Pattern matching using information from theorems

Question

I have the following question, look in the code.

(* Suppose we have type A *)
Variable A : Type.
Interpreting this declaration as if a global
declaration prefixed by "Local", i.e. as a global
declaration which shall not be available without
qualification when imported. [local-declaration,scope]

(* Also we have a function that returns the type (option A) *)
Variable f_opt : A -> option A.
Interpreting this declaration as if a global
declaration prefixed by "Local", i.e. as a global
declaration which shall not be available without
qualification when imported. [local-declaration,scope]

(* Then, I can prove that this function always returns something: *)
Theorem always_some : forall x, exists y, f_opt x = Some y.
forall x : A, exists y : A, f_opt x = Some y
Admitted.

(* Or, equivalently: *)
Theorem always_not_none : forall x, f_opt x <> None.
forall x : A, f_opt x <> None
Admitted.

Now I would like to get a version of f_opt that always returns a value of type A. Something like this:

Definition f x : A :=
  match f_opt x with
  | Some y => y
  end.
Non exhaustive pattern-matching: no clause found for pattern
None

I understand that I need to do some kind of work with types, but I do not understand what exactly i should do.

Answer (Arthur Azevedo De Amorim)

In Coq's underlying theory, every pattern matching must be exhaustive -- that is, it must explicitly consider all the constructors of the inductive type in question. This is why you are getting the error message you saw.

How do we get around this restriction? There are a few solutions. First, let's see how to convince Coq that the None branch can never occur. For this, we will use your always_not_none theorem:

Definition f x : A :=
  match f_opt x as res return res <> None -> A with
  | Some y => fun _ => y
  | None => fun H => match H eq_refl with end
  end (always_not_none x).

This code might look strange at first sight, but it almost performs the pattern match you want. To explain to Coq that the None case never arises, it combines always_not_none with the fact that f_opt x = None on that branch to derive a contradiction. This is the H eq_refl term on that branch. Then, the match on that contradiction suffices to convince Coq that the branch is spurious. A bit more formally, because False, the contradictory proposition, is defined without any constructors, when we match on a term of type False, there are no branches to deal with, and the entire expression can return any type that we want -- in this case, A.

What is strange about this code is the type annotations on the match, and that it returns a function instead of something of type A directly. This is done because of how dependent pattern match works in Coq: whenever we want to make use of information that we obtain from being in a particular branch of a match (here, that f_opt x is equal to None in that branch), we must explicitly make the match return a function -- what Adam Chlipala calls the convoy pattern. This is done so that Coq knows where you plan to use that extra information and check that it is done correctly. Here, we use that f_opt x is None to feed the hypothesis needed by always_not_none x to derive a contradiction.

Although this will solve your problem, I would generally advise you against doing it this way. For instance, if you know that your A type is inhabited by some element a : A, then you can simply make f return a on that branch. This has the benefit of avoiding mentioning proofs inside your functions, which often gets in the way when simplifying and rewriting terms.

Answer (Daniel Schepler)

Using Coq's Program module, you can write an exhaustive pattern match, but annotate that some branches should be impossible to reach and then later provide the proof that this is the case:

Require Import Program.

Program Definition f x : A :=
  match f_opt x with
  | Some a => a
  | None => !
  end.
Next Obligation.
x: A
Heq_anonymous: None = f_opt x
False
  destruct (always_some x).
x: A
Heq_anonymous: None = f_opt x
x0: A
H: f_opt x = Some x0
False congruence.
Qed.

(The Program module does a lot of the work behind the scenes that in a complete explicit definition you would have to write using the "convoy pattern". Do be aware, however, that sometimes Program tends to generate lots of dependencies on JMeq and the axiom JMeq_eq when dependent types are involved, even when it might not be necessary.)

A: To check if f depends on any additional axioms one can use Print Assumptions f. command.

Answer (Vinz)

You will need to put your existential proof always_not_none in Set or Type to do so:

Theorem always_some : forall x, {y : A & f_opt x = Some y}.
forall x : A, {y : A & f_opt x = Some y}
Admitted.

Then you can do the following (or use refine or Program):

Definition f (x : A) : A :=
  let s := always_some x in let (x0, _) := s in x0.