Theory Up

(*  Title:      HOL/HOLCF/Up.thy
    Author:     Franz Regensburger
    Author:     Brian Huffman
*)

section ‹The type of lifted values›

theory Up
  imports Cfun
begin

default_sort cpo


subsection ‹Definition of new type for lifting›

datatype 'a u  ("(_⇩⊥)" [1000] 999) = Ibottom | Iup 'a

primrec Ifup :: "('a → 'b::pcpo) ⇒ 'a u ⇒ 'b"
  where
    "Ifup f Ibottom = ⊥"
  | "Ifup f (Iup x) = f⋅x"


subsection ‹Ordering on lifted cpo›

instantiation u :: (cpo) below
begin

definition below_up_def:
  "(⊑) ≡
    (λx y.
      (case x of
        Ibottom ⇒ True
      | Iup a ⇒ (case y of Ibottom ⇒ False | Iup b ⇒ a ⊑ b)))"

instance ..

end

lemma minimal_up [iff]: "Ibottom ⊑ z"
  by (simp add: below_up_def)

lemma not_Iup_below [iff]: "Iup x \<notsqsubseteq> Ibottom"
  by (simp add: below_up_def)

lemma Iup_below [iff]: "(Iup x ⊑ Iup y) = (x ⊑ y)"
  by (simp add: below_up_def)


subsection ‹Lifted cpo is a partial order›

instance u :: (cpo) po
proof
  fix x :: "'a u"
  show "x ⊑ x"
    by (simp add: below_up_def split: u.split)
next
  fix x y :: "'a u"
  assume "x ⊑ y" "y ⊑ x"
  then show "x = y"
    by (auto simp: below_up_def split: u.split_asm intro: below_antisym)
next
  fix x y z :: "'a u"
  assume "x ⊑ y" "y ⊑ z"
  then show "x ⊑ z"
    by (auto simp: below_up_def split: u.split_asm intro: below_trans)
qed


subsection ‹Lifted cpo is a cpo›

lemma is_lub_Iup: "range S <<| x ⟹ range (λi. Iup (S i)) <<| Iup x"
  by (auto simp: is_lub_def is_ub_def ball_simps below_up_def split: u.split)

lemma up_chain_lemma:
  assumes Y: "chain Y"
  obtains "∀i. Y i = Ibottom"
  | A k where "∀i. Iup (A i) = Y (i + k)" and "chain A" and "range Y <<| Iup (⨆i. A i)"
proof (cases "∃k. Y k ≠ Ibottom")
  case True
  then obtain k where k: "Y k ≠ Ibottom" ..
  define A where "A i = (THE a. Iup a = Y (i + k))" for i
  have Iup_A: "∀i. Iup (A i) = Y (i + k)"
  proof
    fix i :: nat
    from Y le_add2 have "Y k ⊑ Y (i + k)" by (rule chain_mono)
    with k have "Y (i + k) ≠ Ibottom" by (cases "Y k") auto
    then show "Iup (A i) = Y (i + k)"
      by (cases "Y (i + k)", simp_all add: A_def)
  qed
  from Y have chain_A: "chain A"
    by (simp add: chain_def Iup_below [symmetric] Iup_A)
  then have "range A <<| (⨆i. A i)"
    by (rule cpo_lubI)
  then have "range (λi. Iup (A i)) <<| Iup (⨆i. A i)"
    by (rule is_lub_Iup)
  then have "range (λi. Y (i + k)) <<| Iup (⨆i. A i)"
    by (simp only: Iup_A)
  then have "range (λi. Y i) <<| Iup (⨆i. A i)"
    by (simp only: is_lub_range_shift [OF Y])
  with Iup_A chain_A show ?thesis ..
next
  case False
  then have "∀i. Y i = Ibottom" by simp
  then show ?thesis ..
qed

instance u :: (cpo) cpo
proof
  fix S :: "nat ⇒ 'a u"
  assume S: "chain S"
  then show "∃x. range (λi. S i) <<| x"
  proof (rule up_chain_lemma)
    assume "∀i. S i = Ibottom"
    then have "range (λi. S i) <<| Ibottom"
      by (simp add: is_lub_const)
    then show ?thesis ..
  next
    fix A :: "nat ⇒ 'a"
    assume "range S <<| Iup (⨆i. A i)"
    then show ?thesis ..
  qed
qed


subsection ‹Lifted cpo is pointed›

instance u :: (cpo) pcpo
  by intro_classes fast

text ‹for compatibility with old HOLCF-Version›
lemma inst_up_pcpo: "⊥ = Ibottom"
  by (rule minimal_up [THEN bottomI, symmetric])


subsection ‹Continuity of \emph{Iup} and \emph{Ifup}›

text ‹continuity for \<^term>‹Iup››

lemma cont_Iup: "cont Iup"
  apply (rule contI)
  apply (rule is_lub_Iup)
  apply (erule cpo_lubI)
  done

text ‹continuity for \<^term>‹Ifup››

lemma cont_Ifup1: "cont (λf. Ifup f x)"
  by (induct x) simp_all

lemma monofun_Ifup2: "monofun (λx. Ifup f x)"
  apply (rule monofunI)
  apply (case_tac x, simp)
  apply (case_tac y, simp)
  apply (simp add: monofun_cfun_arg)
  done

lemma cont_Ifup2: "cont (λx. Ifup f x)"
proof (rule contI2)
  fix Y
  assume Y: "chain Y" and Y': "chain (λi. Ifup f (Y i))"
  from Y show "Ifup f (⨆i. Y i) ⊑ (⨆i. Ifup f (Y i))"
  proof (rule up_chain_lemma)
    fix A and k
    assume A: "∀i. Iup (A i) = Y (i + k)"
    assume "chain A" and "range Y <<| Iup (⨆i. A i)"
    then have "Ifup f (⨆i. Y i) = (⨆i. Ifup f (Iup (A i)))"
      by (simp add: lub_eqI contlub_cfun_arg)
    also have "… = (⨆i. Ifup f (Y (i + k)))"
      by (simp add: A)
    also have "… = (⨆i. Ifup f (Y i))"
      using Y' by (rule lub_range_shift)
    finally show ?thesis by simp
  qed simp
qed (rule monofun_Ifup2)


subsection ‹Continuous versions of constants›

definition up  :: "'a → 'a u"
  where "up = (Λ x. Iup x)"

definition fup :: "('a → 'b::pcpo) → 'a u → 'b"
  where "fup = (Λ f p. Ifup f p)"

translations
  "case l of XCONST up⋅x ⇒ t" ⇌ "CONST fup⋅(Λ x. t)⋅l"
  "case l of (XCONST up :: 'a)⋅x ⇒ t" ⇀ "CONST fup⋅(Λ x. t)⋅l"
  "Λ(XCONST up⋅x). t" ⇌ "CONST fup⋅(Λ x. t)"

text ‹continuous versions of lemmas for \<^typ>‹('a)u››

lemma Exh_Up: "z = ⊥ ∨ (∃x. z = up⋅x)"
  by (induct z) (simp add: inst_up_pcpo, simp add: up_def cont_Iup)

lemma up_eq [simp]: "(up⋅x = up⋅y) = (x = y)"
  by (simp add: up_def cont_Iup)

lemma up_inject: "up⋅x = up⋅y ⟹ x = y"
  by simp

lemma up_defined [simp]: "up⋅x ≠ ⊥"
  by (simp add: up_def cont_Iup inst_up_pcpo)

lemma not_up_less_UU: "up⋅x \<notsqsubseteq> ⊥"
  by simp (* FIXME: remove? *)

lemma up_below [simp]: "up⋅x ⊑ up⋅y ⟷ x ⊑ y"
  by (simp add: up_def cont_Iup)

lemma upE [case_names bottom up, cases type: u]: "⟦p = ⊥ ⟹ Q; ⋀x. p = up⋅x ⟹ Q⟧ ⟹ Q"
  by (cases p) (simp add: inst_up_pcpo, simp add: up_def cont_Iup)

lemma up_induct [case_names bottom up, induct type: u]: "P ⊥ ⟹ (⋀x. P (up⋅x)) ⟹ P x"
  by (cases x) simp_all

text ‹lifting preserves chain-finiteness›

lemma up_chain_cases:
  assumes Y: "chain Y"
  obtains "∀i. Y i = ⊥"
  | A k where "∀i. up⋅(A i) = Y (i + k)" and "chain A" and "(⨆i. Y i) = up⋅(⨆i. A i)"
  by (rule up_chain_lemma [OF Y]) (simp_all add: inst_up_pcpo up_def cont_Iup lub_eqI)

lemma compact_up: "compact x ⟹ compact (up⋅x)"
  apply (rule compactI2)
  apply (erule up_chain_cases)
   apply simp
  apply (drule (1) compactD2, simp)
  apply (erule exE)
  apply (drule_tac f="up" and x="x" in monofun_cfun_arg)
  apply (simp, erule exI)
  done

lemma compact_upD: "compact (up⋅x) ⟹ compact x"
  unfolding compact_def
  by (drule adm_subst [OF cont_Rep_cfun2 [where f=up]], simp)

lemma compact_up_iff [simp]: "compact (up⋅x) = compact x"
  by (safe elim!: compact_up compact_upD)

instance u :: (chfin) chfin
  apply intro_classes
  apply (erule compact_imp_max_in_chain)
  apply (rule_tac p="⨆i. Y i" in upE, simp_all)
  done

text ‹properties of fup›

lemma fup1 [simp]: "fup⋅f⋅⊥ = ⊥"
  by (simp add: fup_def cont_Ifup1 cont_Ifup2 inst_up_pcpo cont2cont_LAM)

lemma fup2 [simp]: "fup⋅f⋅(up⋅x) = f⋅x"
  by (simp add: up_def fup_def cont_Iup cont_Ifup1 cont_Ifup2 cont2cont_LAM)

lemma fup3 [simp]: "fup⋅up⋅x = x"
  by (cases x) simp_all

end