Bounded and unbounded iterators

Require Import Axioms.
Require Import Coqlib.
Require Import Wfsimpl.

Set Asymmetric Patterns.

This modules defines several Coq encodings of a general "while" loop. The loop is presented in functional style as the iteration of a step function of type A -> B + A:

        let rec iterate step a =
          match step a with
          | inl b -> b
          | inr a' -> iterate step a'

This iteration cannot be defined directly in Coq using Fixpoint, because Coq is a logic of total functions, and therefore we must guarantee termination of the loop.

Terminating iteration

We first implement the case where termination is guaranteed because the current state a decreases at each iteration.

Module WfIter.

Section ITERATION.

Variables A B: Type.
Variable step: A -> B + A.
Variable ord: A -> A -> Prop.
Hypothesis ord_wf: well_founded ord.
Hypothesis step_decr: forall a a', step a = inr _ a' -> ord a' a.

Definition step_info (a: A) : {b | step a = inl _ b} + {a' | step a = inr _ a' & ord a' a}.

Proof.

caseEq (step a); intros. left; exists b; auto. right; exists a0; auto.
Defined.

Definition iterate_F (a: A) (rec: forall a', ord a' a -> B) : B :=
  match step_info a with
  | inl (exist b P) => b
  | inr (exist2 a' P Q) => rec a' Q
  end.

Definition iterate (a: A) : B := Fix ord_wf iterate_F a.

We now prove an invariance property iterate_prop, similar to the Hoare logic rule for "while" loops.

Variable P: A -> Prop.
Variable Q: B -> Prop.

Hypothesis step_prop:
  forall a : A, P a ->
  match step a with inl b => Q b | inr a' => P a' end.

Lemma iterate_prop:
  forall a, P a -> Q (iterate a).

Proof.

  intros a0. pattern a0. apply well_founded_ind with (R := ord). auto.
  intros. unfold iterate; rewrite unroll_Fix. unfold iterate_F.
  destruct (step_info x) as [[b U] | [a' U V]].
  exploit step_prop; eauto. rewrite U; auto.
  apply H. auto. exploit step_prop; eauto. rewrite U; auto.
Qed.

End ITERATION.

End WfIter.

Bounded iteration

The presentation of iteration shown above is predicated on the existence of a well-founded ordering that decreases at each step of the iteration. In several parts of the CompCert development, it is very painful to define such a well-founded ordering and to prove decrease, even though we know our iterations always terminate. In the presentation below, we choose instead to bound the number of iterations by an arbitrary constant. iterate then becomes a function that can fail, of type A -> option B. The None result denotes failure to reach a result in the number of iterations prescribed, or, in other terms, failure to find a solution to the dataflow problem. The compiler passes that exploit dataflow analysis (the Constprop, CSE and Allocation passes) will, in this case, either fail (Allocation) or turn off the optimization pass (Constprop and CSE). Since we know (informally) that our computations terminate, we can take a very large constant as the maximal number of iterations. Failure will therefore never happen in practice, but of course our proofs also cover the failure case and show that nothing bad happens in this hypothetical case either.

Module PrimIter.

Section ITERATION.

Variables A B: Type.
Variable step: A -> B + A.

Definition num_iterations := 1000000000000%positive.

The simple definition of bounded iteration is:

Fixpoint iterate (niter: nat) (a: A) {struct niter} : option B :=
  match niter with
  | O => None
  | S niter' =>
      match step a with
      | inl b => b
      | inr a' => iterate niter' a'
      end
  end.

This function is structural recursive over the parameter niter (number of iterations), represented here as a Peano integer (type nat). However, we want to use very large values of niter. As Peano integers, these values would be much too large to fit in memory. Therefore, we must express iteration counts as a binary integer (type positive). However, Peano induction over type positive is not structural recursion, so we cannot define iterate as a Coq fixpoint and must use Noetherian recursion instead.

Definition iter_step (x: positive)
                     (next: forall y, Plt y x -> A -> option B)
                     (s: A) : option B :=
  match peq x xH with
  | left EQ => None
  | right NOTEQ =>
      match step s with
      | inl res => Some res
      | inr s' => next (Pos.pred x) (Ppred_Plt x NOTEQ) s'
      end
  end.

Definition iter: positive -> A -> option B := Fix Plt_wf iter_step.

The iterate function is defined as iter up to num_iterations through the loop.

Definition iterate := iter num_iterations.

We now prove the invariance property iterate_prop.

Variable P: A -> Prop.
Variable Q: B -> Prop.

Hypothesis step_prop:
  forall a : A, P a ->
  match step a with inl b => Q b | inr a' => P a' end.

Lemma iter_prop:
  forall n a b, P a -> iter n a = Some b -> Q b.

Proof.

  apply (well_founded_ind Plt_wf
         (fun p => forall a b, P a -> iter p a = Some b -> Q b)).
  intros. unfold iter in H1. rewrite unroll_Fix in H1. unfold iter_step in H1.
  destruct (peq x 1). discriminate.
  specialize (step_prop a H0).
  destruct (step a) as [b'|a'] eqn:?.
  inv H1. auto.
  apply H with (Pos.pred x) a'. apply Ppred_Plt; auto. auto. auto.
Qed.

Lemma iterate_prop:
forall a b, iterate a = Some b -> P a -> Q b.

Proof.

intros. apply iter_prop with num_iterations a; assumption.
Qed.

End ITERATION.

End PrimIter.

General iteration

Require Import Classical.
Require Import ClassicalDescription.

Module GenIter.

Section ITERATION.

Variables A B: Type.
Variable step: A -> B + A.

Definition B_le (x y: option B) : Prop := x = None \/ y = x.
Definition F_le (x y: A -> option B) : Prop := forall a, B_le (x a) (y a).

Definition F_iter (next: A -> option B) (a: A) : option B :=
  match step a with
  | inl b => Some b
  | inr a' => next a'
  end.

Lemma F_iter_monot:
forall f g, F_le f g -> F_le (F_iter f) (F_iter g).

Proof.

intros; red; intros. unfold F_iter.
destruct (step a) as [b | a']. red; auto. apply H.
Qed.

Fixpoint iter (n: nat) : A -> option B :=
  match n with
  | O => (fun a => None)
  | S m => F_iter (iter m)
  end.

Lemma iter_monot:
  forall p q, (p <= q)%nat -> F_le (iter p) (iter q).

Proof.

  induction p; intros.
  simpl. red; intros; red; auto.
  destruct q. exfalso; lia.
  simpl. apply F_iter_monot. apply IHp. lia.
Qed.

Lemma iter_either:
  forall a,
  (exists n, exists b, iter n a = Some b) \/
  (forall n, iter n a = None).

Proof.

  intro a. elim (classic (forall n, iter n a = None)); intro.
  right; assumption.
  left. generalize (not_all_ex_not nat (fun n => iter n a = None) H).
  intros [n D]. exists n. generalize D.
  case (iter n a); intros. exists b; auto. congruence.
Qed.

Definition converges_to (a: A) (b: option B) : Prop :=
exists n, forall m, (n <= m)%nat -> iter m a = b.

Lemma converges_to_Some:
forall a n b, iter n a = Some b -> converges_to a (Some b).

Proof.

  intros. exists n. intros.
  assert (B_le (iter n a) (iter m a)). apply iter_monot. auto.
  elim H1; intro; congruence.
Qed.

Lemma converges_to_exists:
forall a, exists b, converges_to a b.

Proof.

  intros. elim (iter_either a).
  intros [n [b EQ]]. exists (Some b). apply converges_to_Some with n. assumption.
  intro. exists (@None B). exists O. intros. auto.
Qed.

Lemma converges_to_unique:
forall a b, converges_to a b -> forall b', converges_to a b' -> b = b'.

Proof.

  intros a b [n C] b' [n' C'].
  rewrite <- (C (max n n')). rewrite <- (C' (max n n')). auto.
  apply Nat.le_max_r. apply Nat.le_max_l.
Qed.

Lemma converges_to_exists_uniquely:
forall a, exists! b, converges_to a b .

Proof.

intro. destruct (converges_to_exists a) as [b CT].
exists b. split. assumption. exact (converges_to_unique _ _ CT).
Qed.

Definition iterate (a: A) : option B :=
proj1_sig (constructive_definite_description (converges_to a) (converges_to_exists_uniquely a)).

Lemma converges_to_iterate:
forall a b, converges_to a b -> iterate a = b.

Proof.

  intros. unfold iterate.
  destruct (constructive_definite_description (converges_to a) (converges_to_exists_uniquely a)) as [b' P].
  simpl. apply converges_to_unique with a; auto.
Qed.

Lemma iterate_converges_to:
forall a, converges_to a (iterate a).

Proof.

  intros. unfold iterate.
  destruct (constructive_definite_description (converges_to a) (converges_to_exists_uniquely a)) as [b' P].
  simpl; auto.
Qed.

Invariance property.

Proof.

  induction n; intros until b; intro H; simpl.
  congruence.
  unfold F_iter. generalize (step_prop a H).
  case (step a); intros. congruence.
  apply IHn with a0; auto.
Qed.

Lemma iterate_prop:
forall a b, iterate a = Some b -> P a -> Q b.

Proof.

intros. destruct (iterate_converges_to a) as [n IT].
rewrite H in IT. apply iter_prop with n a. auto. apply IT. auto.
Qed.

End ITERATION.

End GenIter.

Module Iteration

Terminating iteration

Bounded iteration

General iteration