Inductive prechain_dom_ok (A : Set) : mEnsemble A -> prechain A -> Prop :=
| domok_single :
forall (x : Map A) (T : mEnsemble A),
T x -> prechain_dom_ok A T (single A x)
| domok_concat :
forall (x : Map A) (T : mEnsemble A) (p : prechain A),
T x -> prechain_dom_ok A T p -> prechain_dom_ok A T (concat A p x).
Fixpoint chain_length (A : Set) (p : prechain A) {struct p} : nat :=
match p with
| single x => 1
| concat x y => S (chain_length A x)
end.
Definition prechain_last (A : Set) (p : prechain A) :
Map A := match p with
| single x => x
| concat z x => x
end.
Inductive prechain_incr (A : Set) : mRelation A -> prechain A -> Prop :=
| incr_single :
forall (x : Map A) (r : mRelation A), prechain_incr A r (single A x)
| incr_concat :
forall (x : Map A) (r : mRelation A) (p : prechain A),
r (prechain_last A p) x ->
prechain_incr A r p -> prechain_incr A r (concat A p x).
Inductive chain (A : Set) :
mEnsemble A -> mRelation A -> prechain A -> Prop :=
| chain_single :
forall (x : Map A) (T : mEnsemble A) (r : mRelation A),
T x -> chain A T r (single A x)
| chain_concat_s :
forall (x y : Map A) (T : mEnsemble A) (r : mRelation A),
T x -> T y -> r x y -> chain A T r (concat A (single A x) y)
| chain_concat_m :
forall (x y : Map A) (z : prechain A) (T : mEnsemble A)
(r : mRelation A),
T y ->
r x y ->
chain A T r (concat A z x) -> chain A T r (concat A (concat A z x) y).

Lemma chain_incr :
forall (A : Set) (T : mEnsemble A) (r : mRelation A) (p : prechain A),
chain A T r p -> prechain_incr A r p.