From 47411b955bb6dad4fba1e3a5734341b6efe58d42 Mon Sep 17 00:00:00 2001
From: JJ
Date: Mon, 17 Jun 2024 13:55:41 -0700
Subject: implement the simply-typed lambda calculus with impredicative refs

---
 README.md    |   2 +-
 stlc-imp.rkt | 110 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 111 insertions(+), 1 deletion(-)
 create mode 100644 stlc-imp.rkt

diff --git a/README.md b/README.md
index 7daa864..c02390a 100644
--- a/README.md
+++ b/README.md
@@ -13,7 +13,7 @@ For an introduction, see [*Explaining Lisp's quoting without getting tangled*](q
 * [ ] STLC-rec: Recursive types. Also sums, products, ascryption.
 * [x] STLC-ref: References.
 * [x] STLC-pred: Higher-order *predicative* references. Terminating.
-* [ ] STLC-imp: Higher-order *impredicative* references. Terminating.
+* [x] STLC-imp: Higher-order *impredicative* references. Terminating.
 * [ ] STLC-rc: References with reference counting.
 * [ ] STLC-gc: References with a tracing garbage collector.
 * [ ] STLC-own: References with first-class ownership, Rust-style.
diff --git a/stlc-imp.rkt b/stlc-imp.rkt
new file mode 100644
index 0000000..4d16188
--- /dev/null
+++ b/stlc-imp.rkt
@@ -0,0 +1,110 @@
+#lang racket
+(require "lib.rkt")
+
+;; The Simply-Typed Lambda Calculus with higher-order *impredicative* references
+
+(require (only-in "stlc-ref.rkt" interpret))
+
+;;      (check Expr Type Table[Sym, Type]): Bool
+(define (check expr with [Γ #hash()])
+  (check- (desugar expr) with Γ))
+(define (check- expr with Γ)
+  ;   (check- expr (dict-ref Γ with) Γ)
+  (match* (expr with)
+    [('sole 'Unit) #t]
+    [(n 'Nat) #:when (natural? n) #t]
+    [(x _) #:when (dict-has-key? Γ x)
+      (equal? (dict-ref Γ x) with)]
+
+    [(`(new ,e) `(Ref ,t)) (check- e t Γ)]
+    [(`(! ,e) t) (check- e `(Ref ,t) Γ)]
+    [(`(set ,e1 ,e2) 'Unit)
+      (match (infer- e1 Γ)
+        [`(Ref ,t) (check- e2 t Γ)]
+        [t #f])]
+
+    [(`(λ ,x (: ,t) ,e) `(→ ,k ,t1 ,t2))
+      (and
+        (equal? t t1)
+        (> k (max-level e (dict-set Γ x t1) t1 t2))  ; KNOB
+        (check- e t2 (dict-set Γ x t1)))]
+    [(`(,e1 ,e2) t)
+      (match (infer- e1 Γ)
+        [`(→ ,k ,t1 ,t2)
+          (and (equal? t2 t)
+               (equal? t1 (infer- e2 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)
+  ; (print (format "infer: ~a" (fmt expr)))
+  (match expr
+    ['sole 'Unit]
+    [n #:when (natural? n) 'Nat]
+    [x #:when (dict-has-key? Γ x)
+      (dict-ref Γ x)]
+
+    [`(new ,e) `(Ref ,(infer- e Γ))]
+    [`(! ,e)
+      (match (infer- e Γ)
+        [`(Ref ,t) t]
+        [t (err "attempting to deref term not of Ref type!")])]
+    [`(set ,e1 ,e2)
+      (match (infer- e1 Γ)
+        [`(Ref ,t)
+          (if (check- e2 t Γ) 'Unit
+            (err (format "attempting to update ~a: ~a with term ~a: ~a of differing type"
+              e1 t e2 (infer- e2 Γ))))]
+        [t (err (format "attempting to update non-reference ~a: ~a" e1 t))])]
+
+    [`(λ ,x (: ,t1) ,e)
+      (let ([t2 (infer- e (dict-set Γ x t1))])
+        (let ([k (+ 1 (max-level e (dict-set Γ x t1) t1 t2))])  ; KNOB
+          `(→ ,k ,t1 ,t2)))]
+    [`(,e1 ,e2)
+      (match (infer- e1 Γ)
+        [`(→ ,k ,t1 ,t2)
+          (if (check- e2 t1 Γ) t2
+            (err (format "inferred argument type ~a does not match arg ~a of type ~a" t1 e2 (infer- e2 Γ))))]
+        [t (err (format "expected → type on application body, got ~a" t))])]
+
+    [e (err (format "attempting to infer an unknown expression ~a" e))]))
+
+;;      (max-level Table[Sym, Type] Expr Type Type): Natural
+(define (max-level e Γ t1 t2)
+  (max
+    (level-type t1)
+    (level-type t2)
+    (level-body e Γ)))
+
+;;      (level-type Type): Natural
+(define (level-type t)
+  (match t
+    ['Unit 0]
+    ['Nat 0]
+    [`(→ ,k ,t1 ,t2)
+      (if (or (< k (level-type t1)) (< k (level-type t2)))
+        (err (format "annotated level ~a is less than inferred levels of ~a and ~a!"
+          k t1 t2))
+        k)]
+    [`(Ref ,t)
+      (let ([k (level-type t)])
+      (if (zero? k) 0 ((+ 1 k))))] ; KNOB
+    [t (err (format "attempting to infer the level of unknown type ~a" t))]))
+
+;;      (level-body Expr Table[Sym, Type]): Natural
+(define (level-body e Γ)
+  (match e
+    ['sole 0]
+    [n #:when (natural? n) 0]
+    [x #:when (dict-has-key? Γ x)
+      (level-type (dict-ref Γ x))]
+    [`(new ,e) (level-body e Γ)]
+    [`(! ,e) (level-body e Γ)]
+    [`(set ,e1 ,e2) (max (level-body e1 Γ) (level-body e2 Γ))]
+    [`(λ ,x (: ,t) ,e) (level-body e (dict-set Γ x t))] ; todo: should be 0?
+    [`(,e1 ,e2) (max (level-body e1 Γ) (level-body e2 Γ))]))
-- 
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