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(** #<h1>Strongly-Specified Functions in Coq, part 2: the <code>Program</code> Framework</h1>#

    This is the second article (initially published on #<span
    class="time">January 01, 2017</span>#) of a series of two on how to write
    strongly-specified functions in Coq. You can read the previous part #<a
    href="/posts/StronglySpecifiedFunctions">here</a>#. # *)

(** #<div id="generate-toc"></div>#

    #<div id="history">site/posts/StronglySpecifiedFunctionsProgram.v</div># *)

(** ** 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 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. *)

(* begin hide *)
From Coq Require Import List.
Import ListNotations.
(* end hide *)
#[program]
Definition head {a} (l : list a | l <> [])
  : { x : a | exists l', x :: l' = l }.
(* begin hide *)
Abort.
(* end hide *)

(** 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.

    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 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)
  : { v' : vector a e | forall i : nat,
        i < e -> nth v' 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. *)

From Coq Require Import Extraction.

Extraction Implicit take [a n].
Extraction take.

(**
<<
(** 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.

(*begin hide *)
Extraction Implicit drop [a n].
Extraction drop.
(* end hide *)
(** 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.

(*begin hide *)
Extraction Implicit extract [a n].
Extraction extract.
(* end hide *)
(** 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: Type}
           (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 [i]th element of [w] is the result of applying
    the [i]th 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: Type}
        {n:     nat}
        (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.
>>

<<
Error: 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: Type}
        {n:     nat}
        (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 a a0) x) _).
    intros i.
    induction i.
    * cbv.
      reflexivity.
    * simpl.
      apply (H i).
  + refine (exist _ vnil _).
    cbv.
    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}
     (v : vector a n) (v': vector b n)
     (f:  a -> b -> c) {struct v}
  : vector c n :=
  match v 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".
>> *)
(* begin hide *)
Reset map2.
(* end hide *)