# Theory Compact_Basis

theory Compact_Basis
imports Universal
```(*  Title:      HOL/HOLCF/Compact_Basis.thy
Author:     Brian Huffman
*)

section ‹A compact basis for powerdomains›

theory Compact_Basis
imports Universal
begin

default_sort bifinite

subsection ‹A compact basis for powerdomains›

definition "pd_basis = {S::'a compact_basis set. finite S ∧ S ≠ {}}"

typedef 'a pd_basis = "pd_basis :: 'a compact_basis set set"
unfolding pd_basis_def
apply (rule_tac x="{_}" in exI)
apply simp
done

lemma finite_Rep_pd_basis [simp]: "finite (Rep_pd_basis u)"
by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp

lemma Rep_pd_basis_nonempty [simp]: "Rep_pd_basis u ≠ {}"
by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp

text ‹The powerdomain basis type is countable.›

lemma pd_basis_countable: "∃f::'a pd_basis ⇒ nat. inj f"
proof -
obtain g :: "'a compact_basis ⇒ nat" where "inj g"
using compact_basis.countable ..
hence image_g_eq: "⋀A B. g ` A = g ` B ⟷ A = B"
by (rule inj_image_eq_iff)
have "inj (λt. set_encode (g ` Rep_pd_basis t))"
by (simp add: inj_on_def set_encode_eq image_g_eq Rep_pd_basis_inject)
thus ?thesis by - (rule exI)
(* FIXME: why doesn't ".." or "by (rule exI)" work? *)
qed

subsection ‹Unit and plus constructors›

definition
PDUnit :: "'a compact_basis ⇒ 'a pd_basis" where
"PDUnit = (λx. Abs_pd_basis {x})"

definition
PDPlus :: "'a pd_basis ⇒ 'a pd_basis ⇒ 'a pd_basis" where
"PDPlus t u = Abs_pd_basis (Rep_pd_basis t ∪ Rep_pd_basis u)"

lemma Rep_PDUnit:
"Rep_pd_basis (PDUnit x) = {x}"
unfolding PDUnit_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def)

lemma Rep_PDPlus:
"Rep_pd_basis (PDPlus u v) = Rep_pd_basis u ∪ Rep_pd_basis v"
unfolding PDPlus_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def)

lemma PDUnit_inject [simp]: "(PDUnit a = PDUnit b) = (a = b)"
unfolding Rep_pd_basis_inject [symmetric] Rep_PDUnit by simp

lemma PDPlus_assoc: "PDPlus (PDPlus t u) v = PDPlus t (PDPlus u v)"
unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_assoc)

lemma PDPlus_commute: "PDPlus t u = PDPlus u t"
unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_commute)

lemma PDPlus_absorb: "PDPlus t t = t"
unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_absorb)

lemma pd_basis_induct1:
assumes PDUnit: "⋀a. P (PDUnit a)"
assumes PDPlus: "⋀a t. P t ⟹ P (PDPlus (PDUnit a) t)"
shows "P x"
apply (induct x, unfold pd_basis_def, clarify)
apply (erule (1) finite_ne_induct)
apply (cut_tac a=x in PDUnit)
apply (drule_tac a=x in PDPlus)
Abs_pd_basis_inverse [unfolded pd_basis_def])
done

lemma pd_basis_induct:
assumes PDUnit: "⋀a. P (PDUnit a)"
assumes PDPlus: "⋀t u. ⟦P t; P u⟧ ⟹ P (PDPlus t u)"
shows "P x"
apply (induct x rule: pd_basis_induct1)
apply (rule PDUnit, erule PDPlus [OF PDUnit])
done

subsection ‹Fold operator›

definition
fold_pd ::
"('a compact_basis ⇒ 'b::type) ⇒ ('b ⇒ 'b ⇒ 'b) ⇒ 'a pd_basis ⇒ 'b"
where "fold_pd g f t = semilattice_set.F f (g ` Rep_pd_basis t)"

lemma fold_pd_PDUnit:
assumes "semilattice f"
shows "fold_pd g f (PDUnit x) = g x"
proof -
from assms interpret semilattice_set f by (rule semilattice_set.intro)
show ?thesis by (simp add: fold_pd_def Rep_PDUnit)
qed

lemma fold_pd_PDPlus:
assumes "semilattice f"
shows "fold_pd g f (PDPlus t u) = f (fold_pd g f t) (fold_pd g f u)"
proof -
from assms interpret semilattice_set f by (rule semilattice_set.intro)
show ?thesis by (simp add: image_Un fold_pd_def Rep_PDPlus union)
qed

end

```