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.Inductive prechain (A : Set) : Set :=
  | single : Map A -> prechain A
  | concat : prechain A -> Map A -> prechain A.Inductive prechain (A : Set) : Set :=
  | single : Map A -> prechain A
  | concat : prechain A -> Map A -> prechain A.

(* Specifications: *)
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)).