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Typing of patterns
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Date: -- (:)
From: Pierre Weis <Pierre.Weis@i...>
Subject: Re: Typing of patterns
> I would like to know something about the typing rules of identifiers that
> are bound in pattern matching:

The typing rule specifies that identifiers bound in a pattern get
monomorphic types (or unifiable ``unknowns'', or type variables): it
means that all the type constraints found in the program for this
identifier accumulate on the type of the identifier.

> The problem arises in this line:
>   | Num n as num -> num

> To correct the problem, we can write:
>   | Num n -> Num n
> But isn't this a bit strange that an identifier cannot be used in a context
> where replacing it syntactically with its bound value would be perfectly
> ok?

Yes it is strange, but this is exactly what ML polymorphism specifies:
polymorphism is introduced by the let-binding construct, and only
occurrences of let-bound identifiers can be used with different
types. Since your ``num'' above is not let-bound it is monomorphic and
constraints are accumulating.

In contrast, the Num constructor is considered let-bound (by the type
definition it is defined in), hence Num is assigned a fresh instance
of its type scheme, hence the more general typing of the function.

I first encountered an instance of your problem in a very simple
function, that roughly read as follows:

let ensure_nil = function
  | x :: l -> failwith "Not nil"
  | l -> l;;
val ensure_nil : 'a list -> 'a list = <fun>

Unfortunately here, the result has been assigned the same type as the
argument.  However, writing | [] -> [] instead of | l -> l lead us to
the more general 'a list -> 'b list typing:

let ensure_nil = function
  | x :: l -> failwith "Not nil"
  | [] -> [];;
val ensure_nil : 'a list -> 'b list = <fun>

As in your example, adding a ``as'' synonymous to the pattern argument
once more connects the input and the output type of the function:

let ensure_nil = function
  | x :: l -> failwith "Not nil"
  | [] as l -> l;;
val ensure_nil : 'a list -> 'a list = <fun>

> Or has experience shown that using more general types in such cases leads
> to more programming errors?

No problem in the exact situation you mentioned: num can be safely
generalized because there is nothing bound to its type variable
encapsulated into the value bound to num. More generally this should
be done for type variables of synonymous identifiers that are ``free''
in the type of the pattern that bind them: if the identifier id is
bound in the pattern pat and id has type t, then any type variable 'a
that appears in t can be generalized if it does not occur in the type
of any identifier of the pattern pat (intuitively 'a is free in the
sense that no constraints on identifiers appearing in pat can involve

For instance, in fun [] as l -> e, the type of l is 'a list, and it
can be generalized to all 'a. 'a list during the type-checking of e.
In contrast, in fun x :: l as mylist -> e,
the type 'a list of identifier mylist cannot be generalized, since 'a
is the type of x and also occurs in the type of l. 

> One can, of course, always explicitely restrict the type of an identifier,
> but in the upper example we want to have the opposite, i.e. have it more
> general. One side effect of this problem is that we cannot efficiently
> return the value "as is": we have to construct it again, which may come
> with a not insignficant performance penalty...

You're right, that's the main reason to try to solve this problem.

> Are there any deeper insights behind this rationale? (The given code works
> without problems if polymorphic variants are used instead).

We can observe the same kind of behaviour with polymorphic variants:

type 'a pliste = [`Nil | `Cons of 'a * 'a pliste];;

Connection between input and output:

let (ensure_nil : 'a pliste -> 'b pliste) = function
  | `Cons (x, l) -> failwith "Not nil"
  | l -> l;;
val ensure_nil : 'a pliste -> 'a pliste = <fun>

Deconnection with explicit copy:

let (ensure_nil : 'a pliste -> 'b pliste) = function
  | `Cons (x, l) -> failwith "Not nil"
  | `Nil -> `Nil;;
val ensure_nil : 'a bliste -> 'b bliste = <fun>

However, we get a strange difference in typing since, as you
mentioned, the explicit ``as clause'' does not set up a typing
connection between input and output :

let (ensure_nil : 'a pliste -> 'b pliste) = function
  | `Cons (x, l) -> failwith "Not nil"
  | `Nil as l -> l;;
val ensure_nil : 'a pliste -> 'b pliste = <fun>

Jacques may explain us if the above suggested generalization scheme is
used for identifiers bound in as clauses of patterns (and if not,
which scheme is used ?)...

Conclusion: this strongly suggests to generalize the typing of
synonymous identifiers in patterns, in order to obtain the same typing
assignments for pattern matching on ``closed'' polymorphic variants
and pattern matching on their (semantically equivalent) sum data types

Thank you for your interesting remark and question.

Pierre Weis

INRIA, Projet Cristal,,