Library AdditionalTactics

A library of additional tactics.

Require Export String.
Open Scope string_scope.

Extensions and variations of standard tactics


"remember c as x in |-" replaces the term c by the identifier x in the conclusion of the current goal and introduces the hypothesis x=c into the context. This tactic differs from a similar one in the standard library in that the replacement is made only in the conclusion of the goal; the context is left unchanged.

Tactic Notation "remember" constr(c) "as" ident(x) "in" "|-" :=
  let x := fresh x in
  let H := fresh "Heq" x in
  (set (x := c); assert (H : x = c) by reflexivity; clearbody x).

"unsimpl E" replaces all occurence of X by E, where X is the result that tactic simpl would give when used to evaluate E.

Tactic Notation "unsimpl" constr(E) :=
  let F := (eval simpl in E) in change F with E.

The following tactics call the (e)apply tactic with the first hypothesis that succeeds, "first" meaning the hypothesis that comes earliest in the context, i.e., higher up in the list.

Ltac apply_first_hyp :=
  match reverse goal with
    | H : _ |- _ => apply H
  end.

Ltac eapply_first_hyp :=
  match reverse goal with
    | H : _ |- _ => eapply H
  end.

Variations on auto


The auto* and eauto* tactics are intended to be "stronger" versions of the auto and eauto tactics. Similar to auto and eauto, they each take an optional "depth" argument.

Implementation note: if we declare these tactics using a single string, e.g., "auto*", then the resulting tactics are unusable since they fail to parse.

Tactic Notation "auto" "*" :=
  try solve [ congruence | auto | intuition auto ].

Tactic Notation "auto" "*" integer(n) :=
  try solve [ congruence | auto n | intuition (auto n) ].

Tactic Notation "eauto" "*" :=
  try solve [ congruence | eauto | intuition eauto ].

Tactic Notation "eauto" "*" integer(n) :=
  try solve [ congruence | eauto n | intuition (eauto n) ].

Delineating cases in proofs


Tactic definitions


Ltac move_to_top x :=
  match reverse goal with
  | H : _ |- _ => try move x after H
  end.

Tactic Notation "assert_eq" ident(x) constr(v) :=
  let H := fresh in
  assert (x = v) as H by reflexivity;
  clear H.

Tactic Notation "Case_aux" ident(x) constr(name) :=
  first [
    set (x := name); move_to_top x
  | assert_eq x name
  | fail 1 "because we are working on a different case." ].

Ltac Case name := Case_aux case name.
Ltac SCase name := Case_aux subcase name.
Ltac SSCase name := Case_aux subsubcase name.
Ltac SSSCase name := Case_aux subsubsubcase name.
Ltac SSSSCase name := Case_aux subsubsubsubcase name.

Example



One mode of use for the above tactics is to wrap Coq's induction tactic such that it automatically inserts "case" markers into each branch of the proof. For example:

 Tactic Notation "induction" "nat" ident(n) :=
   induction n; [ Case "O" | Case "S" ].
 Tactic Notation "sub" "induction" "nat" ident(n) :=
   induction n; [ SCase "O" | SCase "S" ].
 Tactic Notation "sub" "sub" "induction" "nat" ident(n) :=
   induction n; [ SSCase "O" | SSCase "S" ].
>>

    If you use such customized versions of the induction tactics, then
    the [Case] tactic will verify that you are working on the case
    that you think you are.  You may also use the [Case] tactic with
    the standard version of [induction], in which case no verification
    is done. *)


(* *********************************************************************** *)
(** * Tactics for working with lists and proof contexts *)

(** [ltac_map] applies a unary, non-dependently typed function [F] to
    everything in the context such that the application type checks.
    The tactic returns a [list] containing the results of the
    applications.

    Implementation note: The check for duplicates in the accumulator
    ([match acc with ...]) is necessary to ensure that the tactic does
    not go into an infinite loop. *)

Ltac ltac_map F :=
  let rec map acc :=
    match goal with
      | H : ?S |- _ =>
        let FH := constr:(F H) in
          match acc with
            | context [FH] => fail 1
            | _ => map (List.cons FH acc)
          end
      | _ => acc
    end
  in
  match type of F with
    | ?A -> ?B => let res := map (@List.nil B) in (eval simpl in res)
  end.

(** [ltac_remove_dups] takes a [list] and removes duplicate items from
    it.  The supplied list must, after simplification via [simpl], be
    built from only [nil] and [cons].  Duplicates are recognized only
    "up to syntax," i.e., the limitations of Ltac's [context]
    check. *)

Ltac ltac_remove_dups xs :=
  let rec remove xs acc :=
    match xs with
      | List.nil => acc
      | List.cons ?x ?xs =>
        match acc with
          | context [x] => remove xs acc
          | _ => remove xs (List.cons x acc)
        end
    end
  in
  match type of xs with
    | List.list ?A =>
      let xs := (eval simpl in xs) in
      let xs := remove xs (@List.nil A) in
      eval simpl in (List.rev xs)
  end.

(** [ltac_fold_and_simpl] is a wrapper for [List.fold_right] that does
    some preliminary simplification and removal of duplicates on the
    supplied list. *)

Ltac ltac_fold_and_simpl f start xs :=
  let xs := (eval simpl in xs) in
  let xs := ltac_remove_dups xs in
  let xs := eval simpl in (List.fold_right f start xs) in
  xs.

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