# Module Libtactics

Require Import Coq.Lists.List.
Require Import Coq.Strings.String.

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.

# Variations on built-in tactics

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.

fold any not folds all occurrences of not in the goal. It's useful for "cleaning up" after the intuition tactic.

Tactic Notation "fold" "any" "not" :=
repeat (
match goal with
| H: context [?P -> False] |- _ =>
fold (~ P) in H
| |- context [?P -> False] =>
fold (~ P)
end).

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.

# Delineating cases in proofs

## Tactic definitions

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.

# Tactics for working with lists and proof contexts

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.

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.