Implementing Strongly-Specified Functions with the Program Framework

The Theory

If I had to explain Program, I would say Program is the heir of the refine tactic. It gives you a convenient way to embed proofs within functional programs that are supposed to fade away during code extraction. But what do I mean when I say "embed proofs" within functional programs? I found two ways to do it.

Invariants

First, we can define a record with one or more fields of type Prop. By doing so, we can constrain the values of other fields. Put another way, we can specify invariant for our type. For instance, in SpecCert , I have defined the memory controller's SMRAMC register as follows:

Record SmramcRegister := {
  d_open: bool;
  d_lock: bool;
  lock_is_close: d_lock = true -> d_open = false;
}.

So lock_is_closed is an invariant I know each instance of SmramcRegister will have to comply with, because every time I will construct a new instance, I will have to prove lock_is_closed holds true. For instance:

Definition lock (reg: SmramcRegister)
  : SmramcRegister.
  refine ({| d_open := false; d_lock := true |}).

Coq leaves us this goal to prove.

reg : SmramcRegister
============================
true = true -> false = false

This sound reasonable enough.

Proof.
  trivial.
Defined.

We have seen in my previous article about strongly specified functions that mixing proofs and regular terms may lead to cumbersome code.

From that perspective, Program helps. Indeed, the lock function can also be defined as follows:

From Coq Require Import Program.

#[program]
Definition lock' (reg: SmramcRegister)
  : SmramcRegister :=
  {| d_open := false
   ; d_lock := true
   |}.

Pre and Post Conditions

Another way to "embed proofs in a program" is by specifying pre- and post-conditions for its component. In Coq, this is done using sigma types.

On the one hand, a precondition is a proposition a function input has to satisfy in order for the function to be applied. For instance, a precondition for head : forall {a}, list a -> a the function that returns the first element of a list l requires l to contain at least one element. We can write that using a sigma-type. The type of head then becomes forall {a} (l: list a | l <> []) : a.

On the other hand, a post condition is a proposition a function output has to satisfy in order for the function to be correctly implemented. In this way, head should in fact return the first element of l and not something else.

Program makes writing this specification straightforward.

#[program]
Definition head {a} (l : list a | l <> [])
  : { x : a | exists l', x :: l' = l }.

We recall that because { l: list a | l <> [] } is not the same as list a, in theory we cannot just compare l with x :: l' (we need to use proj1_sig). One advantage of Program is to deal with it using an implicit coercion.

Note that for the type inference to work as expected, the unwrapped value (here, x :: l') needs to be the left operand of =.

Now that head have been specified, we have to implement it.

#[program]
Definition head {a} (l: list a | l <> [])
  : { x : a | exists l', cons x l' = l } :=
  match l with
  | x :: l' => x
  | [] => !
  end.

Next Obligation.
  exists l'.
  reflexivity.
Qed.

I want to highlight several things here:

• We return x (of type a) rather than a sigma-type, then Program is smart enough to wrap it. To do so, it tries to prove the post condition and because it fails, we have to do it ourselves (this is the Obligation we solve after the function definition.)
• The [] case is absurd regarding the precondition, we tell Coq that using the bang (!) symbol.

We can have a look at the extracted code:

(** val head : 'a1 list -> 'a1 **)
let head = function
| Nil -> assert false (* absurd case *)
| Cons (a, _) -> a

The implementation is pretty straightforward, but the pre- and post conditions have faded away. Also, the absurd case is discarded using an assertion. This means one thing: [head] should not be used directly from the Ocaml world. "Interface" functions have to be total. *)

The Practice

From Coq Require Import Lia.

I have challenged myself to build a strongly specified library. My goal was to define a type vector : nat -> Type -> Type such as vector a n is a list of n instance of a.

Inductive vector (a : Type) : nat -> Type :=
| vcons {n} : a -> vector a n -> vector a (S n)
| vnil : vector a O.

Arguments vcons [a n] _ _.
Arguments vnil {a}.

I had three functions in mind: take, drop and extract. I learned a few lessons. My main takeaway remains: do not use sigma types, Program and dependent types together. From my point of view, Coq is not yet ready for this. Maybe it is possible to make those three work together, but I have to admit I did not find out how. As a consequence, my preconditions are defined as extra arguments.

To be able to specify the post conditions of my three functions and some others, I first defined nth to get the nth element of a vector.

My first attempt to write nth was a failure.

#[program]
Fixpoint nth {a n}
    (v : vector a n) (i : nat) {struct v}
  : option a :=
  match v, i with
  | vcons x _, O => Some x
  | vcons x r, S i => nth r i
  | vnil, _ => None
  end.

raised an anomaly.

#[program]
Fixpoint nth {a n}
    (v : vector a n) (i : nat) {struct v}
  : option a :=
  match v with
  | vcons x r =>
    match i with
    | O => Some x
    | S i => nth r i
    end
  | vnil => None
  end.

With nth, it is possible to give a very precise definition of take:

#[program]
Fixpoint take {a n}
    (v : vector a n) (e : nat | e <= n)
  : { u : vector a e | forall i : nat,
        i < e -> nth u i = nth v i } :=
  match e with
  | S e' => match v with
            | vcons x r => vcons x (take r e')
            | vnil => !
            end
  | O => vnil
  end.

Next Obligation.
  now apply le_S_n.
Defined.

Next Obligation.
  induction i.
  + reflexivity.
  + apply e0.
    now apply Lt.lt_S_n.
Defined.

Next Obligation.
  now apply PeanoNat.Nat.nle_succ_0 in H.
Defined.

Next Obligation.
  now apply PeanoNat.Nat.nlt_0_r in H.
Defined.

As a side note, I wanted to define the post condition as follows: { v': vector A e | forall (i : nat | i < e), nth v' i = nth v i }. However, this made the goals and hypotheses become very hard to read and to use. Sigma types in sigma types: not a good idea.

(** val take : 'a1 vector -> nat -> 'a1 vector **)

let rec take v = function
| O -> Vnil
| S e' ->
  (match v with
   | Vcons (_, x, r) -> Vcons (e', x, (take r e'))
   | Vnil -> assert false (* absurd case *))

Then I could tackle drop in a very similar manner:

#[program]
Fixpoint drop {a n}
    (v : vector a n) (b : nat | b <= n)
  : { v': vector a (n - b) | forall i,
        i < n - b -> nth v' i = nth v (b + i) } :=
  match b with
  | 0 => v
  | S n => (match v with
           | vcons _ r => (drop r n)
           | vnil => !
           end)
  end.

Next Obligation.
  now rewrite <- Minus.minus_n_O.
Defined.

Next Obligation.
  induction n;
    rewrite <- eq_rect_eq;
    reflexivity.
Defined.

Next Obligation.
  now apply le_S_n.
Defined.

Next Obligation.
  now apply PeanoNat.Nat.nle_succ_0 in H.
Defined.

The proofs are easy to write, and the extracted code is exactly what one might want it to be:

(** val drop : 'a1 vector -> nat -> 'a1 vector **)
let rec drop v = function
| O -> v
| S n ->
  (match v with
   | Vcons (_, _, r) -> drop r n
   | Vnil -> assert false (* absurd case *))

But Program really shone when it comes to implementing extract. I just had to combine take and drop. *)

#[program]
Definition extract {a n} (v : vector a n)
    (e : nat | e <= n) (b : nat | b <= e)
  : { v': vector a (e - b) | forall i,
        i < (e - b) -> nth v' i = nth v (b + i) } :=
  take (drop v b) (e - b).


Next Obligation.
  transitivity e; auto.
Defined.

Next Obligation.
  now apply PeanoNat.Nat.sub_le_mono_r.
Defined.

Next Obligation.
  destruct drop; cbn in *.
  destruct take; cbn in *.
  rewrite e1; auto.
  rewrite <- e0; auto.
  lia.
Defined.

The proofs are straightforward because the specifications of drop and take are precise enough, and we do not need to have a look at their implementations. The extracted version of extract is as clean as we can anticipate.

(** val extract : 'a1 vector -> nat -> nat -> 'a1 vector **)
let extract v e b =
  take (drop v b) (sub e b)

I was pretty happy, so I tried some more. Each time, using nth, I managed to write a precise post condition and to prove it holds true. For instance, given map to apply a function f to each element of a vector v:

#[program]
Fixpoint map {a b n} (v : vector a n) (f : a -> b)
  : { v': vector b n | forall i,
        nth v' i = option_map f (nth v i) } :=
  match v with
  | vnil => vnil
  | vcons a v => vcons (f a) (map v f)
  end.

Next Obligation.
  induction i.
  + reflexivity.
  + apply e.
Defined.

I also managed to specify and write append:

#[program]
Fixpoint append {a n m}
    (v : vector a n) (u : vector a m)
  : { w : vector a (n + m) | forall i,
        (i < n -> nth w i = nth v i) /\
        (n <= i -> nth w i = nth u (i - n))
    } :=
  match v with
  | vnil => u
  | vcons a v => vcons a (append v u)
  end.

Next Obligation.
  split.
  + now intro.
  + intros _.
    now rewrite PeanoNat.Nat.sub_0_r.
Defined.

Next Obligation.
  rename wildcard' into n.
  destruct (Compare_dec.lt_dec i (S n)); split.
  + intros _.
    destruct i.
    ++ reflexivity.
    ++ cbn.
       specialize (a1 i).
       destruct a1 as [a1 _].
       apply a1.
       auto with arith.
  + intros false.
    lia.
  + now intros.
  + intros ord.
    destruct i.
    ++ lia.
    ++ cbn.
       specialize (a1 i).
       destruct a1 as [_ a1].
       apply a1.
       auto with arith.
Defined.

Finally, I tried to implement map2 that takes a vector of a, a vector of b (both of the same size) and a function f : a -> b -> c and returns a vector of c.

First, we need to provide a precise specification for map2. To do that, we introduce option_app, a function that Haskellers know all to well as being part of the Applicative type class.

Definition option_app {a b}
    (opf: option (a -> b))
    (opx: option a)
  : option b :=
  match opf, opx with
  | Some f, Some x => Some (f x)
  | _, _ => None
end.

We thereafter use <$> as an infix operator for option_map and <*> as an infix operator for option_app. *)

Infix "<$>" := option_map (at level 50).
Infix "<*>" := option_app (at level 55).

Given two vectors v and u of the same size and a function f, and given w the result computed by map2, then we can propose the following specification for map2:

forall (i : nat), nth w i = f <$> nth v i <*> nth u i

This reads as follows: the ith element of w is the result of applying the ith elements of v and u to f.

It turns out implementing map2 with the Program framework has proven to be harder than I originally expected. My initial attempt was the following:

#[program]
Fixpoint map2 {a b c n}
    (v : vector a n) (u : vector b n)
    (f : a -> b -> c) {struct v}
  : { w: vector c n | forall i,
        nth w i = f <$> nth v i <*> nth u i
    } :=
  match v, u with
  | vcons x rst, vcons x' rst' =>
      vcons (f x x') (map2 rst rst' f)
  | vnil, vnil => vnil
  | _, _ => !
  end.

Illegal application:
The term "@eq" of type "forall A : Type, A -> A -> Prop"
cannot be applied to the terms
 "nat" : "Set"
 "S wildcard'" : "nat"
 "b" : "Type"
The 3rd term has type "Type" which should be coercible
to "nat".

So I had to fallback to defining the function in pure Ltac.

#[program]
Fixpoint map2 {a b c n}
    (v : vector a n) (u : vector b n)
    (f : a -> b -> c) {struct v}
  : { w: vector c n | forall i,
        nth w i = f <$> nth v i <*> nth u i
    } := _.

Next Obligation.
  dependent induction v; dependent induction u.
  + remember (IHv u f) as u'.
    inversion u'.
    refine (exist _ (vcons (f a0 a1) x) _).
    intros i.
    induction i.
    * reflexivity.
    * apply (H i).
  + refine (exist _ vnil _).
    reflexivity.
Qed.

Is It Usable?

This post mostly gives the "happy ends" for each function. I think I tried too hard for what I got in return and therefore I am convinced Program is not ready (at least for a dependent type, I cannot tell for the rest). For instance, I found at least one bug in Program logic (I still have to report it). Have a look at the following code:

#[program]
Fixpoint map2 {a b c n}
     (u : vector a n) (v : vector b n)
     (f : a -> b -> c) {struct v}
  : vector c n :=
  match u with
  | _ => vnil
  end.

It gives the following error:

Error: Illegal application:
The term "@eq" of type "forall A : Type, A -> A -> Prop"
cannot be applied to the terms
 "nat" : "Set"
 "0" : "nat"
 "wildcard'" : "vector A n'"
The 3rd term has type "vector A n'" which should be
coercible to "nat".