In lazy functional programming languages like Haskell, infinite data structures are everywhere. Infinite lists and more exotic datatypes provide convenient abstractions for communication between parts of a program. Achieving similar convenience without infinite lazy structures would, in many cases, require acrobatic inversions of control flow. Laziness is easy to implement in Haskell, where all the definitions in a program may be thought of as mutually recursive. In such an unconstrained setting, it is easy to implement an infinite loop when you really meant to build an infinite list, where any finite prefix of the list should be forceable in finite time. Haskell programmers learn how to avoid such slip-ups. In Coq, such a laissez-faire policy is not good enough. We spent some time in the last chapter discussing the Curry-Howard isomorphism, where proofs are identified with functional programs. In such a setting, infinite loops, intended or otherwise, are disastrous. If Coq allowed the full breadth of definitions that Haskell did, we could code up an infinite loop and use it to prove any proposition vacuously. That is, the addition of general recursion would make CIC inconsistent. For an arbitrary proposition P, we could write:
Fixpoint bad (u : unit) : P := bad u.

This would leave us with bad tt as a proof of P. There are also algorithmic considerations that make universal termination very desirable. We have seen how tactics like reflexivity compare terms up to equivalence under computational rules. Calls to recursive, pattern-matching functions are simplified automatically, with no need for explicit proof steps. It would be very hard to hold onto that kind of benefit if it became possible to write non-terminating programs; we would be running smack into the halting problem. One solution is to use types to contain the possibility of non-termination. For instance, we can create a "non-termination monad," inside which we must write all of our general-recursive programs; several such approaches are surveyed in Chapter 7. This is a heavyweight solution, and so we would like to avoid it whenever possible. Luckily, Coq has special support for a class of lazy data structures that happens to contain most examples found in Haskell. That mechanism, co-inductive types, is the subject of this chapter.

Computing with Infinite Data

Let us begin with the most basic type of infinite data, streams, or lazy lists.

The definition is surprisingly simple. Starting from the definition of list, we just need to change the keyword Inductive to CoInductive. We could have left a Nil constructor in our definition, but we will leave it out to force all of our streams to be infinite. How do we write down a stream constant? Obviously, simple application of constructors is not good enough, since we could only denote finite objects that way. Rather, whereas recursive definitions were necessary to use values of recursive inductive types effectively, here we find that we need co-recursive definitions to build values of co-inductive types effectively. We can define a stream consisting only of zeroes.

We can also define a stream that alternates between true and false.

Co-inductive values are fair game as arguments to recursive functions, and we can use that fact to write a function to take a finite approximation of a stream.

     = 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: 0 :: nil
     : list nat

     = true
       :: false
          :: true
             :: false
                :: true :: false :: true :: false :: true :: false :: nil
     : list bool

So far, it looks like co-inductive types might be a magic bullet, allowing us to import all of the Haskeller's usual tricks. However, there are important restrictions that are dual to the restrictions on the use of inductive types. Fixpoints consume values of inductive types, with restrictions on which arguments may be passed in recursive calls. Dually, co-fixpoints produce values of co-inductive types, with restrictions on what may be done with the results of co-recursive calls. The restriction for co-inductive types shows up as the guardedness condition. First, consider this stream definition, which would be legal in Haskell.
CoFixpoint looper : stream nat := looper.

Error:
Recursive definition of looper is ill-formed.
In environment
looper : stream nat

unguarded recursive call in "looper"

The rule we have run afoul of here is that every co-recursive call must be guarded by a constructor; that is, every co-recursive call must be a direct argument to a constructor of the co-inductive type we are generating. It is a good thing that this rule is enforced. If the definition of looper were accepted, our approx function would run forever when passed looper, and we would have fallen into inconsistency. Some familiar functions are easy to write in co-recursive fashion.

This code is a literal copy of that for the list map function, with the nil case removed and Fixpoint changed to CoFixpoint. Many other standard functions on lazy data structures can be implemented just as easily. Some, like filter, cannot be implemented. Since the predicate passed to filter may reject every element of the stream, we cannot satisfy the guardedness condition. The implications of the condition can be subtle. To illustrate how, we start off with another co-recursive function definition that is legal. The function interleave takes two streams and produces a new stream that alternates between their elements.

Now say we want to write a weird stuttering version of map that repeats elements in a particular way, based on interleaving.

  CoFixpoint map' (s : stream A) : stream B :=
    match s with
      | Cons h t => interleave (Cons (f h) (map' t)) (Cons (f h) (map' t))
    end.
    

We get another error message about an unguarded recursive call.

What is going wrong here? Imagine that, instead of interleave, we had called some other, less well-behaved function on streams. Here is one simpler example demonstrating the essential pitfall. We start by defining a standard function for taking the tail of a stream. Since streams are infinite, this operation is total.

Coq rejects the following definition that uses tl.
CoFixpoint bad : stream nat := tl (Cons 0 bad).

Imagine that Coq had accepted our definition, and consider how we might evaluate approx bad 1. We would be trying to calculate the first element in the stream bad. However, it is not hard to see that the definition of bad "begs the question": unfolding the definition of tl, we see that we essentially say "define bad to equal itself"! Of course such an equation admits no single well-defined solution, which does not fit well with the determinism of Gallina reduction. Coq's complete rule for co-recursive definitions includes not just the basic guardedness condition, but also a requirement about where co-recursive calls may occur. In particular, a co-recursive call must be a direct argument to a constructor, nested only inside of other constructor calls or fun or match expressions. In the definition of bad, we erroneously nested the co-recursive call inside a call to tl, and we nested inside a call to interleave in the definition of map'. Coq helps the user out a little by performing the guardedness check after using computation to simplify terms. For instance, any co-recursive function definition can be expanded by inserting extra calls to an identity function, and this change preserves guardedness. However, in other cases computational simplification can reveal why definitions are dangerous. Consider what happens when we inline the definition of tl in bad:
CoFixpoint bad : stream nat := bad.

This is the same looping definition we rejected earlier. A similar inlining process reveals an alternate view on our failed definition of map':
CoFixpoint map' (s : stream A) : stream B :=
  match s with
    | Cons h t => Cons (f h) (Cons (f h) (interleave (map' t) (map' t)))
  end.

Clearly in this case the map' calls are not immediate arguments to constructors, so we violate the guardedness condition.

A more interesting question is why that condition is the right one. We can make an intuitive argument that the original map' definition is perfectly reasonable and denotes a well-understood transformation on streams, such that every output would behave properly with approx. The guardedness condition is an example of a syntactic check for productivity of co-recursive definitions. A productive definition can be thought of as one whose outputs can be forced in finite time to any finite approximation level, as with approx. If we replaced the guardedness condition with more involved checks, we might be able to detect and allow a broader range of productive definitions. However, mistakes in these checks could cause inconsistency, and programmers would need to understand the new, more complex checks. Coq's design strikes a balance between consistency and simplicity with its choice of guard condition, though we can imagine other worthwhile balances being struck, too.

Infinite Proofs

Let us say we want to give two different definitions of a stream of all ones, and then we want to prove that they are equivalent.

The obvious statement of the equality is this:

However, faced with the initial subgoal, it is not at all clear how this theorem can be proved. In fact, it is unprovable. The eq predicate that we use is fundamentally limited to equalities that can be demonstrated by finite, syntactic arguments. To prove this equivalence, we will need to introduce a new relation.

Co-inductive datatypes make sense by analogy from Haskell. What we need now is a co-inductive proposition. That is, we want to define a proposition whose proofs may be infinite, subject to the guardedness condition. The idea of infinite proofs does not show up in usual mathematics, but it can be very useful (unsurprisingly) for reasoning about infinite data structures. Besides examples from Haskell, infinite data and proofs will also turn out to be useful for modelling inherently infinite mathematical objects, like program executions. We are ready for our first co-inductive predicate.

We say that two streams are equal if and only if they have the same heads and their tails are equal. We use the normal finite-syntactic equality for the heads, and we refer to our new equality recursively for the tails. We can try restating the theorem with stream_eq.

Coq does not support tactical co-inductive proofs as well as it supports tactical inductive proofs. The usual starting point is the cofix tactic, which asks to structure this proof as a co-fixpoint.

  ones_eq : stream_eq ones ones'
  ============================
   stream_eq ones ones'
 

It looks like this proof might be easier than we expected!

Proof completed.

Unfortunately, we are due for some disappointment in our victory lap.
Qed.

Error:
Recursive definition of ones_eq is ill-formed.

In environment
ones_eq : stream_eq ones ones'

unguarded recursive call in "ones_eq"

Via the Curry-Howard correspondence, the same guardedness condition applies to our co-inductive proofs as to our co-inductive data structures. We should be grateful that this proof is rejected, because, if it were not, the same proof structure could be used to prove any co-inductive theorem vacuously, by direct appeal to itself! Thinking about how Coq would generate a proof term from the proof script above, we see that the problem is that we are violating the guardedness condition. During our proofs, Coq can help us check whether we have yet gone wrong in this way. We can run the command Guarded in any context to see if it is possible to finish the proof in a way that will yield a properly guarded proof term.
Guarded.

Running Guarded here gives us the same error message that we got when we tried to run Qed. In larger proofs, Guarded can be helpful in detecting problems before we think we are ready to run Qed. We need to start the co-induction by applying stream_eq's constructor. To do that, we need to know that both arguments to the predicate are Conses. Informally, this is trivial, but simpl is not able to help us.

  ones_eq : stream_eq ones ones'
  ============================
   stream_eq ones ones'
 

It turns out that we are best served by proving an auxiliary lemma.

First, we need to define a function that seems pointless at first glance.

Next, we need to prove a theorem that seems equally pointless.

But, miraculously, this theorem turns out to be just what we needed.

We can use the theorem to rewrite the two streams.

  ones_eq : stream_eq ones ones'
  ============================
   stream_eq (frob ones) (frob ones')
 

Now simpl is able to reduce the streams.

  ones_eq : stream_eq ones ones'
  ============================
   stream_eq (Cons 1 ones)
     (Cons 1
        ((cofix map (s : stream nat) : stream nat :=
            match s with
            | Cons h t => Cons (S h) (map t)
            end) zeroes))
 

Note the cofix notation for anonymous co-recursion, which is analogous to the fix notation we have already seen for recursion. Since we have exposed the Cons structure of each stream, we can apply the constructor of stream_eq.

  ones_eq : stream_eq ones ones'
  ============================
   stream_eq ones
     ((cofix map (s : stream nat) : stream nat :=
         match s with
         | Cons h t => Cons (S h) (map t)
         end) zeroes)
 

Now, modulo unfolding of the definition of map, we have matched our assumption.

Why did this silly-looking trick help? The answer has to do with the constraints placed on Coq's evaluation rules by the need for termination. The cofix-related restriction that foiled our first attempt at using simpl is dual to a restriction for fix. In particular, an application of an anonymous fix only reduces when the top-level structure of the recursive argument is known. Otherwise, we would be unfolding the recursive definition ad infinitum. Fixpoints only reduce when enough is known about the definitions of their arguments. Dually, co-fixpoints only reduce when enough is known about how their results will be used. In particular, a cofix is only expanded when it is the discriminee of a match. Rewriting with our superficially silly lemma wrapped new matches around the two cofixes, triggering reduction. If cofixes reduced haphazardly, it would be easy to run into infinite loops in evaluation, since we are, after all, building infinite objects. One common source of difficulty with co-inductive proofs is bad interaction with standard Coq automation machinery. If we try to prove ones_eq' with automation, like we have in previous inductive proofs, we get an invalid proof.

  Guarded.
  

The standard auto machinery sees that our goal matches an assumption and so applies that assumption, even though this violates guardedness. A correct proof strategy for a theorem like this usually starts by destructing some parameter and running a custom tactic to figure out the first proof rule to apply for each case. Alternatively, there are tricks that can be played with "hiding" the co-inductive hypothesis. Must we always be cautious with automation in proofs by co-induction? Induction seems to have dual versions of the same pitfalls inherent in it, and yet we avoid those pitfalls by encapsulating safe Curry-Howard recursion schemes inside named induction principles. It turns out that we can usually do the same with co-induction principles. Let us take that tack here, so that we can arrive at an induction x; crush-style proof for ones_eq'. An induction principle is parameterized over a predicate characterizing what we mean to prove, as a function of the inductive fact that we already know. Dually, a co-induction principle ought to be parameterized over a predicate characterizing what we mean to prove, as a function of the arguments to the co-inductive predicate that we are trying to prove. To state a useful principle for stream_eq, it will be useful first to define the stream head function.

Now we enter a section for the co-induction principle, based on Park's principle as introduced in a tutorial by Gimenez.

This relation generalizes the theorem we want to prove, defining a set of pairs of streams that we must eventually prove contains the particular pair we care about.

Two hypotheses characterize what makes a good choice of R: it enforces equality of stream heads, and it is hereditary in the sense that an R stream pair passes on "R-ness" to its tails. An established technical term for such a relation is bisimulation. Now it is straightforward to prove the principle, which says that any stream pair in R is equal. The reader may wish to step through the proof script to see what is going on.

To see why this proof is guarded, we can print it and verify that the one co-recursive call is an immediate argument to a constructor.

We omit the output and proceed to proving ones_eq'' again. The only bit of ingenuity is in choosing R, and in this case the most restrictive predicate works.

Note that this proof achieves the proper reduction behavior via hd and tl, rather than frob as in the last proof. All three functions pattern match on their arguments, catalyzing computation steps. Compared to the inductive proofs that we are used to, it still seems unsatisfactory that we had to write out a choice of R in the last proof. An alternate is to capture a common pattern of co-recursion in a more specialized co-induction principle. For the current example, that pattern is: prove stream_eq s1 s2 where s1 and s2 are defined as their own tails.

The proof of the principle includes a choice of R, so that we no longer need to make such choices thereafter.

Let us put stream_eq_coind through its paces a bit more, considering two different ways to compute infinite streams of all factorial values. First, we import the fact factorial function from the standard library.

fact =
fix fact (n : nat) : nat :=
  match n with
  | 0 => 1
  | S n0 => S n0 * fact n0
  end
     : nat -> nat

The simplest way to compute the factorial stream involves calling fact afresh at each position.

A more clever, optimized method maintains an accumulator of the previous factorial, so that each new entry can be computed with a single multiplication.

We can verify that the streams are equal up to particular finite bounds.

     = 1 :: 2 :: 6 :: 24 :: 120 :: nil
     : list nat

     = 1 :: 2 :: 6 :: 24 :: 120 :: nil
     : list nat

Now, to prove that the two versions are equivalent, it is helpful to prove (and add as a proof hint) a quick lemma about the computational behavior of fact. (I intentionally skip explaining its proof at this point.)

With the hint added, it is easy to prove an auxiliary lemma relating fact_iter' and fact_slow'. The key bit of ingenuity is introduction of an existential quantifier for the shared parameter n.

The final theorem is a direct corollary of fact_eq'.

As in the case of ones_eq', we may be unsatisfied that we needed to write down a choice of R that seems to duplicate information already present in a lemma statement. We can facilitate a simpler proof by defining a co-induction principle specialized to goals that begin with single universal quantifiers, and the strategy can be extended in a straightforward way to principles for other counts of quantifiers. (Our stream_eq_loop principle is effectively the instantiation of this technique to zero quantifiers.)

We have the types A, the domain of the one quantifier; and B, the type of data found in the streams.

The two streams we compare must be of the forms f x and g x, for some shared x. Note that this falls out naturally when x is a shared universally quantified variable in a lemma statement.

These conditions are inspired by the bisimulation requirements, with a more general version of the R choice we made for fact_eq' inlined into the hypotheses of stream_eq_coind.

We have arrived at one of our customary automated proofs, thanks to the new principle.

Simple Modeling of Non-Terminating Programs

We close the chapter with a quick motivating example for more complex uses of co-inductive types. We will define a co-inductive semantics for a simple imperative programming language and use that semantics to prove the correctness of a trivial optimization that removes spurious additions by 0. We follow the technique of co-inductive big-step operational semantics . We define a suggestive synonym for nat, as we will consider programs over infinitely many variables, represented as nats.

We define a type vars of maps from variables to values. To define a function set for setting a variable's value in a map, we use the standard library function beq_nat for comparing natural numbers.

We define a simple arithmetic expression language with variables, and we give it a semantics via an interpreter.

Finally, we define a language of commands. It includes variable assignment, sequencing, and a while form that repeats as long as its test expression evaluates to a nonzero value.

We could define an inductive relation to characterize the results of command evaluation. However, such a relation would not capture nonterminating executions. With a co-inductive relation, we can capture both cases. The parameters of the relation are an initial state, a command, and a final state. A program that does not terminate in a particular initial state is related to any final state. For more realistic languages than this one, it is often possible for programs to crash, in which case a semantics would generally relate their executions to no final states; so relating safely non-terminating programs to all final states provides a crucial distinction.

Having learned our lesson in the last section, before proceeding, we build a co-induction principle for evalCmd.

The proof is routine. We make use of a form of destruct that takes an intro pattern in an as clause. These patterns control how deeply we break apart the components of an inductive value, and we refer the reader to the Coq manual for more details.

Now that we have a co-induction principle, we should use it to prove something! Our example is a trivial program optimizer that finds places to replace 0 + e with e.

Before proving correctness of optCmd, we prove a lemma about optExp. This is where we have to do the most work, choosing pattern match opportunities automatically.

The final theorem is easy to establish, using our co-induction principle and a bit of Ltac smarts that we leave unexplained for now. Curious readers can consult the Coq manual, or wait for the later chapters of this book about proof automation. At a high level, we show inclusions between behaviors, going in both directions between original and optimized programs.

In this form, the theorem tells us that the optimizer preserves observable behavior of both terminating and nonterminating programs, but we did not have to do more work than for the case of terminating programs alone. We merely took the natural inductive definition for terminating executions, made it co-inductive, and applied the appropriate co-induction principle. Curious readers might experiment with adding command constructs like if; the same proof script should continue working, after the co-induction principle is extended to the new evaluation rules.