Strongly-Specified Functions in Coq, part 2: the Program Framework
This is the second article (initially published on January 01, 2017) of a series of two on how to write strongly-specified functions in Coq. You can read the previous part here.Revisions
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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 = falseThis sound reasonable enough.
Proof.
trivial.
Defined.
We have witness in my previous article about strongly-specified
functions that mixing proofs and regular terms may leads 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.
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 benefit on
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 gigma-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.
(** val head : 'a1 list -> 'a1 **) let head = function | Nil -> assert false (* absurd case *) | Cons (a, _) -> aThe 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
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 few lessons. My main take-away remains: do not use
gigma-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 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.raises 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 <$> as an infix operator for option_map and <*> as
an infix operator for option_app.
Given two vectors v and u of the same size and a function f, and given
w the result computed by map2, then we can propose the following
specification for map2:
forall (i : nat), nth w i = f <$> 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 <$> nth v i <*> nth u i } := match v, u with | vcons x rst, vcons x' rst' => vcons (f x x') (map2 rst rst' f) | vnil, vnil => vnil | _, _ => ! end.
Illegal application: The term "@eq" of type "forall A : Type, A -> A -> Prop" cannot be applied to the terms "nat" : "Set" "S wildcard'" : "nat" "b" : "Type" The 3rd term has type "Type" which should be coercible to "nat".
#[program]
Fixpoint map2 {a b c n}
(v : vector a n) (u : vector b n)
(f : a -> b -> c) {struct v}
: { w: vector c n | forall i,
nth w i = f <$> 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 to 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".