src/HOLCF/Bifinite.thy
author huffman
Wed Nov 10 08:18:32 2010 -0800 (2010-11-10)
changeset 40494 db8a09daba7b
parent 40493 c45a3f8a86ea
child 40497 d2e876d6da8c
permissions -rw-r--r--
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
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(*  Title:      HOLCF/Bifinite.thy
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    Author:     Brian Huffman
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*)
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header {* Bifinite domains *}
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theory Bifinite
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imports Algebraic Cprod Sprod Ssum Up Lift One Tr Countable
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begin
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subsection {* Class of bifinite domains *}
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text {*
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  We define a bifinite domain as a pcpo that is isomorphic to some
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  algebraic deflation over the universal domain.
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  A predomain is a cpo that, when lifted, becomes bifinite.
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*}
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class predomain = cpo +
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  fixes liftdefl :: "('a::cpo) itself \<Rightarrow> defl"
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  fixes liftemb :: "'a\<^sub>\<bottom> \<rightarrow> udom"
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  fixes liftprj :: "udom \<rightarrow> 'a\<^sub>\<bottom>"
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  assumes predomain_ep: "ep_pair liftemb liftprj"
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  assumes cast_liftdefl: "cast\<cdot>(liftdefl TYPE('a::cpo)) = liftemb oo liftprj"
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syntax "_LIFTDEFL" :: "type \<Rightarrow> logic"  ("(1LIFTDEFL/(1'(_')))")
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translations "LIFTDEFL('t)" \<rightleftharpoons> "CONST liftdefl TYPE('t)"
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class bifinite = predomain + pcpo +
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  fixes emb :: "'a::cpo \<rightarrow> udom"
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  fixes prj :: "udom \<rightarrow> 'a::cpo"
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  fixes defl :: "'a itself \<Rightarrow> defl"
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  assumes ep_pair_emb_prj: "ep_pair emb prj"
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  assumes cast_DEFL: "cast\<cdot>(defl TYPE('a)) = emb oo prj"
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syntax "_DEFL" :: "type \<Rightarrow> defl"  ("(1DEFL/(1'(_')))")
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translations "DEFL('t)" \<rightleftharpoons> "CONST defl TYPE('t)"
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interpretation bifinite: pcpo_ep_pair emb prj
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  unfolding pcpo_ep_pair_def
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  by (rule ep_pair_emb_prj)
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lemmas emb_inverse = bifinite.e_inverse
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lemmas emb_prj_below = bifinite.e_p_below
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lemmas emb_eq_iff = bifinite.e_eq_iff
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lemmas emb_strict = bifinite.e_strict
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lemmas prj_strict = bifinite.p_strict
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subsection {* Bifinite domains have a countable compact basis *}
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text {*
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  Eventually it should be possible to generalize this to an unpointed
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  variant of the bifinite class.
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*}
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interpretation compact_basis:
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  ideal_completion below Rep_compact_basis "approximants::'a::bifinite \<Rightarrow> _"
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proof -
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  obtain Y where Y: "\<forall>i. Y i \<sqsubseteq> Y (Suc i)"
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  and DEFL: "DEFL('a) = (\<Squnion>i. defl_principal (Y i))"
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    by (rule defl.obtain_principal_chain)
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  def approx \<equiv> "\<lambda>i. (prj oo cast\<cdot>(defl_principal (Y i)) oo emb) :: 'a \<rightarrow> 'a"
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  interpret defl_approx: approx_chain approx
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  proof (rule approx_chain.intro)
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    show "chain (\<lambda>i. approx i)"
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      unfolding approx_def by (simp add: Y)
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    show "(\<Squnion>i. approx i) = ID"
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      unfolding approx_def
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      by (simp add: lub_distribs Y DEFL [symmetric] cast_DEFL cfun_eq_iff)
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    show "\<And>i. finite_deflation (approx i)"
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      unfolding approx_def
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      apply (rule bifinite.finite_deflation_p_d_e)
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      apply (rule finite_deflation_cast)
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      apply (rule defl.compact_principal)
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      apply (rule below_trans [OF monofun_cfun_fun])
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      apply (rule is_ub_thelub, simp add: Y)
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      apply (simp add: lub_distribs Y DEFL [symmetric] cast_DEFL)
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      done
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  qed
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  (* FIXME: why does show ?thesis fail here? *)
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  show "ideal_completion below Rep_compact_basis (approximants::'a \<Rightarrow> _)" ..
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qed
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subsection {* Chains of approx functions *}
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definition u_approx :: "nat \<Rightarrow> udom\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"
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  where "u_approx = (\<lambda>i. u_map\<cdot>(udom_approx i))"
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definition cfun_approx :: "nat \<Rightarrow> (udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom)"
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  where "cfun_approx = (\<lambda>i. cfun_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition prod_approx :: "nat \<Rightarrow> udom \<times> udom \<rightarrow> udom \<times> udom"
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  where "prod_approx = (\<lambda>i. cprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition sprod_approx :: "nat \<Rightarrow> udom \<otimes> udom \<rightarrow> udom \<otimes> udom"
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  where "sprod_approx = (\<lambda>i. sprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition ssum_approx :: "nat \<Rightarrow> udom \<oplus> udom \<rightarrow> udom \<oplus> udom"
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  where "ssum_approx = (\<lambda>i. ssum_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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lemma approx_chain_lemma1:
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  assumes "m\<cdot>ID = ID"
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  assumes "\<And>d. finite_deflation d \<Longrightarrow> finite_deflation (m\<cdot>d)"
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  shows "approx_chain (\<lambda>i. m\<cdot>(udom_approx i))"
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by (rule approx_chain.intro)
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   (simp_all add: lub_distribs finite_deflation_udom_approx assms)
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lemma approx_chain_lemma2:
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  assumes "m\<cdot>ID\<cdot>ID = ID"
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  assumes "\<And>a b. \<lbrakk>finite_deflation a; finite_deflation b\<rbrakk>
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    \<Longrightarrow> finite_deflation (m\<cdot>a\<cdot>b)"
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  shows "approx_chain (\<lambda>i. m\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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by (rule approx_chain.intro)
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   (simp_all add: lub_distribs finite_deflation_udom_approx assms)
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lemma u_approx: "approx_chain u_approx"
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using u_map_ID finite_deflation_u_map
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unfolding u_approx_def by (rule approx_chain_lemma1)
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lemma cfun_approx: "approx_chain cfun_approx"
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using cfun_map_ID finite_deflation_cfun_map
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unfolding cfun_approx_def by (rule approx_chain_lemma2)
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lemma prod_approx: "approx_chain prod_approx"
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using cprod_map_ID finite_deflation_cprod_map
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unfolding prod_approx_def by (rule approx_chain_lemma2)
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lemma sprod_approx: "approx_chain sprod_approx"
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using sprod_map_ID finite_deflation_sprod_map
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unfolding sprod_approx_def by (rule approx_chain_lemma2)
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lemma ssum_approx: "approx_chain ssum_approx"
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using ssum_map_ID finite_deflation_ssum_map
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unfolding ssum_approx_def by (rule approx_chain_lemma2)
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subsection {* Type combinators *}
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definition
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  defl_fun1 ::
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    "(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) \<Rightarrow> (defl \<rightarrow> defl)"
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where
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  "defl_fun1 approx f =
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    defl.basis_fun (\<lambda>a.
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      defl_principal (Abs_fin_defl
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        (udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)))"
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definition
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  defl_fun2 ::
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    "(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a))
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      \<Rightarrow> (defl \<rightarrow> defl \<rightarrow> defl)"
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where
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  "defl_fun2 approx f =
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    defl.basis_fun (\<lambda>a.
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      defl.basis_fun (\<lambda>b.
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        defl_principal (Abs_fin_defl
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          (udom_emb approx oo
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            f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx))))"
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lemma cast_defl_fun1:
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  assumes approx: "approx_chain approx"
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  assumes f: "\<And>a. finite_deflation a \<Longrightarrow> finite_deflation (f\<cdot>a)"
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  shows "cast\<cdot>(defl_fun1 approx f\<cdot>A) = udom_emb approx oo f\<cdot>(cast\<cdot>A) oo udom_prj approx"
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proof -
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  have 1: "\<And>a. finite_deflation
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        (udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)"
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    apply (rule ep_pair.finite_deflation_e_d_p)
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    apply (rule approx_chain.ep_pair_udom [OF approx])
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    apply (rule f, rule finite_deflation_Rep_fin_defl)
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    done
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  show ?thesis
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    by (induct A rule: defl.principal_induct, simp)
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       (simp only: defl_fun1_def
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                   defl.basis_fun_principal
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                   defl.basis_fun_mono
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                   defl.principal_mono
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                   Abs_fin_defl_mono [OF 1 1]
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                   monofun_cfun below_refl
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                   Rep_fin_defl_mono
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                   cast_defl_principal
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                   Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
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qed
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lemma cast_defl_fun2:
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  assumes approx: "approx_chain approx"
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  assumes f: "\<And>a b. finite_deflation a \<Longrightarrow> finite_deflation b \<Longrightarrow>
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                finite_deflation (f\<cdot>a\<cdot>b)"
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  shows "cast\<cdot>(defl_fun2 approx f\<cdot>A\<cdot>B) =
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    udom_emb approx oo f\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj approx"
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proof -
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  have 1: "\<And>a b. finite_deflation (udom_emb approx oo
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      f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx)"
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    apply (rule ep_pair.finite_deflation_e_d_p)
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    apply (rule ep_pair_udom [OF approx])
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    apply (rule f, (rule finite_deflation_Rep_fin_defl)+)
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    done
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  show ?thesis
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    by (induct A B rule: defl.principal_induct2, simp, simp)
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       (simp only: defl_fun2_def
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                   defl.basis_fun_principal
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                   defl.basis_fun_mono
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                   defl.principal_mono
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                   Abs_fin_defl_mono [OF 1 1]
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                   monofun_cfun below_refl
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                   Rep_fin_defl_mono
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                   cast_defl_principal
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                   Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
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qed
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definition u_defl :: "defl \<rightarrow> defl"
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  where "u_defl = defl_fun1 u_approx u_map"
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definition cfun_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
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  where "cfun_defl = defl_fun2 cfun_approx cfun_map"
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definition prod_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
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  where "prod_defl = defl_fun2 prod_approx cprod_map"
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definition sprod_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
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  where "sprod_defl = defl_fun2 sprod_approx sprod_map"
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definition ssum_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
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where "ssum_defl = defl_fun2 ssum_approx ssum_map"
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lemma cast_u_defl:
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  "cast\<cdot>(u_defl\<cdot>A) =
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    udom_emb u_approx oo u_map\<cdot>(cast\<cdot>A) oo udom_prj u_approx"
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using u_approx finite_deflation_u_map
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unfolding u_defl_def by (rule cast_defl_fun1)
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lemma cast_cfun_defl:
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  "cast\<cdot>(cfun_defl\<cdot>A\<cdot>B) =
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    udom_emb cfun_approx oo cfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj cfun_approx"
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using cfun_approx finite_deflation_cfun_map
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unfolding cfun_defl_def by (rule cast_defl_fun2)
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lemma cast_prod_defl:
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  "cast\<cdot>(prod_defl\<cdot>A\<cdot>B) = udom_emb prod_approx oo
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    cprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj prod_approx"
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using prod_approx finite_deflation_cprod_map
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unfolding prod_defl_def by (rule cast_defl_fun2)
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lemma cast_sprod_defl:
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  "cast\<cdot>(sprod_defl\<cdot>A\<cdot>B) =
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    udom_emb sprod_approx oo
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      sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo
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        udom_prj sprod_approx"
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using sprod_approx finite_deflation_sprod_map
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unfolding sprod_defl_def by (rule cast_defl_fun2)
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lemma cast_ssum_defl:
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  "cast\<cdot>(ssum_defl\<cdot>A\<cdot>B) =
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    udom_emb ssum_approx oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj ssum_approx"
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using ssum_approx finite_deflation_ssum_map
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unfolding ssum_defl_def by (rule cast_defl_fun2)
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subsection {* Lemma for proving bifinite instances *}
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text {*
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  A class of bifinite domains where @{const liftemb}, @{const liftprj},
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  and @{const liftdefl} are all defined in the standard way.
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*}
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class liftdomain = bifinite +
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  assumes liftemb_eq: "liftemb = udom_emb u_approx oo u_map\<cdot>emb"
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  assumes liftprj_eq: "liftprj = u_map\<cdot>prj oo udom_prj u_approx"
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  assumes liftdefl_eq: "liftdefl TYPE('a::cpo) = u_defl\<cdot>DEFL('a)"
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text {* Temporarily relax type constraints. *}
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setup {*
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  fold Sign.add_const_constraint
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  [ (@{const_name defl}, SOME @{typ "'a::pcpo itself \<Rightarrow> defl"})
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  , (@{const_name emb}, SOME @{typ "'a::pcpo \<rightarrow> udom"})
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  , (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"})
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  , (@{const_name liftdefl}, SOME @{typ "'a::pcpo itself \<Rightarrow> defl"})
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  , (@{const_name liftemb}, SOME @{typ "'a::pcpo u \<rightarrow> udom"})
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  , (@{const_name liftprj}, SOME @{typ "udom \<rightarrow> 'a::pcpo u"}) ]
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*}
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lemma liftdomain_class_intro:
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  assumes liftemb: "(liftemb :: 'a u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
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  assumes liftprj: "(liftprj :: udom \<rightarrow> 'a u) = u_map\<cdot>prj oo udom_prj u_approx"
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  assumes liftdefl: "liftdefl TYPE('a) = u_defl\<cdot>DEFL('a)"
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  assumes ep_pair: "ep_pair emb (prj :: udom \<rightarrow> 'a)"
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  assumes cast_defl: "cast\<cdot>DEFL('a) = emb oo (prj :: udom \<rightarrow> 'a)"
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   287
  shows "OFCLASS('a, liftdomain_class)"
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   288
proof
huffman@40491
   289
  show "ep_pair liftemb (liftprj :: udom \<rightarrow> 'a u)"
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   290
    unfolding liftemb liftprj
huffman@40491
   291
    by (intro ep_pair_comp ep_pair_u_map ep_pair ep_pair_udom u_approx)
huffman@40491
   292
  show "cast\<cdot>LIFTDEFL('a) = liftemb oo (liftprj :: udom \<rightarrow> 'a u)"
huffman@40491
   293
    unfolding liftemb liftprj liftdefl
huffman@40491
   294
    by (simp add: cfcomp1 cast_u_defl cast_defl u_map_map)
huffman@40494
   295
next
huffman@40491
   296
qed fact+
huffman@40491
   297
huffman@40491
   298
text {* Restore original type constraints. *}
huffman@40491
   299
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   300
setup {*
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   301
  fold Sign.add_const_constraint
huffman@40491
   302
  [ (@{const_name defl}, SOME @{typ "'a::bifinite itself \<Rightarrow> defl"})
huffman@40491
   303
  , (@{const_name emb}, SOME @{typ "'a::bifinite \<rightarrow> udom"})
huffman@40491
   304
  , (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::bifinite"})
huffman@40491
   305
  , (@{const_name liftdefl}, SOME @{typ "'a::predomain itself \<Rightarrow> defl"})
huffman@40491
   306
  , (@{const_name liftemb}, SOME @{typ "'a::predomain u \<rightarrow> udom"})
huffman@40491
   307
  , (@{const_name liftprj}, SOME @{typ "udom \<rightarrow> 'a::predomain u"}) ]
huffman@40491
   308
*}
huffman@40491
   309
huffman@39987
   310
subsection {* The universal domain is bifinite *}
huffman@39985
   311
huffman@40494
   312
instantiation udom :: liftdomain
huffman@39985
   313
begin
huffman@39985
   314
huffman@39985
   315
definition [simp]:
huffman@39985
   316
  "emb = (ID :: udom \<rightarrow> udom)"
huffman@39985
   317
huffman@39985
   318
definition [simp]:
huffman@39985
   319
  "prj = (ID :: udom \<rightarrow> udom)"
huffman@25903
   320
huffman@33504
   321
definition
huffman@39989
   322
  "defl (t::udom itself) = (\<Squnion>i. defl_principal (Abs_fin_defl (udom_approx i)))"
huffman@33808
   323
huffman@40491
   324
definition
huffman@40491
   325
  "(liftemb :: udom u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
huffman@40491
   326
huffman@40491
   327
definition
huffman@40491
   328
  "(liftprj :: udom \<rightarrow> udom u) = u_map\<cdot>prj oo udom_prj u_approx"
huffman@40491
   329
huffman@40491
   330
definition
huffman@40491
   331
  "liftdefl (t::udom itself) = u_defl\<cdot>DEFL(udom)"
huffman@40491
   332
huffman@40491
   333
instance
huffman@40491
   334
using liftemb_udom_def liftprj_udom_def liftdefl_udom_def
huffman@40494
   335
proof (rule liftdomain_class_intro)
huffman@39985
   336
  show "ep_pair emb (prj :: udom \<rightarrow> udom)"
huffman@39985
   337
    by (simp add: ep_pair.intro)
huffman@39989
   338
  show "cast\<cdot>DEFL(udom) = emb oo (prj :: udom \<rightarrow> udom)"
huffman@39989
   339
    unfolding defl_udom_def
huffman@39985
   340
    apply (subst contlub_cfun_arg)
huffman@39985
   341
    apply (rule chainI)
huffman@39989
   342
    apply (rule defl.principal_mono)
huffman@39985
   343
    apply (simp add: below_fin_defl_def)
huffman@39985
   344
    apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
huffman@39985
   345
    apply (rule chainE)
huffman@39985
   346
    apply (rule chain_udom_approx)
huffman@39989
   347
    apply (subst cast_defl_principal)
huffman@39985
   348
    apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
huffman@33504
   349
    done
huffman@33504
   350
qed
huffman@33504
   351
huffman@39985
   352
end
huffman@39985
   353
huffman@40491
   354
subsection {* Lifted predomains are bifinite *}
huffman@40491
   355
huffman@40494
   356
instantiation u :: (predomain) liftdomain
huffman@40491
   357
begin
huffman@40491
   358
huffman@40491
   359
definition
huffman@40491
   360
  "emb = liftemb"
huffman@40491
   361
huffman@40491
   362
definition
huffman@40491
   363
  "prj = liftprj"
huffman@40491
   364
huffman@40491
   365
definition
huffman@40491
   366
  "defl (t::'a u itself) = LIFTDEFL('a)"
huffman@40491
   367
huffman@40491
   368
definition
huffman@40491
   369
  "(liftemb :: 'a u u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
huffman@40491
   370
huffman@40491
   371
definition
huffman@40491
   372
  "(liftprj :: udom \<rightarrow> 'a u u) = u_map\<cdot>prj oo udom_prj u_approx"
huffman@40491
   373
huffman@40491
   374
definition
huffman@40491
   375
  "liftdefl (t::'a u itself) = u_defl\<cdot>DEFL('a u)"
huffman@40491
   376
huffman@40491
   377
instance
huffman@40491
   378
using liftemb_u_def liftprj_u_def liftdefl_u_def
huffman@40494
   379
proof (rule liftdomain_class_intro)
huffman@40491
   380
  show "ep_pair emb (prj :: udom \<rightarrow> 'a u)"
huffman@40491
   381
    unfolding emb_u_def prj_u_def
huffman@40491
   382
    by (rule predomain_ep)
huffman@40491
   383
  show "cast\<cdot>DEFL('a u) = emb oo (prj :: udom \<rightarrow> 'a u)"
huffman@40491
   384
    unfolding emb_u_def prj_u_def defl_u_def
huffman@40491
   385
    by (rule cast_liftdefl)
huffman@40491
   386
qed
huffman@40491
   387
huffman@40491
   388
end
huffman@40491
   389
huffman@40491
   390
lemma DEFL_u: "DEFL('a::predomain u) = LIFTDEFL('a)"
huffman@40491
   391
by (rule defl_u_def)
huffman@40491
   392
huffman@39987
   393
subsection {* Continuous function space is a bifinite domain *}
huffman@39985
   394
huffman@40494
   395
instantiation cfun :: (bifinite, bifinite) liftdomain
huffman@39985
   396
begin
huffman@39985
   397
huffman@39985
   398
definition
huffman@39985
   399
  "emb = udom_emb cfun_approx oo cfun_map\<cdot>prj\<cdot>emb"
huffman@39985
   400
huffman@39985
   401
definition
huffman@39985
   402
  "prj = cfun_map\<cdot>emb\<cdot>prj oo udom_prj cfun_approx"
huffman@39985
   403
huffman@39985
   404
definition
huffman@39989
   405
  "defl (t::('a \<rightarrow> 'b) itself) = cfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39985
   406
huffman@40491
   407
definition
huffman@40491
   408
  "(liftemb :: ('a \<rightarrow> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
huffman@40491
   409
huffman@40491
   410
definition
huffman@40491
   411
  "(liftprj :: udom \<rightarrow> ('a \<rightarrow> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx"
huffman@40491
   412
huffman@40491
   413
definition
huffman@40491
   414
  "liftdefl (t::('a \<rightarrow> 'b) itself) = u_defl\<cdot>DEFL('a \<rightarrow> 'b)"
huffman@40491
   415
huffman@40491
   416
instance
huffman@40491
   417
using liftemb_cfun_def liftprj_cfun_def liftdefl_cfun_def
huffman@40494
   418
proof (rule liftdomain_class_intro)
huffman@39985
   419
  show "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
huffman@39985
   420
    unfolding emb_cfun_def prj_cfun_def
huffman@39985
   421
    using ep_pair_udom [OF cfun_approx]
huffman@39985
   422
    by (intro ep_pair_comp ep_pair_cfun_map ep_pair_emb_prj)
huffman@39989
   423
  show "cast\<cdot>DEFL('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
huffman@39989
   424
    unfolding emb_cfun_def prj_cfun_def defl_cfun_def cast_cfun_defl
huffman@40002
   425
    by (simp add: cast_DEFL oo_def cfun_eq_iff cfun_map_map)
huffman@27402
   426
qed
huffman@25903
   427
huffman@39985
   428
end
huffman@33504
   429
huffman@39989
   430
lemma DEFL_cfun:
huffman@39989
   431
  "DEFL('a::bifinite \<rightarrow> 'b::bifinite) = cfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39989
   432
by (rule defl_cfun_def)
brianh@39972
   433
huffman@39987
   434
subsection {* Cartesian product is a bifinite domain *}
huffman@39987
   435
huffman@40493
   436
text {*
huffman@40493
   437
  Types @{typ "('a * 'b) u"} and @{typ "'a u \<otimes> 'b u"} are isomorphic.
huffman@40493
   438
*}
huffman@40493
   439
huffman@40493
   440
definition
huffman@40493
   441
  "encode_prod_u = (\<Lambda>(up\<cdot>(x, y)). (:up\<cdot>x, up\<cdot>y:))"
huffman@40493
   442
huffman@40493
   443
definition
huffman@40493
   444
  "decode_prod_u = (\<Lambda>(:up\<cdot>x, up\<cdot>y:). up\<cdot>(x, y))"
huffman@40493
   445
huffman@40493
   446
lemma decode_encode_prod_u [simp]: "decode_prod_u\<cdot>(encode_prod_u\<cdot>x) = x"
huffman@40493
   447
unfolding encode_prod_u_def decode_prod_u_def
huffman@40493
   448
by (case_tac x, simp, rename_tac y, case_tac y, simp)
huffman@40493
   449
huffman@40493
   450
lemma encode_decode_prod_u [simp]: "encode_prod_u\<cdot>(decode_prod_u\<cdot>y) = y"
huffman@40493
   451
unfolding encode_prod_u_def decode_prod_u_def
huffman@40493
   452
apply (case_tac y, simp, rename_tac a b)
huffman@40493
   453
apply (case_tac a, simp, case_tac b, simp, simp)
huffman@40493
   454
done
huffman@40493
   455
huffman@40493
   456
instantiation prod :: (predomain, predomain) predomain
huffman@40493
   457
begin
huffman@40493
   458
huffman@40493
   459
definition
huffman@40493
   460
  "liftemb =
huffman@40493
   461
    (udom_emb sprod_approx oo sprod_map\<cdot>emb\<cdot>emb) oo encode_prod_u"
huffman@40493
   462
huffman@40493
   463
definition
huffman@40493
   464
  "liftprj =
huffman@40493
   465
    decode_prod_u oo (sprod_map\<cdot>prj\<cdot>prj oo udom_prj sprod_approx)"
huffman@40493
   466
huffman@40493
   467
definition
huffman@40493
   468
  "liftdefl (t::('a \<times> 'b) itself) = sprod_defl\<cdot>DEFL('a u)\<cdot>DEFL('b u)"
huffman@40493
   469
huffman@40493
   470
instance proof
huffman@40493
   471
  have "ep_pair encode_prod_u decode_prod_u"
huffman@40493
   472
    by (rule ep_pair.intro, simp_all)
huffman@40493
   473
  thus "ep_pair liftemb (liftprj :: udom \<rightarrow> ('a \<times> 'b) u)"
huffman@40493
   474
    unfolding liftemb_prod_def liftprj_prod_def
huffman@40493
   475
    apply (rule ep_pair_comp)
huffman@40493
   476
    apply (rule ep_pair_comp)
huffman@40493
   477
    apply (intro ep_pair_sprod_map ep_pair_emb_prj)
huffman@40493
   478
    apply (rule ep_pair_udom [OF sprod_approx])
huffman@40493
   479
    done
huffman@40493
   480
  show "cast\<cdot>LIFTDEFL('a \<times> 'b) = liftemb oo (liftprj :: udom \<rightarrow> ('a \<times> 'b) u)"
huffman@40493
   481
    unfolding liftemb_prod_def liftprj_prod_def liftdefl_prod_def
huffman@40493
   482
    by (simp add: cast_sprod_defl cast_DEFL cfcomp1 sprod_map_map)
huffman@40493
   483
qed
huffman@40493
   484
huffman@40493
   485
end
huffman@40493
   486
huffman@39987
   487
instantiation prod :: (bifinite, bifinite) bifinite
huffman@39987
   488
begin
huffman@39987
   489
huffman@39987
   490
definition
huffman@39987
   491
  "emb = udom_emb prod_approx oo cprod_map\<cdot>emb\<cdot>emb"
huffman@39987
   492
huffman@39987
   493
definition
huffman@39987
   494
  "prj = cprod_map\<cdot>prj\<cdot>prj oo udom_prj prod_approx"
huffman@39987
   495
huffman@39987
   496
definition
huffman@39989
   497
  "defl (t::('a \<times> 'b) itself) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39987
   498
huffman@40493
   499
instance proof
huffman@39987
   500
  show "ep_pair emb (prj :: udom \<rightarrow> 'a \<times> 'b)"
huffman@39987
   501
    unfolding emb_prod_def prj_prod_def
huffman@39987
   502
    using ep_pair_udom [OF prod_approx]
huffman@39987
   503
    by (intro ep_pair_comp ep_pair_cprod_map ep_pair_emb_prj)
huffman@39987
   504
next
huffman@39989
   505
  show "cast\<cdot>DEFL('a \<times> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<times> 'b)"
huffman@39989
   506
    unfolding emb_prod_def prj_prod_def defl_prod_def cast_prod_defl
huffman@40002
   507
    by (simp add: cast_DEFL oo_def cfun_eq_iff cprod_map_map)
huffman@39987
   508
qed
huffman@39987
   509
huffman@26962
   510
end
huffman@39987
   511
huffman@39989
   512
lemma DEFL_prod:
huffman@39989
   513
  "DEFL('a::bifinite \<times> 'b::bifinite) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39989
   514
by (rule defl_prod_def)
huffman@39987
   515
huffman@40491
   516
lemma LIFTDEFL_prod:
huffman@40493
   517
  "LIFTDEFL('a::predomain \<times> 'b::predomain) = sprod_defl\<cdot>DEFL('a u)\<cdot>DEFL('b u)"
huffman@40491
   518
by (rule liftdefl_prod_def)
huffman@40491
   519
huffman@39987
   520
subsection {* Strict product is a bifinite domain *}
huffman@39987
   521
huffman@40494
   522
instantiation sprod :: (bifinite, bifinite) liftdomain
huffman@39987
   523
begin
huffman@39987
   524
huffman@39987
   525
definition
huffman@39987
   526
  "emb = udom_emb sprod_approx oo sprod_map\<cdot>emb\<cdot>emb"
huffman@39987
   527
huffman@39987
   528
definition
huffman@39987
   529
  "prj = sprod_map\<cdot>prj\<cdot>prj oo udom_prj sprod_approx"
huffman@39987
   530
huffman@39987
   531
definition
huffman@39989
   532
  "defl (t::('a \<otimes> 'b) itself) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39987
   533
huffman@40491
   534
definition
huffman@40491
   535
  "(liftemb :: ('a \<otimes> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
huffman@40491
   536
huffman@40491
   537
definition
huffman@40491
   538
  "(liftprj :: udom \<rightarrow> ('a \<otimes> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx"
huffman@40491
   539
huffman@40491
   540
definition
huffman@40491
   541
  "liftdefl (t::('a \<otimes> 'b) itself) = u_defl\<cdot>DEFL('a \<otimes> 'b)"
huffman@40491
   542
huffman@40491
   543
instance
huffman@40491
   544
using liftemb_sprod_def liftprj_sprod_def liftdefl_sprod_def
huffman@40494
   545
proof (rule liftdomain_class_intro)
huffman@39987
   546
  show "ep_pair emb (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
huffman@39987
   547
    unfolding emb_sprod_def prj_sprod_def
huffman@39987
   548
    using ep_pair_udom [OF sprod_approx]
huffman@39987
   549
    by (intro ep_pair_comp ep_pair_sprod_map ep_pair_emb_prj)
huffman@39987
   550
next
huffman@39989
   551
  show "cast\<cdot>DEFL('a \<otimes> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
huffman@39989
   552
    unfolding emb_sprod_def prj_sprod_def defl_sprod_def cast_sprod_defl
huffman@40002
   553
    by (simp add: cast_DEFL oo_def cfun_eq_iff sprod_map_map)
huffman@39987
   554
qed
huffman@39987
   555
huffman@39987
   556
end
huffman@39987
   557
huffman@39989
   558
lemma DEFL_sprod:
huffman@39989
   559
  "DEFL('a::bifinite \<otimes> 'b::bifinite) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39989
   560
by (rule defl_sprod_def)
huffman@39987
   561
huffman@40491
   562
subsection {* Countable discrete cpos are predomains *}
huffman@39987
   563
huffman@40491
   564
definition discr_approx :: "nat \<Rightarrow> 'a::countable discr u \<rightarrow> 'a discr u"
huffman@40491
   565
  where "discr_approx = (\<lambda>i. \<Lambda>(up\<cdot>x). if to_nat (undiscr x) < i then up\<cdot>x else \<bottom>)"
huffman@39987
   566
huffman@40491
   567
lemma chain_discr_approx [simp]: "chain discr_approx"
huffman@40491
   568
unfolding discr_approx_def
huffman@40491
   569
by (rule chainI, simp add: monofun_cfun monofun_LAM)
huffman@39987
   570
huffman@40491
   571
lemma lub_discr_approx [simp]: "(\<Squnion>i. discr_approx i) = ID"
huffman@40002
   572
apply (rule cfun_eqI)
huffman@39987
   573
apply (simp add: contlub_cfun_fun)
huffman@40491
   574
apply (simp add: discr_approx_def)
huffman@39987
   575
apply (case_tac x, simp)
huffman@39987
   576
apply (rule thelubI)
huffman@39987
   577
apply (rule is_lubI)
huffman@39987
   578
apply (rule ub_rangeI, simp)
huffman@39987
   579
apply (drule ub_rangeD)
huffman@39987
   580
apply (erule rev_below_trans)
huffman@39987
   581
apply simp
huffman@39987
   582
apply (rule lessI)
huffman@39987
   583
done
huffman@39987
   584
huffman@40491
   585
lemma inj_on_undiscr [simp]: "inj_on undiscr A"
huffman@40491
   586
using Discr_undiscr by (rule inj_on_inverseI)
huffman@40491
   587
huffman@40491
   588
lemma finite_deflation_discr_approx: "finite_deflation (discr_approx i)"
huffman@39987
   589
proof
huffman@40491
   590
  fix x :: "'a discr u"
huffman@40491
   591
  show "discr_approx i\<cdot>x \<sqsubseteq> x"
huffman@40491
   592
    unfolding discr_approx_def
huffman@39987
   593
    by (cases x, simp, simp)
huffman@40491
   594
  show "discr_approx i\<cdot>(discr_approx i\<cdot>x) = discr_approx i\<cdot>x"
huffman@40491
   595
    unfolding discr_approx_def
huffman@39987
   596
    by (cases x, simp, simp)
huffman@40491
   597
  show "finite {x::'a discr u. discr_approx i\<cdot>x = x}"
huffman@39987
   598
  proof (rule finite_subset)
huffman@40491
   599
    let ?S = "insert (\<bottom>::'a discr u) ((\<lambda>x. up\<cdot>x) ` undiscr -` to_nat -` {..<i})"
huffman@40491
   600
    show "{x::'a discr u. discr_approx i\<cdot>x = x} \<subseteq> ?S"
huffman@40491
   601
      unfolding discr_approx_def
huffman@39987
   602
      by (rule subsetI, case_tac x, simp, simp split: split_if_asm)
huffman@39987
   603
    show "finite ?S"
huffman@39987
   604
      by (simp add: finite_vimageI)
huffman@39987
   605
  qed
huffman@39987
   606
qed
huffman@39987
   607
huffman@40491
   608
lemma discr_approx: "approx_chain discr_approx"
huffman@40491
   609
using chain_discr_approx lub_discr_approx finite_deflation_discr_approx
huffman@39987
   610
by (rule approx_chain.intro)
huffman@39987
   611
huffman@40491
   612
instantiation discr :: (countable) predomain
huffman@39987
   613
begin
huffman@39987
   614
huffman@39987
   615
definition
huffman@40491
   616
  "liftemb = udom_emb discr_approx"
huffman@39987
   617
huffman@39987
   618
definition
huffman@40491
   619
  "liftprj = udom_prj discr_approx"
huffman@39987
   620
huffman@39987
   621
definition
huffman@40491
   622
  "liftdefl (t::'a discr itself) =
huffman@40491
   623
    (\<Squnion>i. defl_principal (Abs_fin_defl (liftemb oo discr_approx i oo liftprj)))"
huffman@39987
   624
huffman@39987
   625
instance proof
huffman@40491
   626
  show "ep_pair liftemb (liftprj :: udom \<rightarrow> 'a discr u)"
huffman@40491
   627
    unfolding liftemb_discr_def liftprj_discr_def
huffman@40491
   628
    by (rule ep_pair_udom [OF discr_approx])
huffman@40491
   629
  show "cast\<cdot>LIFTDEFL('a discr) = liftemb oo (liftprj :: udom \<rightarrow> 'a discr u)"
huffman@40491
   630
    unfolding liftemb_discr_def liftprj_discr_def liftdefl_discr_def
huffman@39987
   631
    apply (subst contlub_cfun_arg)
huffman@39987
   632
    apply (rule chainI)
huffman@39989
   633
    apply (rule defl.principal_mono)
huffman@39987
   634
    apply (simp add: below_fin_defl_def)
huffman@40491
   635
    apply (simp add: Abs_fin_defl_inverse
huffman@40491
   636
        ep_pair.finite_deflation_e_d_p [OF ep_pair_udom [OF discr_approx]]
huffman@40491
   637
        approx_chain.finite_deflation_approx [OF discr_approx])
huffman@39987
   638
    apply (intro monofun_cfun below_refl)
huffman@39987
   639
    apply (rule chainE)
huffman@40491
   640
    apply (rule chain_discr_approx)
huffman@39989
   641
    apply (subst cast_defl_principal)
huffman@40491
   642
    apply (simp add: Abs_fin_defl_inverse
huffman@40491
   643
        ep_pair.finite_deflation_e_d_p [OF ep_pair_udom [OF discr_approx]]
huffman@40491
   644
        approx_chain.finite_deflation_approx [OF discr_approx])
huffman@40491
   645
    apply (simp add: lub_distribs)
huffman@39987
   646
    done
huffman@39987
   647
qed
huffman@39987
   648
huffman@39987
   649
end
huffman@39987
   650
huffman@39987
   651
subsection {* Strict sum is a bifinite domain *}
huffman@39987
   652
huffman@40494
   653
instantiation ssum :: (bifinite, bifinite) liftdomain
huffman@39987
   654
begin
huffman@39987
   655
huffman@39987
   656
definition
huffman@39987
   657
  "emb = udom_emb ssum_approx oo ssum_map\<cdot>emb\<cdot>emb"
huffman@39987
   658
huffman@39987
   659
definition
huffman@39987
   660
  "prj = ssum_map\<cdot>prj\<cdot>prj oo udom_prj ssum_approx"
huffman@39987
   661
huffman@39987
   662
definition
huffman@39989
   663
  "defl (t::('a \<oplus> 'b) itself) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39987
   664
huffman@40491
   665
definition
huffman@40491
   666
  "(liftemb :: ('a \<oplus> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
huffman@40491
   667
huffman@40491
   668
definition
huffman@40491
   669
  "(liftprj :: udom \<rightarrow> ('a \<oplus> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx"
huffman@40491
   670
huffman@40491
   671
definition
huffman@40491
   672
  "liftdefl (t::('a \<oplus> 'b) itself) = u_defl\<cdot>DEFL('a \<oplus> 'b)"
huffman@40491
   673
huffman@40491
   674
instance
huffman@40491
   675
using liftemb_ssum_def liftprj_ssum_def liftdefl_ssum_def
huffman@40494
   676
proof (rule liftdomain_class_intro)
huffman@39987
   677
  show "ep_pair emb (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
huffman@39987
   678
    unfolding emb_ssum_def prj_ssum_def
huffman@39987
   679
    using ep_pair_udom [OF ssum_approx]
huffman@39987
   680
    by (intro ep_pair_comp ep_pair_ssum_map ep_pair_emb_prj)
huffman@39989
   681
  show "cast\<cdot>DEFL('a \<oplus> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
huffman@39989
   682
    unfolding emb_ssum_def prj_ssum_def defl_ssum_def cast_ssum_defl
huffman@40002
   683
    by (simp add: cast_DEFL oo_def cfun_eq_iff ssum_map_map)
huffman@39987
   684
qed
huffman@39987
   685
huffman@39987
   686
end
huffman@39987
   687
huffman@39989
   688
lemma DEFL_ssum:
huffman@39989
   689
  "DEFL('a::bifinite \<oplus> 'b::bifinite) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
huffman@39989
   690
by (rule defl_ssum_def)
huffman@39987
   691
huffman@40491
   692
subsection {* Lifted countable types are bifinite domains *}
huffman@40491
   693
huffman@40494
   694
instantiation lift :: (countable) liftdomain
huffman@40491
   695
begin
huffman@40491
   696
huffman@40491
   697
definition
huffman@40491
   698
  "emb = emb oo (\<Lambda> x. Rep_lift x)"
huffman@40491
   699
huffman@40491
   700
definition
huffman@40491
   701
  "prj = (\<Lambda> y. Abs_lift y) oo prj"
huffman@40491
   702
huffman@40491
   703
definition
huffman@40491
   704
  "defl (t::'a lift itself) = DEFL('a discr u)"
huffman@40491
   705
huffman@40491
   706
definition
huffman@40491
   707
  "(liftemb :: 'a lift u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb"
huffman@40491
   708
huffman@40491
   709
definition
huffman@40491
   710
  "(liftprj :: udom \<rightarrow> 'a lift u) = u_map\<cdot>prj oo udom_prj u_approx"
huffman@40491
   711
huffman@40491
   712
definition
huffman@40491
   713
  "liftdefl (t::'a lift itself) = u_defl\<cdot>DEFL('a lift)"
huffman@40491
   714
huffman@40491
   715
instance
huffman@40491
   716
using liftemb_lift_def liftprj_lift_def liftdefl_lift_def
huffman@40494
   717
proof (rule liftdomain_class_intro)
huffman@40491
   718
  note [simp] = cont_Rep_lift cont_Abs_lift Rep_lift_inverse Abs_lift_inverse
huffman@40491
   719
  have "ep_pair (\<Lambda>(x::'a lift). Rep_lift x) (\<Lambda> y. Abs_lift y)"
huffman@40491
   720
    by (simp add: ep_pair_def)
huffman@40491
   721
  thus "ep_pair emb (prj :: udom \<rightarrow> 'a lift)"
huffman@40491
   722
    unfolding emb_lift_def prj_lift_def
huffman@40491
   723
    using ep_pair_emb_prj by (rule ep_pair_comp)
huffman@40491
   724
  show "cast\<cdot>DEFL('a lift) = emb oo (prj :: udom \<rightarrow> 'a lift)"
huffman@40491
   725
    unfolding emb_lift_def prj_lift_def defl_lift_def cast_DEFL
huffman@40491
   726
    by (simp add: cfcomp1)
huffman@40491
   727
qed
huffman@40491
   728
huffman@39987
   729
end
huffman@40491
   730
huffman@40491
   731
end