Theory Domain_Aux

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

section ‹Domain package support›

theory Domain_Aux
imports Map_Functions Fixrec

subsection ‹Continuous isomorphisms›

text ‹A locale for continuous isomorphisms›

locale iso =
  fixes abs :: "'a  'b"
  fixes rep :: "'b  'a"
  assumes abs_iso [simp]: "rep(absx) = x"
  assumes rep_iso [simp]: "abs(repy) = y"

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

lemma abs_below: "(absx  absy) = (x  y)"
  assume "absx  absy"
  then have "rep(absx)  rep(absy)" by (rule monofun_cfun_arg)
  then show "x  y" by simp
  assume "x  y"
  then show "absx  absy" by (rule monofun_cfun_arg)

lemma rep_below: "(repx  repy) = (x  y)"
  by (rule iso.abs_below [OF swap])

lemma abs_eq: "(absx = absy) = (x = y)"
  by (simp add: po_eq_conv abs_below)

lemma rep_eq: "(repx = repy) = (x = y)"
  by (rule iso.abs_eq [OF swap])

lemma abs_strict: "abs = "
proof -
  have "  rep" ..
  then have "abs  abs(rep)" by (rule monofun_cfun_arg)
  then have "abs  " by simp
  then show ?thesis by (rule bottomI)

lemma rep_strict: "rep = "
  by (rule iso.abs_strict [OF swap])

lemma abs_defin': "absx =   x = "
proof -
  have "x = rep(absx)" by simp
  also assume "absx = "
  also note rep_strict
  finally show "x = " .

lemma rep_defin': "repz =   z = "
  by (rule iso.abs_defin' [OF swap])

lemma abs_defined: "z    absz  "
  by (erule contrapos_nn, erule abs_defin')

lemma rep_defined: "z    repz  "
  by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)

lemma abs_bottom_iff: "(absx = ) = (x = )"
  by (auto elim: abs_defin' intro: abs_strict)

lemma rep_bottom_iff: "(repx = ) = (x = )"
  by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)

lemma casedist_rule: "repx =   P  x =   P"
  by (simp add: rep_bottom_iff)

lemma compact_abs_rev: "compact (absx)  compact x"
proof (unfold compact_def)
  assume "adm (λy. absx \<notsqsubseteq> y)"
  with cont_Rep_cfun2
  have "adm (λy. absx \<notsqsubseteq> absy)" by (rule adm_subst)
  then show "adm (λy. x \<notsqsubseteq> y)" using abs_below by simp

lemma compact_rep_rev: "compact (repx)  compact x"
  by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)

lemma compact_abs: "compact x  compact (absx)"
  by (rule compact_rep_rev) simp

lemma compact_rep: "compact x  compact (repx)"
  by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)

lemma iso_swap: "(x = absy) = (repx = y)"
  assume "x = absy"
  then have "repx = rep(absy)" by simp
  then show "repx = y" by simp
  assume "repx = y"
  then have "abs(repx) = absy" by simp
  then show "x = absy" by simp


subsection ‹Proofs about take functions›

text ‹
  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.

lemma deflation_abs_rep:
  fixes abs and rep and d
  assumes abs_iso: "x. rep(absx) = x"
  assumes rep_iso: "y. abs(repy) = y"
  shows "deflation d  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: "n. deflation (d n)"
  shows "d m(d nx) = d (min m n)x"
proof (rule linorder_le_cases)
  assume "m  n"
  with chain have "d m  d n" by (rule chain_mono)
  then have "d m(d nx) = d mx"
    by (rule deflation_below_comp1 [OF defl defl])
  moreover from m  n have "min m n = m" by simp
  ultimately show ?thesis by simp
  assume "n  m"
  with chain have "d n  d m" by (rule chain_mono)
  then have "d m(d nx) = d nx"
    by (rule deflation_below_comp2 [OF defl defl])
  moreover from n  m have "min m n = n" by simp
  ultimately show ?thesis by simp

lemma lub_ID_take_lemma:
  assumes "chain t" and "(n. t n) = ID"
  assumes "n. t nx = t ny" shows "x = y"
proof -
  have "(n. t nx) = (n. t ny)"
    using assms(3) by simp
  then have "(n. t n)x = (n. t n)y"
    using assms(1) by (simp add: lub_distribs)
  then show "x = y"
    using assms(2) by simp

lemma lub_ID_reach:
  assumes "chain t" and "(n. t n) = ID"
  shows "(n. t nx) = x"
using assms by (simp add: lub_distribs)

lemma lub_ID_take_induct:
  assumes "chain t" and "(n. t n) = ID"
  assumes "adm P" and "n. P (t nx)" shows "P x"
proof -
  from chain t have "chain (λn. t nx)" by simp
  from adm P this n. P (t nx) have "P (n. t nx)" by (rule admD)
  with chain t (n. t n) = ID show "P x" by (simp add: lub_distribs)

subsection ‹Finiteness›

text ‹
  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.

  decisive :: "('a::pcpo  'a)  bool"
  "decisive d  (x. dx = x  dx = )"

lemma decisiveI: "(x. dx = x  dx = )  decisive d"
  unfolding decisive_def by simp

lemma decisive_cases:
  assumes "decisive d" obtains "dx = x" | "dx = "
using assms unfolding decisive_def by auto

lemma decisive_bottom: "decisive "
  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_mapfg)"
  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)

lemma decisive_sprod_map:
  assumes f: "decisive f"
  assumes g: "decisive g"
  shows "decisive (sprod_mapfg)"
  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)

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="reps"])
     apply (simp add: iso.rep_iso [OF iso])
    apply (simp add: iso.abs_strict [OF iso])

lemma lub_ID_finite:
  assumes chain: "chain d"
  assumes lub: "(n. d n) = ID"
  assumes decisive: "n. decisive (d n)"
  shows "n. d nx = x"
proof -
  have 1: "chain (λn. d nx)" using chain by simp
  have 2: "(n. d nx) = x" using chain lub by (rule lub_ID_reach)
  have "n. d nx = x  d nx = "
    using decisive unfolding decisive_def by simp
  hence "range (λn. d nx)  {x, }"
    by auto
  hence "finite (range (λn. d nx))"
    by (rule finite_subset, simp)
  with 1 have "finite_chain (λn. d nx)"
    by (rule finite_range_imp_finch)
  then have "n. (n. d nx) = d nx"
    unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
  with 2 show "n. d nx = x" by (auto elim: sym)

lemma lub_ID_finite_take_induct:
  assumes "chain d" and "(n. d n) = ID" and "n. decisive (d n)"
  shows "(n. P (d nx))  P x"
using lub_ID_finite [OF assms] by metis

subsection ‹Proofs about constructor functions›

text ‹Lemmas for proving nchotomy rule:›

lemma ex_one_bottom_iff:
  "(x. P x  x  ) = P ONE"
by simp

lemma ex_up_bottom_iff:
  "(x. P x  x  ) = (x. P (upx))"
by (safe, case_tac x, auto)

lemma ex_sprod_bottom_iff:
 "(y. P y  y  ) =
  (x y. (P (:x, y:)  x  )  y  )"
by (safe, case_tac y, auto)

lemma ex_sprod_up_bottom_iff:
 "(y. P y  y  ) =
  (x y. P (:upx, y:)  y  )"
by (safe, case_tac y, simp, case_tac x, auto)

lemma ex_ssum_bottom_iff:
 "(x. P x  x  ) =
 ((x. P (sinlx)  x  ) 
  (x. P (sinrx)  x  ))"
by (safe, case_tac x, auto)

lemma exh_start: "p =   (x. p = x  x  )"
  by auto

lemmas ex_bottom_iffs =

text ‹Rules for turning nchotomy into exhaust:›

lemma exh_casedist0: "R; R  P  P" (* like make_elim *)
  by auto

lemma exh_casedist1: "((P  Q  R)  S)  (P  R; Q  R  S)"
  by rule auto

lemma exh_casedist2: "(x. P x  Q)  (x. P x  Q)"
  by rule auto

lemma exh_casedist3: "(P  Q  R)  (P  Q  R)"
  by rule auto

lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3

text ‹Rules for proving constructor properties›

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:
  "sscaseID(sinrx) = "
  "sscaseID(sinlx) = x"
  "sscaseID(sinlx) = "
  "sscaseID(sinrx) = x"
  "fupID(upx) = x"
by (cases "x = ", 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 ‹ML setup›

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

ML_file ‹Tools/Domain/domain_take_proofs.ML›
ML_file ‹Tools/cont_consts.ML›
ML_file ‹Tools/cont_proc.ML›
simproc_setup cont ("cont f") = K ContProc.cont_proc

ML_file ‹Tools/Domain/domain_constructors.ML›
ML_file ‹Tools/Domain/domain_induction.ML›