Module ListAutomationNotations.
Notation "x 'el' L" := (In x L) (at level 70).
Notation "A '<<=' B" := (incl A B) (at level 70).
Notation "( A × B × .. × C )" := (list_prod .. (list_prod A B) .. C) (at level 0, left associativity).
Notation "[ s | p ∈ A ',' P ]" :=
(map (fun p => s) (filter (fun p => Dec P) A)) (p pattern).
Notation "[ s | p ∈ A ]" :=
(map (fun p => s) A) (p pattern).
End ListAutomationNotations.
Instance list_in_dec X (x : X) (A : list X) :
eq_dec X -> dec (x el A).
Ltac in_app n :=
(match goal with
| [ |- In _ (_ ++ _) ] =>
match n with
| 0 => idtac
| 1 => eapply in_app_iff; left
| S ?n => eapply in_app_iff; right; in_app n
end
| [ |- In _ (_ :: _) ] => match n with 0 => idtac | 1 => left | S ?n => right; in_app n end
end) || (repeat (try right; eapply in_app_iff; right)).
Ltac in_collect a :=
eapply in_map_iff; exists a; split; [ eauto | match goal with [ |- In _ (filter _ _) ] =>  eapply in_filter_iff; split; [ try (rewrite !in_prod_iff; repeat split) | eapply Dec_auto; repeat split; eauto ] | _ => try (rewrite !in_prod_iff; repeat split) end ].
Ltac inv_collect :=
repeat
(match goal with
| [ H : ?x el concat _ |- _ ] => eapply in_concat_iff in H as (? & ? & ?)
| [ H : ?x el map _ _ |- _ ] => let x := fresh "x" in eapply in_map_iff in H as (x & ? & ?)
| [ x : ?A * ?B |- _ ] => destruct x; subst
| [ H : ?x el filter _ _ |- _ ] => let H' := fresh "H" in eapply in_filter_iff in H as (? & H' % to_dec)
| [ H : ?x el list_prod _ _ |- _ ] => eapply in_prod_iff in H
| [ H : _ el _ ++ _ |- _ ] => try eapply in_app_iff in H as []
| [H : _ el _ :: _ |- _ ] => destruct H
end; intuition; subst).
Local Set Implicit Arguments.
Local Unset Strict Implicit.
Hint Extern 4 =>
match goal with
|[ H: ?x el nil |- _ ] => destruct H
end : core.
Hint Extern 4 =>
match goal with
|[ H: False |- _ ] => destruct H
|[ H: true=false |- _ ] => discriminate H
|[ H: false=true |- _ ] => discriminate H
end : core.

Lemma cons_incl X (a : X) (A B : list X) : a :: A <<= B -> A <<= B.