This chapter introduces several additional proof strategies and tactics that allow us to begin proving more interesting properties of functional programs. We will see:

• how to use auxiliary lemmas in both "forward-" and "backward-style" proofs;
• how to reason about data constructors -- in particular, how to use the fact that they are injective and disjoint;
• how to strengthen an induction hypothesis, and when such strengthening is required; and
• more details on how to reason by case analysis.

The apply Tactic

We often encounter situations where the goal to be proved is exactly the same as some hypothesis in the context or some previously proved lemma.

Here, we could finish with "rewrite → eq. reflexivity." as we have done several times before. Alternatively, we can finish in a single step by using the apply tactic:

The apply tactic also works with conditional hypotheses and lemmas: if the statement being applied is an implication, then the premises of this implication will be added to the list of subgoals needing to be proved.

Typically, when we use apply H, the statement H will begin with a ∀ that introduces some universally quantified variables. When Coq matches the current goal against the conclusion of H, it will try to find appropriate values for these variables. For example, when we do apply eq₂ in the following proof, the universal variable q in eq₂ gets instantiated with n, and r gets instantiated with m.

Exercise: 2 stars, standard, optional (silly_ex)

Complete the following proof using only intros and apply.

To use the apply tactic, the (conclusion of the) fact being applied must match the goal exactly -- for example, apply will not work if the left and right sides of the equality are swapped.

Here we cannot use apply directly...

but we can use the symmetry tactic, which switches the left and right sides of an equality in the goal.

Exercise: 3 stars, standard (apply_exercise1)

Hint: You can also use apply with previously defined lemmas, not just hypotheses in the context. You may find earlier lemmas like app_nil_r, app_assoc, rev_app_distr, rev_involutive, etc. helpful. Also, remember that Search is your friend (though it may not find earlier lemmas if they were posed as optional problems and you chose not to finish the proofs).

Exercise: 1 star, standard, optional (apply_rewrite)

Briefly explain the difference between the tactics apply and rewrite. What are the situations where both can usefully be applied?

The apply with Tactic

The following silly example uses two rewrites in a row to get from [a;b] to [e;f].

Since this is a common pattern, we might like to pull it out as a lemma that records, once and for all, the fact that equality is transitive.

Now, we should be able to use trans_eq to prove the above example. However, to do this we need a slight refinement of the apply tactic.

If we simply tell Coq apply trans_eq at this point, it can tell (by matching the goal against the conclusion of the lemma) that it should instantiate X with [nat], n with [a,b], and o with [e,f]. However, the matching process doesn't determine an instantiation for m: we have to supply one explicitly by adding "with (m:=[c,d])" to the invocation of apply.

Actually, the name m in the with clause is not required, since Coq is often smart enough to figure out which variable we are instantiating. We could instead simply write apply trans_eq with [c;d]. Coq also has a built-in tactic transitivity that accomplishes the same purpose as applying trans_eq. The tactic requires us to state the instantiation we want, just like apply with does.

Exercise: 3 stars, standard, optional (trans_eq_exercise)

The injection and discriminate Tactics

Recall the definition of natural numbers:
     Inductive nat : Type :=
       | O
       | S (n : nat).

It is obvious from this definition that every number has one of two forms: either it is the constructor O or it is built by applying the constructor S to another number. But there is more here than meets the eye: implicit in the definition are two additional facts:

• The constructor S is injective (or one-to-one). That is, if S n = S m, it must be that n = m.
• The constructors O and S are disjoint. That is, O is not equal to S n for any n.

Similar principles apply to every inductively defined type: all constructors are injective, and the values built from distinct constructors are never equal. For lists, the cons constructor is injective and nil is different from every non-empty list. For booleans, true and false are different. (Since true and false take no arguments, their injectivity is neither here nor there.) And so on. We can prove the injectivity of S by using the pred function defined in Basics.v.

This technique can be generalized to any constructor by writing the equivalent of pred -- i.e., writing a function that "undoes" one application of the constructor. As a more convenient alternative, Coq provides a tactic called injection that allows us to exploit the injectivity of any constructor. Here is an alternate proof of the above theorem using injection:

By writing injection H as Hmn at this point, we are asking Coq to generate all equations that it can infer from H using the injectivity of constructors (in the present example, the equation n = m). Each such equation is added as a hypothesis (with the name Hmn in this case) into the context.

Here's a more interesting example that shows how injection can derive multiple equations at once.

Alternatively, if you just say injection H with no as clause, then all the equations will be turned into hypotheses with names chosen by Coq.

Exercise: 3 stars, standard (injection_ex₃)

So much for injectivity of constructors. What about disjointness? The principle of disjointness says that two terms beginning with different constructors (like O and S, or true and false) can never be equal. This means that, any time we find ourselves in a context where we've assumed that two such terms are equal, we are justified in concluding anything we want, since the assumption is nonsensical. The discriminate tactic embodies this principle: It is used on a hypothesis involving an equality between different constructors (e.g., false = true), and it solves the current goal immediately. Some examples:

These examples are instances of a logical principle known as the principle of explosion, which asserts that a contradictory hypothesis entails anything (even manifestly false things!). If you find the principle of explosion confusing, remember that these proofs are not showing that the conclusion of the statement holds. Rather, they are showing that, if the nonsensical situation described by the premise did somehow arise, then the nonsensical conclusion would also follow, because we'd be living in an inconsistent universe where every statement is true. We'll explore the principle of explosion in more detail in the next chapter.

Exercise: 1 star, standard (discriminate_ex₃)

For a slightly more involved example, we can use discriminate to make a connection between the two different notions of equality (= and =?) on natural numbers.

We can proceed by case analysis on n. The first case is trivial.

However, the second one doesn't look so simple: assuming 0 =? (S n') = true, we must show S n' = 0! The way forward is to observe that the assumption itself is nonsensical:

If we use discriminate on this hypothesis, Coq confirms that the subgoal we are working on is impossible and removes it from further consideration.

The injectivity of constructors allows us to reason that ∀ (n m : nat), S n = S m → n = m. The converse of this implication is an instance of a more general fact about both constructors and functions, which we will find convenient in a few places below:

There is also a tactic named `f_equal` that can prove such theorems. Given a goal of the form f a₁ ... an = g b₁ ... bn, the tactic f_equal will produce subgoals of the form f = g, a₁ = b₁, ..., an = bn. At the same time, any of these subgoals that are simple enough (e.g., immediately provable by reflexivity) will be automatically discharged by f_equal.

Using Tactics on Hypotheses

By default, most tactics work on the goal formula and leave the context unchanged. However, most tactics also have a variant that performs a similar operation on a statement in the context. For example, the tactic "simpl in H" performs simplification on the hypothesis H in the context.

Similarly, apply L in H matches some conditional statement L (of the form X → Y, say) against a hypothesis H in the context. However, unlike ordinary apply (which rewrites a goal matching Y into a subgoal X), apply L in H matches H against X and, if successful, replaces it with Y. In other words, apply L in H gives us a form of "forward reasoning": given X → Y and a hypothesis matching X, it produces a hypothesis matching Y. By contrast, apply L is "backward reasoning": it says that if we know X → Y and we are trying to prove Y, it suffices to prove X. Here is a variant of a proof from above, using forward reasoning throughout instead of backward reasoning.

Forward reasoning starts from what is given (premises, previously proven theorems) and iteratively draws conclusions from them until the goal is reached. Backward reasoning starts from the goal and iteratively reasons about what would imply the goal, until premises or previously proven theorems are reached. The informal proofs that you've seen in math or computer science classes probably tended to use forward reasoning. In general, idiomatic use of Coq favors backward reasoning, but in some situations the forward style can be easier to think about.

Varying the Induction Hypothesis

Sometimes it is important to control the exact form of the induction hypothesis when carrying out inductive proofs in Coq. In particular, we sometimes need to be careful about which of the assumptions we move (using intros) from the goal to the context before invoking the induction tactic. For example, suppose we want to show that double is injective -- i.e., that it maps different arguments to different results:
       Theorem double_injective: ∀ n m,
         double n = double m → n = m.

The way we start this proof is a bit delicate: if we begin it with
       intros n. induction n.

all is well. But if we begin it with introducing both variables
       intros n m. induction n.

we get stuck in the middle of the inductive case...

At this point, the induction hypothesis (IHn') does not give us n' = m' -- there is an extra S in the way -- so the goal is not provable.

What went wrong? The problem is that, at the point we invoke the induction hypothesis, we have already introduced m into the context -- intuitively, we have told Coq, "Let's consider some particular n and m..." and we now have to prove that, if double n = double m for these particular n and m, then n = m. The next tactic, induction n says to Coq: We are going to show the goal by induction on n. That is, we are going to prove, for all n, that the proposition

• P n = "if double n = double m, then n = m"

holds, by showing

• P O
  (i.e., "if double O = double m then O = m") and
• P n → P (S n)
  (i.e., "if double n = double m then n = m" implies "if double (S n) = double m then S n = m").

If we look closely at the second statement, it is saying something rather strange: that, for a particular m, if we know

• "if double n = double m then n = m"

then we can prove

• "if double (S n) = double m then S n = m".

To see why this is strange, let's think of a particular (arbitrary, but fixed) m -- say, 5. The statement is then saying that, if we know

• Q = "if double n = 10 then n = 5"

then we can prove

• R = "if double (S n) = 10 then S n = 5".

But knowing Q doesn't give us any help at all with proving R! If we tried to prove R from Q, we would start with something like "Suppose double (S n) = 10..." but then we'd be stuck: knowing that double (S n) is 10 tells us nothing helpful about whether double n is 10 (indeed, it strongly suggests that double n is not 10!!), so Q is useless. Trying to carry out this proof by induction on n when m is already in the context doesn't work because we are then trying to prove a statement involving every n but just a single m. A successful proof of double_injective leaves m universally quantified in the goal statement at the point where the induction tactic is invoked on n:

Notice that both the goal and the induction hypothesis are different this time: the goal asks us to prove something more general (i.e., we must prove the statement for every m), but the IH is correspondingly more flexible, allowing us to choose any m we like when we apply the IH.

Now we've chosen a particular m and introduced the assumption that double n = double m. Since we are doing a case analysis on n, we also need a case analysis on m to keep the two "in sync."

The 0 case is trivial:

At this point, since we are in the second branch of the destruct m, the m' mentioned in the context is the predecessor of the m we started out talking about. Since we are also in the S branch of the induction, this is perfect: if we instantiate the generic m in the IH with the current m' (this instantiation is performed automatically by the apply in the next step), then IHn' gives us exactly what we need to finish the proof.

What you should take away from all this is that you need to be careful, when using induction, that you are not trying to prove something too specific: When proving a property involving two variables n and m by induction on n, it is sometimes crucial to leave m generic. The following exercise (which further strengthens the link between =? and =) follows the same pattern.

Exercise: 2 stars, standard (eqb_true)

Exercise: 2 stars, advanced (eqb_true_informal)

Give a careful informal proof of eqb_true, being as explicit as possible about quantifiers.

Exercise: 3 stars, standard, especially useful (plus_n_n_injective)

In addition to being careful about how you use intros, practice using "in" variants in this proof. (Hint: use plus_n_Sm.)

The strategy of doing fewer intros before an induction to obtain a more general IH doesn't always work by itself; sometimes some rearrangement of quantified variables is needed. Suppose, for example, that we wanted to prove double_injective by induction on m instead of n.

The problem is that, to do induction on m, we must first introduce n. (And if we simply say induction m without introducing anything first, Coq will automatically introduce n for us!) What can we do about this? One possibility is to rewrite the statement of the lemma so that m is quantified before n. This works, but it's not nice: We don't want to have to twist the statements of lemmas to fit the needs of a particular strategy for proving them! Rather we want to state them in the clearest and most natural way. What we can do instead is to first introduce all the quantified variables and then re-generalize one or more of them, selectively taking variables out of the context and putting them back at the beginning of the goal. The generalize dependent tactic does this.

Let's look at an informal proof of this theorem. Note that the proposition we prove by induction leaves n quantified, corresponding to the use of generalize dependent in our formal proof. Theorem: For any nats n and m, if double n = double m, then n = m. Proof: Let m be a nat. We prove by induction on m that, for any n, if double n = double m then n = m.

• First, suppose m = 0, and suppose n is a number such that double n = double m. We must show that n = 0.
  Since m = 0, by the definition of double we have double n = 0. There are two cases to consider for n. If n = 0 we are done, since m = 0 = n, as required. Otherwise, if n = S n' for some n', we derive a contradiction: by the definition of double, we can calculate double n = S (S (double n')), but this contradicts the assumption that double n = 0.
• Second, suppose m = S m' and that n is again a number such that double n = double m. We must show that n = S m', with the induction hypothesis that for every number s, if double s = double m' then s = m'.
  By the fact that m = S m' and the definition of double, we have double n = S (S (double m')). There are two cases to consider for n.
  If n = 0, then by definition double n = 0, a contradiction.
  Thus, we may assume that n = S n' for some n', and again by the definition of double we have S (S (double n')) = S (S (double m')), which implies by injectivity that double n' = double m'. Instantiating the induction hypothesis with n' thus allows us to conclude that n' = m', and it follows immediately that S n' = S m'. Since S n' = n and S m' = m, this is just what we wanted to show. ☐

Exercise: 3 stars, standard, especially useful (gen_dep_practice)

Prove this by induction on l.

Unfolding Definitions

It sometimes happens that we need to manually unfold a name that has been introduced by a Definition so that we can manipulate the expression it denotes. For example, if we define...

... and try to prove a simple fact about square...

... we appear to be stuck: simpl doesn't simplify anything, and since we haven't proved any other facts about square, there is nothing we can apply or rewrite with. To make progress, we can manually unfold the definition of square:

Now we have plenty to work with: both sides of the equality are expressions involving multiplication, and we have lots of facts about multiplication at our disposal. In particular, we know that it is commutative and associative, and from these it is not hard to finish the proof.

At this point, some deeper discussion of unfolding and simplification is in order. We already have observed that tactics like simpl, reflexivity, and apply will often unfold the definitions of functions automatically when this allows them to make progress. For example, if we define foo m to be the constant 5...

.... then the simpl in the following proof (or the reflexivity, if we omit the simpl) will unfold foo m to (fun x ⇒ 5) m and then further simplify this expression to just 5.

But this automatic unfolding is somewhat conservative. For example, if we define a slightly more complicated function involving a pattern match...

...then the analogous proof will get stuck:

The reason that simpl doesn't make progress here is that it notices that, after tentatively unfolding bar m, it is left with a match whose scrutinee, m, is a variable, so the match cannot be simplified further. It is not smart enough to notice that the two branches of the match are identical, so it gives up on unfolding bar m and leaves it alone. Similarly, tentatively unfolding bar (m+1) leaves a match whose scrutinee is a function application (that cannot itself be simplified, even after unfolding the definition of +), so simpl leaves it alone. At this point, there are two ways to make progress. One is to use destruct m to break the proof into two cases, each focusing on a more concrete choice of m (O vs S _). In each case, the match inside of bar can now make progress, and the proof is easy to complete.

This approach works, but it depends on our recognizing that the match hidden inside bar is what was preventing us from making progress. A more straightforward way forward is to explicitly tell Coq to unfold bar.

Now it is apparent that we are stuck on the match expressions on both sides of the =, and we can use destruct to finish the proof without thinking too hard.

Using destruct on Compound Expressions

We have seen many examples where destruct is used to perform case analysis of the value of some variable. Sometimes we need to reason by cases on the result of some expression. We can also do this with destruct. Here are some examples:

After unfolding sillyfun in the above proof, we find that we are stuck on if (n =? 3) then ... else .... But either n is equal to 3 or it isn't, so we can use destruct (eqb n 3) to let us reason about the two cases. In general, the destruct tactic can be used to perform case analysis of the results of arbitrary computations. If e is an expression whose type is some inductively defined type T, then, for each constructor c of T, destruct e generates a subgoal in which all occurrences of e (in the goal and in the context) are replaced by c.

Exercise: 3 stars, standard (combine_split)

Here is an implementation of the split function mentioned in chapter Poly:

Prove that split and combine are inverses in the following sense:

The eqn: part of the destruct tactic is optional; though we've chosen to include it most of the time, for the sake of documentation, but it can often be omitted without harm. However, when destructing compound expressions, the information recorded by the eqn: can actually be critical: if we leave it out, then destruct can erase information we need to complete a proof. For example, suppose we define a function sillyfun1 like this:

Now suppose that we want to convince Coq that sillyfun1 n yields true only when n is odd. If we start the proof like this (with no eqn: on the destruct)...

... then we are stuck at this point because the context does not contain enough information to prove the goal! The problem is that the substitution performed by destruct is quite brutal -- in this case, it throws away every occurrence of n =? 3, but we need to keep some memory of this expression and how it was destructed, because we need to be able to reason that, since n =? 3 = true in this branch of the case analysis, it must be that n = 3, from which it follows that n is odd. What we want here is to substitute away all existing occurences of n =? 3, but at the same time add an equation to the context that records which case we are in. This is precisely what the eqn: qualifier does.

Now we have the same state as at the point where we got stuck above, except that the context contains an extra equality assumption, which is exactly what we need to make progress.

When we come to the second equality test in the body of the function we are reasoning about, we can use eqn: again in the same way, allowing us to finish the proof.

Exercise: 2 stars, standard (destruct_eqn_practice)

Review

We've now seen many of Coq's most fundamental tactics. We'll introduce a few more in the coming chapters, and later on we'll see some more powerful automation tactics that make Coq help us with low-level details. But basically we've got what we need to get work done. Here are the ones we've seen:

• intros: move hypotheses/variables from goal to context
• reflexivity: finish the proof (when the goal looks like e = e)
• apply: prove goal using a hypothesis, lemma, or constructor
• apply... in H: apply a hypothesis, lemma, or constructor to a hypothesis in the context (forward reasoning)
• apply... with...: explicitly specify values for variables that cannot be determined by pattern matching
• simpl: simplify computations in the goal
• simpl in H: ... or a hypothesis
• rewrite: use an equality hypothesis (or lemma) to rewrite the goal
• rewrite ... in H: ... or a hypothesis
• symmetry: changes a goal of the form t=u into u=t
• symmetry in H: changes a hypothesis of the form t=u into u=t
• transitivity y: prove a goal x=z by proving two new subgoals, x=y and y=z
• unfold: replace a defined constant by its right-hand side in the goal
• unfold... in H: ... or a hypothesis
• destruct... as...: case analysis on values of inductively defined types
• destruct... eqn:...: specify the name of an equation to be added to the context, recording the result of the case analysis
• induction... as...: induction on values of inductively defined types
• injection: reason by injectivity on equalities between values of inductively defined types
• discriminate: reason by disjointness of constructors on equalities between values of inductively defined types
• assert (H: e) (or assert (e) as H): introduce a "local lemma" e and call it H
• generalize dependent x: move the variable x (and anything else that depends on it) from the context back to an explicit hypothesis in the goal formula
• f_equal: change a goal of the form f x = f y into x = y

Additional Exercises

Exercise: 3 stars, standard (eqb_sym)

Exercise: 3 stars, advanced, optional (eqb_sym_informal)

Give an informal proof of this lemma that corresponds to your formal proof above: Theorem: For any nats n m, (n =? m) = (m =? n). Proof:

Exercise: 3 stars, standard, optional (eqb_trans)

Exercise: 3 stars, advanced (split_combine)

We proved, in an exercise above, that for all lists of pairs, combine is the inverse of split. How would you formalize the statement that split is the inverse of combine? When is this property true? Complete the definition of split_combine_statement below with a property that states that split is the inverse of combine. Then, prove that the property holds. (Be sure to leave your induction hypothesis general by not doing intros on more things than necessary. Hint: what property do you need of l₁ and l₂ for split (combine l₁ l₂) = (l₁,l₂) to be true?)

Exercise: 3 stars, advanced (filter_exercise)

This one is a bit challenging. Pay attention to the form of your induction hypothesis.

Exercise: 4 stars, advanced, especially useful (forall_exists_challenge)

Define two recursive Fixpoints, forallb and existsb. The first checks whether every element in a list satisfies a given predicate:
      forallb oddb [1;3;5;7;9] = true

      forallb negb [false;false] = true

      forallb evenb [0;2;4;5] = false

      forallb (eqb 5) [] = true

The second checks whether there exists an element in the list that satisfies a given predicate:
      existsb (eqb 5) [0;2;3;6] = false

      existsb (andb true) [true;true;false] = true

      existsb oddb [1;0;0;0;0;3] = true

      existsb evenb [] = false

Next, define a nonrecursive version of existsb -- call it existsb' -- using forallb and negb. Finally, prove a theorem existsb_existsb' stating that existsb' and existsb have the same behavior.