Syntactic proofs empower metatheory Derivation/proof D is a data - - PowerPoint PPT Presentation

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Dec 28, 2023 32 likes •135 views

Syntactic proofs empower metatheory Derivation/proof D is a data structure Got a fact about all derivations? Its a fact about all terminating evaluations They are in 1 to 1 correspondance Prove meta-theoretic properties by structural

SLIDE 1

Syntactic proofs empower metatheory

Derivation/proof

D is a data structure

Got a fact about all derivations?

It’s a fact about all terminating evaluations
They are in 1 to 1 correspondance

Prove meta-theoretic properties by structural induction over derivations

aka “induction on height of derivation tree”

Example: Evaluating an expression doesn’t create

r destroy any global variables (the set of defined

global variables is invariant)

SLIDE 2

Metatheorems often help implementors

More example metatheorems:

OK to mutate environments if you use a stack

(Impcore)

Interactive browser doesn’t leak space

(POPL 2012)

Device driver can’t harm kernel

(Microsoft Singularity)

SLIDE 3

Metatheorems come in stylized form

For any e,

, , , v, ′, and ′ such that he ;

;

; i + hv ; ′ ; ; ′ i;

METATHEORETIC PROPERTY

SLIDE 4

Metatheorems are proved by induction

Induction over height of derivation trees

These are “math-class proofs” (not derivations) Proof

Goes by case analysis of the last rule in the derivation.
Has one case for each rule
Base cases don’t have proper sub-derivations.
Inductive cases assume the induction hypothesis for any

proper sub-derivation Let’s try it!

SLIDE 5

Example metatheorem

During the evaluation of an Impcore expression, evaluation does not change the set of defined global variables. Formally, for any e,

, , , v, ′, and ′ such that he ;

;

; i + hv ; ′ ; ; ′ i;

dom

( ) = dom (′ )

SLIDE 6

Literal case

Base case:

D=

hLITERAL (v );

;

; i + hv ;

;

; i

Both sides identical!

dom

SLIDE 7

Formal Var case

Another base case:

D=

x

2 dom

hVAR

(x );

;

; i + h(x );

;

; i

Both sides identical!

dom

SLIDE 8

Formal Assign case

Assignment to formal parameter

D=

x

2 dom

he ;

;

; i + hv ; ′ ; ; ′ i hSET (x ;e );

;

; i + hv ; ′ ; ; ′ fx 7! v gi

By induction hypothesis on

Dr, dom

dom ′

Both sides have same domain!

SLIDE 9

IfTrue case

True conditional

D=

D1 he1 ;

;

; i + hv1 ; ′ ; ; ′ i

v1

6= 0 D2 he2 ; ′ ; ; ′ i + hv2 ; ′′ ; ; ′′ i hIF (e1 ;e2 ;e3 );

;

; i + hv2 ; ′′ ; ; ′′ i

By induction hypothesis on

D1, dom

dom ′

By induction hypothesis on

D2, dom ′ = dom ′′

Therefore, both sides have same domain:

dom

dom ′′

SLIDE 10

Global Assign: The only interesting case

x

= 2 dom

2 dom

he ;

;

; i + hv ; ′ ; ; ′ i hSET (x ;e );

;

; i + hv ; ′ fx 7! v g; ; ′ i

Do both sides have same domain?

Does

dom

dom (′ fx 7! v g) ?

By induction hypothesis on

Dr, dom

dom ′

And

dom (′ fx 7! v g) = dom ′ [ fx g = dom

[

fx g

But x

2 dom ! So dom

[

fx g = dom