Gagallium

With Pierre Boutillier, we have been working on a new Coq tactic lately, called invert. From my point of view, it started as a quest to build a replacement to the inversion tactic. inversion is a pain to use, as it generates sub-goals with many (dependent) equalities that must be substituted, which force the use of subst, which in turns also has its quirks, making the mantra inversion H; clear H; subst quite fragile. Furthermore, inversion has efficiency problems, being quite slow and generating big proof terms. From Pierre’s point of view, this work was a good way to implement a better destruct tactic, based on what he did during an internship (report in French (PDF)).

In a nutshell, the idea behind a destruction and an inversion is quite similar: it boils down to a case analysis over a given hypothesis. And there are quite a few tactics that follow this scheme: elim, case, destruct, inversion, dependent destruction, injection and discriminate (it is true that the last two tactics are quite specialized, but fit the bill nevertheless). Why on Earth would we need to add a new element to this list?

Well, it turns out that building on ideas by Jean-Francois Monin to make so called “small inversions”, one can unify the inner-working of most of the aforementioned list: it suffices to build the right return clause for the case analysis.

Let’s take an example.

Variable A : Type.
Inductive vector: nat -> Type :=
| nil : vector 0
| cons: forall n (h:A) (v: vector n), vector (S n).

Inductive P : forall n, vector n -> Prop :=
| Pnil : P 0 nil
| Pcons: forall n h v, P n v -> P (S n) (cons n h v).

Lemma test n h v (H: P (S n) (cons n h v)) : P n v.
Proof.

At this point, doing inversion H generates 4 new hypotheses:

H2 : P n v0
H0 : n0 = n
H1 : h0 = h
H3 : existT (fun n : nat => vector n) n v0 =
       existT (fun n : nat => vector n) n v
 ============================
   P n v

Yuck: first, H0 and H1 are just cruft. Then, the goal isn’t very palatable, because the equality H3 between v and v0 is defined in terms of a dependent equality: in order to go further, one need to assume axioms about dependent equality¹, equivalent to Streicher’s axiom K. (Just to keep tabs, note that running the Show Proof command in Coq outputs a partial proof term that is already 73 lines long at this point.)

If we use dependent destruction H instead of inversion, we get the expected hypothesis H: P n v (which is far better from an usability point of view). Yet, there is no magic here: dependent destruction simply used a dependent equality axiom internally to get rid of the dependent equality, and generates a 64 lines long proof term that is not very pretty.

At this point, one may wonder: what should the proof term look like? and, is it necessary to use the K axiom here?

A black belt Coq user versed in dependent types could write the following one.

let diag :=
   fun n0 : nat =>
   match n0 as n' return (forall v0 : vector n', P n' v0 -> Prop) with
   | 0 => fun (v0 : vector 0) (_ : P 0 v0) => True
   | S m =>
       fun v0 : vector (S m) =>
       match v0 as v1 in (vector m0) return (P m0 v1 -> Prop) with
       | nil => fun _ : P 0 nil => True
       | cons p x v1 => fun _ : P (S p) (cons p x v1) => P p v1
       end
   end in
 match H as H' in (P x y) return (diag x y H') with
 | Pnil => I
 | Pcons n0 h0 v0 Pv => Pv
 end.

Wow, 15 lines long. Let’s demystify it a bit.

First, recall that the return type of a match is dictated by its return clause (the as ... in ... return ... part). This is basically a function that binds the arguments of the inductive (S n as x, cons n h v as y in our case), H' of type P x y, and which body is the return part. Usually, the return part is a constant (e.g., nat for the match in the List.length), but it is not mandatory. Here, the diag term packs some computations, such that diag (S n) (cons n h v) H reduces to P n v, the conclusion of the goal. (In general, this kind of return clauses make it possible to eliminate impossible branches in a match, as done here by marking them with the trivial return type True; we direct the interested readers to the online CPDT book by Adam Chlipala for more informations on this, especially this chapter.)

Then, what is diag? Well, it is a function that follows the structure of the arguments of P to single out impossible cases, and to refine the context in the other ones using dependent pattern matching, in order to reduce to the right type (the type of the initial conclusion of the goal). The idea behind such “small-scale inversions” was described by Monin in 2010 and is out of the scope of this blog post. What is new here is that we have mechanized the construction of the diag functions as a Coq tactic, making this whole approach practical.

All in all, using our new tactic, we can just use the following proof script:

invert H; tauto. 

At this point, Show Proof. outputs the following complete proof term (where invert_subgoal is the type of the subgoal solved by tauto):

let diag :=
  fun n0 : nat =>
  match n0 as n1 return (forall v0 : vector n1, P n1 v0 -> Prop) with
  | 0 => fun (v0 : vector 0) (_ : P 0 v0) => False -> True
  | S x =>
      fun v0 : vector (S x) =>
      match
        v0 as v1 in (vector n1)
        return
          (match n1 as n2 return (vector n2 -> Type) with
           | 0 => fun _ : vector 0 => False -> True
           | S x0 => fun v2 : vector (S x0) => P (S x0) v2 -> Prop
           end v1)
      with
      | nil => fun H0 : False => False_rect True H0
      | cons n1 h0 v1 => fun _ : P (S n1) (cons n1 h0 v1) => P n1 v1
      end
  end in
(fun
   invert_subgoal : forall (n0 : nat) (h0 : A) (v0 : vector n0)
                      (H0 : P n0 v0),
                    diag (S n0) (cons n0 h0 v0) (Pcons n0 h0 v0 H0) =>
 match H as p in (P n0 v0) return (diag n0 v0 p) with
 | Pnil => fun H0 : False => False_rect True H0
 | Pcons x x0 x1 x2 => invert_subgoal x x0 x1 x2
 end) (fun (n0 : nat) (_ : A) (v0 : vector n0) (H0 : P n0 v0) => H0))

Some of the differences with the proof term above come from the fact that we generate it interactively, rather than writing it once at all.

A legitimate question: how do we compare to destruct and inversion and dependent destruction? First, we aim at producing a “better” destruct: that is, we might resolve the situation in which destruct fails, in order to avoid producing ill-typed terms. Then, the situation with respect to inversion and dependent destruction is less clear. Right now, we would rather not assume the K axiom (the right thing to do if homotopy is the future). In that case, we would fail for inversion problems that require K, and inversion and dependent destruction would be more powerful than our tactic. For problems that do no require to use K, invert would be equivalent to dependent destruction with better looking proof terms².

We are still working on our prototype, but we are quite confident that we got the main thing right: mechanizing the construction of the return clause. We will come back to this blog when we need beta-testers!