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#lang racket
(require "lib.rkt")
;; The Simply-Typed Lambda Calculus with higher-order *predicative* references
; Γ, x: τ₁ ⊢ e: τ₂ k ≥ max-level(Γ, τ₁, τ₂)
; ---------------------------------------------
; Γ ⊢ λx:τ₁.e : τ₁ →ᵏ τ₂
; Γ ⊢ e₁: τ₁ →ᵏ τ₂ Γ ⊢ e₂: τ₁
; --------------------------------
; Γ ⊢ (e₁ e₂): τ₂
; --------------------------
; Nat::Type₀, Unit::Type₀
; τ::Typeᵢ
; -------------
; Ref τ :: Typeᵢ₊₁
; τ₁::Typeᵢ, τ₂::Typeⱼ, k ≥ max-level(τ₁, τ₂)
; -----------------------------------------
; τ₁ →ᵏ τ₂ :: Typeₖ
(require (only-in "stlc-ref.rkt" interpret))
;; (check Expr Type Table[Sym, Type]): Bool
(define (check expr with)
(check-core (desugar expr) with #hash()))
(define (check-core expr with Γ)
(match expr
['sole (equal? 'Unit with)]
[n #:when (natural? n) (equal? 'Nat with)]
[x #:when (dict-has-key? Γ x)
(equal? (dict-ref Γ x) with)]
[`(new ,e)
(match with
[`(Ref ,t) (check-core e t Γ)]
[_ #f])]
[`(! ,e) (check-core e `(Ref ,with) Γ)]
[`(set ,e1 ,e2)
(match (infer-core e1 Γ)
[`(Ref ,t)
(and (equal? 'Unit with)
(check-core e2 t Γ))]
[_ #f])]
[`(λ (,x : ,t) ,e)
(match with
[`(,t1 → ,k ,t2)
(and
(equal? t t1)
(>= k (max-level e (dict-set Γ x t1) t1 t2)) ; (KNOB)
(check-core e t2 (dict-set Γ x t1)))])]
[`(,e1 ,e2)
(match (infer-core e1 Γ)
[`(,t1 → ,k ,t2)
(and (equal? t2 with) (equal? t1 (infer-core e2 Γ)))]
[_ #f])]
[_ #f]))
;; (infer Expr Table[Sym, Type]): Type
(define (infer expr)
(infer-core (desugar expr) #hash()))
(define (infer-core expr Γ)
(match expr
['sole 'Unit]
[n #:when (natural? n) 'Nat]
[x #:when (dict-has-key? Γ x)
(dict-ref Γ x)]
[`(new ,e) `(Ref ,(infer-core e Γ))]
[`(! ,e)
(match (infer-core e Γ)
[`(Ref ,t) t]
[t (err "attempting to deref term not of Ref type!")])]
[`(set ,e1 ,e2)
(match (infer-core e1 Γ)
[`(Ref ,t)
(if (check-core e2 t Γ) 'Unit
(err (format "attempting to update ~a: ~a with term ~a: ~a of differing type"
e1 t e2 (infer-core e2 Γ))))]
[t (err (format "attempting to update non-reference ~a: ~a" e1 t))])]
[`(λ (,x : ,t1) ,e)
(let ([t2 (infer-core e (dict-set Γ x t1))])
(let ([k (max-level e (dict-set Γ x t1) t1 t2)]) ; (KNOB)
`(,t1 → ,k ,t2)))]
[`(,e1 ,e2)
(match (infer-core e1 Γ)
[`(,t1 → ,k ,t2)
(if (check-core e2 t1 Γ) t2
(err (format "inferred argument type ~a does not match arg ~a of type ~a" t1 e2 (infer-core e2 Γ))))]
[t (err (format "expected → type on application body, got ~a" t))])]
[e (err (format "attempting to infer an unknown expression ~a" e))]))
;; (max-level Table[Sym, Type] Expr Type Type): Natural
(define (max-level e Γ t1 t2)
(max
(level-type t1)
(level-type t2)
(level-body e Γ)))
;; (level-type Type): Natural
(define (level-type t)
(match t
['Unit 0]
['Nat 0]
[`(,t1 → ,k ,t2)
(if (or (< k (level-type t1)) (< k (level-type t2)))
(err (format "annotated level ~a is less than inferred levels of ~a and ~a!"
k t1 t2))
k)]
[`(Ref ,t) (+ 1 (level-type t))] ; (KNOB)
[t (err (format "attempting to infer the level of unknown type ~a" t))]))
;; (level-body Expr Table[Sym, Type]): Natural
(define (level-body e Γ)
(match e
['sole 0]
[n #:when (natural? n) 0]
[x #:when (dict-has-key? Γ x)
(level-type (dict-ref Γ x))]
[`(new ,e) (level-body e Γ)]
[`(! ,e) (level-body e Γ)]
[`(set ,e1 ,e2) (max (level-body e1 Γ) (level-body e2 Γ))]
[`(λ (,x : ,t) ,e) (level-body e (dict-set Γ x t))] ; todo: should be 0?
[`(,e1 ,e2) (max (level-body e1 Γ) (level-body e2 Γ))]
[e (err (format "attempting to infer the level of unknown expression ~a" e))]))
; simple diverging program in STLC-ref
; https://www.youtube.com/watch?v=XNgE8kBfSz8
#;
(interpret '
(let (id : (Nat → 0 Nat)) (λ x x)
(let (r : (Ref (Nat → 0 Nat))) (new id)
(let (f : (Nat → 1 Nat)) (λ x ((! r) x))
(set r f
(f 0))))))
#;
(print (fmt '
(let (id : (Nat → 0 Nat)) (λ x x)
(let (r : (Ref (Nat → 0 Nat))) (new id)
(let (f : (Nat → 1 Nat)) (λ x ((! r) x))
(set r f
(f 0)))))))
(require rackunit)
(check-exn
exn:fail?
(λ () (infer '
(let (id : (Nat → 0 Nat)) (λ x x)
(let (r : (Ref (Nat → 0 Nat))) (new id)
(let (f : (Nat → 1 Nat)) (λ x ((! r) x))
(set r f
(f 0))))))))
(check-eq?
(infer '
(let (id : (Nat → 0 Nat)) (λ x x)
(let (r : (Ref (Nat → 0 Nat))) (new id)
(let (f : (Nat → 1 Nat)) (λ x ((! r) x))
(f 0)))))
'Nat)
(check-eq?
(check '
(let (id : (Nat → 0 Nat)) (λ x x)
(let (r : (Ref (Nat → 0 Nat))) (new id)
(let (f : (Nat → 1 Nat)) (λ x ((! r) x))
(f 0))))
'Nat)
#t)
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