src/HOL/HOLCF/Domain_Aux.thy

Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context

(*  Title:      HOL/HOLCF/Domain_Aux.thy
    Author:     Brian Huffman
*)

section \<open>Domain package support\<close>

theory Domain_Aux
imports Map_Functions Fixrec
begin

subsection \<open>Continuous isomorphisms\<close>

text \<open>A locale for continuous isomorphisms\<close>

locale iso =
  fixes abs :: "'a \<rightarrow> 'b"
  fixes rep :: "'b \<rightarrow> 'a"
  assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"
  assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"
begin

lemma swap: "iso rep abs"
  by (rule iso.intro [OF rep_iso abs_iso])

lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
proof
  assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
  then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
  then show "x \<sqsubseteq> y" by simp
next
  assume "x \<sqsubseteq> y"
  then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
qed

lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
  by (rule iso.abs_below [OF swap])

lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
  by (simp add: po_eq_conv abs_below)

lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
  by (rule iso.abs_eq [OF swap])

lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
proof -
  have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
  then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
  then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
  then show ?thesis by (rule bottomI)
qed

lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
  by (rule iso.abs_strict [OF swap])

lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
proof -
  have "x = rep\<cdot>(abs\<cdot>x)" by simp
  also assume "abs\<cdot>x = \<bottom>"
  also note rep_strict
  finally show "x = \<bottom>" .
qed

lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
  by (rule iso.abs_defin' [OF swap])

lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
  by (erule contrapos_nn, erule abs_defin')

lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
  by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)

lemma abs_bottom_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
  by (auto elim: abs_defin' intro: abs_strict)

lemma rep_bottom_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
  by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)

lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"
  by (simp add: rep_bottom_iff)

lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"
proof (unfold compact_def)
  assume "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> y)"
  with cont_Rep_cfun2
  have "adm (\<lambda>y. abs\<cdot>x \<notsqsubseteq> abs\<cdot>y)" by (rule adm_subst)
  then show "adm (\<lambda>y. x \<notsqsubseteq> y)" using abs_below by simp
qed

lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
  by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)

lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
  by (rule compact_rep_rev) simp

lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
  by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)

lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
proof
  assume "x = abs\<cdot>y"
  then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
  then show "rep\<cdot>x = y" by simp
next
  assume "rep\<cdot>x = y"
  then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
  then show "x = abs\<cdot>y" by simp
qed

end

subsection \<open>Proofs about take functions\<close>

text \<open>
  This section contains lemmas that are used in a module that supports
  the domain isomorphism package; the module contains proofs related
  to take functions and the finiteness predicate.
\<close>

lemma deflation_abs_rep:
  fixes abs and rep and d
  assumes abs_iso: "\<And>x. rep\<cdot>(abs\<cdot>x) = x"
  assumes rep_iso: "\<And>y. abs\<cdot>(rep\<cdot>y) = y"
  shows "deflation d \<Longrightarrow> deflation (abs oo d oo rep)"
by (rule ep_pair.deflation_e_d_p) (simp add: ep_pair.intro assms)

lemma deflation_chain_min:
  assumes chain: "chain d"
  assumes defl: "\<And>n. deflation (d n)"
  shows "d m\<cdot>(d n\<cdot>x) = d (min m n)\<cdot>x"
proof (rule linorder_le_cases)
  assume "m \<le> n"
  with chain have "d m \<sqsubseteq> d n" by (rule chain_mono)
  then have "d m\<cdot>(d n\<cdot>x) = d m\<cdot>x"
    by (rule deflation_below_comp1 [OF defl defl])
  moreover from \<open>m \<le> n\<close> have "min m n = m" by simp
  ultimately show ?thesis by simp
next
  assume "n \<le> m"
  with chain have "d n \<sqsubseteq> d m" by (rule chain_mono)
  then have "d m\<cdot>(d n\<cdot>x) = d n\<cdot>x"
    by (rule deflation_below_comp2 [OF defl defl])
  moreover from \<open>n \<le> m\<close> have "min m n = n" by simp
  ultimately show ?thesis by simp
qed

lemma lub_ID_take_lemma:
  assumes "chain t" and "(\<Squnion>n. t n) = ID"
  assumes "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"
proof -
  have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"
    using assms(3) by simp
  then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>y"
    using assms(1) by (simp add: lub_distribs)
  then show "x = y"
    using assms(2) by simp
qed

lemma lub_ID_reach:
  assumes "chain t" and "(\<Squnion>n. t n) = ID"
  shows "(\<Squnion>n. t n\<cdot>x) = x"
using assms by (simp add: lub_distribs)

lemma lub_ID_take_induct:
  assumes "chain t" and "(\<Squnion>n. t n) = ID"
  assumes "adm P" and "\<And>n. P (t n\<cdot>x)" shows "P x"
proof -
  from \<open>chain t\<close> have "chain (\<lambda>n. t n\<cdot>x)" by simp
  from \<open>adm P\<close> this \<open>\<And>n. P (t n\<cdot>x)\<close> have "P (\<Squnion>n. t n\<cdot>x)" by (rule admD)
  with \<open>chain t\<close> \<open>(\<Squnion>n. t n) = ID\<close> show "P x" by (simp add: lub_distribs)
qed

subsection \<open>Finiteness\<close>

text \<open>
  Let a ``decisive'' function be a deflation that maps every input to
  either itself or bottom.  Then if a domain's take functions are all
  decisive, then all values in the domain are finite.
\<close>

definition
  decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"
where
  "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"

lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"
  unfolding decisive_def by simp

lemma decisive_cases:
  assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>"
using assms unfolding decisive_def by auto

lemma decisive_bottom: "decisive \<bottom>"
  unfolding decisive_def by simp

lemma decisive_ID: "decisive ID"
  unfolding decisive_def by simp

lemma decisive_ssum_map:
  assumes f: "decisive f"
  assumes g: "decisive g"
  shows "decisive (ssum_map\<cdot>f\<cdot>g)"
  apply (rule decisiveI)
  subgoal for s
    apply (cases s, simp_all)
     apply (rule_tac x=x in decisive_cases [OF f], simp_all)
    apply (rule_tac x=y in decisive_cases [OF g], simp_all)
    done
  done

lemma decisive_sprod_map:
  assumes f: "decisive f"
  assumes g: "decisive g"
  shows "decisive (sprod_map\<cdot>f\<cdot>g)"
  apply (rule decisiveI)
  subgoal for s
    apply (cases s, simp)
    subgoal for x y
      apply (rule decisive_cases [OF f, where x = x], simp_all)
      apply (rule decisive_cases [OF g, where x = y], simp_all)
      done
    done
  done

lemma decisive_abs_rep:
  fixes abs rep
  assumes iso: "iso abs rep"
  assumes d: "decisive d"
  shows "decisive (abs oo d oo rep)"
  apply (rule decisiveI)
  subgoal for s
    apply (rule decisive_cases [OF d, where x="rep\<cdot>s"])
     apply (simp add: iso.rep_iso [OF iso])
    apply (simp add: iso.abs_strict [OF iso])
    done
  done

lemma lub_ID_finite:
  assumes chain: "chain d"
  assumes lub: "(\<Squnion>n. d n) = ID"
  assumes decisive: "\<And>n. decisive (d n)"
  shows "\<exists>n. d n\<cdot>x = x"
proof -
  have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp
  have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)
  have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"
    using decisive unfolding decisive_def by simp
  hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"
    by auto
  hence "finite (range (\<lambda>n. d n\<cdot>x))"
    by (rule finite_subset, simp)
  with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"
    by (rule finite_range_imp_finch)
  then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"
    unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
  with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)
qed

lemma lub_ID_finite_take_induct:
  assumes "chain d" and "(\<Squnion>n. d n) = ID" and "\<And>n. decisive (d n)"
  shows "(\<And>n. P (d n\<cdot>x)) \<Longrightarrow> P x"
using lub_ID_finite [OF assms] by metis

subsection \<open>Proofs about constructor functions\<close>

text \<open>Lemmas for proving nchotomy rule:\<close>

lemma ex_one_bottom_iff:
  "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = P ONE"
by simp

lemma ex_up_bottom_iff:
  "(\<exists>x. P x \<and> x \<noteq> \<bottom>) = (\<exists>x. P (up\<cdot>x))"
by (safe, case_tac x, auto)

lemma ex_sprod_bottom_iff:
 "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
  (\<exists>x y. (P (:x, y:) \<and> x \<noteq> \<bottom>) \<and> y \<noteq> \<bottom>)"
by (safe, case_tac y, auto)

lemma ex_sprod_up_bottom_iff:
 "(\<exists>y. P y \<and> y \<noteq> \<bottom>) =
  (\<exists>x y. P (:up\<cdot>x, y:) \<and> y \<noteq> \<bottom>)"
by (safe, case_tac y, simp, case_tac x, auto)

lemma ex_ssum_bottom_iff:
 "(\<exists>x. P x \<and> x \<noteq> \<bottom>) =
 ((\<exists>x. P (sinl\<cdot>x) \<and> x \<noteq> \<bottom>) \<or>
  (\<exists>x. P (sinr\<cdot>x) \<and> x \<noteq> \<bottom>))"
by (safe, case_tac x, auto)

lemma exh_start: "p = \<bottom> \<or> (\<exists>x. p = x \<and> x \<noteq> \<bottom>)"
  by auto

lemmas ex_bottom_iffs =
   ex_ssum_bottom_iff
   ex_sprod_up_bottom_iff
   ex_sprod_bottom_iff
   ex_up_bottom_iff
   ex_one_bottom_iff

text \<open>Rules for turning nchotomy into exhaust:\<close>

lemma exh_casedist0: "\<lbrakk>R; R \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" (* like make_elim *)
  by auto

lemma exh_casedist1: "((P \<or> Q \<Longrightarrow> R) \<Longrightarrow> S) \<equiv> (\<lbrakk>P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> S)"
  by rule auto

lemma exh_casedist2: "(\<exists>x. P x \<Longrightarrow> Q) \<equiv> (\<And>x. P x \<Longrightarrow> Q)"
  by rule auto

lemma exh_casedist3: "(P \<and> Q \<Longrightarrow> R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> R)"
  by rule auto

lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3

text \<open>Rules for proving constructor properties\<close>

lemmas con_strict_rules =
  sinl_strict sinr_strict spair_strict1 spair_strict2

lemmas con_bottom_iff_rules =
  sinl_bottom_iff sinr_bottom_iff spair_bottom_iff up_defined ONE_defined

lemmas con_below_iff_rules =
  sinl_below sinr_below sinl_below_sinr sinr_below_sinl con_bottom_iff_rules

lemmas con_eq_iff_rules =
  sinl_eq sinr_eq sinl_eq_sinr sinr_eq_sinl con_bottom_iff_rules

lemmas sel_strict_rules =
  cfcomp2 sscase1 sfst_strict ssnd_strict fup1

lemma sel_app_extra_rules:
  "sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinr\<cdot>x) = \<bottom>"
  "sscase\<cdot>ID\<cdot>\<bottom>\<cdot>(sinl\<cdot>x) = x"
  "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinl\<cdot>x) = \<bottom>"
  "sscase\<cdot>\<bottom>\<cdot>ID\<cdot>(sinr\<cdot>x) = x"
  "fup\<cdot>ID\<cdot>(up\<cdot>x) = x"
by (cases "x = \<bottom>", simp, simp)+

lemmas sel_app_rules =
  sel_strict_rules sel_app_extra_rules
  ssnd_spair sfst_spair up_defined spair_defined

lemmas sel_bottom_iff_rules =
  cfcomp2 sfst_bottom_iff ssnd_bottom_iff

lemmas take_con_rules =
  ssum_map_sinl' ssum_map_sinr' sprod_map_spair' u_map_up
  deflation_strict deflation_ID ID1 cfcomp2

subsection \<open>ML setup\<close>

named_theorems domain_deflation "theorems like deflation a ==> deflation (foo_map$a)"
  and domain_map_ID "theorems like foo_map$ID = ID"

ML_file \<open>Tools/Domain/domain_take_proofs.ML\<close>
ML_file \<open>Tools/cont_consts.ML\<close>
ML_file \<open>Tools/cont_proc.ML\<close>
ML_file \<open>Tools/Domain/domain_constructors.ML\<close>
ML_file \<open>Tools/Domain/domain_induction.ML\<close>

end