From b6c33b87b64bcbbd7555849031d1597966716634 Mon Sep 17 00:00:00 2001
From: JJ
Date: Fri, 30 Aug 2024 17:27:21 -0700
Subject: add self-contained implementation as stlc-full

---
 stlc-full.rkt | 739 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 739 insertions(+)
 create mode 100644 stlc-full.rkt

diff --git a/stlc-full.rkt b/stlc-full.rkt
new file mode 100644
index 0000000..3df2fa8
--- /dev/null
+++ b/stlc-full.rkt
@@ -0,0 +1,739 @@
+#lang racket
+(require syntax/location)
+(require (for-syntax syntax/location))
+
+;; The simply-typed lambda calculus, with:
+;; - sums, products, recursive types, ascryption
+;; - bidirectional typechecking
+;; - impredicative references built on a levels system
+;; - explicit level stratification syntax
+;; - automatic level collection
+;;
+;; This system is impredicative. It can be made predicative by tweaking
+;; the sections of the code labeled with "KNOB".
+
+(define-syntax-rule (print msg)
+  (eprintf "~a~%" msg))
+
+(define-syntax-rule (dbg body)
+  (let ([res body])
+    (eprintf "[~a:~a] ~v = ~a~%"
+      (syntax-source-file-name #'body)
+      (syntax-line #'body)
+      'body
+      res)
+    res))
+
+(define-syntax-rule (err msg)
+  (error 'error
+    (format "[~a:~a] ~a"
+      (syntax-source-file-name #'msg)
+      (syntax-line #'msg)
+      msg)))
+
+(define-syntax (todo stx)
+  (with-syntax ([src (syntax-source-file-name stx)] [line (syntax-line stx)])
+    #'(error 'todo (format "[~a:~a] unimplemented" src line))))
+
+(define (any? proc lst)
+  (foldl (λ (x acc) (if (proc x) #t acc)) #f lst))
+
+(define (all? proc lst)
+  (foldl (λ (x acc) (if (proc x) acc #f)) #t lst))
+
+;; Whether a given level is a syntactically valid level.
+;; This does not account for context, and so all symbols are valid levels.
+(define (level? level)
+  (match level
+    [n #:when (natural? n) #t]
+    ; Symbols are only valid if previously bound (by `level`).
+    ; We can't check that here, however.
+    [s #:when (symbol? s) #t]
+    ; Levels may be a list of symbols, or a list of symbols followed by a natural.
+    [`(+ ,l ... ,n) #:when (natural? n)
+      (all? (λ (s) (symbol? s)) l)]
+    [`(+ ,l ...)
+      (all? (λ (s) (symbol? s)) l)]
+    [_ #f]))
+
+;; Whether a given type is a syntactically valid type.
+;; This does not account for context, and so all symbols are valid types.
+(define (type? type)
+  (match type
+    [(or 'Unit 'Bool 'Nat) #t]
+    ; Symbols are only valid if previously bound (by `type` or `μ`).
+    ; We can't check that here, however.
+    [s #:when (symbol? s) #t]
+    [`(Ref ,t) (type? t)]
+    [(or `(,t1 × ,t2) `(,t1 ⊕ ,t2)) (and (type? t1) (type? t2))]
+    [`(,t1 → ,l ,t2) (and (type? t1) (level? l) (type? t2))]
+    [`(μ ,x ,t) (and (symbol? x) (type? t))]
+    [_ #f]))
+
+;; Whether a given expression is a syntactically valid expression.
+;; This does not account for context, and so all symbols are valid exprs.
+(define (expr? expr)
+  (match expr
+    ['sole #t]
+    [n #:when (natural? n) #t]
+    [b #:when (boolean? b) #t]
+    ; Symbols are only valid if previously bound by `λ` or `case`.
+    ; We can't check that here, however.
+    [s #:when (symbol? s) #t]
+
+    [`(,e : ,t) (and (expr? e) (type? t))]
+    [`(type ,t1 ,t2 ,e)
+      (and (type? t1) (type? t2) (expr? e))]
+
+    ; Note that level must only introduce new variables as levels.
+    [`(level ,l ,e)
+      (and (symbol? l) (expr? e))]
+    ; Kind of want to call ⇑ :: on analogy with :
+    [`(,e ⇑ ,l)
+      (and (expr? e) (level? l))]
+
+    [(or `(new ,e) `(! ,e)) (expr? e)]
+    ; [`(ptr ,a) (symbol? a)]
+    [`(set ,e1 ,e2) (and (expr? e1) (expr? e2))]
+
+    [`(inc ,e) (expr? e)]
+    [`(if ,c ,e1 ,e2)
+      (and (expr? c) (expr? e1) (expr? e2))]
+
+    [`(pair ,e1 ,e2)
+      (and (expr? e1) (expr? e2))]
+    [(or `(car ,e) `(cdr ,e)) (expr? e)]
+
+    [(or `(inl ,e) `(inr ,e)) (expr? e)]
+    [`(case ,c (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (and (symbol? x1) (symbol? x2)
+        (expr? c) (expr? e1) (expr? e2))]
+
+    [(or `(fold ,e) `(unfold ,e)) (expr? e)]
+
+    [`(λ (,x : ,t) ,e)
+      (and (symbol? x) (type? t) (expr? e))]
+    [`(,e1 ,e2) (and (expr? e1) (expr? e2))]
+    [_ #f]))
+
+;; Whether a term is a value or a computation. What's that little rhyme about this again?
+;; A value is, a computation/expr/term does?
+(define (value? expr)
+  (match expr
+    ['sole #t]
+    [n #:when (natural? n) #t]
+    [b #:when (boolean? b) #t]
+
+    ; [`(,v : ,t) (and (expr? v) (type? t))]
+    ; [`(type ,t1 ,t2 ,b)
+    ;   (and (type? t1) (type? t2) (value? v))]
+
+    ; [(or `(level ,l ,v) `(,v ⇑ ,l))
+    ;   (and (level? l) (value? v))]
+
+    [`(ptr ,l ,a) (and (level? l) (symbol? a))]
+
+    [`(pair ,v1 ,v2)
+      (and (value? v1) (value? v2))]
+
+    [(or `(inl ,v) `(inr ,v)) (value? v)]
+
+    ; [(or `(fold ,v) `(unfold ,v)) (value? v)]
+
+    [`(λ (,x : ,t) ,e ,env)
+      (and (symbol? x) (type? t) (expr? e) (dict? env))]
+    [_ #f]))
+
+(define (ptr? expr)
+  (match expr
+    [`(ptr ,l ,a) (and (level? l) (symbol? a))]
+    [_ #f]))
+
+;; Checks if a level is at its "base" form.
+(define/contract (level-base? l) (-> level? boolean?)
+  (match l
+    [n #:when (natural? n) (zero? n)]
+    [s #:when (symbol? s) #t]
+    [`(+ ,s ... ,n) #:when (natural? n) (zero? n)] ; should be avoided
+    [`(+ ,s ...) #t]))
+
+;; Syntactic equality between levels is not trivial.
+;; This helper function defines it.
+(define/contract (level-eq? l1 l2) (-> level? level? boolean?)
+  (match* (l1 l2)
+    [(n1 n2) #:when (and (natural? n1) (natural? n2))
+      (equal? n1 n2)]
+    [(`(+ ,s1 ... ,n1) `(+ ,s2 ... ,n2)) #:when (and (natural? n1) (natural? n2))
+      (and (equal? n1 n2) (level-eq? `(+ ,@s1) `(+ ,@s2)))]
+    [(`(+ ,s1 ...) `(+ ,s2 ...))
+      todo] ; need to check for equality as multisets
+    [(_ _) #f]))
+
+;; Levels can carry natural numbers, and so we define a stratification between them.
+;; Note: this returns FALSE if the levels are incomparable (i.e. (level-geq 'a 'b))
+(define/contract (level-geq? l1 l2) (-> level? level? boolean?)
+  (match* (l1 l2)
+    [(n1 n2) #:when (and (natural? n1) (natural? n2))
+      (>= n1 n2)]
+    [(s1 s2) #:when (and (symbol? s1) (symbol? s2))
+      (equal? s1 s2)]
+    [(`(+ ,s1 ... ,n1) `(+ ,s2 ... ,n2)) #:when (and (natural? n1) (natural? n2))
+      (and (level-eq? `(+ ,@s1) `(+ ,@s2)) (level-geq? n1 n2))]
+    [(`(+ ,s1 ... ,n) `(+ ,s2 ...)) #:when (natural? n)
+      (level-eq? `(+ ,@s1) `(+ ,@s2))]
+    [(`(+ ,s1 ...) `(+ ,s2 ...))
+      todo] ; this needs to inspect the levels... ugh
+    [(_ _) #f]))
+
+;; We define a maximum of two levels.
+;; This can return one of the two levels or an entirely new level.
+(define/contract (level-max l1 l2) (-> level? level? level?)
+  (match* (l1 l2)
+    [(n1 n2) #:when (and (natural? n1) (natural? n2))
+      (max n1 n2)]
+    [(s1 s2) #:when (and (symbol? s1) (symbol? s2))
+      (if (equal? s1 s2) s1 `(+ ,s1 ,s2))]
+    [(`(+ ,s1 ... ,n1) `(+ ,s2 ... ,n2)) #:when (and (natural? n1) (natural? n2))
+      (if (equal? s1 s2)
+        `(+ ,@s1 ,(max n1 n2))
+        (level-intersect `(+ ,@s1) `(+ ,@s2)))]
+    [(`(+ ,s1 ... ,n) `(+ ,s2 ...)) #:when (natural? n)
+      (if (level-geq? s1 s2)
+        `(+ ,@s1 ,n)
+        (level-intersect `(+ ,@s1) `(+ ,@s2)))]
+    [(`(+ ,s1 ...) `(+ ,s2 ... ,n)) #:when (natural? n)
+      (if (level-geq? s2 s1)
+        `(+ ,@s2 ,n)
+        (level-intersect `(+ ,@s1) `(+ ,@s2)))]
+    [(`(+ ,s ... ,n1) n2) #:when (and (natural? n1) (natural? n2))
+      `(+ ,s ... ,n1)]
+    [(n1 `(+ ,s ... ,n2)) #:when (and (natural? n1) (natural? n2))
+      `(+ ,s ... ,n2)]
+    [(`(+ ,s ...) n) #:when (natural? n)
+      `(+ ,@s ,n)]
+    [(n `(+ ,s ...)) #:when (natural? n)
+      `(+ ,@s ,n)]
+    [(`(+ ,s1 ...) `(+ ,s2 ...))
+      (level-intersect `(+ ,@s1) `(+ ,@s2))]))
+
+;; A helper function to perform intersection of levels.
+(define/contract (level-intersect l1 l2) (-> level-base? level-base? level-base?)
+  (match* (l1 l2)
+    [(0 l) l]
+    [(l 0) l]
+    [(`(+ ,s1 ...) `(+ ,s2 ...)) todo])) ; we need multisets. this is gross.
+
+;; We define addition in terms of our syntactic constructs.
+(define/contract (level-add l1 l2) (-> level? level? level?)
+  (match* (l1 l2)
+    [(n1 n2) #:when (and (natural? n1) (natural? n2))
+      (+ n1 n2)]
+    [(s1 s2) #:when (and (symbol? s1) (symbol? s2))
+      `(+ ,s1 ,s2)]
+    [(`(+ ,s1 ... ,n1) `(+ ,s2 ... ,n2)) #:when (and (natural? n1) (natural? n2))
+      (level-add (level-add `(+ ,@s1) `(+ ,@s2)) (+ n1 n2))]
+    [(`(+ ,s1 ... ,n) `(+ ,s2 ...)) #:when (natural? n)
+      (level-add (level-add `(+ ,@s1) `(+ ,@s2)) n)]
+    [(`(+ ,s1 ...) `(+ ,s2 ... ,n)) #:when (natural? n)
+      (level-add (level-add `(+ ,@s1) `(+ ,@s2)) n)]
+    [(`(+ ,s ... ,n1) n2) #:when (and (natural? n1) (natural? n2))
+      `(+ ,@s ,(+ n1 n2))]
+    [(n1 `(+ ,s ... ,n2)) #:when (and (natural? n1) (natural? n2))
+      `(+ ,@s ,(+ n1 n2))]
+    [(`(+ ,s ...) n) #:when (natural? n)
+      `(+ ,@s ,n)]
+    [(n `(+ ,s ...)) #:when (natural? n)
+      `(+ ,@s ,n)]
+    [(`(+ ,s1 ...) `(+ ,s2 ...))
+      `(+ ,@s1 ,@s2)]))
+
+;; Decrements a level by 1.
+;; ASSUMPTION: the level is not a base level (i.e. there is some n to dec)
+(define/contract (level-dec l) (-> level? level?)
+  (match l
+    [n #:when (and (natural? n) (not (zero? n))) (- n 1)]
+    [`(+ ,s ... 1) `(+ ,@s)]
+    [`(+ ,s ... ,n) #:when (and (natural? n) (not (zero? n))) `(+ ,@s ,(- n 1))]
+    [_ (err (format "attempting to decrement base level ~a" l))]))
+
+;; Returns the "base" form of a level.
+(define/contract (level-base l) (-> level? level?)
+  (match l
+    [n #:when (natural? n) 0]
+    [s #:when (symbol? s) s]
+    [`(+ ,s ... ,n) #:when (natural? n) `(+ ,@s)]
+    [`(+ ,s ...) `(+ ,@s)]))
+
+;; Returns the "offset" portion of a level.
+(define/contract (level-offset l) (-> level? level?)
+  (match l
+    [n #:when (natural? n) n]
+    [s #:when (symbol? s) 0]
+    [`(+ ,s ... ,n) #:when (natural? n) n]
+    [`(+ ,s ...) 0]))
+
+;; Infers the level of a (well-formed) type in a context.
+;; We need the context for type ascryption, and μ-types.
+;; Otherwise, levels are syntactically inferred.
+;; ASSUMPTION: the type is well-formed in the given context (i.e. all symbols bound).
+;; This is not checked via contracts due to (presumably) massive performance overhead.
+(define/contract (level-type t Γ) (-> type? dict? level?)
+  (match t
+    [(or 'Unit 'Bool 'Nat) 0]
+    [s #:when (symbol? s) 0] ; HACK: μ-type variables, not in Γ
+    [`(Ref ,t) ; KNOB
+      (let ([l (level-type t Γ)])
+        (if (level-base? l) l
+          (level-add l 1)))]
+    [(or `(,t1 × ,t2) `(,t1 ⊕ ,t2))
+      (level-max (level-type t1 Γ) (level-type t2 Γ))]
+    [`(,t1 → ,l ,t2) ; KNOB
+      (if (and (level-geq? l (level-type t1 Γ)) (level-geq? l (level-type t2 Γ))) l
+        (err (format "annotated level ~a is less than inferred levels ~a and ~a!")))]
+    [`(μ ,x ,t)
+      (level-type t (dict-set Γ x `(μ ,x ,t)))]))
+
+;; Infers the level of a (well-formed) expression.
+(define/contract (level-expr e Γ) (-> expr? dict? level?)
+  (match e
+    ['sole 0]
+    [n #:when (natural? n) 0]
+    [b #:when (boolean? b) 0]
+    [x #:when (dict-has-key? Γ x) ; free variables
+      (level-type (expand-whnf (dict-ref Γ x) Γ) Γ)]
+    [s #:when (symbol? s) 0] ; local variables
+
+    [`(,e : ,t)
+      (let ([l1 (level-expr e Γ)] [l2 (level-type t Γ)])
+        (if (level-geq? l1 l2) l1
+          (err (format "annotated level ~a is less than inferred level ~a!" l1 l2))))]
+    [`(type ,t1 ,t2 ,e)
+      (level-expr e (dict-set Γ `(type ,t1) t2))]
+
+    [`(level ,l ,e) ; NOTE: is this correct?
+      (level-expr e Γ)]
+    [`(,e ⇑ ,l)
+      (level-add l (level-expr e Γ))]
+
+    [`(new ,e) ; KNOB
+      (let ([l (level-expr e Γ)])
+        (if (level-base? l) l (level-add l 1)))]
+
+    [`(if ,c ,e1 ,e2)
+      (level-max (level-expr c Γ)
+        (level-expr e1 Γ) (level-expr e2 Γ))]
+    [`(case ,c (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (level-max (level-expr c Γ) ; support shadowing
+        (level-expr e1 (dict-remove Γ x1))
+        (level-expr e2 (dict-remove Γ x2)))]
+    [`(λ (,x : ,_) ,e) ; support shadowing
+      (level-expr e (dict-remove Γ x))]
+
+    [(or `(! ,e)`(inc ,e)
+      `(car ,e) `(cdr ,e)
+      `(inl ,e) `(inr ,e)
+      `(fold ,e) `(unfold ,e))
+      (level-expr e Γ)]
+    [(or `(set ,e1 ,e2) `(pair ,e1 ,e2) `(,e1 ,e2))
+      (level-max (level-expr e1 Γ) (level-expr e2 Γ))]))
+
+;; Whether a given type is a semantically valid type.
+;; We assume any type in Γ is semantically valid.
+;; Otherwise, we would infinitely recurse re: μ.
+(define/contract (well-formed? t Γ) (-> type? dict? boolean?)
+  (match t
+    [(or 'Unit 'Bool 'Nat) #t]
+    [s #:when (symbol? s) (dict-has-key? Γ `(type s))]
+    [`(Ref ,t) (well-formed? t Γ)]
+    [(or `(,t1 × ,t2) `(,t1 ⊕ ,t2)) (and (well-formed? t1 Γ) (well-formed? t2 Γ))]
+    [`(,t1 → ,l ,t2)
+      (and (dict-has-key? Γ `(level ,l))
+        (well-formed? t1 Γ) (well-formed? t2 Γ))]
+    [`(μ ,x ,t) ; check: this might infinitely recurse??
+      (well-formed? t (dict-set Γ `(type ,x) `(μ ,x ,t)))]))
+
+;; Whether a given type at a given level is semantically valid.
+(define/contract (well-kinded? t l Γ) (-> type? level? dict? boolean?)
+  (match t
+    [(or 'Unit 'Bool 'Nat) (level-geq? l 0)]
+    [s #:when (symbol? s)
+      (if (dict-has-key? `(type ,s))
+        (well-kinded? (dict-ref Γ `(type ,t))) #f)]
+    [`(Ref ,t) ; FIXME: this is not entirely correct. hrm.
+      (if (level-base? l)
+        (well-kinded? t l Γ)
+        (well-kinded? t (level-dec l) Γ))]
+    [(or `(,t1 × ,t2) `(,t1 ⊕ ,t2))
+      (and (well-kinded? t1 l Γ) (well-kinded? t1 l Γ))]
+    [`(,t1 → ,k ,t2)
+      (and (level-geq? l k) (well-kinded? t1 k Γ) (well-kinded? t2 k Γ))]
+    [`(μ ,x ,t) ; HACK
+      (well-kinded? t l (dict-set Γ `(type ,x) 'Unit))]))
+
+;; Whether a given structure is the heap, in our model.
+;; This is a quite useless function and is just here to note the model of the heap.
+;; Our heap is a Dict[Level, List[Dict[Addr, Expr]]]. In other words:
+;; - the heap is first stratified by algebraic levels, i.e. α, β, α+β, etc
+;; - those heaps are then stratified by n: the level as a natural number.
+;; - this is
+(define (heap? heap)
+  (match heap
+    [`((,level-var . (,subheap ...)) ...)
+      (and (all? (λ (l) (level? l)) level-var)
+        (all? (λ (n) (dict? n)) subheap))]
+    [_ #f]))
+
+;; Extends a list to have at least n+1 elements. Takes a default-generating procedure.
+(define/contract (list-extend l n default) (-> list? natural? procedure? list?)
+  (if (>= n (length l))
+    (build-list (+ n 1)
+      (λ (k)
+        (if (< k (length l))
+          (list-ref l k)
+          (default))))
+    l))
+
+;; Models the allocation of an (unsized) memory pointer at an arbitrary heap level.
+(define/contract (alloc! Σ l) (-> dict? level? ptr?)
+  (let ([addr (gensym)] [base (level-base l)] [offset (level-offset l)])
+    (if (dict-has-key? Σ base)
+      (let ([base-heap (dict-ref Σ base)])
+        (if (>= offset (length base-heap))
+          (dict-set! Σ base ; FIXME: we probably should error if location is occupied
+            (let ([offset-heap (make-hash)])
+              (dict-set! offset-heap addr 'null) ; FIXME: probably should not be null
+              (list-set (list-extend base-heap offset make-hash) offset offset-heap)))
+          (let ([offset-heap (list-ref base-heap offset)])
+            (dict-set! offset-heap addr 'null))))
+      (dict-set! Σ base
+        (let ([offset-heap (make-hash)])
+          (dict-set! offset-heap addr 'null)
+          (list-set (build-list (+ offset 1) (λ (_) (make-hash))) offset offset-heap))))
+    `(ptr ,l ,addr)))
+
+;; Updates the heap given a pointer to a memory location and a value.
+(define/contract (write! Σ p v) (-> dict? ptr? value? ptr?)
+  (match p
+    [`(ptr ,l ,a)
+      (let ([base (level-base l)] [offset (level-offset l)])
+        (if (dict-has-key? Σ base)
+          (let ([base-heap (dict-ref Σ base)])
+            (if (< offset (length base-heap))
+              (dict-set! (list-ref base-heap offset) a v)
+              (err (format "writing to invalid memory location ~a!" p))))
+          (err (format "writing to invalid memory location ~a!" p))))])
+  p)
+
+;; Returns the corresponding value of a pointer to a memory location on the heap.
+(define/contract (read! Σ p) (-> dict? ptr? value?)
+  (match p
+    [`(ptr ,l ,a)
+      (let ([base (level-base l)] [offset (level-offset l)])
+        (if (dict-has-key? Σ base)
+          (let ([base-heap (dict-ref Σ base)])
+            (if (< offset (length base-heap))
+              (dict-ref (list-ref base-heap offset) a)
+              (err (format "reading from invalid memory location ~a!" p))))
+          (err (format "reading from invalid memory location ~a!" p))))]))
+
+;; Models the deallocation of all memory locations of level `l` or higher.
+;; For complexity and performance purposes, we only support deallocating base levels.
+(define/contract (dealloc! Σ l) (-> dict? level-base? void?)
+  (for-each
+    (λ (key) (if (level-geq? key l) (dict-remove! Σ key) (void)))
+    (dict-keys Σ)))
+
+;; Whether two types are equivalent in a context.
+;; We define equivalence as equivalence up to α-renaming.
+(define/contract (equiv-type t1 t2 Γ) (-> type? type? dict? boolean?)
+  (equiv-type-core t1 t2 Γ Γ))
+(define (equiv-type-core t1 t2 Γ1 Γ2)
+  (match* (t1 t2)
+    ; 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)]))
+
+;; Whether two expressions are equivalent in a context.
+;; We define equivalence as equivalence up to α-renaming.
+; (define/contract (equiv-expr e1 e2 Γ) (-> expr? expr? dict? boolean?)
+;   (equiv-expr-core e1 e2 Γ Γ))
+; (define (equiv-expr-core e1 e2 Γ1 Γ2)
+;   (match* (e1 e2)))
+
+;; Expands a type alias into weak-head normal form, for literal matching.
+(define/contract (expand-whnf t Γ) (-> type? dict? type?)
+  (if (dict-has-key? Γ `(type t))
+    (expand-whnf (dict-ref Γ t) Γ) t))
+
+;; Replaces all references to a type alias with another alias.
+(define/contract (replace-type type key value) (-> type? type? type? type?)
+  (match type
+    ; Do not accidentally replace shadowed bindings
+    [`(μ ,x _) #:when (equal? x key) type]
+    [`(,e ...) `(,@(map (λ (x) (replace-type x key value)) e))]
+    [x #:when (equal? x key) value]
+    [v v]))
+
+;; Evaluates an expression to a value.
+;; Follows the call-by-value evaluation strategy.
+(define/contract (call-by-value expr) (-> expr? value?)
+  (cbv-core expr #hash() (make-hash)))
+(define (cbv-core expr Γ Σ) ; ℓ
+  (match expr
+    ['sole 'sole]
+    [n #:when (natural? n) n]
+    [b #:when (boolean? b) b]
+    [p #:when (dict-has-key? Σ p) p]
+    [x #:when (dict-has-key? Γ x) (dict-ref Γ x)]
+
+    [`(,e : ,t)
+      (cbv-core e Γ Σ)]
+    [`(type ,t1 ,t2 ,e)
+      (cbv-core e (dict-set Γ `(type ,t1) t2) Σ)]
+
+    ; The (level ...) syntax introduces new level *variables*.
+    [`(level ,l ,e)
+      (let ([v (cbv-core e (dict-set Γ `(level ,l) 'level) Σ)])
+        (dealloc! Σ l) v)] ; they are then freed at the end of scope
+    [`(,e ⇑ ,l)
+      (cbv-core e Γ Σ)]
+
+    [`(new ,e)
+      (let ([p (alloc! Σ (level-expr e))])
+        (write! Σ p (cbv-core e Γ Σ)))]
+    [`(! ,e)
+      (match (cbv-core e Γ Σ)
+        [`(ptr ,l ,a) (read! Σ `(ptr ,l ,a))]
+        [e (err (format "attempting to deref unknown expression ~a, expected ptr" e))])]
+    [`(set ,e1 ,e2) ; FIXME: we do NOT check levels before writing here
+      (match (cbv-core e1 Γ Σ)
+        [`(ptr ,l ,a) (write! Σ `(ptr ,l ,a) (cbv-core e2 Γ Σ))]
+        [e (err (format "attempting to write to unknown expression ~a, expected ptr" e))])]
+
+    [`(inc ,e)
+      (match (cbv-core e Γ Σ)
+        [n #:when (natural? n) (+ n 1)]
+        [e (err (format "incrementing an unknown value ~a" e))])]
+    [`(if ,c ,e1 ,e2)
+      (match (cbv-core c Γ Σ)
+        ['#t (cbv-core e1 Γ Σ)]
+        ['#f (cbv-core e2 Γ Σ)]
+        [e (err (format "calling if on unknown expression ~a" e))])]
+
+    [`(pair ,e1 ,e2)
+      `(pair ,(cbv-core e1 Γ Σ) ,(cbv-core e2 Γ Σ))]
+    [`(car ,e)
+      (match (cbv-core e Γ Σ)
+        [`(pair ,e1 ,e2) e1]
+        [e (err (format "calling car on unknown expression ~a" e))])]
+    [`(cdr ,e)
+      (match (cbv-core e Γ Σ)
+        [`(pair ,e1 ,e2) e2]
+        [e (err (format "calling cdr on unknown expression ~a" e))])]
+
+    [`(inl ,e)
+      `(inl ,(cbv-core e Γ Σ))]
+    [`(inr ,e)
+      `(inr ,(cbv-core e Γ Σ))]
+    [`(case ,e (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (cbv-core e Γ Σ)
+        [`(inl ,e) (cbv-core e1 (dict-set Γ x1 e) Σ)]
+        [`(inr ,e) (cbv-core e2 (dict-set Γ x2 e) Σ)]
+        [e (err (format "calling case on unknown expression ~a" e))])]
+
+    [`(fold ,e) `(fold ,(cbv-core e Γ Σ))]
+    [`(unfold ,e)
+      (match (cbv-core e Γ Σ)
+        [`(fold ,e) e]
+        [e (err (format "attempting to unfold unknown expression ~a" e))])]
+
+    [`(λ (,x : ,t) ,e)
+      `(λ (,x : ,t) ,e ,Γ)]
+    [`(,e1 ,e2)
+      (match (cbv-core e1 Γ Σ)
+        [`(λ (,x : ,t) ,e1 ,env)
+          (cbv-core e1 (dict-set env x (cbv-core e2 Γ Σ)) Σ)]
+        [e1
+          (err (format "attempting to interpret arg ~a applied to unknown expression ~a"
+            e2 e1))])]
+    [e (err (format "attempting to interpret unknown expression ~a" e))]))
+
+;; Checks that an expression is of a type, and returns #t or #f, or a bubbled-up error.
+;; `with` must be a type in weak-head normal form for structural matching.
+(define/contract (check expr with) (-> expr? type? boolean?)
+  (check-core expr with #hash()))
+(define (check-core expr with Γ)
+  (match expr
+    ['sole 'Unit]
+    [n #:when (natural? n) 'Nat]
+    [b #:when (boolean? b) 'Bool]
+    ; We expand into weak-head normal form as the binding may be whatever.
+    [x #:when (dict-has-key? Γ x)
+      (expand-whnf (dict-ref Γ x) Γ)]
+
+    [`(type ,t1 ,t2 ,e)
+      (check-core e with (dict-set Γ `(type ,t1) t2))]
+
+    [`(level ,l ,e) ; We add the level to the context just to note it is valid.
+      (check-core e with (dict-set Γ `(level ,l) 'level))]
+
+    [`(new ,e)
+      (match with
+        [`(Ref ,t) (check-core e t Γ)]
+        [_ #f])]
+    [`(! ,e)
+      (check-core e `(Ref ,with) Γ)]
+
+    [`(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])]
+    ; We do not technically need case in check form.
+    ; We keep it here to avoid needing type annotations on `c`.
+    [`(case ,c (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (infer-core c Γ)
+        [`(,a1 ⊕ ,a2)
+          (and (check-core e1 with (dict-set Γ x1 a1))
+               (check-core e2 with (dict-set Γ x2 a2)))]
+        [_ #f])]
+
+    [`(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 → ,l ,t2) ; KNOB
+          (and (equiv-type t t1 Γ) (check-core e t2 (dict-set Γ x t))
+            (> l (level-expr e (dict-set Γ x t1))))]
+        [`(,t1 → ,t2) (err (format "missing level annotation on function type"))]
+        [_ #f])]
+
+    [_ (equiv-type (infer-core expr Γ) with Γ)]))
+
+;; Infers a type from a given expression, if possible. Errors out otherwise.
+;; Returns a type in weak-head normal form for structural matching.
+(define/contract (infer expr) (-> expr? type?)
+  (infer-core expr #hash()))
+;; Γ is our context: a dictionary from symbols to types??? i forget actually
+;; note: our context plays many roles.
+(define (infer-core expr Γ)
+  (match expr
+    ['sole 'Unit]
+    [n #:when (natural? n) 'Nat]
+    [b #:when (boolean? b) 'Bool]
+    ; We expand into weak-head normal form as the binding may be to another binding.
+    [x #:when (dict-has-key? Γ x)
+      (expand-whnf (dict-ref Γ x) Γ)]
+
+    [`(type ,t1 ,t2 ,e)
+      (infer-core e (dict-set Γ `(type ,t1) t2))]
+    [`(,e : ,t)
+      (if (check-core e (expand-whnf t Γ) Γ) (expand-whnf t Γ)
+        (err (format "expression ~a is not of annotated type ~a" e t)))]
+
+    [`(level ,l ,e) ; We add the level to the context just to note it is valid.
+      (infer-core e (dict-set Γ `(level ,l) 'level))]
+    [`(,e ⇑ ,l) ; We retrieve the level from the context to check it is valid.
+      (if (dict-has-key? Γ `(level ,(level-base l)))
+        (infer-core e Γ)
+        (err (format "level ~a not found in context, was it introduced?")))]
+
+    [`(new ,e)
+      `(Ref ,(infer-core e Γ))]
+    [`(ptr ,a)
+      (err (format "cannot infer type from raw pointer ~a" `(ptr ,a)))]
+    [`(! ,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) ; FIXME: this does not account for explicit allocation syntax!
+      (match (infer-core e1 Γ) ; should we check levels?
+        [`(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))])]
+
+    [`(inc ,e)
+      (if (check-core e 'Nat Γ) 'Nat
+        (err (format "calling inc on incorrect type ~a, expected Nat" (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 "if ~a is not of consistent type!"
+              `(if Bool ,t ,(infer-core e2 Γ))))))
+        (err (format "if ~a has incorrect type ~a on condition, expected Bool"
+          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, expected a product" t))])]
+    [`(cdr ,e)
+      (match (infer-core e Γ)
+        [`(,t1 × ,t2) t2]
+        [t (err (format "calling cdr on incorrect type ~a, expected a product" 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 ,c (,x1 ⇒ ,e1) (,x2 ⇒ ,e2))
+      (match (infer-core c 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 "case has incorrect type ~a on condition, expected a sum" t))])]
+
+    [`(unfold ,e)
+      (match (expand-whnf (infer-core e Γ) Γ)
+        [`(μ ,x ,t) (replace-type t x `(μ ,x ,t))]
+        [t (err (format "expected ~a to be of recursive type, got ~a" e t))])]
+
+    [`(λ (,x : ,t1) ,e) ; KNOB
+      (let* ([t2 (infer-core e (dict-set Γ x t1))]
+        [t1 (expand-whnf t1 Γ)] [l (+ 1 (level-expr e (dict-set Γ x t1)))])
+        `(,t1 → ,l ,t2))]
+    [`(,e1 ,e2)
+      (match (infer-core e1 Γ)
+        [`(,t1 → ,l ,t2) ; should we check levels here? probably redundant
+          (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))])]
+
+    [_ (err (format "attempting to infer an unknown expression ~a" expr))]))
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