restructure code, put implementations into broadly categorical foldersHEADmain

5 files changed, 705 insertions, 0 deletions

diff --git a/simple/ext.rkt b/simple/ext.rkt
new file mode 100644
index 0000000..abc8758
--- /dev/null
+++ b/simple/ext.rkt
@@ -0,0 +1,319 @@
+#lang racket
+(require "../lib.rkt")
+(provide (all-defined-out))
+
+;; The Simply-Typed Lambda Calculus, with simple extensions
+;; Unit/String/Natural/Boolean, pairs, sums, lists, ascryption
+
+;; Checks an expression for syntactic well-formedness.
+(define (stlc-ext/expr? expr)
+  (match expr
+    [(or 'sole 'nil) #t]
+    [s #:when (string? s) #t]
+    [n #:when (natural? n) #t]
+    [b #:when (boolean? b) #t]
+    [x #:when (symbol? x) #t]
+
+    [(or
+      `(inc ,e)
+      `(car ,e) `(cdr ,e)
+      `(inl ,e) `(inr ,e)
+      `(head ,e) `(tail ,e) `(nil? ,e))
+      (stlc-ext/expr? e)]
+    [(or
+      `(pair ,e1 ,e2)
+      `(cons ,e1 ,e2)
+      `(,e1 ,e2))
+      (and (stlc-ext/expr? e1) (stlc-ext/expr? e2))]
+
+    [`(if ,c ,e1 ,e2)
+      (and (stlc-ext/expr? c) (stlc-ext/expr? e1) (stlc-ext/expr? e2))]
+    [`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (and (stlc-ext/expr? e) (stlc-ext/expr? e1) (stlc-ext/expr? e2)
+        (symbol? x1) (symbol? x2))]
+    [`(type ,t1 ,t2 ,e)
+      (and (stlc-ext/type? t1) (stlc-ext/type? t2) (stlc-ext/expr? e))]
+    [`(λ (,x : ,t) ,e)
+      (and (symbol? x) (stlc-ext/type? t) (stlc-ext/expr? e))]
+    [_ #f]))
+
+;; Checks a type for syntactic well-formedness.
+(define (stlc-ext/type? type)
+  (match type
+    [t #:when (symbol? t) #t]
+    [`(List ,t) (stlc-ext/type? t)]
+    [(or
+      `(,t1 → ,t2)
+      `(,t1 × ,t2)
+      `(,t1 ⊕ ,t2))
+      (and (stlc-ext/type? t1) (stlc-ext/type? t2))]
+    [_ #f]))
+
+;; Checks a value for syntactic well-formedness.
+(define (stlc-ext/value? value)
+  (match value
+    [(or 'sole 'nil) #t]
+    [s #:when (string? s) #t]
+    [n #:when (natural? n) #t]
+    [b #:when (boolean? b) #t]
+    [x #:when (symbol? x) #t]
+    [(or
+      `(pair ,v1 ,v2)
+      `(cons ,v1 ,v2)
+      `(,v1 ,v2))
+      (and (stlc-ext/value? v1) (stlc-ext/value? v2))]
+    [`(λ ,x ,e ,env)
+      (and (symbol? x) (stlc-ext/expr? e) (dict? env))]
+    [_ #f]))
+
+;; Interprets an expression down to a value, in a given context.
+(define (interpret expr)
+  (interpret/core (desugar expr) #hash()))
+(define/contract (interpret/core expr Γ)
+  (-> stlc-ext/expr? dict? stlc-ext/value?)
+  (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)]
+    [f #:when (symbol? f) f]
+
+    [`(,e : ,t) (interpret/core e Γ)]
+    [`(type ,t1 ,t2 ,e) (interpret/core e Γ)]
+
+    [`(inc ,e)
+      (match (interpret/core e Γ)
+        [n #:when (natural? n) (+ n 1)]
+        [e (format "incrementing an unknown value ~a" e)])]
+    [`(if ,c ,e1 ,e2)
+      (match (interpret/core c Γ)
+        ['#t (interpret/core e1 Γ)]
+        ['#f (interpret/core e2 Γ)]
+        [e (err (format "calling if on unknown expression ~a" e))])]
+
+    [`(pair ,e1 ,e2)
+      `(pair ,(interpret/core e1 Γ) ,(interpret/core e2 Γ))]
+    [`(car ,e)
+      (match (interpret/core e Γ)
+        [`(pair ,e1 ,e2) e1]
+        [e (err (format "calling car on unknown expression ~a" e))])]
+    [`(cdr ,e)
+      (match (interpret/core e Γ)
+        [`(pair ,e1 ,e2) e2]
+        [e (err (format "calling cdr on unknown expression ~a" e))])]
+
+    [`(inl ,e) `(inl ,(interpret/core e Γ))]
+    [`(inr ,e) `(inr ,(interpret/core e Γ))]
+    [`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (interpret/core e Γ)
+        [`(inl ,e) (interpret/core e1 (dict-set Γ x1 e))]
+        [`(inr ,e) (interpret/core e2 (dict-set Γ x2 e))]
+        [e (err (format "calling case on unknown expression ~a" e))])]
+
+    ['nil 'nil]
+    [`(nil? ,e)
+      (match (interpret/core e Γ)
+        ['nil '#t]
+        [`(cons ,e1 ,e2) '#f]
+        [e (err (format "calling isnil on unknown expression ~a" e))])]
+    [`(cons ,e1 ,e2)
+     `(cons ,(interpret/core e1 Γ) ,(interpret/core e2 Γ))]
+    [`(head ,e)
+      (match (interpret/core e Γ)
+        [`(cons ,e1 ,e2) (interpret/core e1 Γ)]
+        [e (err (format "calling head on unknown expression ~a" e))])]
+    [`(tail ,e)
+      (match (interpret/core e Γ)
+        [`(cons ,e1 ,e2) (interpret/core e2 Γ)]
+        [e (err (format "calling tail on unknown expression ~a" e))])]
+
+    [`(λ (,x : ,t) ,e) `(λ ,x ,e ,Γ)]
+    [`(,e1 ,e2)
+      (match (interpret/core e1 Γ)
+        [`(λ ,x ,e ,env)
+          (interpret/core e (dict-set env x (interpret/core e2 Γ)))]
+        [e (err (format "applying arg ~a to unknown expression ~a" e2 e))])]))
+
+;; 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-ext/expr? stlc-ext/type? dict? boolean?)
+  (match expr
+    [`(type ,t1 ,t2 ,in)
+      (check/core in with (dict-set Γ 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)
+        [`(,a1 ⊕ ,a2)
+          (and (check/core e1 with (dict-set Γ x1 a1))
+            (check/core e2 with (dict-set Γ x2 a2)))]
+        [_ #f])]
+
+    ['nil
+      (match with
+        [`(List ,t) #t]
+        [_ #f])]
+    [`(cons ,f1 ,f2)
+      (match with
+        [`(List ,t)
+          (and (check/core f1 t Γ)
+            (check/core f2 `(List ,t) Γ))]
+        [_ #f])]
+
+    [`(λ (,x : ,t) ,e)
+      (match with
+        [`(,t1 → ,t2)
+          (and (equiv-type t1 t Γ) (check/core e t2 (dict-set Γ x t1)))]
+        [_ #f])]
+
+    [_ (equiv-type (infer-core expr Γ) with Γ)]))
+
+;; 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-ext/expr? dict? stlc-ext/type?)
+  (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)
+      (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 t1 t2))]
+    [`(,e : ,t)
+      (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) ; annotations necessary
+      (match (infer-core e Γ)
+        [`(,t1 ⊕ ,t2) `(,t1 ⊕ ,t2)]
+        [t (err (format "calling inl on incorrect type ~a" t))])]
+    [`(inr ,e) ; annotations necessary
+      (match (infer-core e Γ)
+        [`(,t1 ⊕ ,t2) `(,t1 ⊕ ,t2)]
+        [t (err (format "calling inr on incorrect type ~a" t))])]
+    [`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (infer-core e Γ)
+        [`(,a1 ⊕ ,a2)
+          (let ([b1 (infer-core e1 (dict-set Γ x1 (type->whnf a1 Γ)))]
+            [b2 (infer-core e2 (dict-set Γ x2 (type->whnf a2 Γ)))])
+          (if (equiv-type b1 b2 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))])]
+
+    ['nil (err (format "unable to infer type of empty list!"))]
+    [`(cons ,e1 ,e2)
+      (let ([t (infer-core e1 Γ)])
+      (if (check/core e2 `(List ,t) Γ) `(List ,t)
+        (err (format "list ~a is not of consistent type!" `(cons ,e1 ,e2)))))]
+    [`(head ,e)
+      (match (infer-core e Γ)
+        [`(List ,t) t]
+        [t (err (format "calling head on incorrect type ~a" t))])]
+    [`(tail ,e)
+      (match (infer-core e Γ)
+        [`(List ,t) `(List ,t)]
+        [t (err (format "calling tail on incorrect type ~a" t))])]
+
+    [`(λ (,x : ,t) ,e)
+      `(,(type->whnf t Γ) → ,(infer-core e (dict-set Γ x t)))]
+    [`(,e1 ,e2)
+      (match (infer-core e1 Γ)
+        [`(,t1 → ,t2)
+          (if (check/core 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))])]))
+
+;; Expands a type alias into weak-head normal form, for literal matching.
+(define (type->whnf t Γ)
+  (if (dict-has-key? Γ `(type ,t))
+    (type->whnf (dict-ref Γ `(type ,t)) Γ) t))
+
+;; Checks if two types are equivalent up to α-conversion in context
+;;      (equiv-type Expr Expr Table[Sym Expr]): Bool
+(define (equiv-type e1 e2 Γ)
+  (equiv-type/core e1 e2 Γ Γ))
+(define (equiv-type/core e1 e2 Γ1 Γ2)
+  (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)]
+
+    ; 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)]))
+
+;; Checks if two terms are equivalent up to α-conversion in context
+;;      (equiv-term Expr Expr Table[Sym Expr]): Bool
+(define (equiv-term e1 e2 Γ)
+  (equiv-term/core e1 e2 Γ Γ))
+(define (equiv-term/core e1 e2 Γ1 Γ2)
+  (match* (e1 e2)
+    ; bound identifiers: if a key exists in the context, look it up
+    [(x1 x2) #:when (dict-has-key? Γ1 x1)
+      (equiv-term/core (dict-ref Γ1 x1) x2 Γ1 Γ2)]
+    [(x1 x2) #:when (dict-has-key? Γ2 x2)
+      (equiv-term/core x1 (dict-ref Γ2 x2) Γ1 Γ2)]
+
+    ; function expressions: parameter names can be arbitrary
+    [(`(λ (,x1 : ,t1) ,e1) `(λ (,x2 : ,t2) ,e2))
+      (let ([name gensym])
+      (and (equiv-term/core e1 e2 (dict-set Γ1 x1 name) (dict-set Γ2 x2 name))
+        (equiv-term/core t1 t2 Γ1 Γ2)))]
+    [(`(λ ,x1 ,e1) `(λ ,x2 ,e2))
+      (let ([name gensym])
+      (equiv-term/core e1 e2 (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-term/core x1 x2 Γ1 Γ2) acc #f)) #t l1 l2)]
+    [(v1 v2) (equal? v1 v2)]))
diff --git a/simple/fix.rkt b/simple/fix.rkt
new file mode 100644
index 0000000..466a79c
--- /dev/null
+++ b/simple/fix.rkt
@@ -0,0 +1,86 @@
+#lang racket
+(require "../lib.rkt")
+(require (only-in "stlc.rkt" stlc/type?))
+
+;; The Simply-Typed Lambda Calculus, with general recursion
+
+;; Checks an expression for syntactic well-formedness.
+(define/contract (stlc-fix/expr? expr)
+  (-> any/c boolean?)
+  (match expr
+    [x #:when (symbol? x) #t]
+    [`(fix ,e) (stlc-fix/expr? e)]
+    [`(λ (,x : ,t) ,e) (and (symbol? x) (stlc/type? t) (stlc-fix/expr? e))]
+    [`(,e1 ,e2) (and (stlc-fix/expr? e1) (stlc-fix/expr? e2))]
+    [_ #f]))
+
+;; Checks a value for syntactic well-formedness.
+(define (stlc-fix/value? value)
+  (match value
+    [x #:when (symbol? x) #t]
+    [`(,v1 ,v2) (and (stlc-fix/value? v1) (stlc-fix/value? v2))]
+    [`(λ ,x ,e ,env) (and (symbol? x) (stlc-fix/expr? e) (dict? env))]
+    [_ #f]))
+
+;; Interprets an expression down to a value, in a given context.
+(define (interpret expr)
+  (interpret/core (desugar expr) #hash()))
+(define/contract (interpret/core expr Γ)
+  (-> stlc-fix/expr? dict? stlc-fix/value?)
+  (match expr
+    ['sole 'sole]
+    [x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
+    [x #:when (symbol? x) x]
+
+    [`(fix ,e)
+      (match (interpret/core e Γ)
+        [`(λ ,x ,e ,env)
+          ; FIXME: unsure what should be Γ and what should be env
+          (interpret/core e (dict-set env x `(fix (λ ,x ,e ,Γ))))]
+        [e (err (format "applying fix to unknown expression ~a" e))])]
+
+    [`(λ (,id : ,t) ,body) `(λ ,id ,body ,Γ)]
+    [`(,body ,arg)
+      (match (interpret/core body Γ)
+        [`(λ ,id ,body ,env)
+          (interpret/core body (dict-set env id (interpret/core arg Γ)))]
+        [e (err (format "applying arg ~a to unknown expression ~a" arg e))])]))
+
+;; 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) ; FIXME
+  (-> stlc-fix/expr? stlc/type? dict? boolean?)
+  (let ([with (if (dict-has-key? Γ with) (dict-ref Γ with) with)])
+  (match expr
+    [`(fix ,e)
+      (check/core (infer/core e) `(,with → ,with) Γ)]
+    [`(λ (,x : ,t) ,e)
+      (match with
+        [`(,t1 → ,t2)
+          (and (equal? t1 t)
+            (check/core e t2 (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-fix/expr? dict? stlc/type?)
+  (match expr
+    [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))]
+    [`(fix ,e)
+      (match (infer/core e Γ)
+        [`(,t → ,t) t]
+        [t (err (format "fix expects a term of type t → t, got ~a" t))])]
+    [`(λ (,x : ,t) ,e)
+      `(,t → ,(infer/core e (dict-set Γ x t)))]
+    [`(,e1 ,e2)
+      (match (infer/core e1 Γ)
+        [`(,t1 → ,t2)
+          (if (check/core 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))])]))
diff --git a/simple/rec.rkt b/simple/rec.rkt
new file mode 100644
index 0000000..91a50ce
--- /dev/null
+++ b/simple/rec.rkt
@@ -0,0 +1,114 @@
+#lang racket
+(require "../lib.rkt")
+(provide (all-defined-out))
+
+;; The Simply-Typed Lambda Calculus with iso-recursive types
+
+;; Checks an expression for syntactic well-formedness.
+(define (stlc-rec/expr? expr)
+  (match expr
+    [x #:when (symbol? x) #t]
+    [(or `(fold ,e) `(unfold ,e)) (stlc-rec/expr? e)]
+    [`(,e1 ,e2) (and (stlc-rec/expr? e1) (stlc-rec/expr? e2))]
+    [`(λ (,x : ,t) ,e) (and (symbol? x) (stlc-rec/type? t) (stlc-rec/expr? e))]
+    [_ #f]))
+
+;; Checks a type for syntactic well-formedness.
+(define (stlc-rec/type? type)
+  (match type
+    [t #:when (symbol? t) #t]
+    [`(,t1 → ,t2) (and (stlc-rec/type? t1) (stlc-rec/type? t2))]
+    [_ #f]))
+
+;; Checks a value for syntactic well-formedness.
+(define (stlc-rec/value? value)
+  (match value
+    [x #:when (symbol? x) #t]
+    [`(,v1 ,v2) (and (stlc-rec/value? v1) (stlc-rec/value? v2))]
+    [`(λ ,x ,e ,env) (and (symbol? x) (stlc-rec/expr? e) (dict? env))]
+    [_ #f]))
+
+;; Interprets an expression down to a value, in a given context.
+(define (interpret expr)
+  (interpret/core (desugar expr) #hash()))
+(define/contract (interpret/core expr Γ)
+  (-> stlc-rec/expr? dict? stlc-rec/value?)
+  (match expr
+    ['sole 'sole]
+    [x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
+    [n #:when (natural? n) n]
+    [f #:when (symbol? f) f]
+
+    [`(fold ,e) `(fold ,(interpret/core e Γ))]
+    [`(unfold ,e)
+      (match (interpret/core e Γ)
+        [`(fold ,e) e]
+        [e `(unfold e)])]
+
+    [`(λ (,x : ,t) ,e) `(λ ,x ,e ,Γ)]
+    [`(,e1 ,e2)
+      (match (interpret/core e1 Γ)
+        [`(λ ,x ,e ,env)
+          (interpret/core e (dict-set env x (interpret/core e2 Γ)))]
+        [e (err (format "applying arg ~a to unknown expression ~a" e2 e))])]))
+
+;; 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-rec/expr? stlc-rec/type? dict? boolean?)
+  (match expr
+    [`(fold (μ ,x ,t) ,e)
+      (match with
+        [`(μ ,x ,t) (check/core e t (dict-set Γ x `(μ ,x ,t)))]
+        [_ #f])]
+
+    [`(λ (,x : ,t) ,e)
+      (match with
+        [`(,t1 → ,t2)
+          (and (equal? t1 t) (check/core e t2 (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-rec/expr? dict? stlc-rec/type?)
+  (match expr
+    ['sole 'Unit]
+    [x #:when (dict-has-key? Γ x)
+      (dict-ref Γ x)]
+    [b #:when (boolean? b) 'Bool]
+    [n #:when (natural? n) 'Nat]
+    [f #:when (symbol? f)
+      (err (format "attempting to infer type of free variable ~a" f))]
+
+    [`(fold (μ ,x ,t) ,e)
+      (if (check/core e t (dict-set Γ x `(μ ,x ,t))) `(μ ,x ,t)
+        (err (format "expected ~a to be of type ~a, got ~a"
+          e t (infer e (dict-set Γ x `(μ ,x ,t))))))]
+    [`(unfold (μ ,x ,t) ,e)
+      (if (check/core e `(μ ,x ,t)) (replace t x `(μ ,x ,t))
+        (err (format "expected ~a to be of type ~a, got ~a"
+          e `(μ ,x ,t) (infer/core e Γ))))]
+
+    [`(λ (,x : ,t) ,e)
+      `(,t → ,(infer/core e (dict-set Γ x t)))]
+    [`(,e1 ,e2)
+      (match (infer/core e1 Γ)
+        [`(,t1 → ,t2)
+          (if (check/core 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))])]))
+
+;; Replace all references to an expression with a value.
+(define (replace expr key value)
+  (match expr
+    ; do not accidentally replace shadowed bindings
+    [(or `(λ (,x : ,_) ,_) `(λ ,x ,_) `(μ ,x ,_)
+      `(type ,x ,_ ,_)) #:when (equal? x key) expr]
+    [`(,e ...) `(,@(map (λ (x) (replace x key value)) e))]
+    [x #:when (equal? x key) value]
+    [v v]))
diff --git a/simple/ref.rkt b/simple/ref.rkt
new file mode 100644
index 0000000..aff983f
--- /dev/null
+++ b/simple/ref.rkt
@@ -0,0 +1,119 @@
+#lang racket
+(require "../lib.rkt")
+(provide (all-defined-out))
+
+;; The Simply-Typed Lambda Calculus with references
+
+; todo: rewrite to use call-by-reference or call-by-value or call-by-name explicitly
+
+;; Checks an expression for syntactic well-formedness.
+(define (stlc-ref/expr? expr)
+  (match expr
+    [x #:when (symbol? x) #t]
+    [n #:when (natural? n) #t]
+    [(or `(new ,e) `(! ,e)) (stlc-ref/expr? e)]
+    [(or `(set ,e1 ,e2) `(,e1 ,e2)) (and (stlc-ref/expr? e1) (stlc-ref/expr? e2))]
+    [`(λ (,x : ,t) ,e) (and (symbol? x) (stlc-ref/type? t) (stlc-ref/expr? e))]
+    [_ #f]))
+
+;; Checks a type for syntactic well-formedness.
+(define (stlc-ref/type? type)
+  (match type
+    [t #:when (symbol? t) #t]
+    [`(Ref ,t) (stlc-ref/type? t)]
+    [`(,t1 → ,t2) (and (stlc-ref/type? t1) (stlc-ref/type? t2))]
+    [_ #f]))
+
+;; Checks a value for syntactic well-formedness.
+(define (stlc-ref/value? value)
+  (match value
+    [x #:when (symbol? x) #t]
+    [n #:when (natural? n) #t]
+    [`(,v1 ,v2) (and (stlc-ref/value? v1) (stlc-ref/value? v2))]
+    [`(λ ,x ,e ,env) (and (symbol? x) (stlc-ref/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-ref/expr? dict? dict? stlc-ref/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 : ,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 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-ref/expr? stlc-ref/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 → ,t2)
+          (and (equal? t1 t) (check/core e t2 (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-ref/expr? dict? stlc-ref/type?)
+  (match expr
+    ['sole 'Unit]
+    [x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
+    [n #:when (natural? n) n]
+    [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 : ,t) ,e)
+      `(,t → ,(infer/core e (dict-set Γ x t)))]
+    [`(,e1 ,e2)
+      (match (infer/core e1 Γ)
+        [`(,t1 → ,t2)
+          (if (check/core 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))])]))
diff --git a/simple/stlc.rkt b/simple/stlc.rkt
new file mode 100644
index 0000000..1be0f91
--- /dev/null
+++ b/simple/stlc.rkt
@@ -0,0 +1,67 @@
+#lang racket
+(require "../lib.rkt")
+(provide (all-defined-out))
+
+;; The Simply-Typed Lambda Calculus
+
+;; Checks an expression for syntactic well-formedness.
+(define (stlc/expr? expr)
+  (match expr
+    [x #:when (symbol? x) #t]
+    [`(,e1 ,e2) (and (stlc/expr? e1) (stlc/expr? e2))]
+    [`(λ (,x : ,t) ,e) (and (symbol? x) (stlc/type? t) (stlc/expr? e))]
+    [_ #f]))
+
+;; Checks a type for syntactic well-formedness.
+(define (stlc/type? type)
+  (match type
+    [t #:when (symbol? t) #t]
+    [`(,t1 → ,t2) (and (stlc/type? t1) (stlc/type? t2))]
+    [_ #f]))
+
+;; Checks a value for syntactic well-formedness.
+(define (stlc/value? value)
+  (match value
+    [x #:when (symbol? x) #t]
+    [`(,v1 ,v2) (and (stlc/value? v1) (stlc/value? v2))]
+    [`(λ ,x ,e ,env) (and (symbol? x) (stlc/expr? e) (dict? env))]
+    [_ #f]))
+
+;; Interprets an expression down to a value, in a given context.
+(define/contract (interpret expr [Γ #hash()])
+  (->* (stlc/expr?) (dict?) stlc/value?)
+  (match expr
+    [x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
+    [f #:when (symbol? f) f]
+    [`(λ (,x : ,t) ,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 `(,e ,(interpret e2 Γ))])]))
+
+;; Checks an expression against some type, in a given context.
+(define/contract (check expr with [Γ #hash()])
+  (->* (stlc/expr? stlc/type?) (dict?) boolean?)
+  (match expr
+    [`(λ (,x : ,t) ,e)
+      (match with
+        [`(,t1 → ,t2)
+          (and (equal? t t1) (check e t2 (dict-set Γ x t)))]
+        [_ #f])]
+    [_ (equal? with (infer with Γ))]))
+
+;; Infers a type from some expression, in a given context.
+(define/contract (infer expr [Γ #hash()])
+  (->* (stlc/expr?) (dict?) stlc/type?)
+  (match expr
+    [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))]
+    [`(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))])]))