```-Axiom: ---------
-        A |- A
-
-Structural Rules:
-
-          Γ, A, B, Δ |- C
-Exchange: ---------------------------
-          Γ, B, A, Δ |- C
-
-             Γ, A, A |- B
-Contraction: -------------------
-             Γ, A |- B
-
-           Γ |- B
-Weakening: -----------------
-           Γ, A |- B
-
-Logical Rules:
-
-         Γ, A |- B
---> I:   -------------------
-         Γ |- A --> B
-
-         Γ |- A --> B         Γ |- A
---> E:   -----------------------------------
-         Γ |- B
+let div (x:int) (y:int) =
+  match y with 0 -> None |
+               _ -> Some (x / y);;
+
+(*
+val div : int -> int -> int option =
+# 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
+*)
```
-`A`, `B`, etc. are variables over formulas. -Γ, Δ, etc. are variables over (possibly empty) sequences -of formulas. Γ `|- A` is a sequent, and is interpreted as -claiming that if each of the formulas in Γ is true, then `A` -must also be true. - -This logic allows derivations of theorems like the following: +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:
```--------  Id
-A |- A
----------- Weak
-A, B |- A
-------------- --> I
-A |- B --> A
------------------ --> I
-|- A --> B --> A
+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
+*)
```
-Should remind you of simple types. (What was `A --> B --> A` the type -of again?) +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. -The easy way to grasp the Curry-Howard correspondence is to *label* -the proofs. Since we wish to establish a correspondence between this -logic and the lambda calculus, the labels will all be terms from the -simply-typed lambda calculus. Here are the labeling rules: +I prefer to line up the `match` alternatives by using OCAML's +built-in tuple type:
```-Axiom: -----------
-       x:A |- x:A
-
-Structural Rules:
-
-          Γ, x:A, y:B, Δ |- R:C
-Exchange: -------------------------------
-          Γ, y:B, x:A, Δ |- R:C
-
-             Γ, x:A, x:A |- R:B
-Contraction: --------------------------
-             Γ, x:A |- R:B
-
-           Γ |- R:B
-Weakening: ---------------------
-           Γ, x:A |- R:B     [x chosen fresh]
-
-Logical Rules:
+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);;
+```
- Γ, x:A |- R:B ---> I: ------------------------- - Γ |- \xM:A --> B +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: - Γ |- f:(A --> B) Γ |- x:A ---> E: ------------------------------------- - Γ |- (fx):B +
```+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
+*)
```
-In these labeling rules, if a sequence Γ in a premise contains -labeled formulas, those labels remain unchanged in the conclusion. - -What is means for a variable `x` to be chosen *fresh* is that -`x` must be distinct from any other variable in any of the labels -used in the proof. +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. -Using these labeling rules, we can label the proof -just given: +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):
```-------------  Id
-x:A |- x:A
----------------- Weak
-x:A, y:B |- x:A
-------------------------- --> I
-x:A |- (\y.x):(B --> A)
----------------------------- --> I
-|- (\x y. x):A --> B --> A
+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
+*)
```
-We have derived the *K* combinator, and typed it at the same time! - -Need a proof that involves application, and a proof with cut that will -show beta reduction, so "normal" proof. - -[To do: add pairs and destructors; unit and negation...] +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. -Excercise: construct a proof whose labeling is the combinator S. -- 2.11.0