(** * Automation and dedicated tactics. *)
#[global]
Hint Constructors tree MapsTo In bst : core.
#[global]
Hint Unfold lt_tree gt_tree : core.
Tactic Notation "factornode" ident(l) ident(x) ident(d) ident(r) ident(h)
"as" ident(s) :=
set (s:=Node l x d r h) in *; clearbody s; clear l x d r h.
(** A tactic for cleaning hypothesis after use of functional induction. *)
Ltac clearf :=
match goal with
| H := _ |- _ => subst; subst H; clearf
| H : (@Logic.eq (Compare _ _ _ _) _ _) |- _ => clear H; clearf
| H : (@Logic.eq (sumbool _ _) _ _) |- _ => clear H; clearf
| _ => idtac
end.
(** A tactic to repeat [inversion_clear] on all hyps of the form [(f (Node ...))] *)
Ltac inv f :=
match goal with
| H:f (Leaf _) |- _ => inversion_clear H; inv f
| H:f _ (Leaf _) |- _ => inversion_clear H; inv f
| H:f _ _ (Leaf _) |- _ => inversion_clear H; inv f
| H:f _ _ _ (Leaf _) |- _ => inversion_clear H; inv f
| H:f (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| H:f _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| H:f _ _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| H:f _ _ _ (Node _ _ _ _ _) |- _ => inversion_clear H; inv f
| _ => idtac
end.
Ltac inv_all f :=
match goal with
| H: f _ |- _ => inversion_clear H; inv f
| H: f _ _ |- _ => inversion_clear H; inv f
| H: f _ _ _ |- _ => inversion_clear H; inv f
| H: f _ _ _ _ |- _ => inversion_clear H; inv f
| _ => idtac
end.
(** Helper tactic concerning order of elements. *)
Ltac order := match goal with
| U: lt_tree _ ?s, V: In _ ?s |- _ => generalize (U _ V); clear U; order
| U: gt_tree _ ?s, V: In _ ?s |- _ => generalize (U _ V); clear U; order
| _ => MX.order
end.
Ltac intuition_in := repeat (intuition auto; inv In; inv MapsTo).
(* Function/Functional Scheme can't deal with internal fix. Let's do its job by hand: *)
Ltac join_tac :=
intros ?l; induction l as [| ?ll _ ?lx ?ld ?lr ?Hlr ?lh];
[ | intros ?x ?d ?r; induction r as [| ?rl ?Hrl ?rx ?rd ?rr _ ?rh]; unfold join;
[ | destruct (gt_le_dec lh (rh+2)) as [?GT|?LE];
[ match goal with |- context [ bal ?u ?v ?w ?z ] =>
replace (bal u v w z)
with (bal ll lx ld (join lr x d (Node rl rx rd rr rh))); [ | auto]
end
| destruct (gt_le_dec rh (lh+2)) as [?GT'|?LE'];
[ match goal with |- context [ bal ?u ?v ?w ?z ] =>
replace (bal u v w z)
with (bal (join (Node ll lx ld lr lh) x d rl) rx rd rr); [ | auto]
end
| ] ] ] ]; intros.
Section Elt.
Variable elt:Type.
Implicit Types m r : t elt.
(** * Basic results about [MapsTo], [In], [lt_tree], [gt_tree], [height] *)
(** Facts about [MapsTo] and [In]. *)

Lemma In_alt : forall k m, In0 k m <-> In k m.