oplss2024/ahmed/day1.fst
2024-06-06 10:21:04 -04:00

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module Day1
let var = int
type ty =
| TyBool
| TyArrow of ty * ty
type term =
| EVar of int
| ETrue
| EFalse
| EIfThenElse of term * term * term
| ELambda of ty * term
| EApp of term * term
type value =
| VTrue
| VFalse
| VLambda of ty * term
type evalctx =
| ECDot
| ECIfThenElse of evalctx * term * term
| ECAppL of evalctx * term
| ECAppR of value * evalctx
type step : term -> term -> Type =
| StepIfTrue : (e1 : term) -> (e2 : term) -> step (EIfThenElse (ETrue, e1, e2)) e1
| StepIfFalse : (e1 : term) -> (e2 : term) -> step (EIfThenElse (EFalse, e1, e2)) e2
let ctx = var -> option ty
let empty : ctx = fun _ -> None
let extend (t : ty) (g : ctx) : ctx = fun y -> if y = 0 then Some t else g (y - 1)
noeq type typing : ctx -> term -> ty -> Type =
| TypeTrue : (ctx : ctx) -> typing ctx ETrue TyBool
| TypeFalse : (ctx : ctx) -> typing ctx EFalse TyBool
| TypeIfThenElse : (ctx : ctx) -> (ty : ty)
-> (c : term) -> (e1 : term) -> (e2 : term)
-> typing ctx c TyBool
-> typing ctx e1 ty -> typing ctx e2 ty
-> typing ctx (EIfThenElse (c, e1, e2)) ty
let vRelation (ty : ty) (term : term) : prop =
match ty, term with
| (TyBool, ETrue) -> True
| (TyBool, EFalse) -> True
| _ -> False
let irred (term : term) : prop = admit()
let rec stepsToInNSteps (e1 : term) (e2 : term) (n : nat) : prop =
if n = 0 then False
else exists e' . (stepsToInNSteps e' e2 (n - 1) /\ (step e1 e'))
// let stepsTo (e1 : term) (e2 : term) : prop =
let eRelation (ty : ty) (term : term) : prop =
exists e' . irred e' /\ vRelation ty e'
let goodRelation (g : ctx) =
admit()
let subst (#ctx:ctx) (g : goodRelation ctx) (term : term) =
admit()
// let typeSoundness (ctx : ctx) (term : term) (ty : ty) : prop =
// forall (e' : )
let semanticSoundness (ctx : ctx) (term : term) (ty : ty) : prop =
forall (y : goodRelation ctx) . eRelation ty (subst y term)
let fundamentalProperty (#ctx : ctx) (#term : term) (#ty : ty)
(_ : typing ctx term ty) : semanticSoundness ctx term ty =
admit()