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CIS 500
Software Foundations
Fall 2004
6 October
CIS 500, 6 October 1
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The Simply Typed Lambda-Calculus - Software Foundations | CIS 500, Study notes of Computer Science
Material Type: Notes; Professor: Pierce; Class: SOFTWARE FOUNDATIONS; Subject: Computer & Information Science; University: University of Pennsylvania; Term: Fall 2004;
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CIS 500
Software Foundations
Fall 2004
6 October
CIS 500, 6 October
1
Midterm 1 is next Wednesday
Today’s lecture will not be covered by the midterm.
Next Monday, review class.
Old exams and review questions on webpage.
No recitation sections next week.
New office hours next week, watch newsgroup for details.
CIS 500, 6 October
2
Plans
Where we’ve been:
Inductive definitions
abstract syntax
inference rules
Proofs by structural induction
Operational semantics
The lambda-calculus
Typing rules and type soundness
Where we’re going:
“Simple types” for the lambda-calculus
references, exceptions, etc.)Formalizing more features of real-world languages (records, datatypes,
Subtyping
Objects
CIS 500, 6 October
3-a
The Simply Typed Lambda-Calculus
CIS 500, 6 October
4
“Simple Types”
T
types
Bool
type of booleans
T
T
types of functions
CIS 500, 6 October
6
Typing rules
true
Bool
T-True
false
Bool
T-False
t 1
Bool
t 2 : T t 3 : T
if t
1
then t
2
else t
3
T
T-If
CIS 500, 6 October
7
Typing rules
true
Bool
T-True
false
Bool
T-False
t 1
Bool
t 2 : T t 3 : T
if t
1
then t
2
else t
3
T
T-If
x:T
Γ ` x : T (
T-Var
CIS 500, 6 October
7-b
Γ^ Typing rules
`
true
Bool
T-True
`
false
Bool
T-False
`
t 1
Bool
Γ t 2 : T Γ t 3 : T
`
if t
1
then t
2
else t
3
T
T-If
x:T
Γ ` x : T (
T-Var
CIS 500, 6 October
7-c
Γ^ Typing rules
`
true
Bool
T-True
`
false
Bool
T-False
`
t 1
Bool
Γ t 2 : T Γ t 3 : T
`
if t
1
then t
2
else t
3
T
T-If
x:T
Γ ` x : T (
T-Var
x:T
1 ` t 2 : T 2
`
λ x:T
1 .t
2 : T 1 → T 2 (
T-Abs
Γ ` t 1 : T
11
T
12
Γ ` t 2 : T
11
Γ ` t 1 t 2 : T
12
T-App
CIS 500, 6 October
7-e
Typing Derivations
What derivations justify the following typing statements?
`
λ x:Bool.x) true
Bool
f:Bool
Bool
`
f (if false then true else false)
Bool
f:Bool
Bool
`
λ x:Bool. f (if x then false else x)
Bool
Bool
CIS 500, 6 October
8
Properties of
→
soundness As before, the fundamental property of the type system we have just defined is
with respect to the operational semantics.
Progress:
A closed, well-typed term is not stuck
If
`
t
T
, then either
t
is a value or else
t
− →
t (^) ′
for some
t (^) ′ .
Preservation:
Types are preserved by one-step evaluation
If
`
t
T
and
t
− →
t (^) ′ , then
`
t (^) ′
T
CIS 500, 6 October
9-a
Proving progress
Same steps as before...
CIS 500, 6 October
10
Typing rules again (for reference)
`
true
Bool
T-True
`
false
Bool
T-False
`
t 1
Bool
Γ t 2 : T Γ t 3 : T
`
if t
1
then t
2
else t
3
T
T-If
x:T
Γ ` x : T (
T-Var
x:T
1 ` t 2 : T 2
`
λ x:T
1 .t
2 : T 1 → T 2 (
T-Abs
Γ ` t 1 : T
11
T
12
Γ ` t 2 : T
11
Γ ` t 1 t 2 : T
12
T-App
CIS 500, 6 October
11
Inversion
Lemma:
If
`
true
R
, then
R
Bool
If
`
false
R
, then
R
Bool
If
`
if t
1
then t
2
else t
3
R
, then
`
t 1
Bool
and
`
t 2 , (^) t
3
R
CIS 500, 6 October
12