Inductive append : Word -> Word -> Word -> Prop :=
  | append_nil : forall w : Word, append nil w w
  | append_cons :
      forall (w1 w2 w3 : Word) (a : Elt),
      append w1 w2 w3 -> append (cons a w1) w2 (cons a w3).Inductive append : Word -> Word -> Word -> Prop :=
  | append_nil : forall w : Word, append nil w w
  | append_cons :
      forall (w1 w2 w3 : Word) (a : Elt),
      append w1 w2 w3 -> append (cons a w1) w2 (cons a w3).Inductive append : Word -> Word -> Word -> Prop :=
  | append_nil : forall w : Word, append nil w w
  | append_cons :
      forall (w1 w2 w3 : Word) (a : Elt),
      append w1 w2 w3 -> append (cons a w1) w2 (cons a w3).

Lemma Append_inmonoid_d : forall (X : Ensf) (w1 w2 : Word), inmonoid X (Append w1 w2) -> inmonoid X w2.