Working with accumulators

A standard pattern of functional programming involves writting tail-recursive versions of an algorithm using an accumulator (also called wrapper/worker), we show how Equations makes this pattern easy to express and reason about using where clauses, well-founded recursion and function eliminators.

From Equations Require Import Equations.
From Coq Require Import List Program.Syntax Arith Lia.

Worker/wrapper

The most standard example is an efficient implementation of list reversal. Instead of growing the stack by the size of the list, we accumulate a partially reverted list as a new argument of our function.
We implement this using a go auxilliary function defined recursively and pattern matching on the list.
Equations rev_acc {A} (l : list A) : list A :=
  rev_acc l := go [] l

   where go : list A list A list A :=
         go acc [] := acc;
         go acc (hd :: tl) := go (hd :: acc) tl.

A typical issue with such accumulating functions is that one has to write lemmas in two versions, once about the internal go function and then on its wrapper. Using the functional elimination principle associated to rev_acc, we can show both properties simultaneously.

Lemma rev_acc_eq : {A} (l : list A), rev_acc l = rev l.
Proof.
We apply functional elimination on the rev_acc l call. The eliminator expects two predicates: one specifying the wrapper and another for the worker. For the wrapper, we give the expected final goal but for the worker we have to invent a kind of loop invariant: here that the result of the whole go acc l call is equal to rev l ++ acc.
  apply (rev_acc_elim (fun A l revacclrevaccl = rev l)
                      (fun A _ acc l go_resgo_res = rev l ++ acc)); intros; simpl.
Functional elimination provides us with the worker property for the initial go [] l call, i.e. that it is equal to rev l ++ [], hence the proof:
  + now rewrite app_nil_r in H.
For the worker proofs themselves, we use standard reasoning.
  + reflexivity.
  + now rewrite H, <- app_assoc.
Qed.

The worker/wrapper and well-founded recursion

Sometimes the natural expression of an algorithm in the worker/wrapper pattern requires well-founded recursion: here we take an example algorithm from Haskell whose termination is justified by a measure. Note that the worker subprogram's termination measure and implementation depends on the enclosing k argument which is captured in the where clause.

Hint Extern 100 ⇒ progress (simpl in *) : Below.

Equations? isPrime (n : nat) : bool :=
  isPrime 0 := false;
  isPrime 1 := false;
  isPrime 2 := true;
  isPrime 3 := true;
  isPrime k := worker 2
    where worker (n' : nat) : bool by wf (k - n') lt :=
    worker n' with ge_dec n' k :=
      { | left H := true;
        | right H := if Nat.eqb (Nat.modulo k n') 0 then false else worker (S n') }.
Proof. subst k. do 4 (destruct n'; try lia). Defined.

(* Require Import ExtrOcamlBasic. *)
(* Extraction isPrime. *)

Programm equivalence with worker/wrappers

Finally we show how the eliminator can be used to prove program equivalences involving a worker/wrapper definition. Here indexes l computes the list 0..|l|-1 of valid indexes in the list l.

Equations indexes : list nat list nat :=
  indexes l := go [] (length l)

  where go : list nat nat list nat :=
  go acc 0 := acc;
  go acc (S n) := go (n :: acc) n.

Clearly, all indexes in the resulting list should be smaller than length l:
Lemma indexes_spec (l : list nat) : Forall (fun xx < length l) (indexes l).
Proof.
We apply the eliminator, giving a predicate that specifies preservation of the property from the accumulator to the end result for go's specification. The rest of the proof uses simple reasoning.
  apply (indexes_elim (fun l indexeslForall (fun xx < length l) indexesl)
        (fun l acc n indexesln length l
           Forall (fun xx < length l) acc
           Forall (fun xx < length l) indexesl));
    clear l; intros.
  + apply H; constructor.
  + apply H0.
  + apply H. lia. constructor. lia. apply H1.
Qed.

Using well-founded recursion we can also define an interval x y function producing the interval x..y-1

Equations? interval x y : list nat by wf (y - x) lt :=
  interval x y with lt_dec x y :=
    { | left ltxyx :: interval (S x) y;
      | right nltxy[] }.
Proof. lia. Defined.

We prove a simple lemmas on interval:
Lemma interval_large x y : ¬ x < y interval x y = [].
Proof. funelim (interval x y); clear Heq; intros; now try lia. Qed.

One can show that indexes l produces the interval 0..|l|-1 using indexes_elim. The recursion invariant for indexes_go records that acc corresponds to a partial interval n..|l|-1 during the computation, and is finally completed into 0..|l|-1 by the end of the computation. We use the previous lemmas as helpers.
Lemma indexes_interval l : indexes l = interval 0 (length l).
Proof.
  set (P := fun start (l indexesl : list nat) ⇒ indexesl = interval start (length l)).
  revert l.
  apply (indexes_elim (P 0)
          (fun l acc n indexesl
             n length l
             P n l acc P 0 l indexesl)); subst P; simpl.
  intros l.
  + intros H. apply H; auto. rewrite interval_large; trivial; lia.
  + intros; trivial.
  + intros l ? n H Hn →. apply H. lia.
    rewrite (interval_equation_1 n).
    destruct lt_dec. reflexivity. elim n0. lia.
Qed.

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