Propositions

We've spent the last five weeks learning how to program in Gallina. Along the way, we've noticed ways our programs can be right or wrong---give 'correct' answers or subtly err. We'll spend the next four weeks thinking more rigorously about the programs we've written, proving various properties. Why care about proofs? Here are two reasons, one pragmatic and one academic. 1. When you write code, the reasoning you do is about questions of efficacy, correctness, and efficiency. Efficacy asks, "How can I achieve this end?" Correctness asks, "Have I implemented my plan correctly? Do I get the correct output for the inputs I care about?" Efficiency asks, "Am I doing this in the best way possible? Can I achieve the same ends doing less work or using less space?" The process of answering these questions is the same reasoning exercised when doing proof! Our exercises in proof are exercises for your reasoning skills. 2. An algorithm without proof is just a pile of code. Computer science is the rigorous study of computation, and rigor demands proof. If you are a computer scientist, you make claims and rely on some form of proof for justifying those claims. Proofs vary from subjective experimental (e.g., testing a user interface with real users) to objective experimental (e.g., testing code on a variety of computers) to informal (e.g., argumentation in favor of one design over another) to formal (e.g., paper proofs or verified proofs in Coq). You'll learn other forms in other courses; in this course, you learn formal proof. It's the most important for you to learn early: you've likely been exposed to the other forms of proof elsewhere in your life, but formal proof is rarely taught before college. Before diving into details, let's talk a bit about the status of mathematical statements in Coq. Recall that Coq is a typed language, which means that every sensible expression in its world has an associated type. Logical claims are no exception: any statement we might try to prove in Coq has a type, namely Prop, the type of propositions. We can see this with the Check command: Here we assert that 3 = 3, read "3 is equal to 3". This proposition holds, i.e., it's possible to prove it. Not that we know how yet.

Here we assert that plus is commutative. There are several parts to this proposition. The ∀ n m : nat, ... blah ... part is a quantifier. We're saying that "for arbitrary n and m of type nat, it is the case that ... blah ...". Here, our "blah" is n + m = m + n, i.e., for any n and m, the number n + m is equal to m + n, i.e., order doesn't matter for addition, i.e., addition is commutative. This proposition is also provable.

So far we've seen two ways to form a Prop: equality and quantification. We can give a grammar in BNF (Backus-Naur Form):
   PROP ::= EXPR1 = EXPR2
          | ∀ x : TYPE, PROP

where EXPRi refers to an expression and TYPE refers to some type. When you have multiple variables in a quantifier, it's like stringing them together:
   ∀ n m : TYPE, PROP

is the same as
   ∀ n : TYPE, ∀ m : TYPE, PROP

You can of course have variables of different types. Here is a proposition asserting that the length of repeat b n is n, for all b and n. This proposition is also provable.

Note that all syntactically well-formed propositions have type Prop in Coq, regardless of whether they are true. Simply being a proposition is one thing; being provable is something else!

This one seems unlikely. Translating to English, "For any natural number n, it is the case that n is equal to 2." There are definitely natural numbers other than 2. Like, say, 1. Or 0. Or 731. None of those are equal to 2. This proposition does not hold---it is not provable. In fact, it's possible to prove that this proposition does not hold.

Even more baldly wrong. It is not the case that 3 = 4. It is possible to prove that this proposition does not hold.

Indeed, propositions don't just have types: they are first-class objects that can be manipulated in the same ways as the other entities in Coq's world. Propositions can be used in many other ways. For example, we can give a name to a proposition using a Definition, just as we have given names to expressions of other sorts.

We can also write parameterized propositions -- that is, functions that take arguments of some type and return a proposition. For instance, the following function takes a number and returns a proposition asserting that this number is equal to three:

In Coq, functions that return propositions are said to define properties of their arguments. For instance, here's a (polymorphic) property defining the notion of an injective function: no two inputs share the same output. Literally: if f maps x and y to the same value, then x has to be the same as y.

The proposition that S, the successor function, is injective. This property holds.

The proposition that pred, the successor function, is injective. This property does not hold: pred 0 = pred 1 = 0.

The equality operator = is also a function that returns a Prop. That's why it showed up in our BNF for Prop earlier! The expression n = m is syntactic sugar for eq n m (defined using Coq's Notation mechanism). Because eq can be used with elements of any type, it is also polymorphic:

(Notice that we wrote @eq instead of eq: The type argument A to eq is declared as implicit, so we need to turn off implicit arguments to see the full type of eq.)

Simple proofs

Let's get started with some proofs! The general format of a theorem is:
Theorem NAME : PROP.
Proof.
  TACTICS...
Qed.

Where NAME is the name of the theorem and PROP (of type Prop) is the proposition we'll prove. TACTICS... are the way we'll do proofs in Coq. They loosely correspond to the kinds of things people write in paper proofs, but the correspondence isn't one-to-one.

Proof by Simplification

Now that we've defined a few datatypes and functions, let's turn to stating and proving properties of their behavior. A "proof by simplification" can be used to prove a variety of properties. For example, the fact that 0 is a "neutral element" for + on the left can be proved just by observing that 0 + n reduces to n no matter what n is, a fact that can be read directly off the definition of plus.

What's happening here? It's worth stepping through the proof slowly, looking at the various views: there's the main script screen, the screen with the goal and context, and a response screen. We write Theorem and then a name for our theorem---here plus_O_n. Then we write a proposition, in this case ∀ n : nat, 0 + n = n, i.e., for any possible natural number n, it is the case that 0 + n is equal to n. We'll talk more about what counts as a proposition later in the course. After we type the Proof. keyword, we're shown a screen like the following:
1 subgoal, subgoal 1 (ID 50)

  ============================
  ∀ n : nat, 0 + n = n

The first line says how many cases we're considering in our proof at present: there's 1 subgoal, and we're currently working on it. That is: our proof has only one case. Then there's a line. Beneath the line is our *goal*---that's what we're trying to prove. Above the line is our *context*, which is currently empty. The way a proof works is that we try to show that given our context, our goal *holds*, that is, that our goal is a true proposition. In Coq, we use tactics to manipulate the goal and context, where Coq will keep track of the goal and context for us. On paper (or on a chalkboard or in person), we'll use natural language (for CS 54, English) to manipulate the goal and context, which we'll keep track of ourselves. Throughout the course, we'll try to keep parallel tracks in mind: how does proof work in Coq and how does it work on paper? We'll only really be doing proofs in Coq at first, and then we'll switch and only do proofs on paper. After this course, you'll probably only use paper proofs. We'll be using Coq as a tool to help you learn the ropes of paper proof. One of the hardest things about paper proof is that it can be very easy to get confused and make mistakes, breaking the "rules". Coq enforces the rules! Using Coq will help you internalize the rules. After the Proof. keyword comes our *proof script*, a series of *tactics* telling Coq how to manipulate the proof state. Each tactic is followed by a period. Before explaining the proof script, let's see the above proof in English.

• Theorem: For any natural number n,
        0 + n = n.
  Proof: Let n be given. We must show that
       0 + n = n.
  By the definition of plus, we know that 0+n reduces to n. Finally, we have
      n = n
  immediately. Qed.

How does the above proof work? To start out, we needed to show that ∀ n : nat, 0 + n = n. The line Let n be given phrases the proof as a sort of a game. To show that 0 + n = n for any n, we let our imaginary "opponent" give us any n they please. Now we have that n in hand---that is, in our proof's context---we want to show that 0 + n = n, our new goal. Showing this goal is as simple as looking at the definition of plus: the first case of the pattern match says that when the first argument to plus is 0, plus just returns its second argument---in this case n. So 0 + n reduces to n; finally, we observe that n = n immediately---an inarguable fact about natural numbers. How does the proof in Coq work? Our first tactic is intros with the argument n. Our goal is ∀ n : nat, 0 + n = n; in English, our goal is to show that for every possible n that is a natural number 0 + n is equal to n. The intros tactic *introduces* things into the context: our goal was a ∀, so intros will try to introduce whatever's been *quantified* by the for all. Here give a name explicitly: intros n says to name the introduced variable n and put it into our context:
1 subgoal, subgoal 1 (ID 51)
  
  n : nat
  ============================
  0 + n = n

We could have chosen a different name---go back and change it to read intros m and see what goal you get! (It's generally good style to use the name in the quantifier, since it's a little clearer.) Our next tactic is simpl, which asks Coq to simplify our goal, running a few steps of computation:
1 subgoal, subgoal 1 (ID 53)
  
  n : nat
  ============================
  n = n

Our context remains the same, but our new goal is to show that n = n. To do so, we use the reflexivity tactic, named after the property of equality: all things are equal to themselves. When we run the tactic, we get a new readout:
No more subgoals.
(dependent evars: (printing disabled) )

Coq is being a little too lowkey here: we proved it! To celebrate (and tell Coq that we're satisfied with our proof), we say Qed., which is short for quod erat demonstrandum, which is Latin for "that which was to be proved" which is perhaps better said as "and we've proved what we want to" or "and that's the proof" or "I told you so!" or "mic drop". Then Coq acknowledges that we're truly done:
plus_O_n is defined

The paper and Coq proofs are very much in parallel. Here's the paper proof annotated with Coq tactics:

• Theorem: For any natural number n,
        0 + n = n.
  Proof: Let n be given (intros n.). We must show that
       0 + n = n.
  By the definition of plus, we know that 0+n reduces to n (simpl.). Finally, we have
      n = n
  immediately (reflexivity.). Qed.

Later in the course, we'll have to work hard to write good proofs. At first our proofs will follow the Coq tactics quite closely. But just like any other kind of writing, proof writing is a craft to be honed and practiced. (You may notice that the above statement looks different in the .v file in your IDE than it does in the HTML rendition in your browser, if you are viewing both. In .v files, we write the ∀ universal quantifier using the reserved identifier "forall." When the .v files are converted to HTML, this gets transformed into an upside-down-A symbol.) This is a good place to mention that reflexivity is a bit more powerful than we have admitted. In the examples we have seen, the calls to simpl were actually not needed, because reflexivity can perform some simplification automatically when checking that two sides are equal; simpl was just added so that we could see the intermediate state -- after simplification but before finishing the proof. Here is a shorter proof of the theorem:

Moreover, it will be useful later to know that reflexivity does somewhat more simplification than simpl does -- for example, it tries "unfolding" defined terms, replacing them with their right-hand sides. The reason for this difference is that, if reflexivity succeeds, the whole goal is finished and we don't need to look at whatever expanded expressions reflexivity has created by all this simplification and unfolding; by contrast, simpl is used in situations where we may have to read and understand the new goal that it creates, so we would not want it blindly expanding definitions and leaving the goal in a messy state. The form of the theorem we just stated and its proof are almost exactly the same as the simpler examples we saw earlier; there are just a few differences. First, we've used the keyword Theorem instead of Example. This difference is mostly a matter of style; the keywords Example and Theorem (and a few others, including Lemma, Fact, and Remark) mean pretty much the same thing to Coq. Second, we've added the quantifier ∀ n:nat, so that our theorem talks about all natural numbers n. Informally, to prove theorems of this form, we generally start by saying "Suppose n is some number..." Formally, this is achieved in the proof by intros n, which moves n from the quantifier in the goal to a context of current assumptions. The keywords intros, simpl, and reflexivity are examples of tactics. A tactic is a command that is used between Proof and Qed to guide the process of checking some claim we are making. We will see several more tactics in the rest of this chapter and yet more in future chapters. Other similar theorems can be proved with the same pattern.

The _l suffix in the names of these theorems is pronounced "on the left." It is worth stepping through these proofs to observe how the context and the goal change. You may want to add calls to simpl before reflexivity to see the simplifications that Coq performs on the terms before checking that they are equal.

The intros and apply Tactics

Most of the properties we will be interested in proving about programs will begin with some quantifiers (e.g., "for all numbers n, ...") and/or hypothesis ("assuming m=n, ..."). In such situations, we will need to be able to reason by assuming the hypothesis -- i.e., we start by saying "OK, suppose n is some arbitrary number," or "OK, suppose m=n." The intros tactic permits us to do this by moving one or more quantifiers or hypotheses from the goal to a "context" of current assumptions. Here's an example of a trivial use of hypothesis: every proposition implies itself. The arrow symbol → is pronounced "implies." We've already defined a notion of implies on booleans impb; now we have a notion of implies on propositions. The way a proof with implies works is: you have to prove what's to the right of the arrow... but you may assume what's to the left.

Let a proposition A be given such that A holds (call this hypothesis HA). We must prove that A holds.

We find A by HA.

Exercise: 2 stars, standard (implies_PQR)

Remove "Admitted." and fill in the proof. Use intros and apply to show that if P implies Q and Q implies R and P holds, then R holds. Hint: you'll want to use backwards reasoning. To prove that R holds, it suffices to show that Q holds (because Q → R). To prove that Q holds, it suffices to show that P holds (because P → Q). But we know P holds, so...

Logical connectives

Let's look at more kinds of proposition: logical connectives let us form new propositions from others. We've already seen one connective: →, implication. We'll meet many more: ∧, conjunction; ∨, disjunction; ¬, negation; and ↔, if-and-only-if a/k/a iff. For now, we'll look at conjunction, disjunction, and the always-true proposition, True.

Conjunction

For any propositions A and B, if we assume that A is true and we assume that B is true, we can conclude that A ∧ B is also true. Here we have two arrows. We can read the proposition aloud in stilted English as:

• For all propositions A and B, if it is the case that A

holds, then if it is the case that B holds, then A ∧ B (read "A and B") holds. Chains of arrows, like curried functions, are pronounced idiomatically as a single condition, i.e.:

• For all propositions A and B, if A and B hold, then A ∧ B holds.

Let's look at the proof!

In addition to the outlining marks -, +, and ×, you can use curly braces to enter new outline levels. Every time you use curly braces, you reset the tracking of outlines, and you can reuse any outlining symbol.

Formatting matters. Coq will accept proofs in any valid formatting, but humans... less so. Keeping things clean is very important: for now, it helps us grade your work. Later on, clean formatting will make it much easier to work with other people. Many software development projects care so much about clean formatting that they use automatic formatters, and no code can be added to the project unless it meets the formatting guidelines!

To prove a conjunction, use the split tactic. It will generate two subgoals, one for each part of the statement:

Since applying a theorem with hypotheses to some goal has the effect of generating as many subgoals as there are hypotheses for that theorem, we can apply and_intro to achieve the same effect as split.

By the way, the infix notation ∧ is actually just syntactic sugar for and A B. That is, and is a Coq operator that takes two propositions as arguments and yields a proposition.

Exercise: 2 stars, standard (and_PQ)

Some more practice with implications and conjunctions. If P implies Q and you know P, then you know both P and Q.

Disjunction

Another important connective is the disjunction, or logical or of two propositions: A ∨ B is true when either A or B is. (Alternatively, we can write or A B, where or : Prop → Prop → Prop.) To show that a disjunction holds, we need to show that one of its sides does. This is done via two tactics, left and right. As their names imply, the first one requires proving the left side of the disjunction, while the second requires proving its right side. Here is a trivial use...

In natural language 'or' is often interpreted to be exclusive (one or the other) rather than inclusive (at least one, possibly both). In formal work, disjunction is inclusive, meaning 'at least one'. Consider the proposition 2 = 2 ∨ 4 = 4. Both sides are true, but a proof need only show one side. In fact, it would be a little bit strange---not wrong, but strange---for a proof to show both!

Exercise: 1 star, standard (or_which)

Truth

Coq's standard library also defines True, a proposition that is trivially true. To prove it, we use the predefined constant I : True:

If you can't remember that I is the single proof of True, you can also just use the reflexivity tactic.

True is used quite rarely, since it is trivial (and therefore uninteresting) to prove as a goal, and it carries no useful information as a hypothesis. But True can be quite useful when defining complex Props using conditionals or as a parameter to higher-order Props. We will see examples of such uses of True later on.

Proof by Rewriting

This theorem is a bit more interesting than the others we've seen:

Instead of making a universal claim about all numbers n and m, it talks about a more specialized property that only holds when n = m. The arrow symbol is pronounced "implies." We've already defined a notion of implies on booleans impb; now we have a notion of implies on propositions. The way a proof with implies works is: you have to prove what's to the right of the arrow... but you may assume what's to the left. That is, to show that n + n = m + m, we know (in our context) not only that n and m are natural numbers, but in fact it is the case that n = m. As before, we need to be able to reason by assuming we are given such numbers n and m. We also need to assume the hypothesis n = m. The intros tactic will serve to move all three of these from the goal into assumptions in the current context. Since n and m are arbitrary numbers, we can't just use simplification to prove this theorem. Instead, we prove it by observing that, if we are assuming n = m, then we can replace n with m in the goal statement and obtain an equality with the same expression on both sides. The tactic that tells Coq to perform this replacement is called rewrite.

The first line of the proof moves the universally quantified variables n and m into the context. The second moves the hypothesis n = m into the context and gives it the name H. The third tells Coq to rewrite the current goal (n + n = m + m) by replacing the left side of the equality hypothesis H with the right side. (The arrow symbol in the rewrite has nothing to do with implication: it tells Coq to apply the rewrite from left to right. To rewrite from right to left, you can use rewrite <-. Try making this change in the above proof and see what difference it makes.) Here's a paper proof of the same theorem, with the tactics interwoven.

• Theorem: For any natural numbers n and m, if
        n = m
  then
        n + n = m + m.
  Proof: Let n and m be given (intros n m.) such that n = m (intros H). We must show:
        n + n = m + m.
  Since we've assumed that n = m, we can replace each reference to n on the left-hand side with a reference to m, and we have
        m + m = m + m
  immediately (reflexivity.). Qed.

Exercise: 1 star, standard (plus_id_exercise)

The Admitted command tells Coq that we want to skip trying to prove this theorem and just accept it as a given. This can be useful for developing longer proofs, since we can state subsidiary lemmas that we believe will be useful for making some larger argument, use Admitted to accept them on faith for the moment, and continue working on the main argument until we are sure it makes sense; then we can go back and fill in the proofs we skipped. Be careful, though: every time you say Admitted you are leaving a door open for total nonsense to enter Coq's nice, rigorous, formally checked world! We can also use the rewrite tactic with a previously proved theorem instead of a hypothesis from the context. If the statement of the previously proved theorem involves quantified variables, as in the example below, Coq tries to instantiate them by matching with the current goal.

Note that we've just rewritten by an existing theorem: the proof of mult_0_plus uses the proof of plus_O_n. In an informal proof, we might use language by "by plus_0_n, we know...". For example:

• Theorem: For any natural number n and m,
        (0 + n) × m = n × m.
  Proof: Let n and m be given (intros n m.). We must show:
        (0 + n) × m = n × m.
  By plus_0_n, we know that 0 + n = n, so we can replace 0 + n on the left-hand side with just n, and we have
        n × m = n × m
  immediately (reflexivity.). Qed.

Exercise: 2 stars, standard (mult_S_1)

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 → eq₂. reflexivity." as we have done several times before. We can achieve the same effect in a single step by using the apply tactic instead:

silly1 is the first theorem we've seen that has multiple hypotheses. How should we parse the following proposition?
     n = m →
     [n;o] = [n;p] →
     [n;o] = [m;p].

The answer is that implication is right associative, that is, we assume that parentheses go on the right, as in:
     n = m →
     ([n;o] = [n;p] →
      [n;o] = [m;p]).

If we were to read this proposition aloud, we could naively read it as:

    if it is the case that [n = m], 
    then if it is the case that [[n;o] = [n;p]]
         then it is the case that [[n;o] = [m;p]].

But we can read it more naturally as:

    if it is the case that [n = m] 
                       and [[n;o] = [n;p]],
    then it is the case that [[n;o] = [m;p]].

That is, we've translated nested implications to a conjuction, with the word "and". We'll talk more about conjunctions in Logic. 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. If there are no premises (i.e., H isn't an implication), then you're done!

Typically, when we use apply H, the statement H will begin with a ∀ that binds some universal 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 (silly_ex)

Complete the following proof without using simpl.

It's often helpful to look at definitions before doing proofs!

To use the apply tactic, the (conclusion of the) theorem, lemma, hypothesis, or other 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.

To sum up, one can say apply H when H is a hypothesis in the context or a previously proven lemma of the form ∀ x₁ ..., H₁ → H₂ → ... → Hn and the current goal is shape Hn (for some instantatiation of each of the xi). After running apply H in such a state, you will have a subgoal for each of H₁ through Hn-1. In the special case where H isn't an implication, your proof will be completed. The informal analogue of the apply tactic is phrasing like "by H, it suffices to show Hn by proving each of H₁, ..., Hn-1", followed by a proof of those subsidiary parts. In the case where H isn't an implication, you can try using language like "we are done by H" or "which H proves exactly".

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 recording, 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, we usually don't have to include the name m in the with clause; Coq is often smart enough to figure out which instantiation we're giving. We could instead write: apply trans_eq with [c;d].

Exercise: 3 stars, standard (trans_eq_exercise)

Here's a lemma to practice apply ... with ....

The informal analogue of apply H with (x:=m) is to say "by H with m for x" or "which we can see by H on m".

Propositions so far

We've seen quite a few different propositions! Equality, universal quantification, implication, conjunction, disjunction, and the always-true proposition. PROP ::= EXPR1 = EXPR2 | forall x : TYPE, PROP | PROP1 -> PROP2 | PROP1 /\ PROP2 | PROP1 \/ PROP2 | True

Tactics so far

We've seen:

• intros
• simpl
• reflexivity
• apply
• split
• left, right
• rewrite
• symmetry
• apply ... with ...

Zounds! That's a lot. We'll meet even more of them tomorrow. It's a whole other programming language...

What's happening here?

You might be thinking, "What is this crap? I wanted to learn computer programming! Proof? Who cares! Coq and Gallina are for the birds." A few thoughts. 1. This is a computer science department, not a computer programming department. You'll learn to program because it's a critical skill for computer science, but we're not in it just for the programming. (Similarly, astronomers learn to use cool telescopes, and you could hardly have modern astronomy without telescopes... but astronomy isn't just about telescopes.) 2. If you could make your program fast or correct, which would you choose? 3. It is unlikely that your future career in computer science (or tech, or whatever) will involve proving programs correct. But anyone who writes programs has to reason about programs. Learning to prove properties of your programs stretches and strengthens your reasoning muscles. 4. Computer science is a very formal field: to succeed as a computer scientist, you must be comfortable with formalism. This is practice! After Coq, you'll be ready for anything.