- 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 2. Verify that these encodings (zero, succ , rec) typecheck in System F.
+ (Draw a type tree for each term.)
+
+ 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:
+
+ 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
+