Joachim Breitner

Verifying local definitions in Coq

Published 2018-04-22 in sections English, Haskell.

TL;DR: We can give top-level names to local definitions, so that we can state and prove stuff about them without having to rewrite the programs.

When a Haskeller writes Coq

Imagine you teach Coq to a Haskell programmer, and give them the task of pairing each element in a list with its index. The Haskell programmer might have

addIndex :: [a] -> [(Integer, a)]
addIndex xs = go 0 xs
  where go n [] = []
        go n (x:xs) = (n,x) : go (n+1) xs

in mind and write this Gallina function (Gallina is the programming language of Coq):

Require Import Coq.Lists.List.
Import ListNotations.

Definition addIndex {a} (xs : list a) : list (nat * a) :=
  let fix go n xs := match xs with
                     | []    => []
                     | x::xs => (n, x) :: go (S n) xs
                     end
  in go 0 xs.

Alternatively, imagine you are using hs-to-coq to mechanically convert the Haskell definition into Coq.

When a Coq user tries to verify that

Now your task is to prove something about this function, for example

Theorem addIndex_spec:
  forall {a} n (xs : list a),
  nth n (map fst (addIndex xs)) n = n.

If you just have learned Coq, you will think “I can do this, this surely holds by induction on xs.” But if you have a bit more experience, you will already see a problem with this (if you do not see the problem yet, I encourage you to stop reading, copy the few lines above, and try to prove it).

The problem is that – as so often – you have to generalize the statement for the induction to go through. The theorem as stated says something about addIndex or, in other words, about go 0. But in the inductive case, you will need some information about go 1. In fact, you need a lemma like this:

Lemma go_spec:
  forall {a} n m k (xs : list a), k = n + m ->
  nth n (map fst (go m xs)) k = k.

But go is not a (top-level) function! How can we fix that?

  • We can try to awkwardly work-around not having a name for go in our proofs, and essentially prove go_spec inside the proof of addIndex_spec. Might work in this small case, but does not scale up to larger proofs.
  • We can ask the programmer to avoid using local functions, and first define go as a top-level fixed point. But maybe we don’t want to bother them because of that. (Or, more likely, we are using hs-to-coq and that tool stubbornly tries to make the output as similar to the given Haskell code as possible.)
  • We can copy’n’paste the definition of go and make a separate, after-the-fact top-level definition. But this is not nice from a maintenance point of view: If the code changes, we have to update this copy.
  • Or we apply this one weird trick...

The weird trick

We can define go after-the-fact, but instead of copy’n’pasting the definition, we can use Coq’s tactics to define it. Here it goes:

Definition go {a} := ltac:(
  let e := eval cbv beta delta [addIndex] in (@addIndex a []) in
  (* idtac e; *)
  lazymatch e with | let x := ?def in _ =>
    exact def
  end).

Let us take it apart:

  1. We define go, and give the parameters that go depends upon. Note that of the two parameters of addIndex, the definition of go only depends on (“captures”) a, but not xs.
  2. We do not give a type to go. We could, but that would again just be copying information that is already there.
  3. We define go via an ltac expression: Instead of a term we give a tactic that calculates the term.
  4. This tactic first binds e to the body of addIndex. To do so, it needs to pass enough arguments to addIndex. The concrete value of the list argument does not matter, so we pass []. The term @addIndex a [] is now evaluated with the evaluation flags eval cbv beta delta [addIndex], which says “unfold addIndex and do beta reduction, but nothing else”. In particularly, we do not do zeta reduction, which would reduce the let go := … definition. (The user manual very briefly describes these flags.)
  5. The idtac e line can be used to peek at e, for example when the next tactic fails. We can use this to check that e really is of the form let fix go := … in ….
  6. The lazymatch line matches e against the pattern let x := ?def in _, and binds the definition of go to the name def.
  7. And the exact def tactic tells Coq to use def as the definition of go.

We now have defined go, of type go : forall {a}, nat -> list a -> list (nat * a), and can state and prove the auxiliary lemma:

Lemma go_spec:
  forall {a} n m k (xs : list a), k = n + m ->
  nth n (map fst (go m xs)) k = k.
Proof.
  intros ?????.
  revert n m k.
  induction xs; intros; destruct n; subst; simpl.
  1-3:reflexivity.
  apply IHxs; lia.
Qed.

When we come to the theorem about addIndex, we can play a little trick with fold to make the proof goal pretty:

Theorem addIndex_spec:
  forall {a} n (xs : list a),
  nth n (map fst (addIndex xs)) n = n.
Proof.
  intros.
  unfold addIndex.
  fold (@go a).
  (* goal here: nth n (map fst (go 0 xs)) n = n *)
  apply go_spec; lia.
Qed.

Multiple local definitions

The trick extends to multiple local definitions, but needs some extra considerations to ensure that terms are closed. A bit contrived, but let us assume that we have this function definition:

Definition addIndex' {a} (xs : list a) : list (nat * a) :=
  let inc := length xs in
  let fix go n xs := match xs with
                     | []    => []
                     | x::xs => (n, x) :: go (inc + n) xs
                     end in
  go 0 xs.

We now want to give names to inc and to go. I like to use a section to collect the common parameters, but that is not essential here. The trick above works flawlessly for `inc':

Section addIndex'.
Context {a} (xs : list a).

Definition inc := ltac:(
  let e := eval cbv beta delta [addIndex'] in (@addIndex' a xs) in
  lazymatch e with | let x := ?def in _ =>
    exact def
  end).

But if we try it for go', like such:

Definition go' := ltac:(
  let e := eval cbv beta delta [addIndex'] in (@addIndex' a xs) in
  lazymatch e with | let x := _ in let y := ?def in _ =>
    exact def
  end).

we get “Ltac variable def depends on pattern variable name x which is not bound in current context”. To fix this, we use higher-order pattern matchin (@?def) to substitute “our” inc for the local inc:

Definition go' := ltac:(
  let e := eval cbv beta delta [addIndex'] in (@addIndex' a xs) in
  lazymatch e with | let x := _ in let y := @?def x in _ =>
    let def' := eval cbv beta in (def inc) in
    exact def'
  end).

instead. We have now defined both inc and go' and can use them in proofs about addIndex':

Theorem addIndex_spec':
  forall n, nth n (map fst (addIndex' xs)) n = n * length xs.
Proof.
  intros.
  unfold addIndex'.
  fold inc go'. (* order matters! *)
  (* goal here: nth n (map fst (go' 0 xs)) n = n * inc *)

Reaching into a match

This trick also works when the local definition we care about is inside a match statement. Consider:

Definition addIndex_weird {a} (oxs : option (list a))
  := match oxs with
     | None => []
     | Some xs =>
       let fix go n xs := match xs with
                          | []    => []
                          | x::xs => (n, x) :: go (S n) xs
                          end in
       go 0 xs
     end.

Definition go_weird {a} := ltac:(
  let e := eval cbv beta match delta [addIndex_weird]
           in (@addIndex_weird a (Some [])) in
  idtac e;
  lazymatch e with | let x := ?def in _ =>
    exact def
  end).

Note the addition of match to the list of evaluation flags passed to cbv.

Conclusion

While local definitions are idiomatic in Haskell (in particular thanks to the where syntax), they are usually avoided in Coq, because they get in the way of verification. If, for some reason, one is stuck with such definitions, then this trick presents a reasonable way out.

Comments

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