{typ "'a \<rightarrow> 'b"} and @{typ "'a u \<rightarrow>! 'b"} are isomorphic.
-*}
-
-definition
- "encode_cfun = (\<Lambda> f. sfun_abs\<cdot>(fup\<cdot>f))"
-
-definition
- "decode_cfun = (\<Lambda> g x. sfun_rep\<cdot>g\<cdot>(up\<cdot>x))"
-
-lemma decode_encode_cfun [simp]: "decode_cfun\<cdot>(encode_cfun\<cdot>x) = x"
-unfolding encode_cfun_def decode_cfun_def
-by (simp add: eta_cfun)
-
-lemma encode_decode_cfun [simp]: "encode_cfun\<cdot>(decode_cfun\<cdot>y) = y"
-unfolding encode_cfun_def decode_cfun_def
-apply (simp add: sfun_eq_iff strictify_cancel)
-apply (rule cfun_eqI, case_tac x, simp_all)
-done
-
-instantiation cfun :: (predomain, "domain") liftdomain
-begin
-
-definition
- "emb = emb oo encode_cfun"
-
-definition
- "prj = decode_cfun oo prj"
-
-definition
- "defl (t::('a \<rightarrow> 'b) itself) = DEFL('a u \<rightarrow>! '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)
- have "ep_pair encode_cfun decode_cfun"
- by (rule ep_pair.intro, simp_all)
- thus "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
- unfolding emb_cfun_def prj_cfun_def
- using ep_pair_emb_prj by (rule ep_pair_comp)
- 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
- by (simp add: cast_DEFL cfcomp1)
+lemma approx_chain_iso:
+ assumes a: "approx_chain a"
+ assumes [simp]: "\<And>x. f\<cdot>(g\<cdot>x) = x"
+ assumes [simp]: "\<And>y. g\<cdot>(f\<cdot>y) = y"
+ shows "approx_chain (\<lambda>i. f oo a i oo g)"
+proof -
+ have 1: "f oo g = ID" by (simp add: cfun_eqI)
+ have 2: "ep_pair f g" by (simp add: ep_pair_def)
+ from 1 2 show ?thesis
+ using a unfolding approx_chain_def
+ by (simp add: lub_APP ep_pair.finite_deflation_e_d_p)
qed
-end
-
-lemma DEFL_cfun:
- "DEFL('a::predomain \<rightarrow> 'b::domain) = DEFL('a u \<rightarrow>! 'b)"
-by (rule defl_cfun_def)
-
-subsubsection {* Strict product *}
-
-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)"
+lemma approx_chain_u_map:
+ assumes "approx_chain a"
+ shows "approx_chain (\<lambda>i. u_map\<cdot>(a i))"
+ using assms unfolding approx_chain_def
+ by (simp add: lub_APP u_map_ID finite_deflation_u_map)
-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)
-
-subsubsection {* Cartesian product *}
-
-text {*
- Types @{typ "('a * 'b) u"} and @{typ "'a u \<otimes> 'b u"} are isomorphic.
-*}
+lemma approx_chain_sfun_map:
+ assumes "approx_chain a" and "approx_chain b"
+ shows "approx_chain (\<lambda>i. sfun_map\<cdot>(a i)\<cdot>(b i))"
+ using assms unfolding approx_chain_def
+ by (simp add: lub_APP sfun_map_ID finite_deflation_sfun_map)
-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 = emb oo encode_prod_u"
-
-definition
- "liftprj = decode_prod_u oo prj"
-
-definition
- "liftdefl (t::('a \<times> 'b) itself) = DEFL('a\<^sub>\<bottom> \<otimes> 'b\<^sub>\<bottom>)"
+lemma approx_chain_sprod_map:
+ assumes "approx_chain a" and "approx_chain b"
+ shows "approx_chain (\<lambda>i. sprod_map\<cdot>(a i)\<cdot>(b i))"
+ using assms unfolding approx_chain_def
+ by (simp add: lub_APP sprod_map_ID finite_deflation_sprod_map)
-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
- using ep_pair_emb_prj by (rule ep_pair_comp)
- 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_DEFL cfcomp1)
-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 approx_chain_ssum_map:
+ assumes "approx_chain a" and "approx_chain b"
+ shows "approx_chain (\<lambda>i. ssum_map\<cdot>(a i)\<cdot>(b i))"
+ using assms unfolding approx_chain_def
+ by (simp add: lub_APP ssum_map_ID finite_deflation_ssum_map)
-lemma LIFTDEFL_prod:
- "LIFTDEFL('a::predomain \<times> 'b::predomain) = DEFL('a u \<otimes> 'b u)"
-by (rule liftdefl_prod_def)
-
-subsubsection {* Unit type *}
-
-instantiation unit :: liftdomain
-begin
-
-definition
- "emb = (\<bottom> :: unit \<rightarrow> udom)"
-
-definition
- "prj = (\<bottom> :: udom \<rightarrow> unit)"
-
-definition
- "defl (t::unit itself) = \<bottom>"
-
-definition
- "(liftemb :: unit u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
+lemma approx_chain_cfun_map:
+ assumes "approx_chain a" and "approx_chain b"
+ shows "approx_chain (\<lambda>i. cfun_map\<cdot>(a i)\<cdot>(b i))"
+ using assms unfolding approx_chain_def
+ by (simp add: lub_APP cfun_map_ID finite_deflation_cfun_map)
-definition
- "(liftprj :: udom \<rightarrow> unit u) = u_map\<cdot>prj oo udom_prj u_approx"
-
-definition
- "liftdefl (t::unit itself) = u_defl\<cdot>DEFL(unit)"
+lemma approx_chain_prod_map:
+ assumes "approx_chain a" and "approx_chain b"
+ shows "approx_chain (\<lambda>i. prod_map\<cdot>(a i)\<cdot>(b i))"
+ using assms unfolding approx_chain_def
+ by (simp add: lub_APP prod_map_ID finite_deflation_prod_map)
-instance
-using liftemb_unit_def liftprj_unit_def liftdefl_unit_def
-proof (rule liftdomain_class_intro)
- show "ep_pair emb (prj :: udom \<rightarrow> unit)"
- unfolding emb_unit_def prj_unit_def
- by (simp add: ep_pair.intro)
-next
- show "cast\<cdot>DEFL(unit) = emb oo (prj :: udom \<rightarrow> unit)"
- unfolding emb_unit_def prj_unit_def defl_unit_def by simp
-qed
-
-end
-
-subsubsection {* Discrete cpo *}
+text {* Approx chains for countable discrete types. *}
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>)"
@@ -704,123 +152,124 @@
using chain_discr_approx lub_discr_approx finite_deflation_discr_approx
by (rule approx_chain.intro)
-instantiation discr :: (countable) predomain
-begin
+subsection {* Class instance proofs *}
+
+instance bifinite \<subseteq> profinite
+proof
+ show "\<exists>(a::nat \<Rightarrow> 'a\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>). approx_chain a"
+ using bifinite [where 'a='a]
+ by (fast intro!: approx_chain_u_map)
+qed
+
+instance u :: (profinite) bifinite
+ by default (rule profinite)
-definition
- "liftemb = udom_emb discr_approx"
+text {*
+ Types @{typ "'a \<rightarrow> 'b"} and @{typ "'a u \<rightarrow>! 'b"} are isomorphic.
+*}
+
+definition "encode_cfun = (\<Lambda> f. sfun_abs\<cdot>(fup\<cdot>f))"
+
+definition "decode_cfun = (\<Lambda> g x. sfun_rep\<cdot>g\<cdot>(up\<cdot>x))"
+
+lemma decode_encode_cfun [simp]: "decode_cfun\<cdot>(encode_cfun\<cdot>x) = x"
+unfolding encode_cfun_def decode_cfun_def
+by (simp add: eta_cfun)
-definition
- "liftprj = udom_prj discr_approx"
+lemma encode_decode_cfun [simp]: "encode_cfun\<cdot>(decode_cfun\<cdot>y) = y"
+unfolding encode_cfun_def decode_cfun_def
+apply (simp add: sfun_eq_iff strictify_cancel)
+apply (rule cfun_eqI, case_tac x, simp_all)
+done
-definition
- "liftdefl (t::'a discr itself) =
- (\<Squnion>i. defl_principal (Abs_fin_defl (liftemb oo discr_approx i oo liftprj)))"
+instance cfun :: (profinite, bifinite) bifinite
+proof
+ obtain a :: "nat \<Rightarrow> 'a\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>" where a: "approx_chain a"
+ using profinite ..
+ obtain b :: "nat \<Rightarrow> 'b \<rightarrow> 'b" where b: "approx_chain b"
+ using bifinite ..
+ have "approx_chain (\<lambda>i. decode_cfun oo sfun_map\<cdot>(a i)\<cdot>(b i) oo encode_cfun)"
+ using a b by (simp add: approx_chain_iso approx_chain_sfun_map)
+ thus "\<exists>(a::nat \<Rightarrow> ('a \<rightarrow> 'b) \<rightarrow> ('a \<rightarrow> 'b)). approx_chain a"
+ by - (rule exI)
+qed
+
+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)
-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
+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
+
+instance prod :: (profinite, profinite) profinite
+proof
+ obtain a :: "nat \<Rightarrow> 'a\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>" where a: "approx_chain a"
+ using profinite ..
+ obtain b :: "nat \<Rightarrow> 'b\<^sub>\<bottom> \<rightarrow> 'b\<^sub>\<bottom>" where b: "approx_chain b"
+ using profinite ..
+ have "approx_chain (\<lambda>i. decode_prod_u oo sprod_map\<cdot>(a i)\<cdot>(b i) oo encode_prod_u)"
+ using a b by (simp add: approx_chain_iso approx_chain_sprod_map)
+ thus "\<exists>(a::nat \<Rightarrow> ('a \<times> 'b)\<^sub>\<bottom> \<rightarrow> ('a \<times> 'b)\<^sub>\<bottom>). approx_chain a"
+ by - (rule exI)
+qed
+
+instance prod :: (bifinite, bifinite) bifinite
+proof
+ show "\<exists>(a::nat \<Rightarrow> ('a \<times> 'b) \<rightarrow> ('a \<times> 'b)). approx_chain a"
+ using bifinite [where 'a='a] and bifinite [where 'a='b]
+ by (fast intro!: approx_chain_prod_map)
+qed
+
+instance sfun :: (bifinite, bifinite) bifinite
+proof
+ show "\<exists>(a::nat \<Rightarrow> ('a \<rightarrow>! 'b) \<rightarrow> ('a \<rightarrow>! 'b)). approx_chain a"
+ using bifinite [where 'a='a] and bifinite [where 'a='b]
+ by (fast intro!: approx_chain_sfun_map)
+qed
+
+instance sprod :: (bifinite, bifinite) bifinite
+proof
+ show "\<exists>(a::nat \<Rightarrow> ('a \<otimes> 'b) \<rightarrow> ('a \<otimes> 'b)). approx_chain a"
+ using bifinite [where 'a='a] and bifinite [where 'a='b]
+ by (fast intro!: approx_chain_sprod_map)
+qed
+
+instance ssum :: (bifinite, bifinite) bifinite
+proof
+ show "\<exists>(a::nat \<Rightarrow> ('a \<oplus> 'b) \<rightarrow> ('a \<oplus> 'b)). approx_chain a"
+ using bifinite [where 'a='a] and bifinite [where 'a='b]
+ by (fast intro!: approx_chain_ssum_map)
+qed
+
+lemma approx_chain_unit: "approx_chain (\<bottom> :: nat \<Rightarrow> unit \<rightarrow> unit)"
+by (simp add: approx_chain_def cfun_eq_iff finite_deflation_UU)
+
+instance unit :: bifinite
+ by default (fast intro!: approx_chain_unit)
+
+instance discr :: (countable) profinite
+ by default (fast intro!: discr_approx)
+
+instance lift :: (countable) bifinite
+proof
+ note [simp] = cont_Abs_lift cont_Rep_lift Rep_lift_inverse Abs_lift_inverse
+ obtain a :: "nat \<Rightarrow> ('a discr)\<^sub>\<bottom> \<rightarrow> ('a discr)\<^sub>\<bottom>" where a: "approx_chain a"
+ using profinite ..
+ hence "approx_chain (\<lambda>i. (\<Lambda> y. Abs_lift y) oo a i oo (\<Lambda> x. Rep_lift x))"
+ by (rule approx_chain_iso) simp_all
+ thus "\<exists>(a::nat \<Rightarrow> 'a lift \<rightarrow> 'a lift). approx_chain a"
+ by - (rule exI)
qed
end
-
-subsubsection {* Strict sum *}
-
-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)
-
-subsubsection {* Lifted HOL type *}
-
-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