# Theory Cpodef

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

section ‹Subtypes of pcpos›

theory Cpodef
keywords "pcpodef" "cpodef" :: thy_goal_defn
begin

subsection ‹Proving a subtype is a partial order›

text ‹
A subtype of a partial order is itself a partial order,
if the ordering is defined in the standard way.
›

theorem typedef_po:
fixes Abs :: "'a::po ⇒ 'b::type"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "OFCLASS('b, po_class)"
apply (intro_classes, unfold below)
apply (rule below_refl)
apply (erule (1) below_trans)
apply (rule type_definition.Rep_inject [OF type, THEN iffD1])
apply (erule (1) below_antisym)
done

setup ‹Sign.add_const_constraint (\<^const_name>‹Porder.below›, SOME \<^typ>‹'a::below ⇒ 'a::below ⇒ bool›)›

subsection ‹Proving a subtype is finite›

lemma typedef_finite_UNIV:
fixes Abs :: "'a::type ⇒ 'b::type"
assumes type: "type_definition Rep Abs A"
shows "finite A ⟹ finite (UNIV :: 'b set)"
proof -
assume "finite A"
then have "finite (Abs  A)"
by (rule finite_imageI)
then show "finite (UNIV :: 'b set)"
by (simp only: type_definition.Abs_image [OF type])
qed

subsection ‹Proving a subtype is chain-finite›

lemma ch2ch_Rep:
assumes below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "chain S ⟹ chain (λi. Rep (S i))"
unfolding chain_def below .

theorem typedef_chfin:
fixes Abs :: "'a::chfin ⇒ 'b::po"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "OFCLASS('b, chfin_class)"
apply intro_classes
apply (drule ch2ch_Rep [OF below])
apply (drule chfin)
apply (unfold max_in_chain_def)
apply (simp add: type_definition.Rep_inject [OF type])
done

subsection ‹Proving a subtype is complete›

text ‹
A subtype of a cpo is itself a cpo if the ordering is
defined in the standard way, and the defining subset
is closed with respect to limits of chains.  A set is
closed if and only if membership in the set is an
›

lemma typedef_is_lubI:
assumes below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "range (λi. Rep (S i)) <<| Rep x ⟹ range S <<| x"
by (simp add: is_lub_def is_ub_def below)

lemma Abs_inverse_lub_Rep:
fixes Abs :: "'a::cpo ⇒ 'b::po"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "chain S ⟹ Rep (Abs (⨆i. Rep (S i))) = (⨆i. Rep (S i))"
apply (rule type_definition.Abs_inverse [OF type])
apply (rule type_definition.Rep [OF type])
done

theorem typedef_is_lub:
fixes Abs :: "'a::cpo ⇒ 'b::po"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
assumes S: "chain S"
shows "range S <<| Abs (⨆i. Rep (S i))"
proof -
from S have "chain (λi. Rep (S i))"
by (rule ch2ch_Rep [OF below])
then have "range (λi. Rep (S i)) <<| (⨆i. Rep (S i))"
by (rule cpo_lubI)
then have "range (λi. Rep (S i)) <<| Rep (Abs (⨆i. Rep (S i)))"
by (simp only: Abs_inverse_lub_Rep [OF type below adm S])
then show "range S <<| Abs (⨆i. Rep (S i))"
by (rule typedef_is_lubI [OF below])
qed

lemmas typedef_lub = typedef_is_lub [THEN lub_eqI]

theorem typedef_cpo:
fixes Abs :: "'a::cpo ⇒ 'b::po"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "OFCLASS('b, cpo_class)"
proof
fix S :: "nat ⇒ 'b"
assume "chain S"
then have "range S <<| Abs (⨆i. Rep (S i))"
by (rule typedef_is_lub [OF type below adm])
then show "∃x. range S <<| x" ..
qed

subsubsection ‹Continuity of \emph{Rep} and \emph{Abs}›

text ‹For any sub-cpo, the \<^term>‹Rep› function is continuous.›

theorem typedef_cont_Rep:
fixes Abs :: "'a::cpo ⇒ 'b::cpo"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "cont (λx. f x) ⟹ cont (λx. Rep (f x))"
apply (erule cont_apply [OF _ _ cont_const])
apply (rule contI)
apply (simp only: typedef_lub [OF type below adm])
apply (simp only: Abs_inverse_lub_Rep [OF type below adm])
apply (rule cpo_lubI)
apply (erule ch2ch_Rep [OF below])
done

text ‹
For a sub-cpo, we can make the \<^term>‹Abs› function continuous
only if we restrict its domain to the defining subset by
composing it with another continuous function.
›

theorem typedef_cont_Abs:
fixes Abs :: "'a::cpo ⇒ 'b::cpo"
fixes f :: "'c::cpo ⇒ 'a::cpo"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and f_in_A: "⋀x. f x ∈ A"
shows "cont f ⟹ cont (λx. Abs (f x))"
unfolding cont_def is_lub_def is_ub_def ball_simps below
by (simp add: type_definition.Abs_inverse [OF type f_in_A])

subsection ‹Proving subtype elements are compact›

theorem typedef_compact:
fixes Abs :: "'a::cpo ⇒ 'b::cpo"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
shows "compact (Rep k) ⟹ compact k"
proof (unfold compact_def)
have cont_Rep: "cont Rep"
by (rule typedef_cont_Rep [OF type below adm cont_id])
assume "adm (λx. Rep k \<notsqsubseteq> x)"
with cont_Rep have "adm (λx. Rep k \<notsqsubseteq> Rep x)" by (rule adm_subst)
then show "adm (λx. k \<notsqsubseteq> x)" by (unfold below)
qed

subsection ‹Proving a subtype is pointed›

text ‹
A subtype of a cpo has a least element if and only if
the defining subset has a least element.
›

theorem typedef_pcpo_generic:
fixes Abs :: "'a::cpo ⇒ 'b::cpo"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and z_in_A: "z ∈ A"
and z_least: "⋀x. x ∈ A ⟹ z ⊑ x"
shows "OFCLASS('b, pcpo_class)"
apply (intro_classes)
apply (rule_tac x="Abs z" in exI, rule allI)
apply (unfold below)
apply (subst type_definition.Abs_inverse [OF type z_in_A])
apply (rule z_least [OF type_definition.Rep [OF type]])
done

text ‹
As a special case, a subtype of a pcpo has a least element
if the defining subset contains \<^term>‹⊥›.
›

theorem typedef_pcpo:
fixes Abs :: "'a::pcpo ⇒ 'b::cpo"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and bottom_in_A: "⊥ ∈ A"
shows "OFCLASS('b, pcpo_class)"
by (rule typedef_pcpo_generic [OF type below bottom_in_A], rule minimal)

subsubsection ‹Strictness of \emph{Rep} and \emph{Abs}›

text ‹
For a sub-pcpo where \<^term>‹⊥› is a member of the defining
subset, \<^term>‹Rep› and \<^term>‹Abs› are both strict.
›

theorem typedef_Abs_strict:
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and bottom_in_A: "⊥ ∈ A"
shows "Abs ⊥ = ⊥"
apply (rule bottomI, unfold below)
apply (simp add: type_definition.Abs_inverse [OF type bottom_in_A])
done

theorem typedef_Rep_strict:
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and bottom_in_A: "⊥ ∈ A"
shows "Rep ⊥ = ⊥"
apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
apply (rule type_definition.Abs_inverse [OF type bottom_in_A])
done

theorem typedef_Abs_bottom_iff:
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and bottom_in_A: "⊥ ∈ A"
shows "x ∈ A ⟹ (Abs x = ⊥) = (x = ⊥)"
apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst])
apply (simp add: type_definition.Abs_inject [OF type] bottom_in_A)
done

theorem typedef_Rep_bottom_iff:
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and bottom_in_A: "⊥ ∈ A"
shows "(Rep x = ⊥) = (x = ⊥)"
apply (rule typedef_Rep_strict [OF type below bottom_in_A, THEN subst])
apply (simp add: type_definition.Rep_inject [OF type])
done

subsection ‹Proving a subtype is flat›

theorem typedef_flat:
fixes Abs :: "'a::flat ⇒ 'b::pcpo"
assumes type: "type_definition Rep Abs A"
and below: "(⊑) ≡ λx y. Rep x ⊑ Rep y"
and bottom_in_A: "⊥ ∈ A"
shows "OFCLASS('b, flat_class)"
apply (intro_classes)
apply (unfold below)
apply (simp add: type_definition.Rep_inject [OF type, symmetric])
apply (simp add: typedef_Rep_strict [OF type below bottom_in_A])
`