(* Title: HOL/HOLCF/Cpodef.thy
Author: Brian Huffman
*)
section \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 \ \ \x 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"
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 \ \ \x y. Rep x \ Rep y"
shows "chain S \ chain (\i. 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 \ \ \x 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
admissible predicate.
\
lemma typedef_is_lubI:
assumes below: "op \ \ \x y. Rep x \ Rep y"
shows "range (\i. 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 \ \ \x y. Rep x \ Rep y"
and adm: "adm (\x. x \ A)"
shows "chain S \ Rep (Abs (\i. Rep (S i))) = (\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 \ \ \x y. Rep x \ Rep y"
and adm: "adm (\x. x \ A)"
shows "chain S \ range S <<| Abs (\i. Rep (S i))"
proof -
assume S: "chain S"
hence "chain (\i. Rep (S i))" by (rule ch2ch_Rep [OF below])
hence "range (\i. Rep (S i)) <<| (\i. Rep (S i))" by (rule cpo_lubI)
hence "range (\i. Rep (S i)) <<| Rep (Abs (\i. Rep (S i)))"
by (simp only: Abs_inverse_lub_Rep [OF type below adm S])
thus "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: "op \ \ \x y. Rep x \ Rep y"
and adm: "adm (\x. x \ A)"
shows "OFCLASS('b, cpo_class)"
proof
fix S::"nat \ 'b" assume "chain S"
hence "range S <<| Abs (\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 \ \ \x y. Rep x \ Rep y"
and adm: "adm (\x. x \ A)"
shows "cont (\x. f x) \ cont (\x. 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 \ \ \x y. Rep x \ Rep y"
and adm: "adm (\x. x \ A)" (* not used *)
and f_in_A: "\x. f x \ A"
shows "cont f \ cont (\x. 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 \ \ \x y. Rep x \ Rep y"
and adm: "adm (\x. 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 (\x. Rep k \ x)"
with cont_Rep have "adm (\x. Rep k \ Rep x)" by (rule adm_subst)
thus "adm (\x. k \ 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 \ \ \x 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: "op \ \ \x 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: "op \ \ \x 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: "op \ \ \x 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: "op \ \ \x 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: "op \ \ \x 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: "op \ \ \x 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])
apply (simp add: ax_flat)
done
subsection \HOLCF type definition package\
ML_file "Tools/cpodef.ML"
end