We have seen that Coq has mechanisms both for programming, using inductive data types like nat or list and functions over these types, and for proving properties of these programs, using inductive propositions (like ev), implication, universal quantification, and the like. So far, we have mostly treated these mechanisms as if they were quite separate, and for many purposes this is a good way to think. But we have also seen hints that Coq's programming and proving facilities are closely related. For example, the keyword Inductive is used to declare both data types and propositions, and → is used both to describe the type of functions on data and logical implication. This is not just a syntactic accident! In fact, programs and proofs in Coq are almost the same thing. In this chapter we will study how this works. We have already seen the fundamental idea: provability in Coq is represented by concrete evidence. When we construct the proof of a basic proposition, we are actually building a tree of evidence, which can be thought of as a data structure. If the proposition is an implication like A → B, then its proof will be an evidence transformer: a recipe for converting evidence for A into evidence for B. So at a fundamental level, proofs are simply programs that manipulate evidence. Question: If evidence is data, what are propositions themselves? Answer: They are types! Look again at the formal definition of the ev property.

Suppose we introduce an alternative pronunciation of ":". Instead of "has type," we can say "is a proof of." For example, the second line in the definition of ev declares that ev_0 : ev 0. Instead of "ev_0 has type ev 0," we can say that "ev_0 is a proof of ev 0." This pun between types and propositions -- between : as "has type" and : as "is a proof of" or "is evidence for" -- is called the Curry-Howard correspondence. It proposes a deep connection between the world of logic and the world of computation:

                 propositions  ~  types
                 proofs        ~  programs

See [Wadler 2015] for a brief history and up-to-date exposition. Many useful insights follow from this connection. To begin with, it gives us a natural interpretation of the type of the ev_SS constructor:

This can be read "ev_SS is a constructor that takes two arguments -- a number n and evidence for the proposition ev n -- and yields evidence for the proposition ev (S (S n))." Now let's look again at a previous proof involving ev.

As with ordinary data values and functions, we can use the Print command to see the proof term that results from this proof script.

Indeed, we can also write down this proof term directly, without the need for a separate proof script:

The expression ev_SS 2 (ev_SS 0 ev_0) can be thought of as instantiating the parameterized constructor ev_SS with the specific arguments 2 and 0 plus the corresponding proof terms for its premises ev 2 and ev 0. Alternatively, we can think of ev_SS as a primitive "evidence constructor" that, when applied to a particular number, wants to be further applied to evidence that this number is even; its type,
      ∀ n, ev n → ev (S (S n)), expresses this functionality, in the same way that the polymorphic type ∀ X, list X expresses the fact that the constructor nil can be thought of as a function from types to empty lists with elements of that type. We saw in the Logic chapter that we can use function application syntax to instantiate universally quantified variables in lemmas, as well as to supply evidence for assumptions that these lemmas impose. For instance:

Proof Scripts

The proof terms we've been discussing lie at the core of how Coq operates. When Coq is following a proof script, what is happening internally is that it is gradually constructing a proof term -- a term whose type is the proposition being proved. The tactics between Proof and Qed tell it how to build up a term of the required type. To see this process in action, let's use the Show Proof command to display the current state of the proof tree at various points in the following tactic proof.

At any given moment, Coq has constructed a term with a "hole" (indicated by ?Goal here, and so on), and it knows what type of evidence is needed to fill this hole. Each hole corresponds to a subgoal, and the proof is finished when there are no more subgoals. At this point, the evidence we've built is stored in the global context under the name given in the Theorem command. Tactic proofs are useful and convenient, but they are not essential in Coq: in principle, we can always construct the required evidence by hand. Then we can use Definition (rather than Theorem) to give a global name directly to this evidence.

All these different ways of building the proof lead to exactly the same evidence being saved in the global environment.

Exercise: 2 stars, standard (eight_is_even)

Give a tactic proof and a proof term showing that ev 8.

Quantifiers, Implications, Functions

In Coq's computational universe (where data structures and programs live), there are two sorts of values that have arrows in their types: constructors introduced by Inductively defined data types, and functions. Similarly, in Coq's logical universe (where we carry out proofs), there are two ways of giving evidence for an implication: constructors introduced by Inductively defined propositions, and... functions! For example, consider this statement:

What is the proof term corresponding to ev_plus4? We're looking for an expression whose type is ∀ n, ev n → ev (4 + n) -- that is, a function that takes two arguments (one number and a piece of evidence) and returns a piece of evidence! Here it is:

Recall that fun n ⇒ blah means "the function that, given n, yields blah," and that Coq treats 4 + n and S (S (S (S n))) as synonyms. Another equivalent way to write this definition is:

When we view the proposition being proved by ev_plus4 as a function type, one interesting point becomes apparent: The second argument's type, ev n, mentions the value of the first argument, n. While such dependent types are not found in most mainstream programming languages, they can be quite useful in programming too, as the flurry of activity in the functional programming community over the past couple of decades demonstrates. Notice that both implication (→) and quantification (∀) correspond to functions on evidence. In fact, they are really the same thing: → is just a shorthand for a degenerate use of ∀ where there is no dependency, i.e., no need to give a name to the type on the left-hand side of the arrow:
           ∀ (x:nat), nat
        = ∀ (_:nat), nat
        = nat → nat For example, consider this proposition:

A proof term inhabiting this proposition would be a function with two arguments: a number n and some evidence E that n is even. But the name E for this evidence is not used in the rest of the statement of ev_plus2, so it's a bit silly to bother making up a name for it. We could write it like this instead, using the dummy identifier _ in place of a real name:

Or, equivalently, we can write it in a more familiar way:

In general, "P → Q" is just syntactic sugar for "∀ (_:P), Q".

Programming with Tactics

If we can build proofs by giving explicit terms rather than executing tactic scripts, you may be wondering whether we can build programs using tactics rather than by writing down explicit terms. Naturally, the answer is yes!

Notice that we terminate the Definition with a . rather than with := followed by a term. This tells Coq to enter proof scripting mode to build an term of type nat → nat. Also, we terminate the proof with Defined rather than Qed; this makes the definition transparent so that it can be used in computation like a normally-defined function. (Qed-defined terms are opaque during computation.) This feature is mainly useful for writing functions with dependent types, which we won't explore much further in this book. But it does illustrate the uniformity and orthogonality of the basic ideas in Coq.

Logical Connectives as Inductive Types

Inductive definitions are powerful enough to express most of the logical connectives we have seen so far. Indeed, only universal quantification (with implication as a special case) is built into Coq; all the others are defined inductively. We'll see how in this section.

Conjunction

To prove that P ∧ Q holds, we must present evidence for both P and Q. Thus, it makes sense to define a proof term for P ∧ Q as consisting of a pair of two proofs: one for P and another one for Q. This leads to the following definition.

Notice the similarity with the definition of the prod type, given in chapter Poly; the only difference is that prod takes Type arguments, whereas and takes Prop arguments.

This similarity should clarify why destruct and intros patterns can be used on a conjunctive hypothesis. Case analysis allows us to consider all possible ways in which P ∧ Q was proved -- here just one (the conj constructor).

Similarly, the split tactic actually works for any inductively defined proposition with exactly one constructor. In particular, it works for and:

This shows why the inductive definition of and can be manipulated by tactics as we've been doing. We can also use it to build proofs directly, using pattern-matching. For instance:

Exercise: 2 stars, standard (conj_fact)

Construct a proof term for the following proposition. The goal of these exercises is to teach you how to write proof terms by hand, which is also helpful for the exam. So submitting ugly proof terms produced by tactics such as destruct will definitely not earn you full points for such exercises.

Disjunction

The inductive definition of disjunction uses two constructors, one for each side of the disjunct:

This declaration explains the behavior of the destruct tactic on a disjunctive hypothesis, since the generated subgoals match the shape of the or_introl and or_intror constructors. Once again, we can also directly write proof terms for theorems involving or, without resorting to tactics.

Exercise: 2 stars, standard (or_commut')

Construct a proof term for the following proposition.

Existential Quantification

To give evidence for an existential quantifier, we package a witness x together with a proof that x satisfies the property P:

This probably needs a little unpacking. The core definition is for a type former ex that can be used to build propositions of the form ex P, where P itself is a function from witness values in the type A to propositions. The ex_intro constructor then offers a way of constructing evidence for ex P, given a witness x and a proof of P x. The notation in the standard library is a slight extension of the above, enabling syntactic forms such as ∃ x y, P x y. The more familiar form ∃ x, ev x desugars to an expression involving ex:

Here's how to define an explicit proof term involving ex:

Exercise: 2 stars, standard (ex_ev_Sn)

Construct a proof term for the following proposition.

True and False

The inductive definition of the True proposition is simple:

It has one constructor (so every proof of True is the same, so being given a proof of True is not informative.)

Exercise: 1 star, standard (p_implies_true)

Construct a proof term for the following proposition.

False is equally simple -- indeed, so simple it may look syntactically wrong at first glance!

That is, False is an inductive type with no constructors -- i.e., no way to build evidence for it. For example, there is no way to complete the following definition such that it succeeds (rather than fails).

But it is possible to destruct False by pattern matching. There can be no patterns that match it, since it has no constructors. So the pattern match also is so simple it may look syntactically wrong at first glance.

Since there are no branches to evaluate, the match expression can be considered to have any type we want, including 0 = 1. Indeed, it's impossible to ever cause the match to be evaluated, because we can never construct a value of type False to pass to the function.

Exercise: 1 star, standard (ex_falso_quodlibet')

Construct a proof term for the following proposition.

Equality

Even Coq's equality relation is not built in. We can define it ourselves:

The way to think about this definition (which is just a slight variant of the standard library's) is that, given a set X, it defines a family of propositions "x is equal to y," indexed by pairs of values (x and y) from X. There is just one way of constructing evidence for members of this family: applying the constructor eq_refl to a type X and a single value x : X, which yields evidence that x is equal to x. Other types of the form eq x y where x and y are not the same are thus uninhabited. We can use eq_refl to construct evidence that, for example, 2 = 2. Can we also use it to construct evidence that 1 + 1 = 2? Yes, we can. Indeed, it is the very same piece of evidence! The reason is that Coq treats as "the same" any two terms that are convertible according to a simple set of computation rules. These rules, which are similar to those used by Compute, include evaluation of function application, inlining of definitions, and simplification of matches.

The reflexivity tactic that we have used to prove equalities up to now is essentially just shorthand for apply eq_refl. In tactic-based proofs of equality, the conversion rules are normally hidden in uses of simpl (either explicit or implicit in other tactics such as reflexivity). But you can see them directly at work in the following explicit proof terms:

We can also pattern-match on an equality proof:

By pattern-matching against n₁ == n₂, we obtain a term n that is known to be convertible to both n₁ and n₂. The term eq_refl (S n) establishes (S n) == (S n). The first n can be converted to n₁, and the second to n₂, which yields (S n₁) == (S n₂). Coq handles all that conversion for us. A tactic-based proof runs into some difficulties if we try to use our usual repertoire of tactics, such as rewrite and reflexivity. Those work with *setoid* relations that Coq knows about, such as =, but not our ==. We could prove to Coq that == is a setoid, but a simpler way is to use destruct and apply instead.

Exercise: 2 stars, standard (eq_cons)

Construct the proof term for the following theorem. Use pattern matching on the equality hypotheses.

Exercise: 2 stars, standard (equality__leibniz_equality)

The inductive definition of equality implies Leibniz equality: what we mean when we say "x and y are equal" is that every property on P that is true of x is also true of y. Prove that.

Exercise: 2 stars, standard (equality__leibniz_equality_term)

Construct the proof term for the previous exercise. All it requires is anonymous functions and pattern-matching; the large proof term constructed by tactics in the previous exercise is needessly complicated. Hint: pattern-match as soon as possible.

Exercise: 3 stars, standard, optional (leibniz_equality__equality)

Show that, in fact, the inductive definition of equality is equivalent to Leibniz equality. Hint: the proof is quite short; about all you need to do is to invent a clever property P to instantiate the antecedent.

Inversion, Again

We've seen inversion used with both equality hypotheses and hypotheses about inductively defined propositions. Now that we've seen that these are actually the same thing, we're in a position to take a closer look at how inversion behaves. In general, the inversion tactic...

• takes a hypothesis H whose type P is inductively defined, and
• for each constructor C in P's definition,
  • generates a new subgoal in which we assume H was built with C,
  • adds the arguments (premises) of C to the context of the subgoal as extra hypotheses,
  • matches the conclusion (result type) of C against the current goal and calculates a set of equalities that must hold in order for C to be applicable,
  • adds these equalities to the context (and, for convenience, rewrites them in the goal), and
  • if the equalities are not satisfiable (e.g., they involve things like S n = O), immediately solves the subgoal.

Example: If we invert a hypothesis built with or, there are two constructors, so two subgoals get generated. The conclusion (result type) of the constructor (P ∨ Q) doesn't place any restrictions on the form of P or Q, so we don't get any extra equalities in the context of the subgoal. Example: If we invert a hypothesis built with and, there is only one constructor, so only one subgoal gets generated. Again, the conclusion (result type) of the constructor (P ∧ Q) doesn't place any restrictions on the form of P or Q, so we don't get any extra equalities in the context of the subgoal. The constructor does have two arguments, though, and these can be seen in the context in the subgoal. Example: If we invert a hypothesis built with eq, there is again only one constructor, so only one subgoal gets generated. Now, though, the form of the eq_refl constructor does give us some extra information: it tells us that the two arguments to eq must be the same! The inversion tactic adds this fact to the context.

Coq's Trusted Computing Base

One issue that arises with any automated proof assistant is "why trust it?": what if there is a bug in the implementation that renders all its reasoning suspect? While it is impossible to allay such concerns completely, the fact that Coq is based on the Curry-Howard correspondence gives it a strong foundation. Because propositions are just types and proofs are just terms, checking that an alleged proof of a proposition is valid just amounts to type-checking the term. Type checkers are relatively small and straightforward programs, so the "trusted computing base" for Coq -- the part of the code that we have to believe is operating correctly -- is small too. What must a typechecker do? Its primary job is to make sure that in each function application the expected and actual argument types match, that the arms of a match expression are constructor patterns belonging to the inductive type being matched over and all arms of the match return the same type, and so on. There are a few additional wrinkles: First, since Coq types can themselves be expressions, the checker must normalize these (by using the computation rules) before comparing them. Second, the checker must make sure that match expressions are exhaustive. That is, there must be an arm for every possible constructor. To see why, consider the following alleged proof term:

All the types here match correctly, but the match only considers one of the possible constructors for or. Coq's exhaustiveness check will reject this definition. Third, the checker must make sure that each recursive function terminates. It does this using a syntactic check to make sure that each recursive call is on a subexpression of the original argument. To see why this is essential, consider this alleged proof:

Recursive function infinite_loop purports to return a value of any type X that you would like. (The struct annotation on the function tells Coq that it recurses on argument n, not X.) Were Coq to allow infinite_loop, then falso would be definable, thus giving evidence for False. So Coq rejects infinite_loop. Note that the soundness of Coq depends only on the correctness of this typechecking engine, not on the tactic machinery. If there is a bug in a tactic implementation (and this certainly does happen!), that tactic might construct an invalid proof term. But when you type Qed, Coq checks the term for validity from scratch. Only theorems whose proofs pass the type-checker can be used in further proof developments.

More Exercises

Most of the following theorems were already proved with tactics in Logic. Now construct the proof terms for them directly.

Exercise: 2 stars, standard (and_assoc)

Exercise: 3 stars, standard (or_distributes_over_and)

Exercise: 3 stars, standard (negations)

Exercise: 2 stars, standard (currying)

Proof Irrelevance (Advanced, Optional)

In Logic we saw that functional extensionality could be added to Coq. A similar notion about propositions can also be defined (and added as an axiom, if desired):

Propositional extensionality asserts that if two propositions are equivalent -- i.e., each implies the other -- then they are in fact equal. The proof terms for the propositions might be syntactically different terms. But propositional extensionality overlooks that, just as functional extensionality overlooks the syntactic differences between functions.

Exercise: 1 star, advanced, optional (pe_implies_or_eq)

Prove the following consequence of propositional extensionality.

Exercise: 1 star, advanced, optional (pe_implies_true_eq)

Prove that if a proposition P is provable, then it is equal to True -- as a consequence of propositional extensionality.

Exercise: 3 stars, advanced, optional (pe_implies_pi)

Acknowledgment: this theorem and its proof technique are inspired by Gert Smolka's manuscript Modeling and Proving in Computational Type Theory Using the Coq Proof Assistant, 2021. Another, perhaps surprising, consequence of propositional extensionality is that it implies proof irrelevance, which asserts that all proof terms for a proposition are equal.

Prove that fact. Use pe_implies_true_eq to establish that the proposition P in proof_irrelevance is equal to True. Leverage that equality to establish that both proofs terms pf₁ and pf₂ must be just I.