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#lang racket
(require "lib.rkt")
;; The Simply-Typed Lambda Calculus with higher-order *predicative* references
(require (only-in "stlc-ref.rkt" interpret))
;; (check Expr Type Table[Sym, Type]): Bool
(define (check expr with [Γ #hash()])
(check- (desugar expr) with Γ))
(define (check- expr with Γ)
; (print (format "check: ~a" (fmt expr)))
(match* (expr with)
[('sole 'Unit) #t] ; ↝ Γ ⊢ ⟨⟩: Unit
[(n 'Nat) #:when (natural? n) #t] ; ↝ Γ ⊢ n: Nat
[(x _) #:when (dict-has-key? Γ x) ; x: τ ∈ Γ → Γ ⊢ x: τ
(equal? (dict-ref Γ x) with)]
[(`(new ,e) `(Ref ,t)) (check- e t Γ)] ; Γ ⊢ e: τ → Γ ⊢ new e: Ref τ
[(`(! ,e) t) (check- e `(Ref ,t) Γ)] ; Γ ⊢ e: Ref τ → Γ ⊢ !e: τ
[(`(set ,e1 ,e2) 'Unit) ; ↝ Γ ⊢ e1 := e2: Unit
(match (infer- e1 Γ)
[`(Ref ,t) (check- e2 t Γ)] ; Γ ⊢ e1: Ref τ, Γ ⊢ e2: τ
[t (err (format "attempting to update non-reference ~a: ~a" e1 t))])]
[(`(λ ,x (: ,t) ,e) `(→ ,k ,t1 ,t2)) ; ↝ Γ ⊢ λx: τ1.e: τ1 →k τ2
(and
(equal? t t1)
(>= k (max-level e (dict-set Γ x t1) t1 t2)) ; k ≥ max-level(Γ, τ1, τ2) (KNOB)
(check- e t2 (dict-set Γ x t1)))] ; Γ, x: τ1 ⊢ e: τ2
[(`(,e1 ,e2) t) ; ↝ Γ ⊢ (e1 e2): τ2
(match (infer- e1 Γ)
[`(→ ,k ,t1 ,t2) ; Γ ⊢ e1: τ1 →k τ2
(and (equal? t2 t) (equal? t1 (infer- e2 Γ)))] ; Γ ⊢ e2: τ1
[t (err (format "expected → type on application body, got ~a" t))])]
[(`(λ ,x (: ,t) ,e) `(→ ,t1 ,t2)) (err "you forgot to add a level annotation dummy")]
[(e t) (err (format "checking an unknown expression ~a with type ~a" e with))]))
;; (infer Expr Table[Sym, Type]): Type
(define (infer expr [Γ #hash()])
(infer- (desugar expr) Γ))
(define (infer- expr Γ)
; (print (format "infer: ~a" (fmt expr)))
(match expr
['sole 'Unit] ; ↝ Γ ⊢ ⟨⟩: Unit
[n #:when (natural? n) 'Nat] ; ↝ Γ ⊢ n: Nat
[x #:when (dict-has-key? Γ x) ; x: τ ∈ Γ
(dict-ref Γ x)] ; ↝ Γ ⊢ x: τ
[`(new ,e) `(Ref ,(infer- e Γ))] ; Γ ⊢ e: τ → Γ ⊢ new e: Ref τ
[`(! ,e)
(match (infer- e Γ)
[`(Ref ,t) t] ; Γ ⊢ e: Ref τ → Γ ⊢ !e: τ
[t (err "attempting to deref term not of Ref type!")])]
[`(set ,e1 ,e2)
(match (infer- e1 Γ)
[`(Ref ,t) ; Γ ⊢ e1: Ref τ, Γ ⊢ e2: τ
(if (check- e2 t Γ) 'Unit ; ↝ Γ ⊢ e1 := e2: Unit
(err (format "attempting to update ~a: ~a with term ~a: ~a of differing type"
e1 t e2 (infer- e2 Γ))))]
[t (err (format "attempting to update non-reference ~a: ~a" e1 t))])]
[`(λ ,x (: ,t1) ,e)
(let ([t2 (infer- e (dict-set Γ x t1))]) ; Γ, x: τ1 ⊢ e: τ2
(let ([k (max-level e (dict-set Γ x t1) t1 t2)]) ; k ≥ max-level(Γ, τ1, τ2) (KNOB)
`(→ ,k ,t1 ,t2)))] ; ↝ Γ ⊢ λx: τ1.e: τ1 →k τ2
[`(,e1 ,e2)
(match (infer- e1 Γ)
[`(→ ,k ,t1 ,t2) ; Γ ⊢ e1: τ1 →k τ2
(if (check- e2 t1 Γ) t2 ; Γ ⊢ e2: τ1 ↝ Γ ⊢ (e1 e2): τ2
(err (format "inferred argument type ~a does not match arg ~a" t1 e2)))]
[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]
[`(→ ,k ,t1 ,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 (: (→ 0 Nat Nat)) (λ x x)
(let r (: (Ref (→ 0 Nat Nat))) (new id)
(let f (: (→ 1 Nat Nat)) (λ x ((! r) x))
(set r f
(f 0))))))
#;
(print (fmt '
(let id (: (→ 0 Nat Nat)) (λ x x)
(let r (: (Ref (→ 0 Nat Nat))) (new id)
(let f (: (→ 1 Nat Nat)) (λ x ((! r) x))
(set r f
(f 0)))))))
(require rackunit)
(check-exn
exn:fail?
(λ () (infer '
(let id (: (→ 0 Nat Nat)) (λ x x)
(let r (: (Ref (→ 0 Nat Nat))) (new id)
(let f (: (→ 1 Nat Nat)) (λ x ((! r) x))
(set r f
(f 0))))))))
(check-eq?
(infer '
(let id (: (→ 0 Nat Nat)) (λ x x)
(let r (: (Ref (→ 0 Nat Nat))) (new id)
(let f (: (→ 1 Nat Nat)) (λ x ((! r) x))
(f 0)))))
'Nat)
(check-eq?
(check '
(let id (: (→ 0 Nat Nat)) (λ x x)
(let r (: (Ref (→ 0 Nat Nat))) (new id)
(let f (: (→ 1 Nat Nat)) (λ x ((! r) x))
(f 0))))
'Nat)
#t)
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