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#lang racket
(require "lib.rkt")
(require "base.rkt")
(provide (all-defined-out))
;; The Simply-Typed Lambda Calculus with higher-order references
;; Whether the system is *predicative* or *impredicative*.
;; The predicative system disallows quantification over references.
;; The impredicative system allows so by tweaking level rules.
(define impredicative? #f)
;; Checks an expression for syntactic well-formedness.
(define (stlc-hor/expr? expr)
(match expr
[x #:when (symbol? x) #t]
[n #:when (natural? n) #t]
[(or `(new ,e) `(! ,e)) (stlc-hor/expr? e)]
[(or `(set ,e1 ,e2) `(,e1 ,e2)) (and (stlc-hor/expr? e1) (stlc-hor/expr? e2))]
[`(λ (,x : ,t) ,e) (and (symbol? x) (stlc-hor/type? t) (stlc-hor/expr? e))]
[_ #f]))
;; Checks a type for syntactic well-formedness.
(define (stlc-hor/type? type)
(match type
[t #:when (symbol? t) #t]
[`(,t1 → ,k ,t2) (and (natural? k) (stlc-hor/type? t1) (stlc-hor/type? t2))]
[_ #f]))
;; Checks a value for syntactic well-formedness.
(define (stlc-hor/value? value)
(match value
[x #:when (symbol? x) #t]
[n #:when (natural? n) #t]
[`(,v1 ,v2) (and (stlc-hor/value? v1) (stlc-hor/value? v2))]
[`(λ ,x ,e ,env) (and (symbol? x) (stlc-hor/expr? e) (dict? env))]
[_ #f]))
;; Interprets an expression down to a value, in a given context.
(define (interpret expr)
(interpret/core (desugar expr) #hash() (make-hash)))
(define/contract (interpret/core expr Γ Σ)
(-> stlc-hor/expr? dict? dict? stlc-hor/value?)
(match expr
[r #:when (dict-has-key? Σ r) r]
[x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
[f #:when (symbol? f) f]
[`(new ,e)
(let ([r (gensym)])
(dict-set! Σ r e) r)]
[`(! ,e)
(let ([r (interpret/core e Γ Σ)])
(if (dict-has-key? Σ r)
(interpret/core (dict-ref Σ r) Γ Σ)
(err (format "attempting to deref unknown reference ~a" r))))]
[`(set ,e1 ,e2)
(let ([r (interpret/core e1 Γ Σ)])
(if (dict-has-key? Σ r) (dict-set! Σ r (interpret/core e2 Γ Σ))
(err (format "attempting to update unknown reference ~a" r))))
'sole]
[`(λ ,x ,e) `(λ ,x ,e ,Γ)]
[`(,e1 ,e2)
(match (interpret/core e1 Γ Σ)
[`(λ ,x ,e1 ,env)
(interpret/core e1 (dict-set env x (interpret/core e2 Γ Σ)) Σ)]
[e1 (err (format "attempting to interpret arg ~a applied to unknown expression ~a" e2 e1))])]))
;; Checks an expression against some type, in a given context.
(define (check expr with)
(check/core (desugar expr) with #hash()))
(define/contract (check/core expr with Γ)
(-> stlc-hor/expr? stlc-hor/type? dict? boolean?)
(match expr
[`(new ,e)
(match with
[`(Ref ,t) (check/core e t Γ)]
[_ #f])]
[`(! ,e)
(check/core e `(Ref ,with) Γ)]
[`(λ (,x : ,t) ,e)
(match with
[`(,t1 → ,k ,t2)
(and (equal? t t1) (check/core e t2 (dict-set Γ x t))
(if impredicative?
(> k (level-expr e (dict-set Γ x t1)))
(>= k (level-expr e (dict-set Γ x t1)))))]
[_ #f])]
[_ (equal? (infer/core expr Γ) with)]))
;; Infers a type from some expression, in a given context.
(define (infer expr)
(infer/core (desugar expr) #hash()))
(define/contract (infer/core expr Γ)
(-> stlc-hor/expr? dict? stlc-hor/type?)
(match expr
['sole 'Unit]
[n #:when (natural? n) 'Nat]
[x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
[f #:when (symbol? f)
(err (format "attempting to infer type of free variable ~a" f))]
[`(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))]
[k (level-expr e (dict-set Γ x t1))])
`(,t1 → ,(if impredicative? (+ k 1) 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))])]))
;; Get the level associated with a type. This is known at compile time.
(define/contract (level-type type)
(-> stlc-hor/type? natural?)
(match type
[(or 'Unit 'Nat) 0]
[`(Ref ,t)
(let ([k (level-type t)])
(if (and impredicative? (zero? k)) 0 (+ 1 k)))]
[`(,t1 → ,k ,t2)
(if (and (>= k (level-type t1)) (>= k (level-type t2))) k
(err (format "annotated level ~a is less than inferred levels of ~a and ~a!"
k t1 t2)))]
[t (err (format "attempting to infer the level of unknown type ~a" t))]))
;; Get the level of an arbitrary expression. This is not known until runtime.
(define/contract (level-expr expr Γ)
(-> stlc-hor/expr? dict? natural?)
(match expr
['sole 0]
[n #:when (natural? n) 0]
[x #:when (dict-has-key? Γ x) (level-type (dict-ref Γ x))]
[(or `(new ,e) `(! ,e) `(λ (,_ : ,_) ,e)) (level-expr e Γ)]
[(or `(set ,e1 ,e2) `(,e1 ,e2)) (max (level-expr e1 Γ) (level-expr e2 Γ))]
[x #:when (symbol? x) 0])) ; local variables, not in Γ
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