Separate Compilation

Before getting started on this chapter, we need to import all of our definitions from the previous chapter:

For this Require command to work, Rocq needs to be able to find a compiled version of the previous chapter (Basics.v). This compiled version, called Basics.vo, is analogous to the .class files compiled from .java source files and the .o files compiled from .c files. To compile Basics.v and obtain Basics.vo, first make sure that the files Basics.v, Induction.v, and _CoqProject are in the current directory. The _CoqProject file should contain just the following line:
      -Q . LF This maps the current directory (".", which contains Basics.v, Induction.v, etc.) to the prefix (or "logical directory") "LF". Proof General, CoqIDE, and VSCoq read _CoqProject automatically, to find out to where to look for the file Basics.vo corresponding to the library LF.Basics. Once the files are in place, there are various ways to build Basics.vo from an IDE, or you can build it from the command line. From an IDE...

• In Proof General: The compilation can be made to happen automatically when you submit the Require line above to PG, by setting the emacs variable coq-compile-before-require to t. This can also be found in the menu: "Coq" > "Auto Compilation" > "Compile Before Require".
• In CoqIDE: One thing you can do on all platforms is open Basics.v; then, in the "Compile" menu, click on "Compile Buffer".
• For VSCode users, open the terminal pane at the bottom and then follow the command line instructions below. (If you downloaded the project setup .tgz file, just doing `make` should build all the code.)

To compile Basics.v from the command line...

• First, generate a Makefile using the rocq makefile utility, which comes installed with Rocq. (If you obtained the whole volume as a single archive, a Makefile should already exist and you can skip this step.)
           rocq makefile -f _CoqProject *.v -o Makefile You should rerun that command whenever you add or remove Rocq files in this directory.
• Now you can compile Basics.v by running make with the corresponding .vo file as a target:
           make Basics.vo All files in the directory can be compiled by giving no arguments:
           make
• Under the hood, make uses the Rocq compiler, rocq compile. You can also run rocq compile directly:
           rocq compile -Q . LF Basics.v
• Since make also calculates dependencies between source files to compile them in the right order, make should generally be preferred over running rocq compile explicitly. But as a last (but not terrible) resort, you can simply compile each file manually as you go. For example, before starting work on the present chapter, you would need to run the following command:
          rocq compile -Q . LF Basics.v Then, once you've finished this chapter, you'd do
          rocq compile -Q . LF Induction.v to get ready to work on the next one. If you ever remove the .vo files, you'd need to give both commands again (in that order).

Troubleshooting:

• For many of the alternatives above you need to make sure that the rocq executable is in your PATH.
• If you get complaints about missing identifiers, it may be because the "load path" for Rocq is not set up correctly. The Print LoadPath. command may be helpful in sorting out such issues.
• When trying to compile a later chapter, if you see a message like
          Compiled library Induction makes inconsistent assumptions over
          library Basics a common reason is that the library Basics was modified and recompiled without also recompiling Induction which depends on it. Recompile Induction, or everything if too many files are affected (for instance by running make and if even this doesn't work then make clean; make).
• If you get complaints about missing identifiers later in this file it may be because the "load path" for Rocq is not set up correctly. The Print LoadPath. command may be helpful in sorting out such issues.
  In particular, if you see a message like
             Compiled library Foo makes inconsistent assumptions over
             library Bar check whether you have multiple installations of Rocq on your machine. It may be that commands (like rocq compile) that you execute in a terminal window are getting a different version of Rocq than commands executed by Proof General or CoqIDE.
• One more tip for CoqIDE users: If you see messages like Error: Unable to locate library Basics, a likely reason is inconsistencies between compiling things within CoqIDE vs using rocq from the command line. This typically happens when there are two incompatible versions of Rocq installed on your system (one associated with CoqIDE, and one associated with rocq from the terminal). The workaround for this situation is compiling using CoqIDE only (i.e. choosing "make" from the menu), and avoiding using rocq directly at all.

Proof by Induction

We can prove that 0 is a neutral element for + on the left using just reflexivity. But the proof that it is also a neutral element on the right ...

... can't be done in the same simple way. Just applying reflexivity doesn't work, since the n in n + 0 is an arbitrary unknown number, so the match in the definition of + can't be simplified.

And reasoning by cases using destruct n doesn't get us much further: the branch of the case analysis where we assume n = 0 goes through just fine, but in the branch where n = S n' for some n' we get stuck in exactly the same way.

We could use destruct n' to get a bit further, but, since n can be arbitrarily large, we'll never get all the way there if we just go on like this. To prove interesting facts about numbers, lists, and other inductively defined sets, we often need a more powerful reasoning principle: induction. Recall (from a discrete math course, probably) the principle of induction over natural numbers: If P(n) is some proposition involving a natural number n and we want to show that P holds for all numbers n, we can reason like this:

• show that P(O) holds;
• show that, for any n', if P(n') holds, then so does P(S n');
• conclude that P(n) holds for all n.

In Rocq, the steps are the same, except we typically encounter them in reverse order: we begin with the goal of proving P(n) for all n and apply the induction tactic to break it down into two separate subgoals: one where we must show P(O) and another where we must show P(n') → P(S n'). Here's how this works for the theorem at hand...

Like destruct, the induction tactic takes an as... clause that specifies the names of the variables to be introduced in the subgoals. Since there are two subgoals, the as... clause has two parts, separated by a vertical bar, |. (Strictly speaking, we can omit the as... clause and Rocq will choose names for us. In practice, this is a bad practice, as Rocq's automatic choices tend to be confusing.) In the first subgoal, n is replaced by 0. No new variables are introduced (so the first part of the as... is empty), and the goal becomes 0 = 0 + 0, which follows easily by simplification. In the second subgoal, n is replaced by S n', and the assumption n' + 0 = n' is added to the context with the name IHn' (i.e., the Induction Hypothesis for n'). These two names are specified in the second part of the as... clause. The goal in this case becomes S n' = (S n') + 0, which simplifies to S n' = S (n' + 0), which in turn follows from IHn'.

(The use of the intros tactic in these proofs is actually redundant. When applied to a goal that contains quantified variables, the induction tactic will automatically move them into the context as needed.)

Exercise: 2 stars, standard, especially useful (basic_induction)

Prove the following using induction. You might need previously proven results.

Exercise: 2 stars, standard (double_plus)

Consider the following function, which doubles its argument:

Use induction to prove this simple fact about double:

Exercise: 2 stars, standard (eqb_refl)

The following theorem relates the computational equality =? on nat with the definitional equality = on bool.

Exercise: 2 stars, standard, optional (even_S)

One inconvenient aspect of our definition of even n is the recursive call on n - 2. This makes proofs about even n harder when done by induction on n, since we may need an induction hypothesis about n - 2. The following lemma gives an alternative characterization of even (S n) that works better with induction:

Proofs Within Proofs

In Rocq, as in informal mathematics, large proofs are often broken into a sequence of theorems, with later proofs referring to earlier theorems. But sometimes a proof will involve some miscellaneous fact that is too trivial and of too little general interest to bother giving it its own top-level name. In such cases, it is convenient to be able to simply use the required fact "in place" and then prove it as a separate step. The replace tactic allows us to do this.

The tactic replace e₁ with e₂ tactic introduces two subgoals. The first subgoal is the same as the one at the point where we invoke replace, except that e₁ is replaced by e₂. The second subgoal is the equality e₁ = e₂ itself. As another example, suppose we want to prove that (n + m) + (p + q) = (m + n) + (p + q). The only difference between the two sides of the = is that the arguments m and n to the first inner + are swapped, so it seems we should be able to use the commutativity of addition (add_comm) to rewrite one into the other. However, the rewrite tactic is not very smart about where it applies the rewrite. There are three uses of + here, and it turns out that doing rewrite → add_comm will affect only the outer one...

To use add_comm at the point where we need it, we can rewrite n + m to m + n using replace and then prove n + m = m + n using add_comm.

Formal vs. Informal Proof

What constitutes a successful proof of a mathematical claim? The question has challenged philosophers for millennia, but a rough and ready answer could be this: A proof of a mathematical proposition P is a written (or spoken) text that instills in the reader or hearer the certainty that P is true -- an unassailable argument for the truth of P. That is, a proof is an act of communication. Acts of communication may involve different sorts of readers. On one hand, the "reader" can be a program like Rocq, in which case the "belief" that is instilled is that P can be mechanically derived from a certain set of formal logical rules, and the proof is a recipe that guides the program in checking this fact. Such recipes are formal proofs. Alternatively, the reader can be a human being, in which case the proof will probably be written in English or some other natural language and will thus necessarily be informal. Here, the criteria for success are less clearly specified. A "valid" proof is one that makes the reader believe P. But the same proof may be read by many different readers, some of whom may be convinced by a particular way of phrasing the argument, while others may not be. Some readers may be particularly pedantic, inexperienced, or just plain thick-headed; the only way to convince them will be to make the argument in painstaking detail. Other readers, more familiar in the area, may find all this detail so overwhelming that they lose the overall thread; all they want is to be told the main ideas, since it is easier for them to fill in the details for themselves than to wade through a written presentation of them. Ultimately, there is no universal standard, because there is no single way of writing an informal proof that will convince every conceivable reader. In practice, however, mathematicians have developed a rich set of conventions and idioms for writing about complex mathematical objects that -- at least within a certain community -- make communication fairly reliable. The conventions of this stylized form of communication give a reasonably clear standard for judging proofs good or bad. Because we are using Rocq in this course, we will be working heavily with formal proofs. But this doesn't mean we can completely forget about informal ones! Formal proofs are useful in many ways, but they are not very efficient ways of communicating ideas between human beings. For example, here is a proof that addition is associative:

Rocq is perfectly happy with this. For a human, however, it is difficult to make much sense of it. We can use comments and bullets to show the structure a little more clearly...

... and if you're used to Rocq you might be able to step through the tactics one after the other in your mind and imagine the state of the context and goal stack at each point, but if the proof were even a little bit more complicated this would be next to impossible. A (pedantic) mathematician might write the proof something like this:

• Theorem: For any n, m and p,
        n + (m + p) = (n + m) + p. Proof: By induction on n.
  • First, suppose n = 0. We must show that
            0 + (m + p) = (0 + m) + p. This follows directly from the definition of +.
  • Next, suppose n = S n', where
            n' + (m + p) = (n' + m) + p. We must now show that
            (S n') + (m + p) = ((S n') + m) + p. By the definition of +, this follows from
            S (n' + (m + p)) = S ((n' + m) + p), which is immediate from the induction hypothesis. Qed.

The overall form of the proof is basically similar, and of course this is no accident: Rocq has been designed so that its induction tactic generates the same sub-goals, in the same order, as the bullet points that a mathematician would usually write. But there are significant differences of detail: the formal proof is much more explicit in some ways (e.g., the use of reflexivity) but much less explicit in others (in particular, the "proof state" at any given point in the Rocq proof is completely implicit, whereas the informal proof reminds the reader several times where things stand).

Exercise: 2 stars, advanced, optional (add_comm_informal)

Translate your solution for add_comm into an informal proof: Theorem: Addition is commutative. Proof: (* FILL IN HERE *)

Exercise: 2 stars, standard, optional (eqb_refl_informal)

Write an informal proof of the following theorem, using the informal proof of add_assoc as a model. Don't just paraphrase the Rocq tactics into English! Theorem: (n =? n) = true for any n. Proof: (* FILL IN HERE *)

More Exercises

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

Use replace to help prove add_shuffle3. You don't need to use induction yet.

Now prove commutativity of multiplication. You will probably want to look for (or define and prove) a "helper" theorem to be used in the proof of this one. Hint: what is n × (1 + k)?

Exercise: 3 stars, standard, optional (more_exercises)

Take a piece of paper. For each of the following theorems, first think about whether (a) it can be proved using only simplification and rewriting, (b) it also requires case analysis (destruct), or (c) it also requires induction. Write down your prediction. Then fill in the proof. (There is no need to turn in your piece of paper; this is just to encourage you to reflect before you hack!)

Nat to Bin and Back to Nat

Recall the bin type we defined in Basics:

Before you start working on the next exercise, replace the stub definitions of incr and bin_to_nat, below, with your solution from Basics. That will make it possible for this file to be graded on its own.

In Basics, we did some unit testing of bin_to_nat, but we didn't prove its correctness. Now we'll do so.

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

Prove that the following diagram commutes:

                            incr
              bin ----------------------> bin
               |                           |
    bin_to_nat |                           |  bin_to_nat
               |                           |
               v                           v
              nat ----------------------> nat
                             S

That is, incrementing a binary number and then converting it to a (unary) natural number yields the same result as first converting it to a natural number and then incrementing. If you want to change your previous definitions of incr or bin_to_nat to make the property easier to prove, feel free to do so!

Exercise: 3 stars, standard (nat_bin_nat)

Write a function to convert natural numbers to binary numbers.

Prove that, if we start with any nat, convert it to bin, and convert it back, we get the same nat which we started with. Hint: This proof should go through smoothly using the previous exercise about incr as a lemma. If not, revisit your definitions of the functions involved and consider whether they are more complicated than necessary: the shape of a proof by induction will match the recursive structure of the program being verified, so make the recursions as simple as possible.

Bin to Nat and Back to Bin (Advanced)

The opposite direction -- starting with a bin, converting to nat, then converting back to bin -- turns out to be problematic. That is, the following theorem does not hold.

Let's explore why that theorem fails, and how to prove a modified version of it. We'll start with some lemmas that might seem unrelated, but will turn out to be relevant.

Exercise: 2 stars, advanced (double_bin)

Prove this lemma about double, which we defined earlier in the chapter.

Now define a similar doubling function for bin.

Check that your function correctly doubles zero.

Prove this lemma, which corresponds to double_incr.

Let's return to our desired theorem:

The theorem fails because there are some bin such that we won't necessarily get back to the original bin, but instead to an "equivalent" bin. (We deliberately leave that notion undefined here for you to think about.) Explain in a comment, below, why this failure occurs. Your explanation will not be graded, but it's important that you get it clear in your mind before going on to the next part. If you're stuck on this, think about alternative implementations of double_bin that might have failed to satisfy double_bin_zero yet otherwise seem correct.

To solve that problem, we can introduce a normalization function that selects the simplest bin out of all the equivalent bin. Then we can prove that the conversion from bin to nat and back again produces that normalized, simplest bin.

Exercise: 4 stars, advanced (bin_nat_bin)

Define normalize. You will need to keep its definition as simple as possible for later proofs to go smoothly. Do not use bin_to_nat or nat_to_bin, but do use double_bin. Hint: Structure the recursion such that it always reaches the end of the bin and process each bit only once. Do not try to "look ahead" at future bits.

It would be wise to do some Example proofs to check that your definition of normalize works the way you intend before you proceed. They won't be graded, but fill them in below.

Finally, prove the main theorem. The inductive cases could be a bit tricky. Hint: Start by trying to prove the main statement, see where you get stuck, and see if you can find a lemma -- perhaps requiring its own inductive proof -- that will allow the main proof to make progress. We have one lemma for the B₀ case (which also makes use of double_incr_bin) and another for the B₁ case.