The so-called "Curry-Howard correspondence" states a formal connection between functional programs and mathematical proofs. In the last chapter, we snuck in a first introduction to this subject in Coq. Witness the close similarity between the types unit and True from the standard library:

  Inductive unit : Set := tt : unit

  Inductive True : Prop := I : True

Recall that unit is the type with only one value, and True is the proposition that always holds. Despite this superficial difference between the two concepts, in both cases we can use the same inductive definition mechanism. The connection goes further than this. We see that we arrive at the definition of True by replacing unit by True, tt by I, and Set by Prop. The first two of these differences are superficial changes of names, while the third difference is the crucial one for separating programs from proofs. A term T of type Set is a type of programs, and a term of type T is a program. A term T of type Prop is a logical proposition, and its proofs are of type T. Chapter 12 goes into more detail about the theoretical differences between Prop and Set. For now, we will simply follow common intuitions about what a proof is. The type unit has one value, tt. The type True has one proof, I. Why distinguish between these two types? Many people who have read about Curry-Howard in an abstract context but who have not put it to use in proof engineering answer that the two types in fact should not be distinguished. There is a certain aesthetic appeal to this point of view, but I want to argue that it is best to treat Curry-Howard very loosely in practical proving. There are Coq-specific reasons for preferring the distinction, involving efficient compilation and avoidance of paradoxes in the presence of classical math, but I will argue that there is a more general principle that should lead us to avoid conflating programming and proving. The essence of the argument is roughly this: to an engineer, not all functions of type A -> B are created equal, but all proofs of a proposition P -> Q are. This idea is known as proof irrelevance, and its formalizations in logics prevent us from distinguishing between alternate proofs of the same proposition. Proof irrelevance is compatible with, but not derivable in, Gallina. Apart from this theoretical concern, I will argue that it is most effective to do engineering with Coq by employing different techniques for programs versus proofs. Most of this book is organized around that distinction, describing how to program, by applying standard functional programming techniques in the presence of dependent types; and how to prove, by writing custom Ltac decision procedures. With that perspective in mind, this chapter is sort of a mirror image of the last chapter, introducing how to define predicates with inductive definitions. We will point out similarities in places, but much of the effective Coq user's bag of tricks is disjoint for predicates versus "datatypes." This chapter is also a covert introduction to dependent types, which are the foundation on which interesting inductive predicates are built, though we will rely on tactics to build dependently typed proof terms for us for now. A future chapter introduces more manual application of dependent types.

Propositional Logic

Let us begin with a brief tour through the definitions of the connectives for propositional logic. We will work within a Coq section that provides us with a set of propositional variables. In Coq parlance, these are just variables of type Prop.

In Coq, the most basic propositional connective is implication, written ->, which we have already used in almost every proof. Rather than being defined inductively, implication is built into Coq as the function type constructor. We have also already seen the definition of True. For a demonstration of a lower-level way of establishing proofs of inductive predicates, we turn to this trivial theorem.

We may always use the apply tactic to take a proof step based on applying a particular constructor of the inductive predicate that we are trying to establish. Sometimes there is only one constructor that could possibly apply, in which case a shortcut is available:

There is also a predicate False, which is the Curry-Howard mirror image of Empty_set from the last chapter.

  Inductive False : Prop :=

We can conclude anything from False, doing case analysis on a proof of False in the same way we might do case analysis on, say, a natural number. Since there are no cases to consider, any such case analysis succeeds immediately in proving the goal.

In a consistent context, we can never build a proof of False. In inconsistent contexts that appear in the courses of proofs, it is usually easiest to proceed by demonstrating the inconsistency with an explicit proof of False.

At this point, we have an inconsistent hypothesis 2 + 2 = 5, so the specific conclusion is not important. We use the elimtype tactic. For a full description of it, see the Coq manual. For our purposes, we only need the variant elimtype False, which lets us replace any conclusion formula with False, because any fact follows from an inconsistent context.

  H : 2 + 2 = 5
  ============================
   False
 

For now, we will leave the details of this proof about arithmetic to crush.

A related notion to False is logical negation.

    not = fun A : Prop => A -> False
      : Prop -> Prop

We see that not is just shorthand for implication of False. We can use that fact explicitly in proofs. The syntax ~ P expands to not P.

  ============================
   2 + 2 = 5 -> False

We also have conjunction, which we introduced in the last chapter.

    Inductive and (A : Prop) (B : Prop) : Prop := conj : A -> B -> A /\ B

The interested reader can check that and has a Curry-Howard equivalent called prod, the type of pairs. However, it is generally most convenient to reason about conjunction using tactics. An explicit proof of commutativity of and illustrates the usual suspects for such tasks. The operator /\ is an infix shorthand for and.

We start by case analysis on the proof of P /\ Q.

  H : P
  H0 : Q
  ============================
   Q /\ P
   

Every proof of a conjunction provides proofs for both conjuncts, so we get a single subgoal reflecting that. We can proceed by splitting this subgoal into a case for each conjunct of Q /\ P.

2 subgoals
  
  H : P
  H0 : Q
  ============================
   Q

subgoal 2 is

   P
 

In each case, the conclusion is among our hypotheses, so the assumption tactic finishes the process.

Coq disjunction is called or and abbreviated with the infix operator \/.

  Inductive or (A : Prop) (B : Prop) : Prop :=
    or_introl : A -> A \/ B | or_intror : B -> A \/ B

We see that there are two ways to prove a disjunction: prove the first disjunct or prove the second. The Curry-Howard analogue of this is the Coq sum type. We can demonstrate the main tactics here with another proof of commutativity.

As in the proof for and, we begin with case analysis, though this time we are met by two cases instead of one.

2 subgoals
  
  H : P
  ============================
   Q \/ P

subgoal 2 is

 Q \/ P
 

We can see that, in the first subgoal, we want to prove the disjunction by proving its second disjunct. The right tactic telegraphs this intent.

The second subgoal has a symmetric proof.
1 subgoal
  
  H : Q
  ============================
   Q \/ P

It would be a shame to have to plod manually through all proofs about propositional logic. Luckily, there is no need. One of the most basic Coq automation tactics is tauto, which is a complete decision procedure for constructive propositional logic. (More on what "constructive" means in the next section.) We can use tauto to dispatch all of the purely propositional theorems we have proved so far.

Sometimes propositional reasoning forms important plumbing for the proof of a theorem, but we still need to apply some other smarts about, say, arithmetic. The tactic intuition is a generalization of tauto that proves everything it can using propositional reasoning. When some further facts must be established to finish the proof, intuition uses propositional laws to simplify them as far as possible. Consider this example, which uses the list concatenation operator ++ from the standard library.

A lot of the proof structure has been generated for us by intuition, but the final proof depends on a fact about lists. The remaining subgoal hints at what cleverness we need to inject.
  ls1 : list nat
  ls2 : list nat
  H0 : length ls1 + length ls2 = 6
  ============================
   length (ls1 ++ ls2) = 6 \/ length ls1 = length ls2
 

We can see that we need a theorem about lengths of concatenated lists, which we proved last chapter and is also in the standard library.

  ls1 : list nat
  ls2 : list nat
  H0 : length ls1 + length ls2 = 6
  ============================
   length ls1 + length ls2 = 6 \/ length ls1 = length ls2
 

Now the subgoal follows by purely propositional reasoning. That is, we could replace length ls1 + length ls2 = 6 with P and length ls1 = length ls2 with Q and arrive at a tautology of propositional logic.

The intuition tactic is one of the main bits of glue in the implementation of crush, so, with a little help, we can get a short automated proof of the theorem.

Ending the section here has the same effect as always. Each of our propositional theorems becomes universally quantified over the propositional variables that we used.

What Does It Mean to Be Constructive?

One potential point of confusion in the presentation so far is the distinction between bool and Prop. The datatype bool is built from two values true and false, while Prop is a more primitive type that includes among its members True and False. Why not collapse these two concepts into one, and why must there be more than two states of mathematical truth, True and False? The answer comes from the fact that Coq implements constructive or intuitionistic logic, in contrast to the classical logic that you may be more familiar with. In constructive logic, classical tautologies like ~ ~ P -> P and P \/ ~ P do not always hold. In general, we can only prove these tautologies when P is decidable, in the sense of computability theory. The Curry-Howard encoding that Coq uses for or allows us to extract either a proof of P or a proof of ~ P from any proof of P \/ ~ P. Since our proofs are just functional programs which we can run, a general law of the excluded middle would give us a decision procedure for the halting problem, where the instantiations of P would be formulas like "this particular Turing machine halts." A similar paradoxical situation would result if every proposition evaluated to either True or False. Evaluation in Coq is decidable, so we would be limiting ourselves to decidable propositions only. Hence the distinction between bool and Prop. Programs of type bool are computational by construction; we can always run them to determine their results. Many Props are undecidable, and so we can write more expressive formulas with Props than with bools, but the inevitable consequence is that we cannot simply "run a Prop to determine its truth." Constructive logic lets us define all of the logical connectives in an aesthetically appealing way, with orthogonal inductive definitions. That is, each connective is defined independently using a simple, shared mechanism. Constructivity also enables a trick called program extraction, where we write programs by phrasing them as theorems to be proved. Since our proofs are just functional programs, we can extract executable programs from our final proofs, which we could not do as naturally with classical proofs. We will see more about Coq's program extraction facility in a later chapter. However, I think it is worth interjecting another warning at this point, following up on the prior warning about taking the Curry-Howard correspondence too literally. It is possible to write programs by theorem-proving methods in Coq, but hardly anyone does it. It is almost always most useful to maintain the distinction between programs and proofs. If you write a program by proving a theorem, you are likely to run into algorithmic inefficiencies that you introduced in your proof to make it easier to prove. It is a shame to have to worry about such situations while proving tricky theorems, and it is a happy state of affairs that you almost certainly will not need to, with the ideal of extracting programs from proofs being confined mostly to theoretical studies.

First-Order Logic

The forall connective of first-order logic, which we have seen in many examples so far, is built into Coq. Getting ahead of ourselves a bit, we can see it as the dependent function type constructor. In fact, implication and universal quantification are just different syntactic shorthands for the same Coq mechanism. A formula P -> Q is equivalent to forall x : P, Q, where x does not appear in Q. That is, the "real" type of the implication says "for every proof of P, there exists a proof of Q." Existential quantification is defined in the standard library.

  Inductive ex (A : Type) (P : A -> Prop) : Prop :=
    ex_intro : forall x : A, P x -> ex P

(Note that here, as always, each forall quantifier has the largest possible scope, so that the type of ex_intro could also be written forall x : A, (P x -> ex P).) The family ex is parameterized by the type A that we quantify over, and by a predicate P over As. We prove an existential by exhibiting some x of type A, along with a proof of P x. As usual, there are tactics that save us from worrying about the low-level details most of the time. Here is an example of a theorem statement with existential quantification. We use the equality operator =, which, depending on the settings in which they learned logic, different people will say either is or is not part of first-order logic. For our purposes, it is.

We can start this proof with a tactic exists, which should not be confused with the formula constructor shorthand of the same name.

The conclusion is replaced with a version using the existential witness that we announced.
  ============================
   1 + 1 = 2

We can also use tactics to reason about existential hypotheses.

We start by case analysis on the proof of the existential fact.

  n : nat
  m : nat
  x : nat
  H : n + x = m
  ============================
   n <= m
 

The goal has been replaced by a form where there is a new free variable x, and where we have a new hypothesis that the body of the existential holds with x substituted for the old bound variable. From here, the proof is just about arithmetic and is easy to automate.

The tactic intuition has a first-order cousin called firstorder, which proves many formulas when only first-order reasoning is needed, and it tries to perform first-order simplifications in any case. First-order reasoning is much harder than propositional reasoning, so firstorder is much more likely than intuition to get stuck in a way that makes it run for long enough to be useless.

Predicates with Implicit Equality

We start our exploration of a more complicated class of predicates with a simple example: an alternative way of characterizing when a natural number is zero.

We can call isZero a judgment, in the sense often used in the semantics of programming languages. Judgments are typically defined in the style of natural deduction, where we write a number of inference rules with premises appearing above a solid line and a conclusion appearing below the line. In this example, the sole constructor IsZero of isZero can be thought of as the single inference rule for deducing isZero, with nothing above the line and isZero 0 below it. The proof of isZero_zero demonstrates how we can apply an inference rule. (Readers not familiar with formal semantics should not worry about not following this paragraph!) The definition of isZero differs in an important way from all of the other inductive definitions that we have seen in this and the previous chapter. Instead of writing just Set or Prop after the colon, here we write nat -> Prop. We saw examples of parameterized types like list, but there the parameters appeared with names before the colon. Every constructor of a parameterized inductive type must have a range type that uses the same parameter, whereas the form we use here enables us to choose different arguments to the type for different constructors. For instance, our definition isZero makes the predicate provable only when the argument is 0. We can see that the concept of equality is somehow implicit in the inductive definition mechanism. The way this is accomplished is similar to the way that logic variables are used in Prolog (but worry not if not familiar with Prolog), and it is a very powerful mechanism that forms a foundation for formalizing all of mathematics. In fact, though it is natural to think of inductive types as folding in the functionality of equality, in Coq, the true situation is reversed, with equality defined as just another inductive type!

  Inductive eq (A : Type) (x : A) : A -> Prop := eq_refl : x = x

Behind the scenes, uses of infix = are expanded to instances of eq. We see that eq has both a parameter x that is fixed and an extra unnamed argument of the same type. The type of eq allows us to state any equalities, even those that are provably false. However, examining the type of equality's sole constructor eq_refl, we see that we can only prove equality when its two arguments are syntactically equal. This definition turns out to capture all of the basic properties of equality, and the equality-manipulating tactics that we have seen so far, like reflexivity and rewrite, are implemented treating eq as just another inductive type with a well-chosen definition. Another way of stating that definition is: equality is defined as the least reflexive relation. Returning to the example of isZero, we can see how to work with hypotheses that use this predicate.

We want to proceed by cases on the proof of the assumption about isZero.

  n : nat
  ============================
   n + 0 = n
 

Since isZero has only one constructor, we are presented with only one subgoal. The argument m to isZero is replaced with that type's argument from the single constructor IsZero. From this point, the proof is trivial.

Another example seems at first like it should admit an analogous proof, but in fact provides a demonstration of one of the most basic gotchas of Coq proving.

Let us try a proof by cases on the assumption, as in the last proof.

  ============================
   False
 

It seems that case analysis has not helped us much at all! Our sole hypothesis disappears, leaving us, if anything, worse off than we were before. What went wrong? We have met an important restriction in tactics like destruct and induction when applied to types with arguments. If the arguments are not already free variables, they will be replaced by new free variables internally before doing the case analysis or induction. Since the argument 1 to isZero is replaced by a fresh variable, we lose the crucial fact that it is not equal to 0. Why does Coq use this restriction? We will discuss the issue in detail in a future chapter, when we see the dependently typed programming techniques that would allow us to write this proof term manually. For now, we just say that the algorithmic problem of "logically complete case analysis" is undecidable when phrased in Coq's logic. A few tactics and design patterns that we will present in this chapter suffice in almost all cases. For the current example, what we want is a tactic called inversion, which corresponds to the concept of inversion that is frequently used with natural deduction proof systems. (Again, worry not if the semantics-oriented terminology from this last sentence is unfamiliar.)

What does inversion do? Think of it as a version of destruct that does its best to take advantage of the structure of arguments to inductive types. In this case, inversion completed the proof immediately, because it was able to detect that we were using isZero with an impossible argument. Sometimes using destruct when you should have used inversion can lead to confusing results. To illustrate, consider an alternate proof attempt for the last theorem.

  ============================
   1 + 1 = 4
 

What on earth happened here? Internally, destruct replaced 1 with a fresh variable, and, trying to be helpful, it also replaced the occurrence of 1 within the unary representation of each number in the goal. Then, within the O case of the proof, we replace the fresh variable with O. This has the net effect of decrementing each of these numbers.

To see more clearly what is happening, we can consider the type of isZero's induction principle.

isZero_ind
     : forall P : nat -> Prop, P 0 -> forall n : nat, isZero n -> P n

In our last proof script, destruct chose to instantiate P as fun n => S n + S n = S (S (S (S n))). You can verify for yourself that this specialization of the principle applies to the goal and that the hypothesis P 0 then matches the subgoal we saw generated. If you are doing a proof and encounter a strange transmutation like this, there is a good chance that you should go back and replace a use of destruct with inversion.

Recursive Predicates

We have already seen all of the ingredients we need to build interesting recursive predicates, like this predicate capturing even-ness.

Think of even as another judgment defined by natural deduction rules. The rule EvenO has nothing above the line and even O below the line, and EvenSS is a rule with even n above the line and even (S (S n)) below. The proof techniques of the last section are easily adapted.

It is not hard to see that sequences of constructor applications like the above can get tedious. We can avoid them using Coq's hint facility, with a new Hint variant that asks to consider all constructors of an inductive type during proof search. The tactic auto performs exhaustive proof search up to a fixed depth, considering only the proof steps we have registered as hints.

We may also use inversion with even.

  H : even 3
  n : nat
  H1 : even 1
  H0 : n = 1
  ============================
   False
 

The inversion tactic can be a little overzealous at times, as we can see here with the introduction of the unused variable n and an equality hypothesis about it. For more complicated predicates, though, adding such assumptions is critical to dealing with the undecidability of general inversion. More complex inductive definitions and theorems can cause inversion to generate equalities where neither side is a variable.

We can also do inductive proofs about even.

It seems a reasonable first choice to proceed by induction on n.

  n : nat
  IHn : forall m : nat, even n -> even m -> even (n + m)
  m : nat
  H : even (S n)
  H0 : even m
  ============================
   even (S (n + m))
 

We will need to use the hypotheses H and H0 somehow. The most natural choice is to invert H.

  n : nat
  IHn : forall m : nat, even n -> even m -> even (n + m)
  m : nat
  H : even (S n)
  H0 : even m
  n0 : nat
  H2 : even n0
  H1 : S n0 = n
  ============================
   even (S (S n0 + m))
 

Simplifying the conclusion brings us to a point where we can apply a constructor.

  ============================
   even (S (S (n0 + m)))

  ============================
   even (n0 + m)
 

At this point, we would like to apply the inductive hypothesis, which is:
  IHn : forall m : nat, even n -> even m -> even (n + m)
 

Unfortunately, the goal mentions n0 where it would need to mention n to match IHn. We could keep looking for a way to finish this proof from here, but it turns out that we can make our lives much easier by changing our basic strategy. Instead of inducting on the structure of n, we should induct on the structure of one of the even proofs. This technique is commonly called rule induction in programming language semantics. In the setting of Coq, we have already seen how predicates are defined using the same inductive type mechanism as datatypes, so the fundamental unity of rule induction with "normal" induction is apparent. Recall that tactics like induction and destruct may be passed numbers to refer to unnamed lefthand sides of implications in the conclusion, where the argument n refers to the nth such hypothesis.

  m : nat
  ============================
   even m -> even (0 + m)

subgoal 2 is
 even m -> even (S (S n) + m)
 

The first case is easily discharged by crush, based on the hint we added earlier to try the constructors of even.

Now we focus on the second case:

  m : nat
  n : nat
  H : even n
  IHeven : even m -> even (n + m)
  H0 : even m
  ============================
   even (S (S n) + m)
 

We simplify and apply a constructor, as in our last proof attempt.

  ============================
   even (n + m)
 

Now we have an exact match with our inductive hypothesis, and the remainder of the proof is trivial.

In fact, crush can handle all of the details of the proof once we declare the induction strategy.

Induction on recursive predicates has similar pitfalls to those we encountered with inversion in the last section.

  n : nat
  ============================
   False

subgoal 2 is
 False
 

We are already sunk trying to prove the first subgoal, since the argument to even was replaced by a fresh variable internally. This time, we find it easier to prove this theorem by way of a lemma. Instead of trusting induction to replace expressions with fresh variables, we do it ourselves, explicitly adding the appropriate equalities as new assumptions.

At this point, it is useful to consider all cases of n and n0 being zero or nonzero. Only one of these cases has any trickiness to it.

  n : nat
  H : even (S n)
  IHeven : forall n0 : nat, S n = S (n0 + n0) -> False
  n0 : nat
  H0 : S n = n0 + S n0
  ============================
   False
 

At this point it is useful to use a theorem from the standard library, which we also proved with a different name in the last chapter. We can search for a theorem that allows us to rewrite terms of the form x + S y.

  plus_n_Sm : forall n m : nat, S (n + m) = n + S m

The induction hypothesis lets us complete the proof, if we use a variant of apply that has a with clause to give instantiations of quantified variables.

As usual, we can rewrite the proof to avoid referencing any locally generated names, which makes our proof script more readable and more robust to changes in the theorem statement. We use the notation <- to request a hint that does right-to-left rewriting, just like we can with the rewrite tactic.

We write the proof in a way that avoids the use of local variable or hypothesis names, using the match tactic form to do pattern-matching on the goal. We use unification variables prefixed by question marks in the pattern, and we take advantage of the possibility to mention a unification variable twice in one pattern, to enforce equality between occurrences. The hint to rewrite with plus_n_Sm in a particular direction saves us from having to figure out the right place to apply that theorem. The original theorem now follows trivially from our lemma, using a new tactic eauto, a fancier version of auto whose explanation we postpone to Chapter 13.

We use a variant eapply of apply which has the same relationship to apply as eauto has to auto. An invocation of apply only succeeds if all arguments to the rule being used can be determined from the form of the goal, whereas eapply will introduce unification variables for undetermined arguments. In this case, eauto is able to determine the right values for those unification variables, using (unsurprisingly) a variant of the classic algorithm for unification . By considering an alternate attempt at proving the lemma, we can see another common pitfall of inductive proofs in Coq. Imagine that we had tried to prove even_contra' with all of the forall quantifiers moved to the front of the lemma statement.

One subgoal remains:
  n : nat
  H : even (S (n + n))
  IHeven : S (n + n) = S (S (S (n + n))) -> False
  ============================
   False
 

We are out of luck here. The inductive hypothesis is trivially true, since its assumption is false. In the version of this proof that succeeded, IHeven had an explicit quantification over n. This is because the quantification of n appeared after the thing we are inducting on in the theorem statement. In general, quantified variables and hypotheses that appear before the induction object in the theorem statement stay fixed throughout the inductive proof. Variables and hypotheses that are quantified after the induction object may be varied explicitly in uses of inductive hypotheses.

Why should Coq implement induction this way? One answer is that it avoids burdening this basic tactic with additional heuristic smarts, but that is not the whole picture. Imagine that induction analyzed dependencies among variables and reordered quantifiers to preserve as much freedom as possible in later uses of inductive hypotheses. This could make the inductive hypotheses more complex, which could in turn cause particular automation machinery to fail when it would have succeeded before. In general, we want to avoid quantifiers in our proofs whenever we can, and that goal is furthered by the refactoring that the induction tactic forces us to do.