# Theory Up

theory Up
imports Cfun
(*  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:
"(op ⊑) ≡ (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"

lemma not_Iup_below [iff]: "Iup x \<notsqsubseteq> Ibottom"

lemma Iup_below [iff]: "(Iup x ⊑ Iup y) = (x ⊑ y)"

subsection ‹Lifted cpo is a partial order›

instance u :: (cpo) po
proof
fix x :: "'a u"
show "x ⊑ x"
unfolding below_up_def by (simp split: u.split)
next
fix x y :: "'a u"
assume "x ⊑ y" "y ⊑ x" thus "x = y"
unfolding below_up_def
by (auto split: u.split_asm intro: below_antisym)
next
fix x y z :: "'a u"
assume "x ⊑ y" "y ⊑ z" thus "x ⊑ z"
unfolding below_up_def
by (auto 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"
unfolding is_lub_def is_ub_def ball_simps
by (auto simp add: 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)
thus "Iup (A i) = Y (i + k)"
by (cases "Y (i + k)", simp_all add: A_def)
qed
from Y have chain_A: "chain A"
unfolding chain_def Iup_below [symmetric]
hence "range A <<| (⨆i. A i)"
by (rule cpo_lubI)
hence "range (λi. Iup (A i)) <<| Iup (⨆i. A i)"
by (rule is_lub_Iup)
hence "range (λi. Y (i + k)) <<| Iup (⨆i. A i)"
by (simp only: Iup_A)
hence "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"
thus "∃x. range (λi. S i) <<| x"
proof (rule up_chain_lemma)
assume "∀i. S i = Ibottom"
hence "range (λi. S i) <<| Ibottom"
thus ?thesis ..
next
fix A :: "nat ⇒ 'a"
assume "range S <<| Iup (⨆i. A i)"
thus ?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)
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)"
hence "Ifup f (⨆i. Y i) = (⨆i. Ifup f (Iup (A i)))"
also have "… = (⨆i. Ifup f (Y (i + k)))"
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)"
apply (induct z)
done

lemma up_eq [simp]: "(up⋅x = up⋅y) = (x = y)"

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"

lemma upE [case_names bottom up, cases type: u]:
"⟦p = ⊥ ⟹ Q; ⋀x. p = up⋅x ⟹ Q⟧ ⟹ Q"
apply (cases p)
done

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)"
apply (rule up_chain_lemma [OF Y])
apply (simp_all add: inst_up_pcpo up_def cont_Iup lub_eqI)
done

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