```-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
-*)
-```
- -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: - -
```-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 =
-# 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
-*)
-```
- -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: - -
```-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);;
-```
- -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: - -
```-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 =
-# add (Some 12) (Some 4);;
-- : int option = Some 16
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-```
- -This works, but is somewhat disappointing: the `add` operation -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): - -
```-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
-*)
-```
- -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 dangerous presupposition-filled world. Note that the new -definition of `add` does not need to test whether its arguments are -None objects or real numbers---those details are hidden inside of the -`bind` function. - -The definition of `div` shows exactly what extra needs to be said in -order to trigger the no-division-by-zero presupposition. - -For linguists: this is a complete theory of a particularly simply form -of presupposition projection (every predicate is a hole). +This has now been moved to the start of [[week7]]. +