Coq's built-in logic is very small: the only primitives are Inductive definitions, universal quantification (∀), and implication (→), while all the other familiar logical connectives — conjunction, disjunction, negation, existential quantification, even equality — can be encoded using just these. This chapter explains the encodings and shows how the tactics we've seen can be used to carry out standard forms of logical reasoning involving these connectives.

Conjunction

The logical conjunction of propositions P and Q can be represented using an Inductive definition with one constructor.

Note that, like the definition of ev in the previous chapter, this definition is parameterized; however, in this case, the parameters are themselves propositions, rather than numbers. The intuition behind this definition is simple: to construct evidence for and P Q, we must provide evidence for P and evidence for Q. More precisely:

• conj p q can be taken as evidence for and P Q if p is evidence for P and q is evidence for Q; and
• this is the only way to give evidence for and P Q — that is, if someone gives us evidence for and P Q, we know it must have the form conj p q, where p is evidence for P and q is evidence for Q.

Since we'll be using conjunction a lot, let's introduce a more familiar-looking infix notation for it.

(The type_scope annotation tells Coq that this notation will be appearing in propositions, not values.) Consider the "type" of the constructor conj:

Notice that it takes 4 inputs — namely the propositions P and Q and evidence for P and Q — and returns as output the evidence of P ∧ Q. Besides the elegance of building everything up from a tiny foundation, what's nice about defining conjunction this way is that we can prove statements involving conjunction using the tactics that we already know. For example, if the goal statement is a conjuction, we can prove it by applying the single constructor conj, which (as can be seen from the type of conj) solves the current goal and leaves the two parts of the conjunction as subgoals to be proved separately.

Let's take a look at the proof object for the above theorem.

Note that the proof is of the form as you'd expect, given the type of conj. Just for convenience, we can use the tactic split as a shorthand for apply conj.

Conversely, the inversion tactic can be used to take a conjunction hypothesis in the context, calculate what evidence must have been used to build it, and add variables representing this evidence to the proof context.

Exercise: 1 star, optional (proj2)

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Once again, we have commented out the Case tactics to make the proof object for this theorem easy to understand. Examining it shows that all that is really happening is taking apart a record containing evidence for P and Q and rebuilding it in the opposite order:

Exercise: 2 stars (and_assoc)

In the following proof, notice how the nested pattern in the inversion breaks the hypothesis H : P ∧ (Q ∧ R) down into HP: P, HQ : Q, and HR : R. Finish the proof from there:

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Exercise: 2 stars (even__ev)

Now we can prove the other direction of the equivalence of even and ev, which we left hanging in chapter Prop. Notice that the left-hand conjunct here is the statement we are actually interested in; the right-hand conjunct is needed in order to make the induction hypothesis strong enough that we can carry out the reasoning in the inductive step. (To see why this is needed, try proving the left conjunct by itself and observe where things get stuck.)

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Exercise: 2 stars, optional (conj_fact)

Construct a proof object demonstrating the following proposition.

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Iff

The handy "if and only if" connective is just the conjunction of two implications.

Exercise: 1 star, optional (iff_properties)

Using the above proof that ↔ is symmetric (iff_sym) as a guide, prove that it is also reflexive and transitive.

Hint: If you have an iff hypothesis in the context, you can use inversion to break it into two separate implications. (Think about why this works.) ☐

Exercise: 2 stars, advanced (beautiful_iff_gorgeous)

We have seen that the families of propositions beautiful and gorgeous actually characterize the same set of numbers. Prove that beautiful n ↔ gorgeous n for all n. Just for fun, write your proof as an explicit proof object, rather than using tactics. (Hint: if you make use of previously defined theorems, you should only need a single line!)

☐ Some of Coq's tactics treat iff statements specially, thus avoiding the need for some low-level manipulation when reasoning with them. In particular, rewrite can be used with iff statements, not just equalities.

Disjunction

Disjunction ("logical or") can also be defined as an inductive proposition.

Consider the "type" of the constructor or_introl:

It takes 3 inputs, namely the propositions P, Q and evidence of P, and returns, as output, the evidence of P ∨ Q. Next, look at the type of or_intror:

It is like or_introl but it requires evidence of Q instead of evidence of P. Intuitively, there are two ways of giving evidence for P ∨ Q:

• give evidence for P (and say that it is P you are giving evidence for — this is the function of the or_introl constructor), or
• give evidence for Q, tagged with the or_intror constructor.

Since P ∨ Q has two constructors, doing inversion on a hypothesis of type P ∨ Q yields two subgoals.

From here on, we'll use the shorthand tactics left and right in place of apply or_introl and apply or_intror.

Exercise: 2 stars, optional (or_commut'')

Try to write down an explicit proof object for or_commut (without using Print to peek at the ones we already defined!).

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Exercise: 2 stars (or_distributes_over_and_2)

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Exercise: 1 star, optional (or_distributes_over_and)

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Relating ∧ and ∨ with andb and orb (advanced)

We've already seen several places where analogous structures can be found in Coq's computational (Type) and logical (Prop) worlds. Here is one more: the boolean operators andb and orb are clearly analogs of the logical connectives ∧ and ∨. This analogy can be made more precise by the following theorems, which show how to translate knowledge about andb and orb's behaviors on certain inputs into propositional facts about those inputs.

Exercise: 2 stars, optional (bool_prop)

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Falsehood

Logical falsehood can be represented in Coq as an inductively defined proposition with no constructors.

Intuition: False is a proposition for which there is no way to give evidence. Since False has no constructors, inverting an assumption of type False always yields zero subgoals, allowing us to immediately prove any goal.

How does this work? The inversion tactic breaks contra into each of its possible cases, and yields a subgoal for each case. As contra is evidence for False, it has no possible cases, hence, there are no possible subgoals and the proof is done. Conversely, the only way to prove False is if there is already something nonsensical or contradictory in the context:

Actually, since the proof of False_implies_nonsense doesn't actually have anything to do with the specific nonsensical thing being proved; it can easily be generalized to work for an arbitrary P:

The Latin ex falso quodlibet means, literally, "from falsehood follows whatever you please." This theorem is also known as the principle of explosion.

Truth

Since we have defined falsehood in Coq, one might wonder whether it is possible to define truth in the same way. We can.

Exercise: 2 stars, advanced (True)

Define True as another inductively defined proposition. (The intution is that True should be a proposition for which it is trivial to give evidence.)

☐ However, unlike False, which we'll use extensively, True is just a theoretical curiosity: it is trivial (and therefore uninteresting) to prove as a goal, and it carries no useful information as a hypothesis.

Negation

The logical complement of a proposition P is written not P or, for shorthand, ~P:

The intuition is that, if P is not true, then anything at all (even False) follows from assuming P.

It takes a little practice to get used to working with negation in Coq. Even though you can see perfectly well why something is true, it can be a little hard at first to get things into the right configuration so that Coq can see it! Here are proofs of a few familiar facts about negation to get you warmed up.

Exercise: 2 stars, advanced (double_neg_inf)

Write an informal proof of double_neg: Theorem: P implies ~~P, for any proposition P. Proof: (* FILL IN HERE *)
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Exercise: 2 stars (contrapositive)

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Exercise: 1 star (not_both_true_and_false)

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Exercise: 1 star, advanced (informal_not_PNP)

Write an informal proof (in English) of the proposition ∀ P : Prop, ~(P ∧ ~P).

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Exercise: 1 star (ev_not_ev_S)

Theorem five_not_even confirms the unsurprising fact that five is not an even number. Prove this more interesting fact:

☐ Note that some theorems that are true in classical logic are not provable in Coq's (constructive) logic. E.g., let's look at how this proof gets stuck...

Exercise: 5 stars, advanced, optional (classical_axioms)

For those who like a challenge, here is an exercise taken from the Coq'Art book (p. 123). The following five statements are often considered as characterizations of classical logic (as opposed to constructive logic, which is what is "built in" to Coq). We can't prove them in Coq, but we can consistently add any one of them as an unproven axiom if we wish to work in classical logic. Prove that these five propositions are equivalent.

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Inequality

Saying x <> y is just the same as saying ~(x = y).

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

Exercise: 2 stars (not_eq_beq_false)

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Exercise: 2 stars, optional (beq_false_not_eq)

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Existential Quantification

Another critical logical connective is existential quantification. We can express it with the following definition:

That is, ex is a family of propositions indexed by a type X and a property P over X. In order to give evidence for the assertion "there exists an x for which the property P holds" we must actually name a witness — a specific value x — and then give evidence for P x, i.e., evidence that x has the property P. For example, consider this existentially quantified proposition:

To prove this proposition, we need to choose a particular number as witness — say, 4 — and give some evidence that that number is even.

Coq's Notation facility can be used to introduce more familiar notation for writing existentially quantified propositions, exactly parallel to the built-in syntax for universally quantified propositions. Instead of writing ex nat ev to express the proposition that there exists some number that is even, for example, we can write ∃ x:nat, ev x. (It is not necessary to understand exactly how the Notation definition works.)

We can use the usual set of tactics for manipulating existentials. For example, to prove an existential, we can apply the constructor ex_intro. Since the premise of ex_intro involves a variable (witness) that does not appear in its conclusion, we need to explicitly give its value when we use apply.

Note, again, that we have to explicitly give the witness. Or, instead of writing apply ex_intro with (witness:=e) all the time, we can use the convenient shorthand ∃ e, which means the same thing.

Conversely, if we have an existential hypothesis in the context, we can eliminate it with inversion. Note the use of the as... pattern to name the variable that Coq introduces to name the witness value and get evidence that the hypothesis holds for the witness. (If we don't explicitly choose one, Coq will just call it witness, which makes proofs confusing.)

Exercise: 1 star, optional (english_exists)

In English, what does the proposition mean?

Complete the definition of the following proof object:

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Exercise: 1 star (dist_not_exists)

Prove that "P holds for all x" and "there is no x for which P does not hold" are equivalent assertions.

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Exercise: 3 stars, optional (not_exists_dist)

(The other direction of this theorem requires the classical "law of the excluded middle".)

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Exercise: 2 stars (dist_exists_or)

Prove that existential quantification distributes over disjunction.

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Equality

Even Coq's equality relation is not built in. It has (roughly) the following inductive definition.

Standard infix notation:

The definition of = is a bit subtle. The way to think about it is that, given a set X, it defines a family of propositions "x is equal to y," indexed by pairs of values (x and y) from X. There is just one way of constructing evidence for members of this family: applying the constructor refl_equal to a type X and a value x : X yields evidence that x is equal to x.

Exercise: 2 stars (leibniz_equality)

The inductive definitions of equality corresponds to Leibniz equality: what we mean when we say "x and y are equal" is that every property on P that is true of x is also true of y.

☐ We can use refl_equal to construct evidence that, for example, 2 = 2. Can we also use it to construct evidence that 1 + 1 = 2? Yes: indeed, it is the very same piece of evidence! The reason is that Coq treats as "the same" any two terms that are convertible according to a simple set of computation rules. These rules, which are similar to those used by Eval simpl, include evaluation of function application, inlining of definitions, and simplification of matches. In tactic-based proofs of equality, the conversion rules are normally hidden in uses of simpl (either explicit or implicit in other tactics such as reflexivity). But you can see them directly at work in the following explicit proof objects:

Inversion, Again (Advanced)

We've seen inversion used with both equality hypotheses and hypotheses about inductively defined propositions. Now that we've seen that these are actually the same thing, we're in a position to take a closer look at how inversion behaves... In general, the inversion tactic

• takes a hypothesis H whose type P is inductively defined, and
• for each constructor C in P's definition,
  • generates a new subgoal in which we assume H was built with C,
  • adds the arguments (premises) of C to the context of the subgoal as extra hypotheses,
  • matches the conclusion (result type) of C against the current goal and calculates a set of equalities that must hold in order for C to be applicable,
  • adds these equalities to the context (and, for convenience, rewrites them in the goal), and
  • if the equalities are not satisfiable (e.g., they involve things like S n = O), immediately solves the subgoal.

Example: If we invert a hypothesis built with or, there are two constructors, so two subgoals get generated. The conclusion (result type) of the constructor (P ∨ Q) doesn't place any restrictions on the form of P or Q, so we don't get any extra equalities in the context of the subgoal. Example: If we invert a hypothesis built with and, there is only one constructor, so only one subgoal gets generated. Again, the conclusion (result type) of the constructor (P ∧ Q) doesn't place any restrictions on the form of P or Q, so we don't get any extra equalities in the context of the subgoal. The constructor does have two arguments, though, and these can be seen in the context in the subgoal. Example: If we invert a hypothesis built with eq, there is again only one constructor, so only one subgoal gets generated. Now, though, the form of the refl_equal constructor does give us some extra information: it tells us that the two arguments to eq must be the same! The inversion tactic adds this fact to the context.

Exercise: 1 star, optional (dist_and_or_eq_implies_and)

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Quantification and Implication

In fact, the built-in logic is even smaller than it appears, since → and ∀ are actually the same primitive! The → notation is actually just a shorthand for a degenerate use of ∀. For example, consider this proposition:

A proof term inhabiting this proposition would be a function with two arguments: a number n and some evidence E that n is beautiful. But the name E for this evidence is not used in the rest of the statement of funny_prop1, so it's a bit silly to bother making up a name for it. We could write it like this instead, using the dummy identifier _ in place of a real name:

Or, equivalently, we can write it in more familiar notation:

In general, "P → Q" is just syntactic sugar for "∀ (_:P), Q".

Relations

A proposition parameterized by a number (such as ev or beautiful) can be thought of as a property — i.e., it defines a subset of nat, namely those numbers for which the proposition is provable. In the same way, a two-argument proposition can be thought of as a relation — i.e., it defines a set of pairs for which the proposition is provable.

We've seen an inductive definition of one fundamental relation: equality. Another useful one is the "less than or equal to" relation on numbers: The following definition should be fairly intuitive. It says that there are two ways to give evidence that one number is less than or equal to another: either observe that they are the same number, or give evidence that the first is less than or equal to the predecessor of the second.

Proofs of facts about <= using the constructors le_n and le_S follow the same patterns as proofs about properties, like ev in chapter Prop. We can apply the constructors to prove <= goals (e.g., to show that 3<=3 or 3<=6), and we can use tactics like inversion to extract information from <= hypotheses in the context (e.g., to prove that ~(2 <= 1).) Here are some sanity checks on the definition. (Notice that, although these are the same kind of simple "unit tests" as we gave for the testing functions we wrote in the first few lectures, we must construct their proofs explicitly — simpl and reflexivity don't do the job, because the proofs aren't just a matter of simplifying computations.)

The "strictly less than" relation n < m can now be defined in terms of le.

Here are a few more simple relations on numbers:

Exercise: 2 stars (total_relation)

Define an inductive binary relation total_relation that holds between every pair of natural numbers.

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Exercise: 2 stars (empty_relation)

Define an inductive binary relation empty_relation (on numbers) that never holds.

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Exercise: 3 stars (R_provability)

We can define three-place relations, four-place relations, etc., in just the same way as binary relations. For example, consider the following three-place relation on numbers:

• Which of the following propositions are provable?
  • R 1 1 2
  • R 2 2 6
• If we dropped constructor c5 from the definition of R, would the set of provable propositions change? Briefly (1 sentence) explain your answer.
• If we dropped constructor c4 from the definition of R, would the set of provable propositions change? Briefly (1 sentence) explain your answer.

(* FILL IN HERE *)
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Exercise: 3 stars, optional (R_fact)

State and prove an equivalent characterization of the relation R. That is, if R m n o is true, what can we say about m, n, and o, and vice versa?

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Additional Exercises

Exercise: 3 stars (all_forallb)

Inductively define a property all of lists, parameterized by a type X and a property P : X → Prop, such that all X P l asserts that P is true for every element of the list l.

Recall the function forallb, from the exercise forall_exists_challenge in chapter Poly:

Using the property all, write down a specification for forallb, and prove that it satisfies the specification. Try to make your specification as precise as possible. Are there any important properties of the function forallb which are not captured by your specification?

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Exercise: 4 stars, advanced (filter_challenge)

One of the main purposes of Coq is to prove that programs match their specifications. To this end, let's prove that our definition of filter matches a specification. Here is the specification, written out informally in English. Suppose we have a set X, a function test: X→bool, and a list l of type list X. Suppose further that l is an "in-order merge" of two lists, l1 and l2, such that every item in l1 satisfies test and no item in l2 satisfies test. Then filter test l = l1. A list l is an "in-order merge" of l1 and l2 if it contains all the same elements as l1 and l2, in the same order as l1 and l2, but possibly interleaved. For example, is an in-order merge of and Your job is to translate this specification into a Coq theorem and prove it. (Hint: You'll need to begin by defining what it means for one list to be a merge of two others. Do this with an inductive relation, not a Fixpoint.)

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Exercise: 5 stars, advanced, optional (filter_challenge_2)

A different way to formally characterize the behavior of filter goes like this: Among all subsequences of l with the property that test evaluates to true on all their members, filter test l is the longest. Express this claim formally and prove it.

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Exercise: 4 stars, advanced (no_repeats)

The following inductively defined proposition...

...gives us a precise way of saying that a value a appears at least once as a member of a list l. Here's a pair of warm-ups about appears_in.

Now use appears_in to define a proposition disjoint X l1 l2, which should be provable exactly when l1 and l2 are lists (with elements of type X) that have no elements in common.

Next, use appears_in to define an inductive proposition no_repeats X l, which should be provable exactly when l is a list (with elements of type X) where every member is different from every other. For example, no_repeats nat [1,2,3,4] and no_repeats bool [] should be provable, while no_repeats nat [1,2,1] and no_repeats bool [true,true] should not be.

Finally, state and prove one or more interesting theorems relating disjoint, no_repeats and ++ (list append).

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Exercise: 2 stars, optional (le_exercises)

Here are a number of facts about the <= and < relations that we are going to need later in the course. The proofs make good practice exercises.

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Exercise: 3 stars (nostutter)

Formulating inductive definitions of predicates is an important skill you'll need in this course. Try to solve this exercise without any help at all (except from your study group partner, if you have one). We say that a list of numbers "stutters" if it repeats the same number consecutively. The predicate "nostutter mylist" means that mylist does not stutter. Formulate an inductive definition for nostutter. (This is different from the no_repeats predicate in the exercise above; the sequence 1,4,1 repeats but does not stutter.)

Make sure each of these tests succeeds, but you are free to change the proof if the given one doesn't work for you. Your definition might be different from mine and still correct, in which case the examples might need a different proof. The suggested proofs for the examples (in comments) use a number of tactics we haven't talked about, to try to make them robust with respect to different possible ways of defining nostutter. You should be able to just uncomment and use them as-is, but if you prefer you can also prove each example with more basic tactics.

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Exercise: 4 stars, advanced (pigeonhole principle)

The "pigeonhole principle" states a basic fact about counting: if you distribute more than n items into n pigeonholes, some pigeonhole must contain at least two items. As is often the case, this apparently trivial fact about numbers requires non-trivial machinery to prove, but we now have enough... First a pair of useful lemmas (we already proved these for lists of naturals, but not for arbitrary lists).

Now define a predicate repeats (analogous to no_repeats in the exercise above), such that repeats X l asserts that l contains at least one repeated element (of type X).

Now here's a way to formalize the pigeonhole principle. List l2 represents a list of pigeonhole labels, and list l1 represents an assignment of items to labels: if there are more items than labels, at least two items must have the same label. You will almost certainly need to use the excluded_middle hypothesis.

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