The apply tactic can also be used with ↔. When given an equivalence as its argument, apply tries to guess which side of the equivalence to use.

Existential Quantification

Another important logical connective is existential quantification. To say that there is some x of type T such that some property P holds of x, we write ∃x : T, P. As with ∀, the type annotation : T can be omitted if Coq is able to infer from the context what the type of x should be. To prove a statement of the form ∃x, P, we must show that P holds for some specific choice of value for x, known as the witness of the existential. This is done in two steps: First, we explicitly tell Coq which witness t we have in mind by invoking the tactic ∃t. Then we prove that P holds after all occurrences of x are replaced by t.

Conversely, if we have an existential hypothesis ∃x, P in the context, we can destruct it to obtain a witness x and a hypothesis stating that P holds of x.

Exercise: 1 star, recommended (dist_not_exists)

Prove that "P holds for all x" implies "there is no x for which P does not hold." (Hint: destruct H as [x E] works on existential assumptions!)

Programming with Propositions

The logical connectives that we have seen provide a rich vocabulary for defining complex propositions from simpler ones. To illustrate, let's look at how to express the claim that an element x occurs in a list l. Notice that this property has a simple recursive structure:

• If l is the empty list, then x cannot occur on it, so the property "x appears in l" is simply false.

• Otherwise, l has the form x' :: l'. In this case, x occurs in l if either it is equal to x' or it occurs in l'.

We can translate this directly into a straightforward recursive function taking an element and a list and returning a proposition:

When In is applied to a concrete list, it expands into a concrete sequence of nested disjunctions.

(Notice the use of the empty pattern to discharge the last case en passant.) We can also prove more generic, higher-level lemmas about In. Note, in the next, how In starts out applied to a variable and only gets expanded when we do case analysis on this variable:

This way of defining propositions recursively, though convenient in some cases, also has some drawbacks. In particular, it is subject to Coq's usual restrictions regarding the definition of recursive functions, e.g., the requirement that they be "obviously terminating." In the next chapter, we will see how to define propositions inductively, a different technique with its own set of strengths and limitations.

Exercise: 2 stars, recommended (In_map_iff)

To test your definition, prove the following facts:

Coq vs. Set Theory

Coq's logical core, the Calculus of Inductive Constructions, differs in some important ways from other formal systems that are used by mathematicians for writing down precise and rigorous proofs. For example, in the most popular foundation for mainstream paper-and-pencil mathematics, Zermelo-Fraenkel Set Theory (ZFC), a mathematical object can potentially be a member of many different sets; a term in Coq's logic, on the other hand, is a member of at most one type. This difference often leads to slightly different ways of capturing informal mathematical concepts, but these are, by and large, quite natural and easy to work with. For example, instead of saying that a natural number n belongs to the set of even numbers, we would say in Coq that ev n holds, where ev : nat → Prop is a property describing even numbers. However, there are some cases where translating standard mathematical reasoning into Coq can be either cumbersome or sometimes even impossible, unless we enrich the core logic with additional axioms. We conclude this chapter with a brief discussion of some of the most significant differences between the two worlds.

Functional Extensionality

In common mathematical practice, two functions f and g are considered equal if they produce the same outputs: This is known as the principle of functional extensionality. Informally speaking, an "extensional property" is one that pertains to an object's observable behavior. Thus, functional extensionality simply means that a function's identity is completely determined by what we can observe from it — i.e., in Coq terms, the results we obtain after applying it. Functional extensionality is not part of Coq's basic axioms. This means that some "reasonable" propositions are not provable.

However, we can add functional extensionality to Coq's core logic using the Axiom command.

Using Axiom has the same effect as stating a theorem and skipping its proof using Admitted, but it alerts the reader that this isn't just something we're going to come back and fill in later! We can now invoke functional extensionality in proofs:

Naturally, we must be careful when adding new axioms into Coq's logic, as they may render it inconsistent — that is, they may make it possible to prove every proposition, including False! Unfortunately, there is no simple way of telling whether an axiom is safe to add: hard work is generally required to establish the consistency of any particular combination of axioms. Fortunately, it is known that adding functional extensionality, in particular, is consistent. To check whether a particular proof relies on any additional axioms, use the Print Assumptions command.

Exercise: 4 stars, optional (tr_rev_correct)

One problem with the definition of the list-reversing function rev that we have is that it performs a call to app on each step; running app takes time asymptotically linear in the size of the list, which means that rev has quadratic running time. We can improve this with the following definition:

This version is said to be tail-recursive, because the recursive call to the function is the last operation that needs to be performed (i.e., we don't have to execute ++ after the recursive call); a decent compiler will generate very efficient code in this case. Prove that the two definitions are indeed equivalent.

Exercise: 3 stars (evenb_double_conv)

In view of this theorem, we say that the boolean computation evenb n is reflected in the truth of the proposition ∃k, n = double k. Similarly, to state that two numbers n and m are equal, we can say either (1) that n =? m returns true or (2) that n = m. Again, these two notions are equivalent.

However, even when the boolean and propositional formulations of a claim are equivalent from a purely logical perspective, they need not be equivalent operationally. Equality provides an extreme example: knowing that n =? m = true is generally of little direct help in the middle of a proof involving n and m; however, if we convert the statement to the equivalent form n = m, we can rewrite with it. The case of even numbers is also interesting. Recall that, when proving the backwards direction of even_bool_prop (i.e., evenb_double, going from the propositional to the boolean claim), we used a simple induction on k. On the other hand, the converse (the evenb_double_conv exercise) required a clever generalization, since we can't directly prove (∃k, n = double k) → evenb n = true. For these examples, the propositional claims are more useful than their boolean counterparts, but this is not always the case. For instance, we cannot test whether a general proposition is true or not in a function definition; as a consequence, the following code fragment is rejected:

Coq complains that n = 2 has type Prop, while it expects an elements of bool (or some other inductive type with two elements). It would be nice if Coq provided a mechanism for testing the truth value of arbitrary propositions but unfortunately computably theory and Godel's incompleteness theorems say that this is impossible. (Complexity theory would add that if we limited that mechanism to computable propositions, it would still take forever.) Although general non-computable properties cannot be phrased as boolean computations, it is worth noting that even many computable properties are easier to express using Prop than bool, since recursive function definitions are subject to significant restrictions in Coq. For instance, the next chapter shows how to define the property that a regular expression matches a given string using Prop. Doing the same with bool would amount to writing a regular expression matcher, which would be more complicated, harder to understand, and harder to reason about. Conversely, an important side benefit of stating facts using booleans is enabling some proof automation through computation with Coq terms, a technique known as proof by reflection. Consider the following statement:

The most direct proof of this fact is to give the value of k explicitly.

On the other hand, the proof of the corresponding boolean statement is even simpler:

What is interesting is that, since the two notions are equivalent, we can use the boolean formulation to prove the other one without mentioning the value 500 explicitly:

Although we haven't gained much in terms of proof size in this case, larger proofs can often be made considerably simpler by the use of reflection. As an extreme example, the Coq proof of the famous 4-color theorem uses reflection to reduce the analysis of hundreds of different cases to a boolean computation. We won't cover reflection in great detail, but it serves as a good example showing the complementary strengths of booleans and general propositions.

Exercise: 2 stars (logical_connectives)

The following lemmas relate the propositional connectives to the corresponding boolean operations.

Prove the theorem below, which relates forallb to the All property of the above exercise.

Are there any important properties of the function forallb which are not captured by this specification?

However, if we happen to know that P is reflected in some boolean term b, then knowing whether it holds or not is trivial: we just have to check the value of b.

In particular, the excluded middle is valid for equations n = m, between natural numbers n and m.

It may seem strange that the general excluded middle is not available by default in Coq; after all, any given claim must be either true or false. Nonetheless, there is an advantage in not assuming the excluded middle: statements in Coq can make stronger claims than the analogous statements in standard mathematics. Notably, if there is a Coq proof of ∃x, P x, it is possible to explicitly exhibit a value of x for which we can prove P x — in other words, every proof of existence is necessarily constructive. Logics like Coq's, which do not assume the excluded middle, are referred to as constructive logics. More conventional logical systems such as ZFC, in which the excluded middle does hold for arbitrary propositions, are referred to as classical. The following example illustrates why assuming the excluded middle may lead to non-constructive proofs: Claim: There exist irrational numbers a and b such that a ^ b is rational. Proof: It is not difficult to show that sqrt 2 is irrational. If sqrt 2 ^ sqrt 2 is rational, it suffices to take a = b = sqrt 2 and we are done. Otherwise, sqrt 2 ^ sqrt 2 is irrational. In this case, we can take a = sqrt 2 ^ sqrt 2 and b = sqrt 2, since a ^ b = sqrt 2 ^ (sqrt 2 * sqrt 2) = sqrt 2 ^ 2 = 2. ☐ Do you see what happened here? We used the excluded middle to consider separately the cases where sqrt 2 ^ sqrt 2 is rational and where it is not, without knowing which one actually holds! Because of that, we wind up knowing that such a and b exist but we cannot determine what their actual values are (at least, using this line of argument). As useful as constructive logic is, it does have its limitations: There are many statements that can easily be proven in classical logic but that have much more complicated constructive proofs, and there are some that are known to have no constructive proof at all! Fortunately, like functional extensionality, the excluded middle is known to be compatible with Coq's logic, allowing us to add it safely as an axiom. However, we will not need to do so in this book: the results that we cover can be developed entirely within constructive logic at negligible extra cost. It takes some practice to understand which proof techniques must be avoided in constructive reasoning, but arguments by contradiction, in particular, are infamous for leading to non-constructive proofs. Here's a typical example: suppose that we want to show that there exists x with some property P, i.e., such that P x. We start by assuming that our conclusion is false; that is, ¬ ∃x, P x. From this premise, it is not hard to derive ∀x, ¬ P x. If we manage to show that this intermediate fact results in a contradiction, we arrive at an existence proof without ever exhibiting a value of x for which P x holds! The technical flaw here, from a constructive standpoint, is that we claimed to prove ∃x, P x using a proof of ¬ ¬ (∃x, P x). Allowing ourselves to remove double negations from arbitrary statements is equivalent to assuming the excluded middle, as shown in one of the exercises below. Thus, this line of reasoning cannot be encoded in Coq without assuming additional axioms.

Exercise: 3 stars (excluded_middle_irrefutable)

Proving the consistency of Coq with the general excluded middle axiom requires complicated reasoning that cannot be carried out within Coq itself. However, the following theorem implies that it is always safe to assume a decidability axiom (i.e., an instance of excluded middle) for any particular Prop P. Why? Because we cannot prove the negation of such an axiom. If we could, we would have both ¬ (P ∨ ¬P) and ¬ ¬ (P ∨ ¬P) (since P implies ¬ ¬ P, by the exercise below), which would be a contradiction. But since we can't, it is safe to add P ∨ ¬P as an axiom.