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+#lang racket
+(require "../lib.rkt")
+(require (only-in "../simple/rec.rkt" replace))
+(require (only-in "../simple/ext.rkt" type->whnf))
+(provide (all-defined-out))
+
+;; The Simply-Typed Lambda Calculus with higher-order *impredicative* references,
+;; plus sums products booleans ascryption etc, to implement doubly-linked lists
+
+;; Checks an expression for syntactic well-formedness.
+(define (stlc-dll/expr? expr)
+  (match expr
+    [x #:when (symbol? x) #t]
+    [n #:when (natural? n) #t]
+    [b #:when (boolean? b) #t]
+    [`(,e : ,t)
+      (and (stlc-dll/expr? e) (stlc-dll/type? t))]
+    [`(type ,t1 ,t2 ,e)
+      (and (stlc-dll/type? t1) (stlc-dll/type? t2) (stlc-dll/expr? e))]
+    [(or
+      `(inc ,e)
+      `(car ,e) `(cdr ,e)
+      `(inl ,e) `(inr ,e)
+      `(new ,e)  `(! ,e)
+      `(fold ,e) `(unfold ,e))
+      (stlc-dll/expr? e)]
+    [(or
+      `(pair ,e1 ,e2)
+      `(set ,e1 ,e2)
+      `(,e1 ,e2))
+      (and (stlc-dll/expr? e1) (stlc-dll/expr? e2))]
+    [`(if ,c ,e1 ,e2)
+      (and (stlc-dll/expr? c) (stlc-dll/expr? e1) (stlc-dll/expr? e2))]
+    [`(case ,c (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (and (symbol? x1) (symbol? x2)
+        (stlc-dll/expr? c) (stlc-dll/expr? e1) (stlc-dll/expr? e2))]
+    [`(λ (,x : ,t) ,e)
+      (and (symbol? x) (stlc-dll/type? t) (stlc-dll/expr? e))]
+    [_ #f]))
+
+;; Checks a type for syntactic well-formedness.
+(define (stlc-dll/type? type)
+  (match type
+    ; Symbols are only valid if previously bound (by `type` or `μ`).
+    ; We can't check that here, however.
+    [x #:when (symbol? x) #t]
+    [`(Ref ,t) (stlc-dll/type? t)]
+    [(or `(,t1 × ,t2) `(,t1 ⊕ ,t2))
+      (and (stlc-dll/type? t1) (stlc-dll/type? t2))]
+    [`(,t1 → ,k ,t2)
+      (and (stlc-dll/type? t1) (natural? k) (stlc-dll/type? t2))]
+    [`(μ ,x ,t)
+      (and (symbol? x) (stlc-dll/type? t))]
+    [_ #f]))
+
+;; Checks a value for syntactic well-formedness.
+(define (stlc-dll/value? expr)
+  (match expr
+    [x #:when (symbol? x) #t]
+    [n #:when (natural? n) #t]
+    [b #:when (boolean? b) #t]
+    [(or `(inl ,v) `(inr ,v))
+      (stlc-dll/value? v)]
+    [(or `(pair ,v1 ,v2) `(,v1 ,v2))
+      (and (stlc-dll/value? v1) (stlc-dll/value? v2))]
+    [`(λ (,x : ,t) ,e ,env)
+      (and (symbol? x) (stlc-dll/type? t) (stlc-dll/expr? e) (dict? env))]
+    [_ #f]))
+
+
+(define (interpret expr)
+  (interpret/core (desugar expr) #hash() (make-hash)))
+;; Γ: a Table[Symbol, Expr] representing the context:
+;;   the current bindings in scope introduced by λx.[]
+;; Σ: a Table[Symbol, Expr] representing the heap:
+;;   the current references on the heap generated by (gensym). mutable
+;; Interprets a *desugared* expression *stripped* of type annotations.
+(define/contract (interpret/core expr Γ Σ)
+  (-> stlc-dll/expr? dict? dict? stlc-dll/value?)
+  (match expr
+    ['sole 'sole]
+    [n #:when (natural? n) n]
+    [b #:when (boolean? b) b]
+    [r #:when (dict-has-key? Σ r) r]
+    [x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
+    [f #:when (symbol? f) f]
+
+    [`(inc ,e)
+      (match (interpret/core e Γ Σ)
+        [n #:when (natural? n) (+ n 1)]
+        [e (format "incrementing an unknown value ~a" e)])]
+    [`(if ,c ,e1 ,e2)
+      (match (interpret/core c Γ Σ)
+        ['#t (interpret/core e1 Γ Σ)]
+        ['#f (interpret/core e2 Γ Σ)]
+        [e (err (format "calling if on unknown expression ~a" e))])]
+
+    [`(pair ,e1 ,e2)
+      `(pair ,(interpret/core e1 Γ Σ) ,(interpret/core e2 Γ Σ))]
+    [`(car ,e)
+      (match (interpret/core e Γ Σ)
+        [`(pair ,e1 ,e2) e1]
+        [e (err (format "calling car on unknown expression ~a" e))])]
+    [`(cdr ,e)
+      (match (interpret/core e Γ Σ)
+        [`(pair ,e1 ,e2) e2]
+        [e (err (format "calling cdr on unknown expression ~a" e))])]
+
+    [`(inl ,e) `(inl ,(interpret/core e Γ Σ))]
+    [`(inr ,e) `(inr ,(interpret/core e Γ Σ))]
+    [`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (interpret/core e Γ Σ)
+        [`(inl ,e) (interpret/core e1 (dict-set Γ x1 e) Σ)]
+        [`(inr ,e) (interpret/core e2 (dict-set Γ x2 e) Σ)]
+        [e (err (format "calling case on unknown expression ~a" e))])]
+
+    [`(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]
+
+    [`(fold ,e) `(fold ,(interpret/core e Γ Σ))]
+    [`(unfold ,e)
+      (match (interpret/core e Γ Σ)
+        [`(fold ,e) e]
+        [e (err (format "attempting to unfold unknown expression ~a" e))])]
+
+    [`(λ (,x : ,t) ,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 that an expression is of a type, and returns #t or #f (or a bubbled-up error)
+;; with: a type in weak-head normal form (!!)
+;; Γ: a Table[Symbol, Expr ⊕ Type] representing the context:
+;;   the current bindings in scope introduced by λx.[] and μx.[] and τx.[]
+(define (check expr with)
+  (check/core (desugar expr) with #hash()))
+(define/contract (check/core expr with Γ)
+  (-> stlc-dll/expr? stlc-dll/type? dict? boolean?)
+  (match expr
+    [`(type ,t1 ,t2 ,in)
+      (check/core in with (dict-set Γ `(type ,t1) t2))]
+
+    [`(if ,c ,e1 ,e2)
+      (and (check/core c 'Bool Γ)
+        (check/core e1 with Γ) (check/core e2 with Γ))]
+
+    [`(pair ,e1 ,e2)
+      (match with
+        [`(,t1 × ,t2) (and (check/core e1 t1 Γ) (check/core e2 t2 Γ))]
+        [_ #f])]
+
+    [`(inl ,e)
+      (match with
+        [`(,t1 ⊕ ,t2) (check/core e t1 Γ)]
+        [_ #f])]
+    [`(inr ,e)
+      (match with
+        [`(,t1 ⊕ ,t2) (check/core e t2 Γ)]
+        [_ #f])]
+    [`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (infer/core e Γ) ; avoid needing type annotation on e
+        [`(,a1 ⊕ ,a2)
+          (and (check/core e1 with (dict-set Γ x1 a1))
+            (check/core e2 with (dict-set Γ x2 a2)))]
+        [_ #f])]
+
+    [`(new ,e)
+      (match with
+        [`(Ref ,t) (check/core e t Γ)]
+        [_ #f])]
+    [`(! ,e)
+      (check/core e `(Ref ,with) Γ)]
+
+    [`(fold ,e)
+      (match with
+        [`(μ ,x ,t) (check/core e t (dict-set Γ `(type ,x) `(μ ,x ,t)))]
+        [_ #f])]
+
+    [`(λ (,x : ,t) ,e)
+      (match with
+        [`(,t1 → ,k ,t2)
+          (and (equiv-type t t1 Γ) (check/core e t2 (dict-set Γ x t))
+            (> k (level-body e (dict-set Γ x t1))))] ; KNOB
+        [`(,t1 → ,t2) (err (format "missing level annotation on function type"))]
+        [_ #f])]
+
+    [_ (equiv-type (infer/core expr Γ) with Γ)]))
+
+;; Checks if two types are equivalent up to α-conversion in context
+(define (equiv-type e1 e2 Γ)
+  (equiv-type/core e1 e2 Γ Γ))
+(define/contract (equiv-type/core e1 e2 Γ1 Γ2)
+  (-> stlc-dll/type? stlc-dll/type? dict? dict? boolean?)
+  (match* (e1 e2)
+    ; bound identifiers: if a key exists in the context, look it up
+    [(x1 x2) #:when (dict-has-key? Γ1 x1)
+      (equiv-type/core (dict-ref Γ1 x1) x2 Γ1 Γ2)]
+    [(x1 x2) #:when (dict-has-key? Γ2 x2)
+      (equiv-type/core x1 (dict-ref Γ2 x2) Γ1 Γ2)]
+
+    ; recursive types: self-referential names can be arbitrary
+    [(`(μ ,x1 ,t1) `(μ ,x2 ,t2))
+      (let ([name gensym])
+      (equiv-type/core t1 t2 (dict-set Γ1 x1 name) (dict-set Γ2 x2 name)))]
+
+    ; check for syntactic equivalence on remaining forms
+    [(`(,l1 ...) `(,l2 ...))
+      (foldl (λ (x1 x2 acc) (if (equiv-type/core x1 x2 Γ1 Γ2) acc #f)) #t l1 l2)]
+    [(v1 v2) (equal? v1 v2)]))
+
+;; Infers a type from a given expression, if possible, or errors out.
+;; Returns a type in weak-head normal form for structural matching.
+(define (infer expr)
+  (infer/core (desugar expr) #hash()))
+(define/contract (infer/core expr Γ)
+  (-> stlc-dll/expr? dict? stlc-dll/type?)
+  (match expr
+    ['sole 'Unit]
+    [n #:when (natural? n) 'Nat]
+    [b #:when (boolean? b) 'Bool]
+    [x #:when (dict-has-key? Γ x)
+      (type->whnf (dict-ref Γ x) Γ)]
+    [f #:when (symbol? f)
+      (err (format "attempting to infer type of free variable ~a" f))]
+
+    [`(type ,t1 ,t2 ,in)
+      (infer/core in (dict-set Γ `(type ,t1) t2))]
+    [`(,e : ,t) ; we have a manual type annotation, so we must expand to weak-head normal form
+      (if (check/core e (type->whnf t Γ) Γ) (type->whnf t Γ)
+        (err (format "annotated expression ~a is not of annotated type ~a" e t)))]
+
+    [`(inc ,e)
+      (if (check/core e 'Nat Γ) 'Nat
+        (err (format "calling inc on incorrect type ~a" (infer/core e Γ))))]
+    [`(if ,c ,e1 ,e2)
+      (if (check/core c 'Bool Γ)
+        (let ([t (infer/core e1 Γ)])
+        (if (check/core e2 t Γ) t
+          (err (format "condition has branches of differing types ~a and ~a"
+            t (infer/core e2 Γ)))))
+        (err (format "condition ~a has incorrect type ~a" c (infer/core c Γ))))]
+
+    [`(pair ,e1 ,e2)
+      `(,(infer/core e1 Γ) × ,(infer/core e2 Γ))]
+    [`(car ,e)
+      (match (infer/core e Γ)
+        [`(,t1 × ,t2) t1]
+        [t (err (format "calling car on incorrect type ~a" t))])]
+    [`(cdr ,e)
+      (match (infer/core e Γ)
+        [`(,t1 × ,t2) t2]
+        [t (err (format "calling cdr on incorrect type ~a" t))])]
+
+    [`(inl ,e)
+      (err (format "unable to infer the type of a raw inl"))]
+    [`(inr ,e)
+      (err (format "unable to infer the type of a raw inr"))]
+    [`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (infer/core e Γ)
+        [`(,a1 ⊕ ,a2)
+          (let ([b1 (infer/core e1 (dict-set Γ x1 a1))]
+                [b2 (infer/core e2 (dict-set Γ x2 a2))])
+            (if (equiv-type b1 b2 Γ) b1
+              (err (format "case ~a is not of consistent type!"
+                `(case (,a1 ⊕ ,a2) (,x1 ⇒ ,b1) (,x2 ⇒ ,b2))))))]
+        [t (err (format "calling case on incorrect type ~a" t))])]
+
+    [`(new ,e)
+      `(Ref ,(infer/core e Γ))]
+    [`(! ,e)
+      (match (infer/core e Γ)
+        [`(Ref ,t) t]
+        [t (err (format "attempting to deref term ~a of type ~a" e t))])]
+    [`(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))])]
+
+    [`(unfold ,e)
+      (match (infer/core e Γ)
+        [`(μ ,x ,t) (replace t x `(μ ,x ,t))]
+        [t (err (format "expected ~a to be recursive, got ~a" e t))])]
+
+    [`(λ (,x : ,t1) ,e)
+      (let* ([t2 (infer/core e (dict-set Γ x t1))]
+        [t1 (type->whnf t1 Γ)] ; type annotation, must expand
+        [k (+ 1 (level-body e (dict-set Γ x t1)))]) ; 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))))]
+        [`(,t1 → ,t2) (err (format "missing level annotation on function type"))]
+        [t (err (format "expected → type on application body, got ~a" t))])]))
+
+;; Checks if a type is well-formed in the current context.
+;; BIG ASSUMPTION: types in the current context are well-formed
+(define/contract (well-formed t Γ)
+  (-> stlc-dll/type? dict? boolean?)
+  (match t
+    [x #:when (dict-has-key? Γ x) #t]
+    [(or 'Unit 'Nat 'Bool) #t]
+    [`(Ref ,t) (well-formed t Γ)]
+    [`(μ ,x ,t) (well-formed t (dict-set Γ x `(μ ,x ,t)))]
+    [`(type ,x ,t) (well-formed t (dict-set Γ x `(μ ,x ,t)))]
+    [(or `(,t1 → ,_ ,t2) `(,t1 × ,t2) `(,t1 ⊕ ,t2))
+      (and (well-formed t1 Γ) (well-formed t2 Γ))]
+    [_ #f]))
+
+;; Checks if a type is well-kinded with respect to a level in the current context
+;; BIG ASSUMPTION: types in the current context are well-formed
+(define/contract (well-kinded t l Γ)
+  (-> stlc-dll/type? natural? dict? boolean?)
+  (match t
+    [x #:when (dict-has-key? Γ x) #t]
+    [(or 'Unit 'Nat 'Bool) (>= l 0)]
+    [`(Ref ,t)
+      (if (zero? l)
+        (well-kinded t l Γ)
+        (well-kinded t (- l 1) Γ))]
+    [`(μ ,x ,t)
+      (well-kinded t l (dict-set Γ x `(μ ,x ,t)))]
+    [(or `(,t1 × ,t2) `(,t1 ⊕ ,t2))
+      (and (well-kinded t1 l Γ) (well-kinded t2 l Γ))]
+    [`(,t1 → ,k ,t2)
+      (and (>= l k) (well-kinded t1 k Γ) (well-kinded t2 k Γ))]
+    [_ #f]))
+
+;; Infers the level of a (well-formed) type.
+(define/contract (level-type t Γ)
+  (-> stlc-dll/type? dict? natural?)
+  (match t
+    [x #:when (dict-has-key? Γ x)
+      (level-type (dict-ref Γ x) Γ)]
+    [(or 'Unit 'Nat) 0]
+    [(or `(,t1 × ,t2) `(,t1 ⊕ ,t2))
+      (max (level-type t1 Γ) (level-type t2 Γ))]
+    [`(μ ,x ,t) ; note: correct but VERY WEIRD
+      (level-type t Γ)]
+    [`(,t1 → ,k ,t2)
+      (if (and (>= k (level-type t1 Γ)) (>= k (level-type t2 Γ))) k ; KNOB
+        (err (format "annotated level ~a is less than inferred levels of ~a and ~a!" k t1 t2)))]
+    [`(Ref ,t)
+      (let ([k (level-type t Γ)])
+      (if (zero? k) 0 (+ 1 k)))] ; KNOB
+    [t #:when (symbol? t) 0])) ; μ-type variables, not in Γ
+
+;; Infers the level of a (well-formed) expression.
+(define/contract (level-body e Γ)
+  (-> stlc-dll/expr? dict? natural?)
+  (match e
+    ['sole 0]
+    [n #:when (natural? n) 0]
+    [x #:when (dict-has-key? Γ x) ; free variables, get their level
+      (level-type (type->whnf (dict-ref Γ x) Γ) Γ)]
+    [(or `(,e : ,_) `(λ (,_ : ,_) ,e)
+      `(inc ,e) `(new ,e) `(! ,e) `(car ,e) `(cdr ,e) `(inl ,e) `(inr ,e)
+      `(fold ,e) `(unfold ,e) `(fold (μ ,_ ,_) ,e) `(unfold (μ ,_ ,_) ,e))
+      (level-body e Γ)]
+    [(or `(set ,e1 ,e2) `(pair ,e1 ,e2) `(,e1 ,e2))
+      (max (level-body e1 Γ) (level-body e2 Γ))]
+    [(or `(if ,c ,e1 ,e2) `(case ,c (,_ ⇒ ,e1) (,_ ⇒ ,e2)))
+      (max (level-body c Γ) (level-body e1 Γ) (level-body e2 Γ))]
+    [x #:when (symbol? x) 0])) ; local variables, not in Γ