src/HOL/HOLCF/Representable.thy
author wenzelm
Sat, 07 Apr 2012 16:41:59 +0200
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/HOLCF/Representable.thy
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    Author:     Brian Huffman
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*)
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header {* Representable domains *}
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theory Representable
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imports Algebraic Map_Functions "~~/src/HOL/Library/Countable"
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begin
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default_sort cpo
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subsection {* Class of representable domains *}
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text {*
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  We define a ``domain'' as a pcpo that is isomorphic to some
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  algebraic deflation over the universal domain; this is equivalent
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  to being omega-bifinite.
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  A predomain is a cpo that, when lifted, becomes a domain.
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  Predomains are represented by deflations over a lifted universal
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  domain type.
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*}
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class predomain_syn = cpo +
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  fixes liftemb :: "'a\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"
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  fixes liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>"
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  fixes liftdefl :: "'a itself \<Rightarrow> udom u defl"
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class predomain = predomain_syn +
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  assumes predomain_ep: "ep_pair liftemb liftprj"
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  assumes cast_liftdefl: "cast\<cdot>(liftdefl TYPE('a)) = 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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definition liftdefl_of :: "udom defl \<rightarrow> udom u defl"
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  where "liftdefl_of = defl_fun1 ID ID u_map"
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lemma cast_liftdefl_of: "cast\<cdot>(liftdefl_of\<cdot>t) = u_map\<cdot>(cast\<cdot>t)"
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by (simp add: liftdefl_of_def cast_defl_fun1 ep_pair_def finite_deflation_u_map)
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class "domain" = predomain_syn + pcpo +
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  fixes emb :: "'a \<rightarrow> udom"
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  fixes prj :: "udom \<rightarrow> 'a"
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  fixes defl :: "'a itself \<Rightarrow> udom 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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  assumes liftemb_eq: "liftemb = u_map\<cdot>emb"
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  assumes liftprj_eq: "liftprj = u_map\<cdot>prj"
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  assumes liftdefl_eq: "liftdefl TYPE('a) = liftdefl_of\<cdot>(defl TYPE('a))"
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syntax "_DEFL" :: "type \<Rightarrow> logic"  ("(1DEFL/(1'(_')))")
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translations "DEFL('t)" \<rightleftharpoons> "CONST defl TYPE('t)"
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instance "domain" \<subseteq> predomain
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proof
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  show "ep_pair liftemb (liftprj::udom\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>)"
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    unfolding liftemb_eq liftprj_eq
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    by (intro ep_pair_u_map ep_pair_emb_prj)
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  show "cast\<cdot>LIFTDEFL('a) = liftemb oo (liftprj::udom\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>)"
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    unfolding liftemb_eq liftprj_eq liftdefl_eq
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    by (simp add: cast_liftdefl_of cast_DEFL u_map_oo)
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qed
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text {*
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  Constants @{const liftemb} and @{const liftprj} imply class predomain.
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*}
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setup {*
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  fold Sign.add_const_constraint
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  [(@{const_name liftemb}, SOME @{typ "'a::predomain u \<rightarrow> udom u"}),
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   (@{const_name liftprj}, SOME @{typ "udom u \<rightarrow> 'a::predomain u"}),
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   (@{const_name liftdefl}, SOME @{typ "'a::predomain itself \<Rightarrow> udom u defl"})]
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*}
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interpretation predomain: pcpo_ep_pair liftemb liftprj
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  unfolding pcpo_ep_pair_def by (rule predomain_ep)
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interpretation "domain": pcpo_ep_pair emb prj
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  unfolding pcpo_ep_pair_def by (rule ep_pair_emb_prj)
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lemmas emb_inverse = domain.e_inverse
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lemmas emb_prj_below = domain.e_p_below
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lemmas emb_eq_iff = domain.e_eq_iff
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lemmas emb_strict = domain.e_strict
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lemmas prj_strict = domain.p_strict
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subsection {* Domains are bifinite *}
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lemma approx_chain_ep_cast:
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  assumes ep: "ep_pair (e::'a::pcpo \<rightarrow> 'b::bifinite) (p::'b \<rightarrow> 'a)"
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  assumes cast_t: "cast\<cdot>t = e oo p"
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  shows "\<exists>(a::nat \<Rightarrow> 'a::pcpo \<rightarrow> 'a). approx_chain a"
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proof -
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  interpret ep_pair e p by fact
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  obtain Y where Y: "\<forall>i. Y i \<sqsubseteq> Y (Suc i)"
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  and t: "t = (\<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. (p oo cast\<cdot>(defl_principal (Y i)) oo e) :: 'a \<rightarrow> 'a"
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  have "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 t [symmetric] cast_t 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 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 t [symmetric] cast_t)
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      done
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  qed
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  thus "\<exists>(a::nat \<Rightarrow> 'a \<rightarrow> 'a). approx_chain a" by - (rule exI)
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qed
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instance "domain" \<subseteq> bifinite
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by default (rule approx_chain_ep_cast [OF ep_pair_emb_prj cast_DEFL])
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instance predomain \<subseteq> profinite
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by default (rule approx_chain_ep_cast [OF predomain_ep cast_liftdefl])
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subsection {* Universal domain ep-pairs *}
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definition "u_emb = udom_emb (\<lambda>i. u_map\<cdot>(udom_approx i))"
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definition "u_prj = udom_prj (\<lambda>i. u_map\<cdot>(udom_approx i))"
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definition "prod_emb = udom_emb (\<lambda>i. prod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition "prod_prj = udom_prj (\<lambda>i. prod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition "sprod_emb = udom_emb (\<lambda>i. sprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition "sprod_prj = udom_prj (\<lambda>i. sprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition "ssum_emb = udom_emb (\<lambda>i. ssum_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition "ssum_prj = udom_prj (\<lambda>i. ssum_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition "sfun_emb = udom_emb (\<lambda>i. sfun_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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definition "sfun_prj = udom_prj (\<lambda>i. sfun_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))"
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lemma ep_pair_u: "ep_pair u_emb u_prj"
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  unfolding u_emb_def u_prj_def
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  by (simp add: ep_pair_udom approx_chain_u_map)
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lemma ep_pair_prod: "ep_pair prod_emb prod_prj"
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  unfolding prod_emb_def prod_prj_def
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  by (simp add: ep_pair_udom approx_chain_prod_map)
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lemma ep_pair_sprod: "ep_pair sprod_emb sprod_prj"
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  unfolding sprod_emb_def sprod_prj_def
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  by (simp add: ep_pair_udom approx_chain_sprod_map)
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lemma ep_pair_ssum: "ep_pair ssum_emb ssum_prj"
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  unfolding ssum_emb_def ssum_prj_def
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  by (simp add: ep_pair_udom approx_chain_ssum_map)
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lemma ep_pair_sfun: "ep_pair sfun_emb sfun_prj"
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  unfolding sfun_emb_def sfun_prj_def
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  by (simp add: ep_pair_udom approx_chain_sfun_map)
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subsection {* Type combinators *}
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definition u_defl :: "udom defl \<rightarrow> udom defl"
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  where "u_defl = defl_fun1 u_emb u_prj u_map"
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definition prod_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl"
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  where "prod_defl = defl_fun2 prod_emb prod_prj prod_map"
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definition sprod_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl"
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  where "sprod_defl = defl_fun2 sprod_emb sprod_prj sprod_map"
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definition ssum_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl"
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where "ssum_defl = defl_fun2 ssum_emb ssum_prj ssum_map"
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definition sfun_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl"
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  where "sfun_defl = defl_fun2 sfun_emb sfun_prj sfun_map"
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lemma cast_u_defl:
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  "cast\<cdot>(u_defl\<cdot>A) = u_emb oo u_map\<cdot>(cast\<cdot>A) oo u_prj"
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using ep_pair_u finite_deflation_u_map
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unfolding u_defl_def by (rule cast_defl_fun1)
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lemma cast_prod_defl:
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  "cast\<cdot>(prod_defl\<cdot>A\<cdot>B) =
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    prod_emb oo prod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo prod_prj"
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using ep_pair_prod finite_deflation_prod_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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    sprod_emb oo sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo sprod_prj"
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using ep_pair_sprod 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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    ssum_emb oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo ssum_prj"
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using ep_pair_ssum finite_deflation_ssum_map
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unfolding ssum_defl_def by (rule cast_defl_fun2)
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lemma cast_sfun_defl:
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  "cast\<cdot>(sfun_defl\<cdot>A\<cdot>B) =
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    sfun_emb oo sfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo sfun_prj"
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using ep_pair_sfun finite_deflation_sfun_map
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unfolding sfun_defl_def by (rule cast_defl_fun2)
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text {* Special deflation combinator for unpointed types. *}
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definition u_liftdefl :: "udom u defl \<rightarrow> udom defl"
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  where "u_liftdefl = defl_fun1 u_emb u_prj ID"
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lemma cast_u_liftdefl:
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  "cast\<cdot>(u_liftdefl\<cdot>A) = u_emb oo cast\<cdot>A oo u_prj"
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unfolding u_liftdefl_def by (simp add: cast_defl_fun1 ep_pair_u)
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lemma u_liftdefl_liftdefl_of:
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  "u_liftdefl\<cdot>(liftdefl_of\<cdot>A) = u_defl\<cdot>A"
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by (rule cast_eq_imp_eq)
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   (simp add: cast_u_liftdefl cast_liftdefl_of cast_u_defl)
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subsection {* Class instance proofs *}
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4c5363173f88 section headings
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subsubsection {* Universal domain *}
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instantiation udom :: "domain"
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begin
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definition [simp]:
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  "emb = (ID :: udom \<rightarrow> udom)"
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definition [simp]:
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  "prj = (ID :: udom \<rightarrow> udom)"
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definition
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  "defl (t::udom itself) = (\<Squnion>i. defl_principal (Abs_fin_defl (udom_approx i)))"
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definition
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  "(liftemb :: udom u \<rightarrow> udom u) = u_map\<cdot>emb"
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definition
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  "(liftprj :: udom u \<rightarrow> udom u) = u_map\<cdot>prj"
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definition
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  "liftdefl (t::udom itself) = liftdefl_of\<cdot>DEFL(udom)"
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2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
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instance proof
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  show "ep_pair emb (prj :: udom \<rightarrow> udom)"
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    by (simp add: ep_pair.intro)
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  show "cast\<cdot>DEFL(udom) = emb oo (prj :: udom \<rightarrow> udom)"
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    unfolding defl_udom_def
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    apply (subst contlub_cfun_arg)
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    apply (rule chainI)
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    apply (rule defl.principal_mono)
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    apply (simp add: below_fin_defl_def)
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    apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
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    apply (rule chainE)
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    apply (rule chain_udom_approx)
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    apply (subst cast_defl_principal)
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    apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
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    done
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qed (fact liftemb_udom_def liftprj_udom_def liftdefl_udom_def)+
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end
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4c5363173f88 section headings
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subsubsection {* Lifted cpo *}
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instantiation u :: (predomain) "domain"
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begin
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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definition
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  "emb = u_emb oo liftemb"
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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definition
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  "prj = liftprj oo u_prj"
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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definition
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  "defl (t::'a u itself) = u_liftdefl\<cdot>LIFTDEFL('a)"
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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definition
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  "(liftemb :: 'a u u \<rightarrow> udom u) = u_map\<cdot>emb"
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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definition
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  "(liftprj :: udom u \<rightarrow> 'a u u) = u_map\<cdot>prj"
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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definition
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  "liftdefl (t::'a u itself) = liftdefl_of\<cdot>DEFL('a u)"
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instance proof
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  show "ep_pair emb (prj :: udom \<rightarrow> 'a u)"
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    unfolding emb_u_def prj_u_def
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    by (intro ep_pair_comp ep_pair_u predomain_ep)
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   295
  show "cast\<cdot>DEFL('a u) = emb oo (prj :: udom \<rightarrow> 'a u)"
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   296
    unfolding emb_u_def prj_u_def defl_u_def
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    by (simp add: cast_u_liftdefl cast_liftdefl assoc_oo)
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qed (fact liftemb_u_def liftprj_u_def liftdefl_u_def)+
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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end
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lemma DEFL_u: "DEFL('a::predomain u) = u_liftdefl\<cdot>LIFTDEFL('a)"
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by (rule defl_u_def)
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subsubsection {* Strict function space *}
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instantiation sfun :: ("domain", "domain") "domain"
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begin
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310f98585107 move stuff from Algebraic.thy to Bifinite.thy and elsewhere
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definition
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  "emb = sfun_emb oo sfun_map\<cdot>prj\<cdot>emb"
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definition
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  "prj = sfun_map\<cdot>emb\<cdot>prj oo sfun_prj"
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f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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definition
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  "defl (t::('a \<rightarrow>! 'b) itself) = sfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
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f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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definition
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  "(liftemb :: ('a \<rightarrow>! 'b) u \<rightarrow> udom u) = u_map\<cdot>emb"
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definition
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  "(liftprj :: udom u \<rightarrow> ('a \<rightarrow>! 'b) u) = u_map\<cdot>prj"
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definition
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  "liftdefl (t::('a \<rightarrow>! 'b) itself) = liftdefl_of\<cdot>DEFL('a \<rightarrow>! 'b)"
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2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
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instance proof
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  show "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow>! 'b)"
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    unfolding emb_sfun_def prj_sfun_def
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    by (intro ep_pair_comp ep_pair_sfun ep_pair_sfun_map ep_pair_emb_prj)
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  show "cast\<cdot>DEFL('a \<rightarrow>! 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow>! 'b)"
f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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   333
    unfolding emb_sfun_def prj_sfun_def defl_sfun_def cast_sfun_defl
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   334
    by (simp add: cast_DEFL oo_def sfun_eq_iff sfun_map_map)
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qed (fact liftemb_sfun_def liftprj_sfun_def liftdefl_sfun_def)+
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f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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end
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f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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lemma DEFL_sfun:
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  "DEFL('a::domain \<rightarrow>! 'b::domain) = sfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
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   341
by (rule defl_sfun_def)
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f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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subsubsection {* Continuous function space *}
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instantiation cfun :: (predomain, "domain") "domain"
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begin
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definition
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  "emb = emb oo encode_cfun"
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definition
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  "prj = decode_cfun oo prj"
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definition
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   355
  "defl (t::('a \<rightarrow> 'b) itself) = DEFL('a u \<rightarrow>! 'b)"
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   356
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definition
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  "(liftemb :: ('a \<rightarrow> 'b) u \<rightarrow> udom u) = u_map\<cdot>emb"
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6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
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   360
definition
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  "(liftprj :: udom u \<rightarrow> ('a \<rightarrow> 'b) u) = u_map\<cdot>prj"
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41292
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   363
definition
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   364
  "liftdefl (t::('a \<rightarrow> 'b) itself) = liftdefl_of\<cdot>DEFL('a \<rightarrow> 'b)"
41292
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2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
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instance proof
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  have "ep_pair encode_cfun decode_cfun"
f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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   368
    by (rule ep_pair.intro, simp_all)
f432973ce0f6 move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
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   369
  thus "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
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   370
    unfolding emb_cfun_def prj_cfun_def
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   371
    using ep_pair_emb_prj by (rule ep_pair_comp)
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
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   372
  show "cast\<cdot>DEFL('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
40830
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   373
    unfolding emb_cfun_def prj_cfun_def defl_cfun_def
158d18502378 simplify predomain instances
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   374
    by (simp add: cast_DEFL cfcomp1)
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   375
qed (fact liftemb_cfun_def liftprj_cfun_def liftdefl_cfun_def)+
25903
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parents:
diff changeset
   376
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   377
end
33504
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huffman
parents: 31113
diff changeset
   378
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
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   379
lemma DEFL_cfun:
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diff changeset
   380
  "DEFL('a::predomain \<rightarrow> 'b::domain) = DEFL('a u \<rightarrow>! 'b)"
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
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diff changeset
   381
by (rule defl_cfun_def)
39972
4244ff4f9649 add lemmas finite_deflation_imp_compact, cast_below_cast_iff
Brian Huffman <brianh@cs.pdx.edu>
parents: 37678
diff changeset
   382
40506
4c5363173f88 section headings
huffman
parents: 40502
diff changeset
   383
subsubsection {* Strict product *}
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   384
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   385
instantiation sprod :: ("domain", "domain") "domain"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   386
begin
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   387
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   388
definition
41290
e9c9577d88b5 replace foo_approx functions with foo_emb, foo_prj functions for universal domain embeddings
huffman
parents: 41287
diff changeset
   389
  "emb = sprod_emb oo sprod_map\<cdot>emb\<cdot>emb"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   390
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   391
definition
41290
e9c9577d88b5 replace foo_approx functions with foo_emb, foo_prj functions for universal domain embeddings
huffman
parents: 41287
diff changeset
   392
  "prj = sprod_map\<cdot>prj\<cdot>prj oo sprod_prj"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   393
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   394
definition
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   395
  "defl (t::('a \<otimes> 'b) itself) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   396
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   397
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   398
  "(liftemb :: ('a \<otimes> 'b) u \<rightarrow> udom u) = u_map\<cdot>emb"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   399
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   400
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   401
  "(liftprj :: udom u \<rightarrow> ('a \<otimes> 'b) u) = u_map\<cdot>prj"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   402
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   403
definition
41436
480978f80eae rename constant pdefl to liftdefl_of
huffman
parents: 41297
diff changeset
   404
  "liftdefl (t::('a \<otimes> 'b) itself) = liftdefl_of\<cdot>DEFL('a \<otimes> 'b)"
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   405
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   406
instance proof
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   407
  show "ep_pair emb (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   408
    unfolding emb_sprod_def prj_sprod_def
41290
e9c9577d88b5 replace foo_approx functions with foo_emb, foo_prj functions for universal domain embeddings
huffman
parents: 41287
diff changeset
   409
    by (intro ep_pair_comp ep_pair_sprod ep_pair_sprod_map ep_pair_emb_prj)
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   410
  show "cast\<cdot>DEFL('a \<otimes> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   411
    unfolding emb_sprod_def prj_sprod_def defl_sprod_def cast_sprod_defl
40002
c5b5f7a3a3b1 new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents: 39989
diff changeset
   412
    by (simp add: cast_DEFL oo_def cfun_eq_iff sprod_map_map)
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   413
qed (fact liftemb_sprod_def liftprj_sprod_def liftdefl_sprod_def)+
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   414
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   415
end
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   416
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   417
lemma DEFL_sprod:
40497
d2e876d6da8c rename class 'bifinite' to 'domain'
huffman
parents: 40494
diff changeset
   418
  "DEFL('a::domain \<otimes> 'b::domain) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   419
by (rule defl_sprod_def)
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   420
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   421
subsubsection {* Cartesian product *}
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   422
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   423
definition prod_liftdefl :: "udom u defl \<rightarrow> udom u defl \<rightarrow> udom u defl"
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   424
  where "prod_liftdefl = defl_fun2 (u_map\<cdot>prod_emb oo decode_prod_u)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   425
    (encode_prod_u oo u_map\<cdot>prod_prj) sprod_map"
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   426
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   427
lemma cast_prod_liftdefl:
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   428
  "cast\<cdot>(prod_liftdefl\<cdot>a\<cdot>b) =
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   429
    (u_map\<cdot>prod_emb oo decode_prod_u) oo sprod_map\<cdot>(cast\<cdot>a)\<cdot>(cast\<cdot>b) oo
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   430
      (encode_prod_u oo u_map\<cdot>prod_prj)"
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   431
unfolding prod_liftdefl_def
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   432
apply (rule cast_defl_fun2)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   433
apply (intro ep_pair_comp ep_pair_u_map ep_pair_prod)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   434
apply (simp add: ep_pair.intro)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   435
apply (erule (1) finite_deflation_sprod_map)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   436
done
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   437
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   438
instantiation prod :: (predomain, predomain) predomain
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   439
begin
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   440
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   441
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   442
  "liftemb = (u_map\<cdot>prod_emb oo decode_prod_u) oo
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   443
    (sprod_map\<cdot>liftemb\<cdot>liftemb oo encode_prod_u)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   444
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   445
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   446
  "liftprj = (decode_prod_u oo sprod_map\<cdot>liftprj\<cdot>liftprj) oo
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   447
    (encode_prod_u oo u_map\<cdot>prod_prj)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   448
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   449
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   450
  "liftdefl (t::('a \<times> 'b) itself) = prod_liftdefl\<cdot>LIFTDEFL('a)\<cdot>LIFTDEFL('b)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   451
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   452
instance proof
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   453
  show "ep_pair liftemb (liftprj :: udom u \<rightarrow> ('a \<times> 'b) u)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   454
    unfolding liftemb_prod_def liftprj_prod_def
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   455
    by (intro ep_pair_comp ep_pair_sprod_map ep_pair_u_map
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   456
       ep_pair_prod predomain_ep, simp_all add: ep_pair.intro)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   457
  show "cast\<cdot>LIFTDEFL('a \<times> 'b) = liftemb oo (liftprj :: udom u \<rightarrow> ('a \<times> 'b) u)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   458
    unfolding liftemb_prod_def liftprj_prod_def liftdefl_prod_def
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   459
    by (simp add: cast_prod_liftdefl cast_liftdefl cfcomp1 sprod_map_map)
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   460
qed
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   461
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   462
end
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   463
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   464
instantiation prod :: ("domain", "domain") "domain"
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   465
begin
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   466
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   467
definition
41297
01b2de947cff rename function cprod_map to prod_map
huffman
parents: 41292
diff changeset
   468
  "emb = prod_emb oo prod_map\<cdot>emb\<cdot>emb"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   469
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   470
definition
41297
01b2de947cff rename function cprod_map to prod_map
huffman
parents: 41292
diff changeset
   471
  "prj = prod_map\<cdot>prj\<cdot>prj oo prod_prj"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   472
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   473
definition
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   474
  "defl (t::('a \<times> 'b) itself) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   475
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   476
instance proof
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   477
  show 1: "ep_pair emb (prj :: udom \<rightarrow> 'a \<times> 'b)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   478
    unfolding emb_prod_def prj_prod_def
41297
01b2de947cff rename function cprod_map to prod_map
huffman
parents: 41292
diff changeset
   479
    by (intro ep_pair_comp ep_pair_prod ep_pair_prod_map ep_pair_emb_prj)
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   480
  show 2: "cast\<cdot>DEFL('a \<times> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<times> 'b)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   481
    unfolding emb_prod_def prj_prod_def defl_prod_def cast_prod_defl
41297
01b2de947cff rename function cprod_map to prod_map
huffman
parents: 41292
diff changeset
   482
    by (simp add: cast_DEFL oo_def cfun_eq_iff prod_map_map)
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   483
  show 3: "liftemb = u_map\<cdot>(emb :: 'a \<times> 'b \<rightarrow> udom)"
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   484
    unfolding emb_prod_def liftemb_prod_def liftemb_eq
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   485
    unfolding encode_prod_u_def decode_prod_u_def
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   486
    by (rule cfun_eqI, case_tac x, simp, clarsimp)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   487
  show 4: "liftprj = u_map\<cdot>(prj :: udom \<rightarrow> 'a \<times> 'b)"
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   488
    unfolding prj_prod_def liftprj_prod_def liftprj_eq
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   489
    unfolding encode_prod_u_def decode_prod_u_def
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   490
    apply (rule cfun_eqI, case_tac x, simp)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   491
    apply (rename_tac y, case_tac "prod_prj\<cdot>y", simp)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   492
    done
41436
480978f80eae rename constant pdefl to liftdefl_of
huffman
parents: 41297
diff changeset
   493
  show 5: "LIFTDEFL('a \<times> 'b) = liftdefl_of\<cdot>DEFL('a \<times> 'b)"
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   494
    by (rule cast_eq_imp_eq)
41436
480978f80eae rename constant pdefl to liftdefl_of
huffman
parents: 41297
diff changeset
   495
      (simp add: cast_liftdefl cast_liftdefl_of cast_DEFL 2 3 4 u_map_oo)
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   496
qed
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   497
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   498
end
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   499
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   500
lemma DEFL_prod:
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   501
  "DEFL('a::domain \<times> 'b::domain) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   502
by (rule defl_prod_def)
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   503
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   504
lemma LIFTDEFL_prod:
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   505
  "LIFTDEFL('a::predomain \<times> 'b::predomain) =
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   506
    prod_liftdefl\<cdot>LIFTDEFL('a)\<cdot>LIFTDEFL('b)"
40830
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   507
by (rule liftdefl_prod_def)
158d18502378 simplify predomain instances
huffman
parents: 40774
diff changeset
   508
41034
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   509
subsubsection {* Unit type *}
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   510
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   511
instantiation unit :: "domain"
41034
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   512
begin
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   513
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   514
definition
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   515
  "emb = (\<bottom> :: unit \<rightarrow> udom)"
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   516
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   517
definition
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   518
  "prj = (\<bottom> :: udom \<rightarrow> unit)"
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   519
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   520
definition
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   521
  "defl (t::unit itself) = \<bottom>"
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   522
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   523
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   524
  "(liftemb :: unit u \<rightarrow> udom u) = u_map\<cdot>emb"
41034
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   525
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   526
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   527
  "(liftprj :: udom u \<rightarrow> unit u) = u_map\<cdot>prj"
41034
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   528
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   529
definition
41436
480978f80eae rename constant pdefl to liftdefl_of
huffman
parents: 41297
diff changeset
   530
  "liftdefl (t::unit itself) = liftdefl_of\<cdot>DEFL(unit)"
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   531
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   532
instance proof
41034
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   533
  show "ep_pair emb (prj :: udom \<rightarrow> unit)"
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   534
    unfolding emb_unit_def prj_unit_def
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   535
    by (simp add: ep_pair.intro)
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   536
  show "cast\<cdot>DEFL(unit) = emb oo (prj :: udom \<rightarrow> unit)"
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   537
    unfolding emb_unit_def prj_unit_def defl_unit_def by simp
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   538
qed (fact liftemb_unit_def liftprj_unit_def liftdefl_unit_def)+
41034
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   539
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   540
end
ce5d9e73fb98 instance unit :: domain
huffman
parents: 40830
diff changeset
   541
40506
4c5363173f88 section headings
huffman
parents: 40502
diff changeset
   542
subsubsection {* Discrete cpo *}
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   543
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   544
instantiation discr :: (countable) predomain
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   545
begin
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   546
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   547
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   548
  "(liftemb :: 'a discr u \<rightarrow> udom u) = strictify\<cdot>up oo udom_emb discr_approx"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   549
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   550
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   551
  "(liftprj :: udom u \<rightarrow> 'a discr u) = udom_prj discr_approx oo fup\<cdot>ID"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   552
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   553
definition
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   554
  "liftdefl (t::'a discr itself) =
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   555
    (\<Squnion>i. defl_principal (Abs_fin_defl (liftemb oo discr_approx i oo (liftprj::udom u \<rightarrow> 'a discr u))))"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   556
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   557
instance proof
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   558
  show 1: "ep_pair liftemb (liftprj :: udom u \<rightarrow> 'a discr u)"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   559
    unfolding liftemb_discr_def liftprj_discr_def
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   560
    apply (intro ep_pair_comp ep_pair_udom [OF discr_approx])
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   561
    apply (rule ep_pair.intro)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   562
    apply (simp add: strictify_conv_if)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   563
    apply (case_tac y, simp, simp add: strictify_conv_if)
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   564
    done
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   565
  show "cast\<cdot>LIFTDEFL('a discr) = liftemb oo (liftprj :: udom u \<rightarrow> 'a discr u)"
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   566
    unfolding liftdefl_discr_def
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   567
    apply (subst contlub_cfun_arg)
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   568
    apply (rule chainI)
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   569
    apply (rule defl.principal_mono)
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   570
    apply (simp add: below_fin_defl_def)
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   571
    apply (simp add: Abs_fin_defl_inverse
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   572
        ep_pair.finite_deflation_e_d_p [OF 1]
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   573
        approx_chain.finite_deflation_approx [OF discr_approx])
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   574
    apply (intro monofun_cfun below_refl)
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   575
    apply (rule chainE)
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   576
    apply (rule chain_discr_approx)
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   577
    apply (subst cast_defl_principal)
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   578
    apply (simp add: Abs_fin_defl_inverse
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   579
        ep_pair.finite_deflation_e_d_p [OF 1]
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   580
        approx_chain.finite_deflation_approx [OF discr_approx])
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   581
    apply (simp add: lub_distribs)
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   582
    done
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   583
qed
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   584
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   585
end
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   586
40506
4c5363173f88 section headings
huffman
parents: 40502
diff changeset
   587
subsubsection {* Strict sum *}
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   588
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   589
instantiation ssum :: ("domain", "domain") "domain"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   590
begin
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   591
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   592
definition
41290
e9c9577d88b5 replace foo_approx functions with foo_emb, foo_prj functions for universal domain embeddings
huffman
parents: 41287
diff changeset
   593
  "emb = ssum_emb oo ssum_map\<cdot>emb\<cdot>emb"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   594
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   595
definition
41290
e9c9577d88b5 replace foo_approx functions with foo_emb, foo_prj functions for universal domain embeddings
huffman
parents: 41287
diff changeset
   596
  "prj = ssum_map\<cdot>prj\<cdot>prj oo ssum_prj"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   597
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   598
definition
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   599
  "defl (t::('a \<oplus> 'b) itself) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   600
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   601
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   602
  "(liftemb :: ('a \<oplus> 'b) u \<rightarrow> udom u) = u_map\<cdot>emb"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   603
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   604
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   605
  "(liftprj :: udom u \<rightarrow> ('a \<oplus> 'b) u) = u_map\<cdot>prj"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   606
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   607
definition
41436
480978f80eae rename constant pdefl to liftdefl_of
huffman
parents: 41297
diff changeset
   608
  "liftdefl (t::('a \<oplus> 'b) itself) = liftdefl_of\<cdot>DEFL('a \<oplus> 'b)"
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   609
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   610
instance proof
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   611
  show "ep_pair emb (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   612
    unfolding emb_ssum_def prj_ssum_def
41290
e9c9577d88b5 replace foo_approx functions with foo_emb, foo_prj functions for universal domain embeddings
huffman
parents: 41287
diff changeset
   613
    by (intro ep_pair_comp ep_pair_ssum ep_pair_ssum_map ep_pair_emb_prj)
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   614
  show "cast\<cdot>DEFL('a \<oplus> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   615
    unfolding emb_ssum_def prj_ssum_def defl_ssum_def cast_ssum_defl
40002
c5b5f7a3a3b1 new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents: 39989
diff changeset
   616
    by (simp add: cast_DEFL oo_def cfun_eq_iff ssum_map_map)
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   617
qed (fact liftemb_ssum_def liftprj_ssum_def liftdefl_ssum_def)+
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   618
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   619
end
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   620
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   621
lemma DEFL_ssum:
40497
d2e876d6da8c rename class 'bifinite' to 'domain'
huffman
parents: 40494
diff changeset
   622
  "DEFL('a::domain \<oplus> 'b::domain) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
39989
ad60d7311f43 renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents: 39987
diff changeset
   623
by (rule defl_ssum_def)
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   624
40506
4c5363173f88 section headings
huffman
parents: 40502
diff changeset
   625
subsubsection {* Lifted HOL type *}
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   626
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   627
instantiation lift :: (countable) "domain"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   628
begin
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   629
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   630
definition
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   631
  "emb = emb oo (\<Lambda> x. Rep_lift x)"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   632
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   633
definition
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   634
  "prj = (\<Lambda> y. Abs_lift y) oo prj"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   635
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   636
definition
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   637
  "defl (t::'a lift itself) = DEFL('a discr u)"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   638
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   639
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   640
  "(liftemb :: 'a lift u \<rightarrow> udom u) = u_map\<cdot>emb"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   641
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   642
definition
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   643
  "(liftprj :: udom u \<rightarrow> 'a lift u) = u_map\<cdot>prj"
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   644
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   645
definition
41436
480978f80eae rename constant pdefl to liftdefl_of
huffman
parents: 41297
diff changeset
   646
  "liftdefl (t::'a lift itself) = liftdefl_of\<cdot>DEFL('a lift)"
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   647
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   648
instance proof
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   649
  note [simp] = cont_Rep_lift cont_Abs_lift Rep_lift_inverse Abs_lift_inverse
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   650
  have "ep_pair (\<Lambda>(x::'a lift). Rep_lift x) (\<Lambda> y. Abs_lift y)"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   651
    by (simp add: ep_pair_def)
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   652
  thus "ep_pair emb (prj :: udom \<rightarrow> 'a lift)"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   653
    unfolding emb_lift_def prj_lift_def
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   654
    using ep_pair_emb_prj by (rule ep_pair_comp)
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   655
  show "cast\<cdot>DEFL('a lift) = emb oo (prj :: udom \<rightarrow> 'a lift)"
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   656
    unfolding emb_lift_def prj_lift_def defl_lift_def cast_DEFL
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   657
    by (simp add: cfcomp1)
41292
2b7bc8d9fd6e use deflations over type 'udom u' to represent predomains;
huffman
parents: 41290
diff changeset
   658
qed (fact liftemb_lift_def liftprj_lift_def liftdefl_lift_def)+
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   659
39987
8c2f449af35a move all bifinite class instances to Bifinite.thy
huffman
parents: 39986
diff changeset
   660
end
40491
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   661
6de5839e2fb3 add 'predomain' class: unpointed version of bifinite
huffman
parents: 40484
diff changeset
   662
end