Section list_util.
Variables A B C : Type.
Hypothesis A_eq_dec : forall x y : A, {x = y} + {x <> y}.
Proof using.
intuition.
symmetry in H.
induction l;
now inversion H.
Qed.
Proof using.
induction xs; intros.
- intuition.
- simpl in *.
concludes.
intuition; break_if; subst; try congruence; intuition.
Qed.
Proof using.
induction xs; intros.
- auto.
- simpl in *. break_if; simpl in *; intuition.
Qed.
Proof using.
induction xs; intros; simpl; break_if; congruence.
Qed.
Proof using.
intros. induction xs; simpl in *; try break_if; intuition congruence.
Qed.
Proof using.
induction xs; intros.
- auto.
- simpl. rewrite IHxs. break_if; auto.
Qed.
Proof using.
induction xs; intros.
- auto.
- simpl. rewrite H by intuition. rewrite IHxs by intuition. auto.
Qed.
Proof using.
induction xs; intros.
- constructor.
- simpl. invc_NoDup. constructor.
+ intro. do_in_map.
assert (x = a) by intuition.
congruence.
+ intuition.
Qed.
Proof using.
induction l; intros.
- auto.
- simpl. invc_NoDup. constructor.
+ intro. do_in_app. intuition eauto with *.
+ intuition eauto with *.
Qed.
Proof using.
induction xs; simpl; intros; destruct xs'.
- auto.
- subst. simpl in *. find_inversion.
invc H. exfalso. rewrite map_app in *. simpl in *.
repeat find_rewrite. intuition.
- subst. simpl in *. find_inversion.
invc H. exfalso. rewrite map_app in *. simpl in *.
repeat find_rewrite. intuition.
- subst. simpl in *. find_injection. intros. subst.
f_equal. eapply IHxs; eauto. solve_by_inversion.
Qed.
Proof using.
induction l; intros.
- auto.
- invc_NoDup. simpl. break_if; auto.
constructor; auto.
intro. apply filter_In in H. intuition.
Qed.
Proof using.
intros. induction l; simpl in *.
- constructor.
- invc_NoDup. concludes.
break_if; simpl in *; auto.
constructor; auto.
intro. do_in_map.
find_apply_lem_hyp filter_In. intuition.
match goal with | H : _ -> False |- False => apply H end.
apply in_map_iff. eauto.
Qed.
Proof using.
induction xs; intros.
- auto.
- simpl. now rewrite H, IHxs by intuition.
Qed.

Lemma map_of_map : forall (f : A -> B) (g : B -> C) xs,
map g (map f xs) = map (fun x => g (f x)) xs.