- Conveniently, this encoding “is” its own elimination form, in a sense:
- rec(e, e0, x:τ. e1) := e [τ ] e0 (λx:τ. e1)
- The case analysis is baked into the very definition of the type.
- Exercise 2. Verify that these encodings (zero, succ , rec) typecheck in System F. Write down the typing
- derivations for the terms.
- 1
-
- ══════════════════════════════════════════════════════════════════════════
-
- As mentioned in class, System F can express any inductive datatype. Consider the following list type:
- datatype ’a list =
- Nil
- | Cons of ’a * ’a list
- We can encode τ lists, lists of elements of type τ as follows:1
- τ list := ∀α. α → (τ → α → α) → α
- nilτ := Λα. λn:α. λc:τ → α → α. n
- consτ := λh:τ. λt:τ list. Λα. λn:α. λc:τ → α → α. c h (t [α] n c)
- As with nats, The τ list type’s case analyzing elimination form is just application. We can write functions
- like map:
- map : (σ → τ ) → σ list → τ list
- := λf :σ → τ. λl:σ list. l [τ list] nilτ (λx:σ. λy:τ list. consτ (f x) y
- Exercise 3. Consider the following simple binary tree type:
- datatype ’a tree =
- Leaf
- | Node of ’a tree * ’a * ’a tree
- (a) Give a System F encoding of binary trees, including a definition of the type τ tree and definitions of
- the constructors leaf : τ tree and node : τ tree → τ → τ tree → τ tree.
- (b) Write a function height : τ tree → nat. You may assume the above encoding of nat as well as definitions
- of the functions plus : nat → nat → nat and max : nat → nat → nat.
- (c) Write a function in-order : τ tree → τ list that computes the in-order traversal of a binary tree. You
- may assume the above encoding of lists; define any auxiliary functions you need.
-
---
-Jim Pryor
-jim@jimpryor.net
+
+ **Excercise**: get booleans and Church numbers working in OCaml,
+ including OCaml versions of bool, true, false, zero, succ, and pred.
+ It's especially useful to do a version of pred, starting with one
+ of the (untyped) versions available in the lambda library
+ accessible from the main wiki page. The point of the excercise
+ is to do these things on your own, so avoid using the built-in
+ OCaml booleans and list predicates.
+
+ Consider the following list type:
+
+ type ’a list = Nil | Cons of ’a * ’a list
+
+ We can encode τ lists, lists of elements of type τ as follows:
+
+ τ list := ∀α. α → (τ → α → α) → α
+ nilτ := Λα. λn:α. λc:τ → α → α. n
+ makeListτ := λh:τ. λt:τ list. Λα. λn:α. λc:τ → α → α. c h (t [α] n c)
+
+ As with nats, recursion is built into the datatype.
+
+ We can write functions like head, isNil, and map:
+
+ map : (σ → τ ) → σ list → τ list
+
+ We've given you the type for map, you only need to give the term.
+
+ With regard to `head`, think about what value to give back if the
+ argument is the empty list. Ultimately, we might want to make use
+ of our `'a option` technique, but for this assignment, just pick a
+ strategy, no matter how clunky.
+
+ Please provide both the terms and the types for each item.