Section sec_map_modalop.
Variable A B: Type.
Proof using.
unfold and_tl; intuition.
Qed.
Proof using.
unfold and_tl; intuition.
Qed.
Proof using.
unfold or_tl; intuition.
Qed.
Proof using.
unfold or_tl; intuition.
Qed.
Proof using.
unfold impl_tl; intuition.
Qed.
Proof using.
unfold impl_tl; intuition.
Qed.
Proof using.
unfold not_tl; intuition.
Qed.
Proof using.
unfold not_tl; intuition.
Qed.
Proof using.
intros f P (x, s) nP; assumption.
Qed.
Proof using.
intros f P (x, s) nP; assumption.
Qed.
Proof using.
intros f P Q PQ [x s]; apply PQ.
Qed.
Proof using.
intros f P Q QP [x s]; apply QP.
Qed.
Proof using.
intros f P (x, (y, s)) nP; assumption.
Qed.

Lemma consecutive_map_conv :
forall (f: A->B) (P: B->B->Prop) (s: infseq A),
consecutive P (map f s) -> consecutive (fun x y => P (f x) (f y)) s.