Theory Domain_Aux

theory Domain_Aux
imports Map_Functions Fixrec
(*  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⋅(abs⋅x) = x"
  assumes rep_iso [simp]: "abs⋅(rep⋅y) = y"

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

lemma abs_below: "(abs⋅x ⊑ abs⋅y) = (x ⊑ y)"
  assume "abs⋅x ⊑ abs⋅y"
  then have "rep⋅(abs⋅x) ⊑ rep⋅(abs⋅y)" by (rule monofun_cfun_arg)
  then show "x ⊑ y" by simp
  assume "x ⊑ y"
  then show "abs⋅x ⊑ abs⋅y" by (rule monofun_cfun_arg)

lemma rep_below: "(rep⋅x ⊑ rep⋅y) = (x ⊑ y)"
  by (rule iso.abs_below [OF swap])

lemma abs_eq: "(abs⋅x = abs⋅y) = (x = y)"
  by (simp add: po_eq_conv abs_below)

lemma rep_eq: "(rep⋅x = rep⋅y) = (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': "abs⋅x = ⊥ ⟹ x = ⊥"
proof -
  have "x = rep⋅(abs⋅x)" by simp
  also assume "abs⋅x = ⊥"
  also note rep_strict
  finally show "x = ⊥" .

lemma rep_defin': "rep⋅z = ⊥ ⟹ z = ⊥"
  by (rule iso.abs_defin' [OF swap])

lemma abs_defined: "z ≠ ⊥ ⟹ abs⋅z ≠ ⊥"
  by (erule contrapos_nn, erule abs_defin')

lemma rep_defined: "z ≠ ⊥ ⟹ rep⋅z ≠ ⊥"
  by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)

lemma abs_bottom_iff: "(abs⋅x = ⊥) = (x = ⊥)"
  by (auto elim: abs_defin' intro: abs_strict)

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

lemma casedist_rule: "rep⋅x = ⊥ ∨ P ⟹ x = ⊥ ∨ P"
  by (simp add: rep_bottom_iff)

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

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

lemma compact_abs: "compact x ⟹ compact (abs⋅x)"
  by (rule compact_rep_rev) simp

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

lemma iso_swap: "(x = abs⋅y) = (rep⋅x = y)"
  assume "x = abs⋅y"
  then have "rep⋅x = rep⋅(abs⋅y)" by simp
  then show "rep⋅x = y" by simp
  assume "rep⋅x = y"
  then have "abs⋅(rep⋅x) = abs⋅y" by simp
  then show "x = abs⋅y" 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⋅(abs⋅x) = x"
  assumes rep_iso: "⋀y. abs⋅(rep⋅y) = 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 n⋅x) = 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 n⋅x) = d m⋅x"
    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 n⋅x) = d n⋅x"
    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 n⋅x = t n⋅y" shows "x = y"
proof -
  have "(⨆n. t n⋅x) = (⨆n. t n⋅y)"
    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 n⋅x) = 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 n⋅x)" shows "P x"
proof -
  from ‹chain t› have "chain (λn. t n⋅x)" by simp
  from ‹adm P› this ‹⋀n. P (t n⋅x)› have "P (⨆n. t n⋅x)" 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. d⋅x = x ∨ d⋅x = ⊥)"

lemma decisiveI: "(⋀x. d⋅x = x ∨ d⋅x = ⊥) ⟹ decisive d"
  unfolding decisive_def by simp

lemma decisive_cases:
  assumes "decisive d" obtains "d⋅x = x" | "d⋅x = ⊥"
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_map⋅f⋅g)"
apply (rule decisiveI, rename_tac s)
apply (case_tac 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_map⋅f⋅g)"
apply (rule decisiveI, rename_tac s)
apply (case_tac 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_abs_rep:
  fixes abs rep
  assumes iso: "iso abs rep"
  assumes d: "decisive d"
  shows "decisive (abs oo d oo rep)"
apply (rule decisiveI)
apply (rule_tac x="rep⋅x" in decisive_cases [OF d])
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 n⋅x = x"
proof -
  have 1: "chain (λn. d n⋅x)" using chain by simp
  have 2: "(⨆n. d n⋅x) = x" using chain lub by (rule lub_ID_reach)
  have "∀n. d n⋅x = x ∨ d n⋅x = ⊥"
    using decisive unfolding decisive_def by simp
  hence "range (λn. d n⋅x) ⊆ {x, ⊥}"
    by auto
  hence "finite (range (λn. d n⋅x))"
    by (rule finite_subset, simp)
  with 1 have "finite_chain (λn. d n⋅x)"
    by (rule finite_range_imp_finch)
  then have "∃n. (⨆n. d n⋅x) = d n⋅x"
    unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
  with 2 show "∃n. d n⋅x = 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 n⋅x)) ⟹ 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 (up⋅x))"
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 (:up⋅x, y:) ∧ y ≠ ⊥)"
by (safe, case_tac y, simp, case_tac x, auto)

lemma ex_ssum_bottom_iff:
 "(∃x. P x ∧ x ≠ ⊥) =
 ((∃x. P (sinl⋅x) ∧ x ≠ ⊥) ∨
  (∃x. P (sinr⋅x) ∧ 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:
  "sscase⋅ID⋅⊥⋅(sinr⋅x) = ⊥"
  "sscase⋅ID⋅⊥⋅(sinl⋅x) = x"
  "sscase⋅⊥⋅ID⋅(sinl⋅x) = ⊥"
  "sscase⋅⊥⋅ID⋅(sinr⋅x) = x"
  "fup⋅ID⋅(up⋅x) = 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"
ML_file "Tools/Domain/domain_constructors.ML"
ML_file "Tools/Domain/domain_induction.ML"