Require Import subsequences.
Require Import Sorting.Permutation.

Require Import Nat.
Require Import PeanoNat.
Require Import List.


Import ListNotations.



Definition complete_sequence {X: Type} (u base : list X) :=
  forall v, Permutation base v -> subsequence u v.

Definition shortest_complete_sequence {X: Type} (u base : list X) :=
  complete_sequence u base
  /\ forall v, length v < length u -> ~ complete_sequence v base.


Theorem complete_sequence_rev {X: Type} :
  forall (u base : list X),
  complete_sequence u base -> complete_sequence (rev u) base.
Proof.
  intros u base. intro H.
  intro p. intro I. apply subsequence_rev. rewrite rev_involutive.
  apply H. apply Permutation_trans with (l' := p). assumption.
  apply Permutation_rev.
Qed.