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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