# Multiplicities

Idris 2 is based on Quantitative Type Theory (QTT), a core language developed by Bob Atkey and Conor McBride. In practice, this means that every variable in Idris 2 has a quantity associated with it. A quantity is one of:

• `0`, meaning that the variable is erased at run time

• `1`, meaning that the variable is used exactly once at run time

• Unrestricted, which is the same behaviour as Idris 1

We can see the multiplicities of variables by inspecting holes. For example, if we have the following skeleton definition of `append` on vectors…

```append : Vect n a -> Vect m a -> Vect (n + m) a
append xs ys = ?append_rhs
```

…we can look at the hole `append_rhs`:

```Main> :t append_rhs
0 m : Nat
0 a : Type
0 n : Nat
ys : Vect m a
xs : Vect n a
-------------------------------------
append_rhs : Vect (plus n m) a
```

The `0` next to `m`, `a` and `n` mean that they are in scope, but there will be `0` occurrences at run-time. That is, it is guaranteed that they will be erased at run-time.

Multiplicities can be explicitly written in function types as follows:

• `ignoreN : (0 n : Nat) -> Vect n a -> Nat` - this function cannot look at `n` at run time

• `duplicate : (1 x : a) -> (a, a)` - this function must use `x` exactly once (so good luck implementing it, by the way. There is no implementation because it would need to use `x` twice!)

If there is no multiplicity annotation, the argument is unrestricted. If, on the other hand, a name is implicitly bound (like `a` in both examples above) the argument is erased. So, the above types could also be written as:

• `ignoreN : {0 a : _} -> (0 n : Nat) -> Vect n a -> Nat`

• `duplicate : {0 a : _} -> (1 x : a) -> (a, a)`

This section describes what this means for your Idris 2 programs in practice, with several examples. In particular, it describes:

The most important of these for most programs, and the thing you’ll need to know about if converting Idris 1 programs to work with Idris 2, is erasure. The most interesting, however, and the thing which gives Idris 2 much more expressivity, is linearity, so we’ll start there.

## Linearity

The `1` multiplicity expresses that a variable must be used exactly once. By “used” we mean either:

• if the variable is a data type or primitive value, it is pattern matched against, ex. by being the subject of a case statement, or a function argument that is pattern matched against, etc.,

• if the variable is a function, that function is applied (i.e. ran with an argument)

First, we’ll see how this works on some small examples of functions and data types, then see how it can be used to encode resource protocols.

Above, we saw the type of `duplicate`. Let’s try to write it interactively, and see what goes wrong. We’ll start by giving the type and a skeleton definition with a hole

```duplicate : (1 x : a) -> (a, a)
duplicate x = ?help
```

Checking the type of a hole tells us the multiplicity of each variable in scope. If we check the type of `?help` we’ll see that we can’t use `a` at run time, and we have to use `x` exactly once:

```Main> :t help
0 a : Type
1 x : a
-------------------------------------
help : (a, a)
```

If we use `x` for one part of the pair…

```duplicate : (1 x : a) -> (a, a)
duplicate x = (x, ?help)
```

…then the type of the remaining hole tells us we can’t use it for the other:

```Main> :t help
0 a : Type
0 x : a
-------------------------------------
help : a
```

The same happens if we try defining `duplicate x = (?help, x)` (try it!).

In order to avoid parsing ambiguities, if you give an explicit multiplicity for a variable as with the argument to `duplicate`, you need to give it a name too. But, if the name isn’t used in the scope of the type, you can use `_` instead of a name, as follows:

```duplicate : (1 _ : a) -> (a, a)
```

The intution behind multiplicity `1` is that if we have a function with a type of the following form…

```f : (1 x : a) -> b
```

…then the guarantee given by the type system is that if `f x` is used exactly once, then `x` is used exactly once. So, if we insist on trying to define `duplicate`…:

```duplicate x = (x, x)
```

…then Idris will complain:

```pmtype.idr:2:15--8:1:While processing right hand side of Main.duplicate at pmtype.idr:2:1--8:1:
There are 2 uses of linear name x
```

A similar intuition applies for data types. Consider the following types, `Lin` which wraps an argument that must be used once, and `Unr` which wraps an argument with unrestricted use

```data Lin : Type -> Type where
MkLin : (1 _ : a) -> Lin a

data Unr : Type -> Type where
MkUnr : a -> Unr a
```

If `MkLin x` is used once, then `x` is used once. But if `MkUnr x` is used once, there is no guarantee on how often `x` is used. We can see this a bit more clearly by starting to write projection functions for `Lin` and `Unr` to extract the argument

```getLin : (1 _ : Lin a) -> a
getLin (MkLin x) = ?howmanyLin

getUnr : (1 _ : Unr a) -> a
getUnr (MkUnr x) = ?howmanyUnr
```

Checking the types of the holes shows us that, for `getLin`, we must use `x` exactly once (Because the `val` argument is used once, by pattern matching on it as `MkLin x`, and if `MkLin x` is used once, `x` must be used once):

```Main> :t howmanyLin
0 a : Type
1 x : a
-------------------------------------
howmanyLin : a
```

For `getUnr`, however, we still have to use `val` once, again by pattern matching on it, but using `MkUnr x` once doesn’t place any restrictions on `x`. So, `x` has unrestricted use in the body of `getUnr`:

```Main> :t howmanyUnr
0 a : Type
x : a
-------------------------------------
howmanyUnr : a
```

If `getLin` has an unrestricted argument…

```getLin : Lin a -> a
getLin (MkLin x) = ?howmanyLin
```

…then `x` is unrestricted in `howmanyLin`:

```Main> :t howmanyLin
0 a : Type
x : a
-------------------------------------
howmanyLin : a
```

Remember the intuition from the type of `MkLin` is that if `MkLin x` is used exactly once, `x` is used exactly once. But, we didn’t say that `MkLin x` would be used exactly once, so there is no restriction on `x`.

### Resource protocols

One way to take advantage of being able to express linear usage of an argument is in defining resource usage protocols, where we can use linearity to ensure that any unique external resource has only one instance, and we can use functions which are linear in their arguments to represent state transitions on that resource. A door, for example, can be in one of two states, `Open` or `Closed`

```data DoorState = Open | Closed

data Door : DoorState -> Type where
MkDoor : (doorId : Int) -> Door st
```

(Okay, we’re just pretending here - imagine the `doorId` is a reference to an external resource!)

We can define functions for opening and closing the door which explicitly describe how they change the state of a door, and that they are linear in the door

```openDoor : (1 d : Door Closed) -> Door Open
closeDoor : (1 d : Door Open) -> Door Closed
```

Remember, the intuition is that if `openDoor d` is used exactly once, then `d` is used exactly once. So, provided that a door `d` has multiplicity `1` when it’s created, we know that once we call `openDoor` on it, we won’t be able to use `d` again. Given that `d` is an external resource, and `openDoor` has changed its state, this is a good thing!

We can ensure that any door we create has multiplicity `1` by creating them with a `newDoor` function with the following type

```newDoor : (1 p : (1 d : Door Closed) -> IO ()) -> IO ()
```

That is, `newDoor` takes a function, which it runs exactly once. That function takes a door, which is used exactly once. We’ll run it in `IO` to suggest that there is some interaction with the outside world going on when we create the door. Since the multiplicity `1` means the door has to be used exactly once, we need to be able to delete the door when we’re finished

```deleteDoor : (1 d : Door Closed) -> IO ()
```

So an example correct door protocol usage would be

```doorProg : IO ()
doorProg
= newDoor \$ \d =>
let d' = openDoor d
d'' = closeDoor d' in
deleteDoor d''
```

It’s instructive to build this program interactively, with holes along the way, and see how the multiplicities of `d`, `d'` etc change. For example

```doorProg : IO ()
doorProg
= newDoor \$ \d =>
let d' = openDoor d in
?whatnow
```

Checking the type of `?whatnow` shows that `d` is now spent, but we still have to use `d'` exactly once:

```Main> :t whatnow
0 d : Door Closed
1 d' : Door Open
-------------------------------------
whatnow : IO ()
```

Note that the `0` multiplicity for `d` means that we can still talk about it - in particular, we can still reason about it in types - but we can’t use it again in a relevant position in the rest of the program. It’s also fine to shadow the name `d` throughout

```doorProg : IO ()
doorProg
= newDoor \$ \d =>
let d = openDoor d
d = closeDoor d in
deleteDoor d
```

If we don’t follow the protocol correctly - create the door, open it, close it, then delete it - then the program won’t type check. For example, we can try not to delete the door before finishing

```doorProg : IO ()
doorProg
= newDoor \$ \d =>
let d' = openDoor d
d'' = closeDoor d' in
putStrLn "What could possibly go wrong?"
```

This gives the following error:

```Door.idr:15:19--15:38:While processing right hand side of Main.doorProg at Door.idr:13:1--17:1:
There are 0 uses of linear name d''
```

There’s a lot more to be said about the details here! But, this shows at a high level how we can use linearity to capture resource usage protocols at the type level. If we have an external resource which is guaranteed to be used linearly, like `Door`, we don’t need to run operations on that resource in an `IO` monad, since we’re already enforcing an ordering on operations and don’t have access to any out of date resource states. This is similar to the way interactive programs work in the Clean programming language, and in fact is how `IO` is implemented internally in Idris 2, with a special `%World` type for representing the state of the outside world that is always used linearly

```public export
data IORes : Type -> Type where
MkIORes : (result : a) -> (1 x : %World) -> IORes a

export
data IO : Type -> Type where
MkIO : (1 fn : (1 x : %World) -> IORes a) -> IO a
```

Having multiplicities in the type system raises several interesting questions, such as:

• Can we use linearity information to inform memory management and, for example, have type level guarantees about functions which will not need to perform garbage collection?

• How should multiplicities be incorporated into interfaces such as `Functor`, `Applicative` and `Monad`?

• If we have `0`, and `1` as multiplicities, why stop there? Why not have `2`, `3` and more (like Granule)

• Even without all of that, what can we do now?

## Erasure

The `1` multiplicity give us many possibilities in the kinds of properties we can express. But, the `0` multiplicity is perhaps more important in that it allows us to be precise about which values are relevant at run time, and which are compile time only (that is, which are erased). Using the `0` multiplicity means a function’s type now tells us exactly what it needs at run time.

For example, in Idris 1 you could get the length of a vector as follows

```vlen : Vect n a -> Nat
vlen {n} xs = n
```

This is fine, since it runs in constant time, but the trade off is that `n` has to be available at run time, so at run time we always need the length of the vector to be available if we ever call `vlen`. Idris 1 can infer whether the length is needed, but there’s no easy way for a programmer to be sure.

In Idris 2, we need to state explicitly that `n` is needed at run time

```vlen : {n : Nat} -> Vect n a -> Nat
vlen xs = n
```

(Incidentally, also note that in Idris 2, names bound in types are also available in the definition without explicitly rebinding them.)

This also means that when you call `vlen`, you need the length available. For example, this will give an error

```sumLengths : Vect m a -> Vect n a —> Nat
sumLengths xs ys = vlen xs + vlen ys
```

Idris 2 reports:

```vlen.idr:7:20--7:28:While processing right hand side of Main.sumLengths at vlen.idr:7:1--10:1:
m is not accessible in this context
```

This means that it needs to use `m` as an argument to pass to `vlen xs`, where it needs to be available at run time, but `m` is not available in `sumLengths` because it has multiplicity `0`.

We can see this more clearly by replacing the right hand side of `sumLengths` with a hole…

```sumLengths : Vect m a -> Vect n a -> Nat
sumLengths xs ys = ?sumLengths_rhs
```

…then checking the hole’s type at the REPL:

```Main> :t sumLengths_rhs
0 n : Nat
0 a : Type
0 m : Nat
ys : Vect n a
xs : Vect m a
-------------------------------------
sumLengths_rhs : Nat
```

Instead, we need to give bindings for `m` and `n` with unrestricted multiplicity

```sumLengths : {m, n : _} -> Vect m a -> Vect n a —> Nat
sumLengths xs ys = vlen xs + vlen xs
```

Remember that giving no multiplicity on a binder, as with `m` and `n` here, means that the variable has unrestricted usage.

If you’re converting Idris 1 programs to work with Idris 2, this is probably the biggest thing you need to think about. It is important to note, though, that if you have bound implicits, such as…

```excitingFn : {t : _} -> Coffee t -> Moonbase t
```

…then it’s a good idea to make sure `t` really is needed, or performance might suffer due to the run time building the instance of `t` unnecessarily!

One final note on erasure: it is an error to try to pattern match on an argument with multiplicity `0`, unless its value is inferrable from elsewhere. So, the following definition is rejected

```badNot : (0 x : Bool) -> Bool
```

This is rejected with the error:

```badnot.idr:2:1--3:1:Attempt to match on erased argument False in
```

The following, however, is fine, because in `sNot`, even though we appear to match on the erased argument `x`, its value is uniquely inferrable from the type of the second argument

```data SBool : Bool -> Type where
SFalse : SBool False
STrue  : SBool True

sNot : (0 x : Bool) -> SBool x -> Bool
sNot False SFalse = True
sNot True  STrue  = False
```

Experience with Idris 2 so far suggests that, most of the time, as long as you’re using unbound implicits in your Idris 1 programs, they will work without much modification in Idris 2. The Idris 2 type checker will point out where you require an unbound implicit argument at run time - sometimes this is both surprising and enlightening!

## Pattern Matching on Types

One way to think about dependent types is to think of them as “first class” objects in the language, in that they can be assigned to variables, passed around and returned from functions, just like any other construct. But, if they’re truly first class, we should be able to pattern match on them too! Idris 2 allows us to do this. For example

```showType : Type -> String
showType Int = "Int"
showType (List a) = "List of " ++ showType a
showType _ = "something else"
```

We can try this as follows:

```Main> showType Int
"Int"
Main> showType (List Int)
"List of Int"
Main> showType (List Bool)
"List of something else"
```

Pattern matching on function types is interesting, because the return type may depend on the input value. For example, let’s add a case to `showType`

```showType (Nat -> a) = ?help
```

Inspecting the type of `help` tells us:

```Main> :t help
a : Nat -> Type
-------------------------------------
help : String
```

So, the return type `a` depends on the input value of type `Nat`, and we’ll need to come up with a value to use `a`, for example

```showType (Nat -> a) = "Function from Nat to " ++ showType (a Z)
```

Note that multiplicities on the binders, and the ability to pattern match on non-erased types mean that the following two types are distinct

```id : a -> a
notId : {a : Type} -> a -> a
```

In the case of `notId`, we can match on `a` and get a function which is certainly not the identity function

```notId {a = Integer} x = x + 1
notId x = x
```
```Main> notId 93
94
Main> notId "???"
"???"
```

There is an important consequence of being able to distinguish between relevant and irrelevant type arguments, which is that a function is only parametric in `a` if `a` has multiplicity `0`. So, in the case of `notId`, `a` is not a parameter, and so we can’t draw any conclusions about the way the function will behave because it is polymorphic, because the type tells us it might pattern match on `a`.

On the other hand, it is merely a coincidence that, in non-dependently typed languages, types are irrelevant and get erased, and values are relevant and remain at run time. Idris 2, being based on QTT, allows us to make the distinction between relevant and irrelevant arguments precise. Types can be relevant, values (such as the `n` index to vectors) can be irrelevant.

For more details on multiplicities, see Idris 2: Quantitative Type Theory in Action.