From: chris
Date: Thu, 26 Feb 2015 02:04:38 +0000 (-0500)
Subject: (no commit message)
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=5cab83962241676d710c788561ac107a3563a3e8
---
diff --git a/topics/_week5_system_F.mdwn b/topics/_week5_system_F.mdwn
index 4afb43ba..a80cc58e 100644
--- a/topics/_week5_system_F.mdwn
+++ b/topics/_week5_system_F.mdwn
@@ -39,22 +39,22 @@ match up with usage in O'Caml, whose type system is based on System F):
System F:
---------
- types Ï ::= c | 'a | Ï1 -> Ï2 | â'a. Ï
- expressions e ::= x | Î»x:Ï. e | e1 e2 | Î'a. e | e [Ï]
+ types Ï ::= c | Î± | Ï1 -> Ï2 | â'a. Ï
+ expressions e ::= x | Î»x:Ï. e | e1 e2 | ÎÎ±. e | e [Ï]
In the definition of the types, "`c`" is a type constant. Type
constants play the role in System F that base types play in the
simply-typed lambda calculus. So in a lingusitics context, type
-constants might include `e` and `t`. "`'a`" is a type variable. The
+constants might include `e` and `t`. "Î±" is a type variable. The
tick mark just indicates that the variable ranges over types rather
than over values; in various discussion below and later, type variable
can be distinguished by using letters from the greek alphabet
(α, β, etc.), or by using capital roman letters (X, Y,
etc.). "`Ï1 -> Ï2`" is the type of a function from expressions of
-type `Ï1` to expressions of type `Ï2`. And "`â'a. Ï`" is called a
+type `Ï1` to expressions of type `Ï2`. And "`âÎ±. Ï`" is called a
universal type, since it universally quantifies over the type variable
-`'a`. You can expect that in `â'a. Ï`, the type `Ï` will usually
-have at least one free occurrence of `'a` somewhere inside of it.
+`'a`. You can expect that in `âÎ±. Ï`, the type `Ï` will usually
+have at least one free occurrence of `Î±` somewhere inside of it.
In the definition of the expressions, we have variables "`x`" as usual.
Abstracts "`Î»x:Ï. e`" are similar to abstracts in the simply-typed lambda
@@ -62,7 +62,7 @@ calculus, except that they have their shrug variable annotated with a
type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
In addition to variables, abstracts, and applications, we have two
-additional ways of forming expressions: "`Î'a. e`" is called a *type
+additional ways of forming expressions: "`ÎÎ±. e`" is called a *type
abstraction*, and "`e [Ï]`" is called a *type application*. The idea
is that `Λ`

is a capital `λ`

: just
like the lower-case `λ`

, `Λ`

binds
@@ -72,7 +72,7 @@ variables. So in the expression
`Λ Î± (λ x:Î± . x)`

-the `Λ`

binds the type variable `'a` that occurs in
+the `Λ`

binds the type variable `Î±` that occurs in
the `λ`

abstract. Of course, as long as type
variables are carefully distinguished from expression variables (by
tick marks, Grecification, or capitalization), there is no need to
@@ -85,27 +85,27 @@ be adapted for use with expressions of any type. In order to get it
ready to apply this identity function to, say, a variable of type
boolean, just do this:
-`(Λ 'a (λ x:'a . x)) [t]`

+`(Λ Î± (λ x:Î± . x)) [t]`

This type application (where `t` is a type constant for Boolean truth
-values) specifies the value of the type variable `'a`. Not
+values) specifies the value of the type variable `Î±`. Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:
-`((Λ 'a (λ x:'a . x)) [t]): (b -> b)`

+`((Λ Î± (λ x:Î± . x)) [t]): (b -> b)`

Likewise, if we had instantiated the type variable as an entity (base
type `e`), the resulting identity function would have been a function
of type `e -> e`:
-`((Λ 'a (λ x:'a . x)) [e]): (e -> e)`

+`((Λ Î± (λ x:Î± . x)) [e]): (e -> e)`

-Clearly, for any choice of a type `'a`, the identity function can be
-instantiated as a function from expresions of type `'a` to expressions
-of type `'a`. In general, then, the type of the uninstantiated
+Clearly, for any choice of a type `Î±`, the identity function can be
+instantiated as a function from expresions of type `Î±` to expressions
+of type `Î±`. In general, then, the type of the uninstantiated
(polymorphic) identity function is
-`(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)`

+`(Λ Î± (λ x:Î± . x)): (∀ Î± . Î± -> Î±)`

Pred in System F
----------------