Theory Up

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theory Up
imports Bifinite

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


header {* The type of lifted values *}

theory Up
imports Bifinite
begin


default_sort cpo

subsection {* Definition of new type for lifting *}

datatype 'a u = Ibottom | Iup 'a

type_notation (xsymbols)
u ("(_)" [1000] 999)


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 \<sqsubseteq>) ≡ (λx y. case x of Ibottom => True | Iup a =>
(case y of Ibottom => False | Iup b => a \<sqsubseteq> b))"


instance ..
end

lemma minimal_up [iff]: "Ibottom \<sqsubseteq> z"
by (simp add: below_up_def)

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

lemma Iup_below [iff]: "(Iup x \<sqsubseteq> Iup y) = (x \<sqsubseteq> 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 \<sqsubseteq> x"
unfolding below_up_def by (simp split: u.split)
next
fix x y :: "'a u"
assume "x \<sqsubseteq> y" "y \<sqsubseteq> 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 \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
unfolding below_up_def
by (auto split: u.split_asm intro: below_trans)
qed

lemma u_UNIV: "UNIV = insert Ibottom (range Iup)"
by (auto, case_tac x, auto)

instance u :: (finite_po) finite_po
by (intro_classes, simp add: u_UNIV)


subsection {* Lifted cpo is a cpo *}

lemma is_lub_Iup:
"range S <<| x ==> range (λi. Iup (S i)) <<| Iup x"

apply (rule is_lubI)
apply (rule ub_rangeI)
apply (subst Iup_below)
apply (erule is_ub_lub)
apply (case_tac u)
apply (drule ub_rangeD)
apply simp
apply simp
apply (erule is_lub_lub)
apply (rule ub_rangeI)
apply (drule_tac i=i in ub_rangeD)
apply simp
done

text {* Now some lemmas about chains of @{typ "'a u"} elements *}

lemma up_lemma1: "z ≠ Ibottom ==> Iup (THE a. Iup a = z) = z"
by (case_tac z, simp_all)

lemma up_lemma2:
"[|chain Y; Y j ≠ Ibottom|] ==> Y (i + j) ≠ Ibottom"

apply (erule contrapos_nn)
apply (drule_tac i="j" and j="i + j" in chain_mono)
apply (rule le_add2)
apply (case_tac "Y j")
apply assumption
apply simp
done

lemma up_lemma3:
"[|chain Y; Y j ≠ Ibottom|] ==> Iup (THE a. Iup a = Y (i + j)) = Y (i + j)"

by (rule up_lemma1 [OF up_lemma2])

lemma up_lemma4:
"[|chain Y; Y j ≠ Ibottom|] ==> chain (λi. THE a. Iup a = Y (i + j))"

apply (rule chainI)
apply (rule Iup_below [THEN iffD1])
apply (subst up_lemma3, assumption+)+
apply (simp add: chainE)
done

lemma up_lemma5:
"[|chain Y; Y j ≠ Ibottom|] ==>
(λi. Y (i + j)) = (λi. Iup (THE a. Iup a = Y (i + j)))"

by (rule ext, rule up_lemma3 [symmetric])

lemma up_lemma6:
"[|chain Y; Y j ≠ Ibottom|]
==> range Y <<| Iup (\<Squnion>i. THE a. Iup a = Y(i + j))"

apply (rule_tac j1 = j in is_lub_range_shift [THEN iffD1])
apply assumption
apply (subst up_lemma5, assumption+)
apply (rule is_lub_Iup)
apply (rule cpo_lubI)
apply (erule (1) up_lemma4)
done

lemma up_chain_lemma:
"chain Y ==>
(∃A. chain A ∧ (\<Squnion>i. Y i) = Iup (\<Squnion>i. A i) ∧
(∃j. ∀i. Y (i + j) = Iup (A i))) ∨ (Y = (λi. Ibottom))"

apply (rule disjCI)
apply (simp add: expand_fun_eq)
apply (erule exE, rename_tac j)
apply (rule_tac x="λi. THE a. Iup a = Y (i + j)" in exI)
apply (simp add: up_lemma4)
apply (simp add: up_lemma6 [THEN thelubI])
apply (rule_tac x=j in exI)
apply (simp add: up_lemma3)
done

lemma cpo_up: "chain (Y::nat => 'a u) ==> ∃x. range Y <<| x"
apply (frule up_chain_lemma, safe)
apply (rule_tac x="Iup (\<Squnion>i. A i)" in exI)
apply (erule_tac j="j" in is_lub_range_shift [THEN iffD1, standard])
apply (simp add: is_lub_Iup cpo_lubI)
apply (rule exI, rule lub_const)
done

instance u :: (cpo) cpo
by intro_classes (rule cpo_up)

subsection {* Lifted cpo is pointed *}

lemma least_up: "∃x::'a u. ∀y. x \<sqsubseteq> y"
apply (rule_tac x = "Ibottom" in exI)
apply (rule minimal_up [THEN allI])
done

instance u :: (cpo) pcpo
by intro_classes (rule least_up)

text {* for compatibility with old HOLCF-Version *}
lemma inst_up_pcpo: "⊥ = Ibottom"
by (rule minimal_up [THEN UU_I, 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)"
apply (rule contI)
apply (frule up_chain_lemma, safe)
apply (rule_tac j="j" in is_lub_range_shift [THEN iffD1, standard])
apply (erule monofun_Ifup2 [THEN ch2ch_monofun])
apply (simp add: cont_cfun_arg)
apply (simp add: lub_const)
done

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"
"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)
apply (simp add: inst_up_pcpo)
apply (simp add: up_def cont_Iup)
done

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 \<sqsubseteq> ⊥"
by simp (* FIXME: remove? *)

lemma up_below [simp]: "up·x \<sqsubseteq> up·y <-> x \<sqsubseteq> 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"

apply (cases p)
apply (simp add: inst_up_pcpo)
apply (simp add: up_def cont_Iup)
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:
"chain Y ==>
(∃A. chain A ∧ (\<Squnion>i. Y i) = up·(\<Squnion>i. A i) ∧
(∃j. ∀i. Y (i + j) = up·(A i))) ∨ Y = (λi. ⊥)"

by (simp add: inst_up_pcpo up_def cont_Iup up_chain_lemma)

lemma compact_up: "compact x ==> compact (up·x)"
apply (rule compactI2)
apply (drule up_chain_cases, safe)
apply (drule (1) compactD2, simp)
apply (erule exE, rule_tac x="i + j" in exI)
apply simp
apply simp
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="\<Squnion>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)

subsection {* Map function for lifted cpo *}

definition
u_map :: "('a -> 'b) -> 'a u -> 'b u"
where
"u_map = (Λ f. fup·(up oo f))"


lemma u_map_strict [simp]: "u_map·f·⊥ = ⊥"
unfolding u_map_def by simp

lemma u_map_up [simp]: "u_map·f·(up·x) = up·(f·x)"
unfolding u_map_def by simp

lemma u_map_ID: "u_map·ID = ID"
unfolding u_map_def by (simp add: expand_cfun_eq eta_cfun)

lemma u_map_map: "u_map·f·(u_map·g·p) = u_map·(Λ x. f·(g·x))·p"
by (induct p) simp_all

lemma ep_pair_u_map: "ep_pair e p ==> ep_pair (u_map·e) (u_map·p)"
apply default
apply (case_tac x, simp, simp add: ep_pair.e_inverse)
apply (case_tac y, simp, simp add: ep_pair.e_p_below)
done

lemma deflation_u_map: "deflation d ==> deflation (u_map·d)"
apply default
apply (case_tac x, simp, simp add: deflation.idem)
apply (case_tac x, simp, simp add: deflation.below)
done

lemma finite_deflation_u_map:
assumes "finite_deflation d" shows "finite_deflation (u_map·d)"

proof (intro finite_deflation.intro finite_deflation_axioms.intro)
interpret d: finite_deflation d by fact
have "deflation d" by fact
thus "deflation (u_map·d)" by (rule deflation_u_map)
have "{x. u_map·d·x = x} ⊆ insert ⊥ ((λx. up·x) ` {x. d·x = x})"
by (rule subsetI, case_tac x, simp_all)
thus "finite {x. u_map·d·x = x}"
by (rule finite_subset, simp add: d.finite_fixes)
qed

subsection {* Lifted cpo is a bifinite domain *}

instantiation u :: (profinite) bifinite
begin


definition
approx_up_def:
"approx = (λn. u_map·(approx n))"


instance proof
fix i :: nat and x :: "'a u"
show "chain (approx :: nat => 'a u -> 'a u)"
unfolding approx_up_def by simp
show "(\<Squnion>i. approx i·x) = x"
unfolding approx_up_def
by (induct x, simp, simp add: lub_distribs)
show "approx i·(approx i·x) = approx i·x"
unfolding approx_up_def
by (induct x) simp_all
show "finite {x::'a u. approx i·x = x}"
unfolding approx_up_def
by (intro finite_deflation.finite_fixes
finite_deflation_u_map
finite_deflation_approx)

qed

end

lemma approx_up [simp]: "approx i·(up·x) = up·(approx i·x)"
unfolding approx_up_def by simp

end