Theory Cpodef

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

header {* Subtypes of pcpos *}

theory Cpodef
imports Adm
keywords "pcpodef" "cpodef" :: thy_goal
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.
*}

setup {* Sign.add_const_constraint (@{const_name Porder.below}, NONE) *}

theorem typedef_po:
  fixes Abs :: "'a::po => 'b::type"
  assumes type: "type_definition Rep Abs A"
    and below: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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"
  hence "finite (Abs ` A)" by (rule finite_imageI)
  thus "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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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
  admissible predicate.
*}

lemma typedef_is_lubI:
  assumes below: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
  shows "range (λi. Rep (S i)) <<| Rep x ==> range S <<| x"
unfolding is_lub_def is_ub_def below by simp

lemma Abs_inverse_lub_Rep:
  fixes Abs :: "'a::cpo => 'b::po"
  assumes type: "type_definition Rep Abs A"
    and below: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
    and adm:  "adm (λx. x ∈ A)"
  shows "chain S ==> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))"
 apply (rule type_definition.Abs_inverse [OF type])
 apply (erule admD [OF adm ch2ch_Rep [OF below]])
 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (λx. x ∈ A)"
  shows "chain S ==> range S <<| Abs (\<Squnion>i. Rep (S i))"
proof -
  assume S: "chain S"
  hence "chain (λi. Rep (S i))" by (rule ch2ch_Rep [OF below])
  hence "range (λi. Rep (S i)) <<| (\<Squnion>i. Rep (S i))" by (rule cpo_lubI)
  hence "range (λi. Rep (S i)) <<| Rep (Abs (\<Squnion>i. Rep (S i)))"
    by (simp only: Abs_inverse_lub_Rep [OF type below adm S])
  thus "range S <<| Abs (\<Squnion>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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (λx. x ∈ A)"
  shows "OFCLASS('b, cpo_class)"
proof
  fix S::"nat => 'b" assume "chain S"
  hence "range S <<| Abs (\<Squnion>i. Rep (S i))"
    by (rule typedef_is_lub [OF type below adm])
  thus "∃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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (λx. x ∈ A)"
  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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (λx. x ∈ A)" (* not used *)
    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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
    and adm: "adm (λx. x ∈ A)"
  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)
  thus "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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> Rep y"
    and z_in_A: "z ∈ A"
    and z_least: "!!x. x ∈ A ==> z \<sqsubseteq> 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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: "op \<sqsubseteq> ≡ λx y. Rep x \<sqsubseteq> 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])
 apply (simp add: ax_flat)
done

subsection {* HOLCF type definition package *}

ML_file "Tools/cpodef.ML"

end