Library AdditionalTactics
A library of additional tactics.
Implementation note: We want string_scope to be available for
the Case tactics below, but we want "++" to denote list
concatenation.
Open Scope string_scope.
Open Scope list_scope.
"unsimpl E" replaces all occurences of X in the goal by E,
where X is the result that tactic simpl would give when used
to reduce E.
Tactic Notation "unsimpl" constr(E) :=
let F := (eval simpl in E) in change F with E.
The following tactics call (e)apply 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.
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.
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) ].
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 x at top
| 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.
ltac_map applies a function F, with return type T and
exactly one non-implicit argument, 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.
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 : _ |- _ ⇒
let FH := constr:(F H) in
match acc with
| context [FH] ⇒ fail 1
| _ ⇒ map (List.cons FH acc)
end
| _ ⇒ acc
end
in
let rec ret T :=
match T with
| _ → ?T' ⇒ ret T'
| ?T' ⇒ T'
end
in
let T := ret ltac:(type of F) in
let res := map (@List.nil T) in
eval simpl in res.
ltac_map_list tac xs applies tac to each element of xs,
where xs is a Coq list.
Ltac ltac_map_list tac xs :=
match xs with
| List.nil ⇒ idtac
| List.cons ?x ?xs ⇒ tac x; ltac_map_list tac xs
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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