-So the integer division operation presupposes that its second argument
-(the divisor) is not zero, upon pain of presupposition failure.
-Here's what my OCAML interpreter says:
-
- # 12/0;;
- Exception: Division_by_zero.
-
-So we want to explicitly allow for the possibility that
-division will return something other than a number.
-We'll use OCAML's option type, which works like this:
-
- # type 'a option = None | Some of 'a;;
- # None;;
- - : 'a option = None
- # Some 3;;
- - : int option = Some 3
-
-So if a division is normal, we return some number, but if the divisor is
-zero, we return None:
-
-<pre>
-let div (x:int) (y:int) =
- match y with 0 -> None |
- _ -> Some (x / y);;
-
-(*
-val div : int -> int -> int option = fun
-# div 12 3;;
-- : int option = Some 4
-# div 12 0;;
-- : int option = None
-# div (div 12 3) 2;;
-Characters 4-14:
- div (div 12 3) 2;;
- ^^^^^^^^^^
-Error: This expression has type int option
- but an expression was expected of type int
-*)
-</pre>
-
-This starts off well: dividing 12 by 3, no problem; dividing 12 by 0,
-just the behavior we were hoping for. But we want to be able to use
-the output of the safe division function as input for further division
-operations. So we have to jack up the types of the inputs:
-
-<pre>
-let div (x:int option) (y:int option) =
- match y with None -> None |
- Some 0 -> None |
- Some n -> (match x with None -> None |
- Some m -> Some (m / n));;
-
-(*
-val div : int option -> int option -> int option = <fun>
-# div (Some 12) (Some 4);;
-- : int option = Some 3
-# div (Some 12) (Some 0);;
-- : int option = None
-# div (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-Beautiful, just what we need: now we can try to divide by anything we
-want, without fear that we're going to trigger any system errors.
-
-I prefer to line up the `match` alternatives by using OCAML's
-built-in tuple type:
-
-<pre>
-let div (x:int option) (y:int option) =
- match (x, y) with (None, _) -> None |
- (_, None) -> None |
- (_, Some 0) -> None |
- (Some m, Some n) -> Some (m / n);;
-</pre>
-
-So far so good. But what if we want to combine division with
-other arithmetic operations? We need to make those other operations
-aware of the possibility that one of their arguments will trigger a
-presupposition failure:
-
-<pre>
-let add (x:int option) (y:int option) =
- match (x, y) with (None, _) -> None |
- (_, None) -> None |
- (Some m, Some n) -> Some (m + n);;
-
-(*
-val add : int option -> int option -> int option = <fun>
-# add (Some 12) (Some 4);;
-- : int option = Some 16
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-This works, but is somewhat disappointing: the `add` prediction
-doesn't trigger any presupposition of its own, so it is a shame that
-it needs to be adjusted because someone else might make trouble.
-
-But we can automate the adjustment. The standard way in OCAML,
-Haskell, etc., is to define a `bind` operator (the name `bind` is not
-well chosen to resonate with linguists, but what can you do):
-
-<pre>
-let bind (x: int option) (f: int -> (int option)) =
- match x with None -> None | Some n -> f n;;
-
-let add (x: int option) (y: int option) =
- bind x (fun x -> bind y (fun y -> Some (x + y)));;
-
-let div (x: int option) (y: int option) =
- bind x (fun x -> bind y (fun y -> if (0 = y) then None else Some (x / y)));;
-
-(*
-# div (div (Some 12) (Some 2)) (Some 4);;
-- : int option = Some 1
-# div (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-Compare the new definitions of `add` and `div` closely: the definition
-for `add` shows what it looks like to equip an ordinary operation to
-survive in a presupposition-filled world, and the definition of `div`
-shows exactly what extra needs to be added in order to trigger the
-no-division-by-zero presupposition.