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#lang racket
(require "lib.rkt")
(require rackunit)
(provide (all-defined-out))
;; The Simply-Typed Lambda Calculus, with simple extensions
;; Unit/String/Natural/Boolean, pairs, sums, lists, ascryption
;; (interpret Expr Table[Sym, Expr]): Value
(define (interpret expr [Γ #hash()])
(interpret- (strip (desugar expr)) Γ))
(define (interpret- expr Γ)
(match expr
['sole 'sole]
[s #:when (string? s) s]
[n #:when (natural? n) n]
[b #:when (boolean? b) b]
[x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
[`(inc ,e)
(match (interpret- e Γ)
[n #:when (natural? n) (+ n 1)]
[e (format "incrementing an unknown value ~a" e)])]
[`(if ,c ,e1 ,e2)
(match (interpret- c Γ)
['#t (interpret- e1 Γ)]
['#f (interpret- e2 Γ)]
[e (err (format "calling if on unknown expression ~a" e))])]
[`(pair ,e1 ,e2)
`(pair ,(interpret- e1 Γ) ,(interpret- e2 Γ))]
[`(car ,e)
(match (interpret- e Γ)
[`(pair ,e1 ,e2) e1]
[e (err (format "calling car on unknown expression ~a" e))])]
[`(cdr ,e)
(match (interpret- e Γ)
[`(pair ,e1 ,e2) e2]
[e (err (format "calling cdr on unknown expression ~a" e))])]
[`(inl ,e) `(inl ,(interpret- e Γ))]
[`(inr ,e) `(inr ,(interpret- e Γ))]
[`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
(match (interpret- e Γ)
[`(inl ,e) (interpret- e1 (dict-set Γ x1 e))]
[`(inr ,e) (interpret- e2 (dict-set Γ x2 e))]
[e (err (format "calling case on unknown expression ~a" e))])]
['nil 'nil]
[`(nil? ,e)
(match (interpret- e Γ)
['nil '#t]
[`(cons ,e1 ,e2) '#f]
[e (err (format "calling isnil on unknown expression ~a" e))])]
[`(cons ,e1 ,e2)
`(cons ,(interpret- e1 Γ) ,(interpret- e2 Γ))]
[`(head ,e)
(match (interpret- e Γ)
[`(cons ,e1 ,e2) (interpret- e1 Γ)]
[e (err (format "calling head on unknown expression ~a" e))])]
[`(tail ,e)
(match (interpret- e Γ)
[`(cons ,e1 ,e2) (interpret- e2 Γ)]
[e (err (format "calling tail on unknown expression ~a" e))])]
[`(λ ,x ,e) `(λ ,x ,e ,Γ)]
[`(,e1 ,e2)
(match (interpret- e1 Γ)
[`(λ ,x ,e ,env)
(interpret- e (dict-set env x (interpret- e2 Γ)))]
[e (err (format "applying arg ~a to unknown expression ~a" e2 e))])]
[e (err (format "interpreting an unknown expression ~a" e))]))
;; (check Expr Type Table[Sym, Type]): Bool
(define (check expr with [Γ #hash()])
(check- (desugar expr) with Γ))
(define (check- expr with Γ)
(let ([with (expand with Γ)])
(match* (expr with)
[('sole 'Unit) #t]
[(s 'Str) #:when (string? s) #t]
[(n 'Nat) #:when (natural? n) #t]
[(b 'Bool) #:when (boolean? b) #t]
[(e `(,t1 ⊕ ,t2))
(or (equiv? (infer- e Γ) with) (check- e t1 Γ) (check- e t2 Γ))]
[(x _) #:when (dict-has-key? Γ x)
(equiv? (dict-ref Γ x) with Γ Γ)]
[(`(type ,t1 ,t2 ,in) with)
(check- in with (dict-set Γ t1 t2))]
[(`(inc ,e) 'Nat)
(check- e 'Nat Γ)]
[(`(if ,c ,e1 ,e2) with)
(and (check- c 'Bool Γ)
(check- e1 with Γ) (check e2 with Γ))]
[(`(pair ,e1 ,e2) `(,t1 × ,t2))
(and (check- e1 t1 Γ) (check- e2 t2 Γ))]
[(`(car ,e) with)
(match (infer- e Γ)
[`(,t1 × ,t2) (equiv? t1 with Γ Γ)]
[t #f])]
[(`(cdr ,e) with)
(match (infer- e Γ)
[`(,t1 × ,t2) (equiv? t2 with Γ Γ)]
[t #f])]
[(`(inl ,e) with)
(match (infer- e Γ)
[`(,t1 ⊕ ,t2) (equiv? t1 with Γ Γ)]
[t #f])]
[(`(inr ,e) with)
(match (infer- e Γ)
[`(,t1 ⊕ ,t2) (equiv? t2 with Γ Γ)]
[t #f])]
[(`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2)) with)
(equiv? with (infer- `(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2)) Γ) Γ Γ)]
[(`(,e : ,t) with)
(and (equiv? t with Γ Γ) (check- e t Γ))]
[('nil `(List ,t)) #t]
[(`(cons ,f1 ,f2) `(List ,t))
(and (check- f1 t Γ) (check- f2 `(List ,t) Γ))]
[(`(head ,e) with)
(match (infer- e)
[`(List ,t) (equiv? t with Γ Γ)]
[t #f])]
[(`(tail ,e) with)
(equiv? (infer- e Γ) with Γ Γ)]
[(`(λ (,x : ,t) ,e) `(,t1 → ,t2))
(and (equiv? t t1 Γ Γ) (check- e t2 (dict-set Γ x t1)))]
[(`(,e1 ,e2) t)
(match (infer- e1 Γ)
[`(,t1 → ,t2)
(and (equiv? t2 t Γ Γ) (equiv? t1 (infer- e2 Γ) Γ Γ))]
[t #f])]
[(e t) #f])))
;; (infer Expr Table[Sym, Type]): Type
(define (infer expr [Γ #hash()])
(infer- (desugar expr) Γ))
(define (infer- expr Γ)
(match expr
['sole 'Unit]
[s #:when (string? s) 'Str]
[n #:when (natural? n) 'Nat]
[b #:when (boolean? b) 'Bool]
[x #:when (dict-has-key? Γ x)
(dict-ref Γ x)]
[`(type ,t1 ,t2 ,in)
(infer in (dict-set Γ t1 t2))]
[`(inc ,e)
(if (check- e 'Nat Γ) 'Nat
(err (format "calling inc on incorrect type ~a" (infer- e Γ))))]
[`(if ,c ,e1 ,e2)
(if (check- c 'Bool Γ)
(let ([t (infer- e1 Γ)])
(if (check- e2 t Γ) t
(err (format "condition has branches of differing types ~a and ~a"
t (infer- e2 Γ)))))
(err (format "condition ~a has incorrect type ~a" c (infer- c Γ))))]
[`(pair ,e1 ,e2)
`(,(infer- e1 Γ) × ,(infer- e2 Γ))]
[`(car ,e)
(match (infer- e Γ)
[`(,t1 × ,t2) t1]
[t (err (format "calling car on incorrect type ~a" t))])]
[`(cdr ,e)
(match (infer- e Γ)
[`(,t1 × ,t2) t2]
[t (err (format "calling cdr on incorrect type ~a" t))])]
[`(inl ,e) ; annotations necessary
(match (infer- e Γ)
[`(,t1 ⊕ ,t2) `(,t1 ⊕ ,t2)]
[t (err (format "calling inl on incorrect type ~a" t))])]
[`(inr ,e) ; annotations necessary
(match (infer- e Γ)
[`(,t1 ⊕ ,t2) `(,t1 ⊕ ,t2)]
[t (err (format "calling inr on incorrect type ~a" t))])]
[`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
(match (infer- e Γ)
[`(,a1 ⊕ ,a2)
(let ([b1 (infer- e1 (dict-set Γ x1 (expand a1 Γ)))] [b2 (infer- e2 (dict-set Γ x2 (expand a2 Γ)))])
(if (equiv? b1 b2 Γ Γ) b1
(err (format "case ~a is not of consistent type!" `(case (,a1 ⊕ ,a2) b1 b2)))))]
[t (err (format "calling case on incorrect type ~a" t))])]
[`(,e : ,t)
(if (check- e t Γ) t
(err (format "annotated expression ~a is not of annotated type ~a" e t)))]
['nil (err (format "unable to infer type of empty list!"))]
[`(cons ,e1 ,e2)
(let ([t (infer- e1 Γ)])
(if (check- e2 `(List ,t) Γ) `(List ,t)
(err (format "list ~a is not of consistent type!" `(cons ,e1 ,e2)))))]
[`(head ,e)
(match (infer- e Γ)
[`(List ,t) t]
[t (err (format "calling head on incorrect type ~a" t))])]
[`(tail ,e)
(match (infer- e Γ)
[`(List ,t) `(List ,t)]
[t (err (format "calling tail on incorrect type ~a" t))])]
[`(λ (,x : ,t) ,e)
`(,t → ,(infer- e (dict-set Γ x t)))]
[`(,e1 ,e2)
(match (infer- e1 Γ)
[`(,t1 → ,t2)
(if (check- e2 t1 Γ) t2
(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 "inferring an unknown expression ~a" e))]))
(define (expand t Γ)
(if (dict-has-key? Γ t) (dict-ref Γ t) t))
;; checks if two expressions are equivalent up to α-renaming and ascryption
(define (equiv? e1 e2 [Γ1 #hash()] [Γ2 #hash()])
(match* (e1 e2)
[(x1 x2) #:when (dict-has-key? Γ1 x1)
(equiv? (dict-ref Γ1 x1) x2 Γ1 Γ2)]
[(x1 x2) #:when (dict-has-key? Γ2 x2)
(equiv? x1 (dict-ref Γ2 x2) Γ1 Γ2)]
[(`(λ (,x1 : ,t1) ,e1) `(λ (,x2 : ,t2) ,e2)) ; todo: merge these into one
(let ([name gensym])
(and (equiv? e1 e2 (dict-set Γ1 x1 name) (dict-set Γ2 x2 name))
(equiv? t1 t2 Γ1 Γ2)))]
[(`(λ ,x1 ,e1) `(λ ,x2 ,e2))
(let ([name gensym])
(equiv? e1 e2 (dict-set Γ1 x1 name) (dict-set Γ2 x2 name)))]
[(`(,l1 ...) `(,l2 ...))
(foldl (λ (x1 x2 acc) (if (equiv? x1 x2 Γ1 Γ2) acc #f)) #t l1 l2)]
[(v1 v2) (equal? v1 v2)]))
(check-true (equiv? '(λ a a) '(λ b b)))
(check-true (equiv? '(λ a (λ b a)) '(λ b (λ a b))))
(check-true (equiv? '(λ a (λ b (λ c (a (b c))))) '(λ c (λ a (λ b (c (a b)))))))
(check-eq? (interpret '(if #t 1 0)) 1)
(check-eq? (interpret '(type Natural Nat ((λ (x : Natural) (inc x)) 1))) 2)
(check-eq? (infer '(type Natural Nat ((λ (x : Natural) (inc x)) 1))) 'Nat)
(check-true (check '(type Natural Nat ((λ (x : Natural) (inc x)) 1)) 'Nat))
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