# The Maybe monad in Coq

The Maybe monad is pretty useful whenever you’re dealing with partial functions, especially in a language like Coq (or rather Gallina if you want to be exact) which requires all functions to be total.

The first idea is a to lift the codomain of a partial function $f : A \rightharpoonup B$ by including a special value $\bot$ such that $f^\prime : A \rightarrow B \cup \{\bot\}$ is a total function. In Haskell you’ll find that the type Maybe B plays the role of the lifted codomain $B \cup \{\bot\}$. But enough mathematical foreplay, let’s get to it:

data Maybe t = Just t | Nothing   instance Monad Maybe where return = Just (Just x) >>= k = k x Nothing >>= k = Nothing

Coq’s standard library also includes the datatype Maybe, but there it is called option.

Inductive option (A : Type) := | Some : A -> option A | None : option A.

In order to make working with this type a little more enjoyable and the resulting code more readable, we declare the type argument of its constructors to be implicit which means that they should be inferred from the context:

Arguments Some [A] _. Arguments None [A].

Now we can define the monad operations return (which we’ll call ret to avoid confusion with the keyword of the same name) and >>= (which is also called bind).

Definition ret {A : Type} (x : A) := Some x.   Definition bind {A B : Type} (a : option A) (f : A -> option B) : option B := match a with | Some x => f x | None => None end.

We can also define the usual notations for use in monadic code. First, the familiar infix operator for the bind operation.

Notation "A >>= F" := (bind A F) (at level 42, left associativity).

Followed by the beloved do-notation from Haskell.

Notation "'do' X <- A ; B" := (bind A (fun X => B)) (at level 200, X ident, A at level 100, B at level 200).

We could stop here, but because Coq is also a thereom prover and not only a fancy dependently typed functional programming language, we go right on and prove the three laws that each monad has to satisfy in order to be called a monad at all. Recall from

• left identity: $\mathrm{return}\;a \gg\!= f = f\;a$
• right identity: $a \gg\!= \mathrm{return} = a$
• associativity: $(a \gg\!= f) \gg\!= g = a \gg\!= (\lambda x\,.\, f\;x \gg\!= g)$

These laws correspond directly to the Coq lemmata mon_left_id, mon_right_id, and
mon_assoc below. As you can see, the proofs in all three cases are rather trivial.

Lemma mon_left_id : forall (A B : Type) (a : A) (f : A -> option B), ret a >>= f = f a. intros. reflexivity. Qed.   Lemma mon_right_id : forall (A : Type) (a : option A), a >>= ret = a. intros. induction a; repeat reflexivity. Qed.   Lemma mon_assoc : forall (A B C : Type) (a : option A) (f : A -> option B) (g : B -> option C), (a >>= f) >>= g = a >>= (fun x => f x >>= g). intros. induction a; repeat reflexivity. Qed.

That’s is, have some fun with the Maybe monad in Coq!

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### 8 Responses to The Maybe monad in Coq

1. mikerosss says:

nice post. thanks.

2. erus says:

Definition ret {A : Type} (x : A) := Some x. is a type error

• Thomas Strathmann says:

This definition is perfectly typeable as it is. Perhaps you could elaborate on the specifics of your problem. Perhaps you are using an older Coq version than 8.3?

• Chris Perivolaropoulos says:

Same here. I am getting:

Error:
In environment
A : Type
x : A
The term “x” has type “A” while it is expected to have type “Type”.

• Thomas Strathmann says:

You probably forgot to declare the arguments to the constructors of option implicit.

3. mdgeorge says:

Shouldn’t bind read “… | Some x -> Some f x …”?

• Thomas Strathmann says:

No, Some f x is not even type correct in that context and Some (f x) has type option (option B) but this is not what we want. Actually, the code I presented is just a transliteration from Haskell to Gallina with the proofs added just because we can. 😉

But your comment made me reexamine the code and I found that I did forget to set the argument to the constructors of option as implicit which breaks the definition of bind. Should work now (tested with the current version Coq 8.4).

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