Definition mEnsemble (A : Set) := Map A -> Prop.
Definition mRelation (A : Set) := Map A -> Map A -> Prop.
Definition r_symmetric (A : Set) (r : mRelation A) :=
forall x y : Map A, r x y -> r y x.
Definition r_antisymmetric (A : Set) (r : mRelation A) :=
forall x y : Map A, r x y -> r y x -> x = y.
Definition r_transitive (A : Set) (r : mRelation A) :=
forall x y z : Map A, r x y -> r y z -> r x z.
Definition r_reflexive (A : Set) (r : mRelation A) := forall x : Map A, r x x.
Definition r_order (A : Set) (r : mRelation A) :=
r_reflexive A r /\ r_antisymmetric A r /\ r_transitive A r.
Definition mini (A : Set) (r : mRelation A) (T : mEnsemble A)
(e : Map A) := T e /\ (forall x : Map A, T x -> r e x).
Definition maxi (A : Set) (r : mRelation A) (T : mEnsemble A)
(e : Map A) := T e /\ (forall x : Map A, T x -> r x e).
Definition mLattice (A : Set) (r : mRelation A) (T : mEnsemble A)
(e f : Map A) := r_order A r /\ mini A r T e /\ maxi A r T f.
Inductive prechain (A : Set) : Set :=
| single : Map A -> prechain A
| concat : prechain A -> Map A -> prechain A.

Lemma prechain_sum :
forall (A : Set) (p : prechain A),
(exists x : Map A, p = single A x) \/
(exists x : Map A, (exists y : prechain A, p = concat A y x)).