Module Iteration

Require Import Coqlib.
Require Import Classical.
Require Import Max.

Module Type ITER.
Variable iterate
: forall A B : Type, (A -> B + A) -> A -> option B.
Hypothesis iterate_prop
: forall (A B : Type) (step : A -> B + A) (P : A -> Prop) (Q : B -> Prop),
(forall a : A, P a ->
match step a with inl b => Q b | inr a' => P a' end) ->
forall (a : A) (b : B), iterate A B step a = Some b -> P a -> Q b.
End ITER.

Axiom
dependent_description' :
forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop),
(forall x:A,
exists y : B x, R x y /\ (forall y':B x, R x y' -> y = y')) ->
sigT (fun f : forall x:A, B x => (forall x:A, R x (f x))).

Module PrimIter: ITER.

Section ITERATION.

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

The step parameter represents one step of the iteration. From a current iteration state a: A, it either returns a value of type B, meaning that iteration is over and that this B value is the final result of the iteration, or a value a' : A which is the next state of the iteration. The naive way to define the iteration is:
```Fixpoint iterate (a: A) : B :=
match step a with
| inl b => b
| inr a' => iterate a'
end.```
However, this is a general recursion, not guaranteed to terminate, and therefore not expressible in Coq. The standard way to work around this difficulty is to use Noetherian recursion (Coq module Wf). This requires that we equip the type A with a well-founded ordering < (no infinite ascending chains) and we demand that step satisfies step a = inr a' -> a < a'. For the types A that are of interest to us in this development, it is however very painful to define adequate well-founded orderings, even though we know our iterations always terminate. Instead, we choose 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.

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 (Ppred x) (Ppred_Plt x NOTEQ) s'
end
end.

Definition iter: positive -> A -> option B :=
Fix Plt_wf (fun _ => A -> option B) iter_step.

We then prove the expected unrolling equations for iter.

Remark unroll_iter:
forall x, iter x = iter_step x (fun y _ => iter y).
Proof.
unfold iter; apply (Fix_eq Plt_wf (fun _ => A -> option B) iter_step).
intros. unfold iter_step. apply extensionality. intro s.
case (peq x xH); intro. auto.
rewrite H. auto.
Qed.

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 until b. intro. rewrite unroll_iter.
unfold iter_step. case (peq x 1); intro. congruence.
generalize (step_prop a H0).
case (step a); intros. congruence.
apply H with (Ppred x) a0. 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.

Module GenIter: ITER.

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. elimtype False; omega.
simpl. apply F_iter_monot. apply IHp. omega.
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 le_max_r. apply le_max_l.
Qed.

Lemma converges_to_exists_uniquely:
forall a, exists b, converges_to a b /\ forall b', converges_to a b' -> b = b'.
Proof.
intro. destruct (converges_to_exists a) as [b CT].
exists b. split. assumption. exact (converges_to_unique _ _ CT).
Qed.

Definition exists_iterate :=
dependent_description' A (fun _ => option B)
converges_to converges_to_exists_uniquely.

Definition iterate : A -> option B :=
match exists_iterate with existT f P => f end.

Lemma converges_to_iterate:
forall a b, converges_to a b -> iterate a = b.
Proof.
intros. unfold iterate. destruct exists_iterate as [f P].
apply converges_to_unique with a. apply P. auto.
Qed.

Lemma iterate_converges_to:
forall a, converges_to a (iterate a).
Proof.
intros. unfold iterate. destruct exists_iterate as [f P]. apply P.
Qed.

Invariance property.

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.
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.