Simply#peda-cai.us

WE ( xx . xx) N-

- cow

types A. B '

i. = 2 I A- → B [ Ax B / I]

Terry M ,N : := x / ( xx :A. M) / MN

( M.nl I it.M Ita Nl *

Valves ii. = Xx : A. M

( M ,N) ← ebn

( V , W) ← cbv

H) ← cbnlcbv

"

stuck"

IT, ( xx : A. x) -17HI 4*4*7 -17

CX x : z.X) : 2 -72

-

IT,IM

,N) →bn M¥cmm→j 'THE

The type system-

Contexts T " = . I T,

x : A

T, ,T,

to"

append" T

.,

Tz

dom (T) = { x ) x : A appears in T}⇐ i. A) E T to mean x: A appears in T

> donk?) n domed = 0

Typing : /PtM: judgment

( var)- ( x :A ET)Mt x : A

T,x : A t M : B

labs)- ( x # dome))P t Xx :Aim : A → B

Tt M : A → B Tt N : A(app)-

Tt MN : B

Q : does there ex is a typing derivation

for

+

It Caxias . xx) : B- ?[ AIs : No ?

there is→Hype

.

T such that

Ti- M : A Ti- N :B-

Tt SM,N) : AXB

Tt M : AXB Tt M : AXB- -

T t IT,M : A T '- Iz M : B-

Tt Ca ) : I

-

7x:A3xC *"

wrong"

T t It,(Xx : A.x) : B or

-

"

stuck"

-

Type Safety : If • I- M : A and- ( M -7mm

' /brr

n

then. th

'

: A.

Milner 's"

well - typed programs don't

go wrong"

Wright a Tellefsen 1994 Syntactic Typesafety

• (R ) (Xx :A. M) N →p MENK]

(3) M -70M'

-7nA.M→p Xx :A.M'

-

Lemmy Subject Reduction

[If T th : A and M -7ns M'

then Tt M': A

.

Proof-

By induction on the structure of M .

• case M - X then x -17ps ✓

• case M= Xx : B. M '

o Fx :B TM ':c

o M' -7M

" by-

Pt Xx :B.

M'

:B-7Cin

=A

Thx : Bt M"

:c ( by I. H)

by abs Mt Xx :B . M"

:B -7C ✓-

=A

• Ease µ , Mz) a) B M,= Xx :B

.

M,

'

n' I M.

'

[ Mix]

④ m

'? 7in a conga ]

•Lemmy Substitution

If T,x : A

,Th t M : B and

T,t N : A

thenp

, , Pa t M [Nlx] :B .

Proof : By induction on the structure of M.

• case M=y .

P,x : A

,Th t y : B

-

- (x : Ae. .)

-

x=y I? x : A, Th t x : B

- A -- B

x CN = N

to show : 17,17 t N : A -

- B V

FYI weakening if I , t M : A=

and I , Th is well formed

then 17,17 I- M : A"I ''Fasten:X xiH⇒⇒×.

Rest by-

* case x # yM[NIX] = y ENID = y

17,x : A

, they :B IT,,PztY:BJ toshow

(Einsteins Efa In "B

then T., Th t M :B

Proof By induction on th.

1-• case M= Xy :C .

M'

have 17,x : A

,Rt Ty : C.M

': CTD

by inversion T. ,x : A , Tz , y :C

t M'

:D

and x# y .

Hy :c .M

' ) CNH] = Xy :C .( MININ)

By IH .

with th : - th, y :C and B :-D

we hare n, th

, y :c1-MIND :D

use abs . M,

tht Ay :c .CM'

B)i. GD

ngV

• Case M= M, Mz

by .

IH . twice,we have

17,17 + M,[ NAT :C -713

I? ,RtMzCNlx3 :c-

l?,tht (M . Ma) (Nld :B V

-

-• Preservation . TM : A and M -7

,

M'

-

other . I- M'

: A

[ ''

"

iii.ar.

E there exists M 's. E

.

M → M !

Corollary : ifA then Mis-

not stuck ,

Mh→