The apply Tactic

We can use apply when some hypothesis or an earlier lemma exactly matches the goal:

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:

apply also works with conditional hypotheses:

Observe how Coq picks appropriate values for the ∀-quantified variables of the hypothesis:

The goal must match the hypothesis exactly for apply to work:

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.

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.

Applying this lemma to the example above requires a slight refinement of apply:

Doing apply trans_eq doesn't work! But...

does.

transitivity is also a tactic.

The injection and discriminate Tactics

In Coq, the constructors of inductive types are injective (or one-to-one) and disjoint. E.g., for nat...

• if S n = S m then it must be that n = m
• O is not equal to S n for any n

We can prove the injectivity of S by using the pred function defined in Basics.v.

As a convenience, the injection tactic allows us to exploit injectivity of any constructor.

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

So much for injectivity of constructors. What about disjointness? Two terms beginning with different constructors (like O and S, or true and false) can never be equal!

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!).

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.

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:

Or we can use the f_equal tactic.

Using Tactics on Hypotheses

Many tactics come with "... in ..." variants that work on hypotheses instead of goals.

The ordinary apply tactic is a form of "backward reasoning": it says "We're trying to prove X and we know Y → X, so if we can prove Y we'll be done." By contrast, the variant apply... in... is "forward reasoning": it says "We know Y and we know Y → X, so we also know X."

Varying the Induction Hypothesis

Recall this function for doubling a natural number (from the Induction chapter):

Suppose we want to show that double is injective -- i.e., that it maps different arguments to different results. The way we start this proof is a little bit delicate:

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? 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:

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 theorem illustrates the same point:

Exercise: 2 stars, standard (eqb_true)

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.

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

The destruct tactic can be used on expressions as well as variables:

The eqn: part of the destruct tactic is optional; although we've chosen to include it most of the time, for the sake of documentation, 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.

Adding the eqn: qualifier saves this information so we can use it.

Micro Sermon

Mindless proof-hacking is a terrible temptation... Try to resist!