Module Iteration

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.

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.

We then prove the expected unrolling equations for iter.

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

We now prove the invariance property iterate_prop.

Invariance property.