header {* Domain package *}
theory Domain
imports Ssum Sprod Up One Tr Fixrec
begin
defaultsort pcpo
subsection {* Continuous isomorphisms *}
text {* A locale for continuous isomorphisms *}
locale iso =
fixes abs :: "'a -> 'b"
fixes rep :: "'b -> 'a"
assumes abs_iso [simp]: "rep·(abs·x) = x"
assumes rep_iso [simp]: "abs·(rep·y) = y"
begin
lemma swap: "iso rep abs"
by (rule iso.intro [OF rep_iso abs_iso])
lemma abs_less: "(abs·x \<sqsubseteq> abs·y) = (x \<sqsubseteq> y)"
proof
assume "abs·x \<sqsubseteq> abs·y"
then have "rep·(abs·x) \<sqsubseteq> rep·(abs·y)" by (rule monofun_cfun_arg)
then show "x \<sqsubseteq> y" by simp
next
assume "x \<sqsubseteq> y"
then show "abs·x \<sqsubseteq> abs·y" by (rule monofun_cfun_arg)
qed
lemma rep_less: "(rep·x \<sqsubseteq> rep·y) = (x \<sqsubseteq> y)"
by (rule iso.abs_less [OF swap])
lemma abs_eq: "(abs·x = abs·y) = (x = y)"
by (simp add: po_eq_conv abs_less)
lemma rep_eq: "(rep·x = rep·y) = (x = y)"
by (rule iso.abs_eq [OF swap])
lemma abs_strict: "abs·⊥ = ⊥"
proof -
have "⊥ \<sqsubseteq> rep·⊥" ..
then have "abs·⊥ \<sqsubseteq> abs·(rep·⊥)" by (rule monofun_cfun_arg)
then have "abs·⊥ \<sqsubseteq> ⊥" by simp
then show ?thesis by (rule UU_I)
qed
lemma rep_strict: "rep·⊥ = ⊥"
by (rule iso.abs_strict [OF swap])
lemma abs_defin': "abs·x = ⊥ ==> x = ⊥"
proof -
have "x = rep·(abs·x)" by simp
also assume "abs·x = ⊥"
also note rep_strict
finally show "x = ⊥" .
qed
lemma rep_defin': "rep·z = ⊥ ==> z = ⊥"
by (rule iso.abs_defin' [OF swap])
lemma abs_defined: "z ≠ ⊥ ==> abs·z ≠ ⊥"
by (erule contrapos_nn, erule abs_defin')
lemma rep_defined: "z ≠ ⊥ ==> rep·z ≠ ⊥"
by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
lemma abs_defined_iff: "(abs·x = ⊥) = (x = ⊥)"
by (auto elim: abs_defin' intro: abs_strict)
lemma rep_defined_iff: "(rep·x = ⊥) = (x = ⊥)"
by (rule iso.abs_defined_iff [OF iso.swap]) (rule iso_axioms)
lemma (in iso) compact_abs_rev: "compact (abs·x) ==> compact x"
proof (unfold compact_def)
assume "adm (λy. ¬ abs·x \<sqsubseteq> y)"
with cont_Rep_CFun2
have "adm (λy. ¬ abs·x \<sqsubseteq> abs·y)" by (rule adm_subst)
then show "adm (λy. ¬ x \<sqsubseteq> y)" using abs_less by simp
qed
lemma compact_rep_rev: "compact (rep·x) ==> compact x"
by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
lemma compact_abs: "compact x ==> compact (abs·x)"
by (rule compact_rep_rev) simp
lemma compact_rep: "compact x ==> compact (rep·x)"
by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
lemma iso_swap: "(x = abs·y) = (rep·x = y)"
proof
assume "x = abs·y"
then have "rep·x = rep·(abs·y)" by simp
then show "rep·x = y" by simp
next
assume "rep·x = y"
then have "abs·(rep·x) = abs·y" by simp
then show "x = abs·y" by simp
qed
end
subsection {* Casedist *}
lemma ex_one_defined_iff:
"(∃x. P x ∧ x ≠ ⊥) = P ONE"
apply safe
apply (rule_tac p=x in oneE)
apply simp
apply simp
apply force
done
lemma ex_up_defined_iff:
"(∃x. P x ∧ x ≠ ⊥) = (∃x. P (up·x))"
apply safe
apply (rule_tac p=x in upE)
apply simp
apply fast
apply (force intro!: up_defined)
done
lemma ex_sprod_defined_iff:
"(∃y. P y ∧ y ≠ ⊥) =
(∃x y. (P (:x, y:) ∧ x ≠ ⊥) ∧ y ≠ ⊥)"
apply safe
apply (rule_tac p=y in sprodE)
apply simp
apply fast
apply (force intro!: spair_defined)
done
lemma ex_sprod_up_defined_iff:
"(∃y. P y ∧ y ≠ ⊥) =
(∃x y. P (:up·x, y:) ∧ y ≠ ⊥)"
apply safe
apply (rule_tac p=y in sprodE)
apply simp
apply (rule_tac p=x in upE)
apply simp
apply fast
apply (force intro!: spair_defined)
done
lemma ex_ssum_defined_iff:
"(∃x. P x ∧ x ≠ ⊥) =
((∃x. P (sinl·x) ∧ x ≠ ⊥) ∨
(∃x. P (sinr·x) ∧ x ≠ ⊥))"
apply (rule iffI)
apply (erule exE)
apply (erule conjE)
apply (rule_tac p=x in ssumE)
apply simp
apply (rule disjI1, fast)
apply (rule disjI2, fast)
apply (erule disjE)
apply force
apply force
done
lemma exh_start: "p = ⊥ ∨ (∃x. p = x ∧ x ≠ ⊥)"
by auto
lemmas ex_defined_iffs =
ex_ssum_defined_iff
ex_sprod_up_defined_iff
ex_sprod_defined_iff
ex_up_defined_iff
ex_one_defined_iff
text {* Rules for turning exh into casedist *}
lemma exh_casedist0: "[|R; R ==> P|] ==> P"
by auto
lemma exh_casedist1: "((P ∨ Q ==> R) ==> S) ≡ ([|P ==> R; Q ==> R|] ==> S)"
by rule auto
lemma exh_casedist2: "(∃x. P x ==> Q) ≡ (!!x. P x ==> Q)"
by rule auto
lemma exh_casedist3: "(P ∧ Q ==> R) ≡ (P ==> Q ==> R)"
by rule auto
lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3
end