ProofObjectsThe Curry-Howard Correspondence

"Algorithms are the computational content of proofs." —Robert Harper

Require Export IndProp.

We have seen that Coq has mechanisms both for programming, using inductive data types like nat or list and functions over these types, and for proving properties of these programs, using inductive propositions (like ev), implication, universal quantification, and the like. So far, we have mostly treated these mechanisms as if they were quite separate, and for many purposes this is a good way to think. But we have also seen hints that Coq's programming and proving facilities are closely related. For example, the keyword Inductive is used to declare both data types and propositions, and is used both to describe the type of functions on data and logical implication. This is not just a syntactic accident! In fact, programs and proofs in Coq are almost the same thing. In this chapter we will study how this works.
We have already seen the fundamental idea: provability in Coq is represented by concrete evidence. When we construct the proof of a basic proposition, we are actually building a tree of evidence, which can be thought of as a data structure.
If the proposition is an implication like A B, then its proof will be an evidence transformer: a recipe for converting evidence for A into evidence for B. So at a fundamental level, proofs are simply programs that manipulate evidence.
Question: If evidence is data, what are propositions themselves?
Answer: They are types!
Look again at the formal definition of the ev property.

Print ev.
(* ==>
  Inductive ev : nat -> Prop :=
    | ev_0 : ev 0
    | ev_SS : forall n, ev n -> ev (S (S n)).
*)


Suppose we introduce an alternative pronunciation of ":". Instead of "has type," we can say "is a proof of." For example, the second line in the definition of ev declares that ev_0 : ev 0. Instead of "ev_0 has type ev 0," we can say that "ev_0 is a proof of ev 0."
This pun between types and propositions — between : as "has type" and : as "is a proof of" or "is evidence for" — is called the Curry-Howard correspondence. It proposes a deep connection between the world of logic and the world of computation:
                 propositions  ~  types
                 proofs        ~  data values
Many useful insights follow from this connection. To begin with, it gives us a natural interpretation of the type of the ev_SS constructor:

Check ev_SS.
(* ===> ev_SS : forall n,
                  ev n ->
                  ev (S (S n)) *)


This can be read "ev_SS is a constructor that takes two arguments — a number n and evidence for the proposition ev n — and yields evidence for the proposition ev (S (S n))."
Now let's look again at a previous proof involving ev.

Theorem ev_4 : ev 4.
Proof.
  apply ev_SS. apply ev_SS. apply ev_0. Qed.

As with ordinary data values and functions, we can use the Print command to see the proof object that results from this proof script.

Print ev_4.
(* ===> ev_4 = ev_SS 2 (ev_SS 0 ev_0)
     : ev 4  *)


As a matter of fact, we can also write down this proof object directly, without the need for a separate proof script:

Check (ev_SS 2 (ev_SS 0 ev_0)).
(* ===> ev 4 *)

The expression ev_SS 2 (ev_SS 0 ev_0) can be thought of as instantiating the parameterized constructor ev_SS with the specific arguments 2 and 0 plus the corresponding proof objects for its premises ev 2 and ev 0. Alternatively, we can think of ev_SS as a primitive "evidence constructor" that, when applied to a particular number, wants to be further applied to evidence that that number is even; its type,
    nev n  ev (S (S n)),
expresses this functionality, in the same way that the polymorphic type X, list X expresses the fact that the constructor nil can be thought of as a function from types to empty lists with elements of that type.
You may recall (as seen in the Logic chapter) that we can use function application syntax to instantiate universally quantified variables in lemmas, as well as to supply evidence for assumptions that these lemmas impose. For instance,

Theorem ev_4': ev 4.
Proof.
  apply (ev_SS 2 (ev_SS 0 ev_0)).
Qed.

We can now see that this feature is a trivial consequence of the status the Coq grants to proofs and propositions: Lemmas and hypotheses can be combined in expressions (i.e., proof objects) according to the same basic rules used for programs in the language.

Proof Scripts

The proof objects we've been discussing lie at the core of how Coq operates. When Coq is following a proof script, what is happening internally is that it is gradually constructing a proof object — a term whose type is the proposition being proved. The tactics between Proof and Qed tell it how to build up a term of the required type. To see this process in action, let's use the Show Proof command to display the current state of the proof tree at various points in the following tactic proof.

Theorem ev_4'' : ev 4.
Proof.
  Show Proof.
  apply ev_SS.
  Show Proof.
  apply ev_SS.
  Show Proof.
  apply ev_0.
  Show Proof.
Qed.

At any given moment, Coq has constructed a term with some "holes" (indicated by ?1, ?2, and so on), and it knows what type of evidence is needed at each hole.
Each of the holes corresponds to a subgoal, and the proof is finished when there are no more subgoals. At this point, the evidence we've built stored in the global context under the name given in the Theorem command.
Tactic proofs are useful and convenient, but they are not essential: in principle, we can always construct the required evidence by hand, as shown above. Then we can use Definition (rather than Theorem) to give a global name directly to a piece of evidence.

Definition ev_4''' : ev 4 :=
  ev_SS 2 (ev_SS 0 ev_0).

All these different ways of building the proof lead to exactly the same evidence being saved in the global environment.

Print ev_4.
(* ===> ev_4    =   ev_SS 2 (ev_SS 0 ev_0) : ev 4 *)
Print ev_4'.
(* ===> ev_4'   =   ev_SS 2 (ev_SS 0 ev_0) : ev 4 *)
Print ev_4''.
(* ===> ev_4''  =   ev_SS 2 (ev_SS 0 ev_0) : ev 4 *)
Print ev_4'''.
(* ===> ev_4''' =   ev_SS 2 (ev_SS 0 ev_0) : ev 4 *)

Exercise: 1 star (eight_is_even)

Give a tactic proof and a proof object showing that ev 8.

Theorem ev_8 : ev 8.
Proof.
  (* FILL IN HERE *) Admitted.

Definition ev_8' : ev 8 :=
  (* FILL IN HERE *) admit.

Quantifiers, Implications, Functions

In Coq's computational universe (where data structures and programs live), there are two sorts of values with arrows in their types: constructors introduced by Inductive-ly defined data types, and functions.
Similarly, in Coq's logical universe (where we carry out proofs), there are two ways of giving evidence for an implication: constructors introduced by Inductive-ly defined propositions, and... functions!
For example, consider this statement:

Theorem ev_plus4 : n, ev n ev (4 + n).
Proof.
  intros n H. simpl.
  apply ev_SS.
  apply ev_SS.
  apply H.
Qed.

What is the proof object corresponding to ev_plus4?
We're looking for an expression whose type is n, ev n ev (4 + n) — that is, a function that takes two arguments (one number and a piece of evidence) and returns a piece of evidence! Here it is:

Definition ev_plus4' : n, ev n ev (4 + n) :=
  fun (n : nat) ⇒ fun (H : ev n) ⇒
    ev_SS (S (S n)) (ev_SS n H).

Check ev_plus4'.
(* ===> ev_plus4' : forall n : nat, ev n -> ev (4 + n) *)

Recall that fun n blah means "the function that, given n, yields blah," and that Coq treats 4 + n and S (S (S (S n))) as synonyms. Another equivalent way to write this definition is:

Definition ev_plus4'' (n : nat) (H : ev n) : ev (4 + n) :=
  ev_SS (S (S n)) (ev_SS n H).

Check ev_plus4''.
(* ===> ev_plus4'' : forall n : nat, ev n -> ev (4 + n) *)

When we view the proposition being proved by ev_plus4 as a function type, one aspect of it may seem a little unusual. The second argument's type, ev n, mentions the value of the first argument, n. While such dependent types are not found in conventional programming languages, they can be useful in programming too, as recent work in the functional programming community demonstrates.
Notice that both implication () and quantification () correspond to functions on evidence. In fact, they are really the same thing: is just a shorthand for a degenerate use of where there is no dependency, i.e., no need to give a name to the type on the LHS of the arrow.
For example, consider this proposition:

Definition ev_plus2 : Prop :=
  n, (E : ev n), ev (n + 2).

A proof term inhabiting this proposition would be a function with two arguments: a number n and some evidence E that n is even. But the name E for this evidence is not used in the rest of the statement of ev_plus2, 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:

Definition ev_plus2' : Prop :=
  n, (_ : ev n), ev (n + 2).

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

Definition ev_plus2'' : Prop :=
  n, ev n ev (n + 2).

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

Connectives as Inductive Types

Inductive definitions are powerful enough to express most of the connectives and quantifiers we have seen so far. Indeed, only universal quantification (and thus implication) is built into Coq; all the others are defined inductively. We study these definitions in this section.

Module Props.

Conjunction

To prove that P Q holds, we must present evidence for both P and Q. Thus, it makes sense to define a proof object for P Q as consisting of a pair of two proofs: one for P and another one for Q. This leads to the following definition.

Module And.

Inductive and (P Q : Prop) : Prop :=
| conj : P Q and P Q.

End And.

Notice the similarity with the definition of the prod type, given in chapter Poly; the only difference is that prod takes Type arguments, whereas and takes Prop arguments.

Print prod.
(* ===>
   Inductive prod (X Y : Type) : Type :=
   | pair : X -> Y -> X * Y. *)


This should clarify why destruct and intros patterns can be used on a conjunctive hypothesis. Case analysis allows us to consider all possible ways in which P Q was proved — here just one (the conj constructor). Similarly, the split tactic actually works for any inductively defined proposition with only one constructor. In particular, it works for and:

Lemma and_comm : P Q : Prop, P Q Q P.
Proof.
  intros P Q. split.
  - intros [HP HQ]. split.
    + apply HQ.
    + apply HP.
  - intros [HP HQ]. split.
    + apply HQ.
    + apply HP.
Qed.

This shows why the inductive definition of and can be manipulated by tactics as we've been doing. We can also use it to build proofs directly, using pattern-matching. For instance:

Definition and_comm'_aux P Q (H : P Q) :=
  match H with
  | conj HP HQconj HQ HP
  end.

Definition and_comm' P Q : P Q Q P :=
  conj (and_comm'_aux P Q) (and_comm'_aux Q P).

Exercise: 2 stars, optional (conj_fact)

Construct a proof object demonstrating the following proposition.

Definition conj_fact : P Q R, P Q Q R P R :=
  (* FILL IN HERE *) admit.

Disjunction

The inductive definition of disjunction uses two constructors, one for each side of the disjunct:

Module Or.

Inductive or (P Q : Prop) : Prop :=
| or_introl : P or P Q
| or_intror : Q or P Q.

End Or.

This declaration explains the behavior of the destruct tactic on a disjunctive hypothesis, since the generated subgoals match the shape of the or_introl and or_intror constructors.
Once again, we can also directly write proof objects for theorems involving or, without resorting to tactics.

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!).

Definition or_comm : P Q, P Q Q P :=
  (* FILL IN HERE *) admit.

Existential Quantification

To give evidence for an existential quantifier, we package a witness x together with a proof that x satisfies the property P:

Module Ex.

Inductive ex {A : Type} (P : A Prop) : Prop :=
| ex_intro : x : A, P x ex P.

End Ex.

This may benefit from a little unpacking. The core definition is for a type former ex that can be used to build propositions of the form ex P, where P itself is a function from witness values in the type A to propositions. The ex_intro constructor then offers a way of constructing evidence for ex P, given a witness x and a proof of P x.
The more familiar form x, P x desugars to an expression involving ex:

Check ex (fun nev n).
(* ===> exists n : nat, ev n
        : Prop *)


Here's how to define an explicit proof object involving ex:

Definition some_nat_is_even : n, ev n :=
  ex_intro ev 4 (ev_SS 2 (ev_SS 0 ev_0)).

Exercise: 2 stars, optional (ex_ev_Sn)

Complete the definition of the following proof object:

Definition ex_ev_Sn : ex (fun nev (S n)) :=
(* FILL IN HERE *) admit.

True and False

The inductive definition of the True proposition is simple:

Inductive True : Prop :=
  I : True.

It has one constructor (so every proof of True is the same, so being given a proof of True is not informative.)
False is equally simple — indeed, so simple it may look syntactically wrong at first glance!

Inductive False : Prop :=.

That is, False is an inductive type with no constructors — i.e., no way to build evidence for it.

End Props.

Programming with Tactics

If we can build proofs by giving explicit terms rather than executing tactic scripts, you may be wondering whether we can build programs using tactics rather than explicit terms. Naturally, the answer is yes!

Definition add1 : nat nat.
intro n.
Show Proof.
apply S.
Show Proof.
apply n. Defined.

Print add1.
(* ==>
    add1 = fun n : nat => S n
         : nat -> nat
*)


Compute add1 2.
(* ==> 3 : nat *)

Notice that we terminate the Definition with a . rather than with := followed by a term. This tells Coq to enter proof scripting mode to build an object of type nat nat. Also, we terminate the proof with Defined rather than Qed; this makes the definition transparent so that it can be used in computation like a normally-defined function. (Qed-defined objects are opaque during computation.)
This feature is mainly useful for writing functions with dependent types, which we won't explore much further in this book. But it does illustrate the uniformity and orthogonality of the basic ideas in Coq.

Equality

Even Coq's equality relation is not built in. It has the following inductive definition. (Actually, the definition in the standard library is a small variant of this, which gives an induction principle that is slightly easier to use.)

Module MyEquality.

Inductive eq {X:Type} : X X Prop :=
| eq_refl : x, eq x x.

Notation "x = y" := (eq x y)
                    (at level 70, no associativity)
                    : type_scope.

The way to think about this definition 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 each member of this family: applying the constructor eq_refl to a type X and a value x : X yields evidence that x is equal to x.

Exercise: 2 stars (leibniz_equality)

The inductive definition 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.

Lemma leibniz_equality : (X : Type) (x y: X),
  x = y P:XProp, P x P y.
Proof.
(* FILL IN HERE *) Admitted.
We can use eq_refl to construct evidence that, for example, 2 = 2. Can we also use it to construct evidence that 1 + 1 = 2? Yes, we can. 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 Compute, include evaluation of function application, inlining of definitions, and simplification of matches.

Lemma four: 2 + 2 = 1 + 3.
Proof.
  apply eq_refl.
Qed.

The reflexivity tactic that we have used to prove equalities up to now is essentially just short-hand for apply refl_equal.
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:

Definition four' : 2 + 2 = 1 + 3 :=
  eq_refl 4.

Definition singleton : (X:Set) (x:X), []++[x] = x::[] :=
  fun (X:Set) (x:X) ⇒ eq_refl [x].

End MyEquality.

Definition quiz6 : x, x + 3 = 4
  := ex_intro (fun z ⇒ (z + 3 = 4)) 1 (refl_equal 4).

Inversion, Again

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.

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