ladder-calculus/coq/AdditionalTactics.v

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(** A library of additional tactics. *)
Require Export String.
Open Scope string_scope.
(* *********************************************************************** *)
(** * Extensions of the standard library *)
(** "[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 replacmement 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 tactic calls the [apply] tactic with the first
hypothesis that succeeds, "first" meaning the hypothesis that
comes earlist in the context (i.e., higher up in the list). *)
Ltac apply_first_hyp :=
match reverse goal with
| H : _ |- _ => apply 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. Note that
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 *)
(** This section was taken from the POPLmark Wiki
( http://alliance.seas.upenn.edu/~plclub/cgi-bin/poplmark/ ). *)
(** ** 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.
(** ** Example
One mode of use for the above tactics is to wrap Coq's [induction]
tactic such that 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. *)