So far, we have seen...

• propositions: factual claims
  • equality propositions (e₁ = e₂)
  • implications (P → Q)
  • quantified propositions (∀ x, P)
• proofs: ways of presenting evidence for the truth of a proposition

In this chapter we will introduce several more flavors of both propositions and proofs.

Like everything in Coq, propositions have a type:

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 a different thing!

So far, we've seen one primary place that propositions can appear: in Theorem (and Lemma and Example) declarations.

Propositions are first-class entities in Coq. For example, we can name them:

We can also write parameterized propositions -- that is, functions that take arguments of some type and return a proposition.

In Coq, functions that return propositions are said to define properties of their arguments. For instance, here's a (polymorphic) property defining the familiar notion of an injective function.

The equality operator = is a (binary) function that returns a Prop. The expression n = m is syntactic sugar for eq n m (defined in Coq's standard library using the Notation mechanism). Because eq can be used with elements of any type, it is also polymorphic:

Logical Connectives

Conjunction

The conjunction, or logical and, of propositions A and B is written A ∧ B, representing the claim that both A and B are true.

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

For any propositions A and B, if we assume that A is true and that B is true, we can conclude that A ∧ B is also true.

So much for proving conjunctive statements. To go in the other direction -- i.e., to use a conjunctive hypothesis to help prove something else -- we employ the destruct tactic. When the current proof context contains a hypothesis H of the form A ∧ B, writing destruct H as [HA HB] will remove H from the context and replace it with two new hypotheses: HA, stating that A is true, and HB, stating that B is true.

As usual, we can also destruct H right when we introduce it, instead of introducing and then destructing it:

You may wonder why we bothered packing the two hypotheses n = 0 and m = 0 into a single conjunction, since we could have also stated the theorem with two separate premises:

For the present example, both ways work. But in other situations we may wind up with a conjunctive hypothesis in the middle of a proof...

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.

Disjunction

Another important connective is the disjunction, or logical or, of two propositions: A ∨ B is true when either A or B is. (This infix notation stands for or A B, where or : Prop → Prop → Prop.)

To use a disjunctive hypothesis in a proof, we proceed by case analysis -- which, as with other data types like nat, can be done explicitly with destruct or implicitly with an intros pattern:

We can see in this example that, when we perform case analysis on a disjunction A ∨ B, we must separately satisfy two proof obligations, each showing that the conclusion holds under a different assumption -- A in the first subgoal and B in the second. Note that the case analysis pattern [Hn | Hm] allows us to name the hypotheses that are generated for the subgoals.

Conversely, to show that a disjunction holds, it suffices to show that one of its sides holds. This can be 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...

... and here is a slightly more interesting example requiring both left and right:

Falsehood and Negation

So far, we have mostly been concerned with proving "positive" statements -- addition is commutative, appending lists is associative, etc. Of course, we may also be interested in negative results, demonstrating that some given proposition is not true. Such statements are expressed with the logical negation operator ¬.

To see how negation works, recall the principle of explosion from the Tactics chapter, which asserts that, if we assume a contradiction, then any other proposition can be derived. Following this intuition, we could define ¬ P ("not P") as ∀ Q, P → Q. Coq actually makes a slightly different (but equivalent) choice, defining ¬ P as P → False, where False is a specific contradictory proposition defined in the standard library.

Since False is a contradictory proposition, the principle of explosion also applies to it. If we get False into the proof context, we can use destruct on it to complete any goal:

Inequality is a frequent enough form of negated statement that there is a special notation for it, x ≠ y:
      Notation "x <> y" := (~(x = y)). We can use not to state that 0 and 1 are different elements of nat:

It takes a little practice to get used to working with negation in Coq. Even though you can see perfectly well why a statement involving negation is true, it can be a little tricky at first to make Coq understand it! Here are proofs of a few familiar facts to get you warmed up.

Since inequality involves a negation, it also requires a little practice to be able to work with it fluently. Here is one useful trick: If you are trying to prove a goal that is nonsensical (e.g., the goal state is false = true), apply ex_falso_quodlibet to change the goal to False. This makes it easier to use assumptions of the form ¬P that may be available in the context -- in particular, assumptions of the form x≠y.

Truth

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

Unlike False, which is used extensively, True is used relatively rarely, since it is trivial (and therefore uninteresting) to prove as a goal, and conversely it provides no interesting information when used as a hypothesis.

Logical Equivalence

The handy "if and only if" connective, which asserts that two propositions have the same truth value, is simply the conjunction of two implications.

Example: using apply with ↔.

Setoids and Logical Equivalence

Some of Coq's tactics treat iff statements specially, avoiding the need for some low-level proof-state manipulation. In particular, rewrite and reflexivity can be used with iff statements, not just equalities. To enable this behavior, we have to import the Coq library that supports it:

A "setoid" is a set equipped with an equivalence relation, such as = or ↔.

Here is a simple example demonstrating how these tactics work with iff. First, let's prove a couple of basic iff equivalences.

Existential Quantification

To prove a statement of the form ∃ x, P, we must show that P holds for some specific choice of value for x, known as the witness of the existential. This is done in two steps: First, we explicitly tell Coq which witness t we have in mind by invoking the tactic ∃ t. Then we prove that P holds after all occurrences of x are replaced by t.

Conversely, if we have an existential hypothesis ∃ x, P in the context, we can destruct it to obtain a witness x and a hypothesis stating that P holds of x.

Exercise: 3 stars, standard, optional (leb_plus_exists)

Programming with Propositions

What does it mean to say that "an element x occurs in a list l"?

• If l is the empty list, then x cannot occur in it, so the property "x appears in l" is simply false.

• Otherwise, l has the form x' :: l'. In this case, x occurs in l if it is equal to x' or if it occurs in l'.

We can translate this directly into a straightforward recursive function taking an element and a list and returning a proposition (!):

When In is applied to a concrete list, it expands into a concrete sequence of nested disjunctions.

We can also prove more generic, higher-level lemmas about In.

Applying Theorems to Arguments

Coq also treats proofs as first-class objects! We have seen that we can use Check to ask Coq to print the type of an expression. We can also use it to ask what theorem a particular identifier refers to.

Coq checks the statement of the add_comm theorem (or prints it for us, if we leave off the part beginning with the colon) in the same way that it checks the type of any term (e.g., plus) that we ask it to Check. Why?

The reason is that the identifier add_comm actually refers to a proof object, which represents a logical derivation establishing of the truth of the statement ∀ n m : nat, n + m = m + n. The type of this object is the proposition that it is a proof of.

The type of an ordinary function tells us what we can do with it.

• e.g., if we have a term of type nat → nat → nat, we can give it two nats as arguments and get a nat back.

Similarly, the statement of a theorem tells us what we can use that theorem for.

• if we have a term of type n = m → n + n = m + m and we provide it two numbers n and m plus an "argument" of type n = m, we can derive n + n = m + m.

Coq actually allows us to apply a theorem as if it were a function. This is often handy in proof scripts -- e.g., suppose we want too prove the following:

It appears at first sight that we ought to be able to prove this by rewriting with add_comm twice to make the two sides match. The problem, however, is that the second rewrite will undo the effect of the first.

We can fix this by applying add_comm to the arguments we want it be to instantiated with. Then the rewrite can only happen in one place.

Let's see another example of using a theorem like a function. The following theorem says: any list l containing some element must be nonempty.

We should be able to use this theorem to prove the special case where x is 42. However, naively, the tactic apply in_not_nil will fail because it cannot infer the value of x.

There are several ways to work around this: Use apply ... with ...

Use apply ... in ...

Explicitly apply the lemma to the value for x.

Explicitly apply the lemma to a hypothesis.

Coq vs. Set Theory

Coq's logical core, the Calculus of Inductive Constructions, is a "metalanguage for mathematics" in the same sense as familiar foundations for paper-and-pencil math, like Zermelo-Fraenkel Set Theory (ZFC). Mostly, the differences are not too important. But there are a few points worth discussing...

Functional Extensionality

Coq's logic is intentionally quite minimal. This means that there are occasionally some cases where translating standard mathematical reasoning into Coq can be cumbersome or even impossible, unless we enrich the core logic with additional axioms. For example, we can write an equality proposition stating that two functions are equal to each other:

In common mathematical practice, two functions f and g are considered equal if they produce the same output on every input:
    (∀ x, f x = g x) → f = g This is known as the principle of functional extensionality.

However, functional extensionality is not part of Coq's built-in logic. This means that some apparently "obvious" propositions are not provable.

However, if we like, we can add functional extensionality to Coq's core using the Axiom command.

Defining something as an Axiom has the same effect as stating a theorem and skipping its proof using Admitted, but it alerts the reader that this isn't just something we're going to come back and fill in later!

We can now invoke functional extensionality in proofs:

Naturally, we must be careful when adding new axioms into Coq's logic, as this can render it inconsistent -- that is, it may become possible to prove every proposition, including False, 2+2=5, etc.! Unfortunately, there is no simple way of telling whether an axiom is safe to add: hard work by highly trained mathematicians is often required to establish the consistency of any particular combination of axioms. Fortunately, it is known that adding functional extensionality, in particular, is consistent.

To check whether a particular proof relies on any additional axioms, use the Print Assumptions command: Print Assumptions function_equality_ex₂.

(If you try this yourself, you may also see add_comm listed as an assumption, depending on whether the copy of Tactics.v in the local directory has the proof of add_comm filled in.)

Propositions vs. Booleans

We've seen two different ways of expressing logical claims in Coq: with booleans (of type bool), and with propositions (of type Prop). Here are the key differences between bool and Prop:

                                           bool     Prop
                                           ====     ====
           decidable?                       yes       no
           useable with match?             yes       no
           equalities rewritable?          no        yes

Working with Decidable Properties

Since every function terminates on all inputs in Coq, a function of type nat → bool is a decision procedure -- i.e., it yields true or false on all inputs.

• For example, even : nat → bool is a decision procedure for the property "is even".

It follows that there are some properties of numbers that we cannot express as functions of type nat → bool.

• For example, the property "is the code of a halting Turing machine" is undecidable, so there is no way to write it as a function of type nat → bool.

On the other hand, nat→Prop is the type of all properties of numbers that can be expressed in Coq's logic, including both decidable and undecidable ones.

• For example, "being the code of a halting Turing machine" is a perfectly legitimate mathematical property, and we can absolutely represent it as a Coq expression of type nat → Prop.

Since Prop includes both decidable and undecidable properties, we have two choices when we want to formalize a property that happens to be decidable: we can express it as a boolean computation or as a function into Prop. For instance, to claim that a number n is even, we can say either... ... that even n evaluates to true...

... or that there exists some k such that n = double k.

Of course, it would be pretty strange if these two characterizations of evenness did not describe the same set of natural numbers! Fortunately, we can prove that they do...

Now the main theorem:

In view of this theorem, we say that the boolean computation even n is reflected in the truth of the proposition ∃ k, n = double k.

Similarly, to state that two numbers n and m are equal, we can say either

• (1) that n =? m returns true, or
• (2) that n = m.

Again, these two notions are equivalent.

Even when the boolean and propositional formulations of a claim are equivalent from a purely logical perspective, they are often not equivalent from the point of view of convenience for some specific purpose.

For example, we cannot test whether or not a Prop is true in a function definition; as a consequence, the following code fragment is rejected:

An important side benefit of stating facts using booleans is enabling some proof automation through computation with Coq terms, a technique known as proof by reflection. Consider the following statement:

The most direct way to prove this is to give the value of k explicitly.

The proof of the corresponding boolean statement is even simpler, because we don't have to invent the witness: Coq's computation mechanism does it for us!

What is interesting is that, since the two notions are equivalent, we can use the boolean formulation to prove the other one without mentioning the value 500 explicitly:

Another notable difference is that the negation of a "boolean fact" is straightforward to state and prove: simply flip the expected boolean result.

In contrast, propositional negation can be more difficult to work with directly.

Equality provides a complementary example, where it is sometimes easier to work in the propositional world. Knowing that (n =? m) = true is generally of little direct help in the middle of a proof involving n and m; however, if we convert the statement to the equivalent form n = m, we can rewrite with it.

We won't discuss reflection any further for the moment, but it serves as a good example showing the different strengths of booleans and general propositions; being able to cross back and forth between the boolean and propositional worlds will often be convenient in later chapters.

Classical vs. Constructive Logic

The following reasoning principle is not derivable in Coq (though, again, it can consistently be added):

Logics like Coq's, which do not assume the excluded middle, are referred to as constructive logics. More conventional logical systems such as ZFC, in which the excluded middle does hold for arbitrary propositions, are referred to as classical.