section {* The accessible part of a relation *}
theory Acc imports Main begin
text {*
Inductive definition of @{text "acc(r)"}; see @{cite "paulin-tlca"}.
*}
consts
acc :: "i => i"
inductive
domains "acc(r)" ⊆ "field(r)"
intros
vimage: "[| r-``{a}: Pow(acc(r)); a ∈ field(r) |] ==> a ∈ acc(r)"
monos Pow_mono
text {*
The introduction rule must require @{prop "a ∈ field(r)"},
otherwise @{text "acc(r)"} would be a proper class!
\medskip
The intended introduction rule:
*}
lemma accI: "[| !!b. <b,a>:r ==> b ∈ acc(r); a ∈ field(r) |] ==> a ∈ acc(r)"
by (blast intro: acc.intros)
lemma acc_downward: "[| b ∈ acc(r); <a,b>: r |] ==> a ∈ acc(r)"
by (erule acc.cases) blast
lemma acc_induct [consumes 1, case_names vimage, induct set: acc]:
"[| a ∈ acc(r);
!!x. [| x ∈ acc(r); ∀y. <y,x>:r --> P(y) |] ==> P(x)
|] ==> P(a)"
by (erule acc.induct) (blast intro: acc.intros)
lemma wf_on_acc: "wf[acc(r)](r)"
apply (rule wf_onI2)
apply (erule acc_induct)
apply fast
done
lemma acc_wfI: "field(r) ⊆ acc(r) ==> wf(r)"
by (erule wf_on_acc [THEN wf_on_subset_A, THEN wf_on_field_imp_wf])
lemma acc_wfD: "wf(r) ==> field(r) ⊆ acc(r)"
apply (rule subsetI)
apply (erule wf_induct2, assumption)
apply (blast intro: accI)+
done
lemma wf_acc_iff: "wf(r) <-> field(r) ⊆ acc(r)"
by (rule iffI, erule acc_wfD, erule acc_wfI)
end