(* This file is distributed under the terms of the MIT License, also known as the X11 Licence.  A copy of this license is in the README file that accompanied the original distribution of this file.  Based on code written by: Brian Aydemir *)
(** A library of code for supporting LNgen. *)
Require Export Metalib.LibDefaultSimp.
(* ********************************************************************** *)
(** * Assorted functions not in the standard library *)
(** [lt_ge_dec] is a useful comparison operation when defining the "close" operation on locally nameless terms. *)
Definition lt_ge_dec (n m : nat) : {n < m} + {n >= m} :=
match Compare_dec.le_gt_dec m n with
| left pf => right pf
| right pf => left pf
end.
(* *********************************************************************** *)
(** * Tactics *)
(** [generalize_wrt x] is an ad hoc tactic that generalizes the goal with respect to everything that [x] does not depend on.  It seems to work only if [x] is the most recently introduced item into the context. *)
Ltac generalize_wrt x :=
repeat (progress (match goal with
| J : _ |- _ => move J after x; generalize dependent J
end)).
(** [apply_mutual_ind] applies to the current goal a mutual induction principle that is stated in the form generated by [Combined Scheme].  It works even in degenerate cases, i.e., when there is no mutual induction, only simple induction.  It is intended to be used only at the start of a proof, and the argument(s) to induct over should come first. *)
Ltac apply_mutual_ind ind :=
let H := fresh in
first [ (* apply ind
| *) intros H; induction H using ind
| intros ? H; induction H using ind
| intros ? ? H; induction H using ind
| intros ? ? ? H; induction H using ind
| intros ? ? ? ? H; induction H using ind
| intros ? ? ? ? ? H; induction H using ind
| intros ? ? ? ? ? ? H; induction H using ind
| intros ? ? ? ? ? ? ? H; induction H using ind
| intros ? ? ? ? ? ? ? ? H; induction H using ind
].
(** Renames the last hypothesis to the given identifier. *)
Ltac rename_last_into H :=
match goal with
| J : _ |- _ => rename J into H
end.
(** Specializes every possible hypothesis with the given term. *)
Ltac specialize_all x :=
repeat (match goal with
| H : _ |- _ => specialize (H x)
end).
(* *********************************************************************** *)
(** * Facts about finite sets *)

(* Specifications: *)
Lemma remove_union_distrib : forall (s1 s2 : atoms) (x : atom),
remove x (union s1 s2) [=] union (remove x s1) (remove x s2).