The apply Tactic

The apply tactic is useful 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. Or we can finish in a single step by using apply:

apply also works with conditional hypotheses:

Observe how Rocq 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 available as a tactic.

The injection and discriminate Tactics

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 (not just S).

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 more useful example, we can use discriminate to make a connection between the two different notions of equality (= and =?) that we have seen for 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 useful below:

Rocq also provides f_equal as a 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."

Specializing Hypotheses

Another handy tactic for manipulating hypotheses is specialize. It is essentially just a combination of assert and apply, but it often provides a pleasingly smooth way to nail down overly general assumptions. It works like this: If H is a quantified hypothesis in the current context -- i.e., H : ∀ (x:T), P -- then specialize H with (x := e) will change H so that it looks like P with x replaced by e. For example:

Using specialize before apply gives us yet another way to control where apply does its work.

Things to note:

• We can specialize facts in the global context, not just local hypotheses.
• The as... clause at the end tells specialize how to name the new hypothesis in this case.

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., 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 particular m.

A successful proof of double_injective keeps m universally quantified in the goal statement at the point where the induction tactic is invoked on n:

The thing to 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 quantified over variables n and m by induction on n, it is sometimes crucial to leave m "generic." The following theorem, which further strengthens the link between =? and =, follows the same pattern.

The strategy of doing fewer intros before an induction to obtain a more general IH doesn't always work; 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. (If we simply say induction m without introducing anything first, Rocq will automatically introduce n for us!)

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.

Rewriting with conditional statements

Suppose that we want to show that plus is the inverse of minus. Since we are working with natural numbers, we need an assumption to prevent minus from truncating its result. With this assumption, the induction hypothesis becomes ∀ m, n' <=? m = true → (m - n') + n' = m. The beginning of the proof uses techniques we have already seen -- in particular, notice how we induct on n before introducing m, so that the induction hypothesis becomes sufficiently general.

We could use the assert tactic to prove (m' - n') + n' = m' from the IH. However, we can also just use rewrite directly...

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 stands for. 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, a bit 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 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:

There are now two ways make progress. First, we can use destruct m to break the proof into two cases:

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 Rocq 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 so 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. One example where it cannot _ be omitted is when we are destructing compound expressions; here, the information recorded by the eqn: can actually be critical, and, if we leave it out, then destruct can erase information we need to complete a proof.