implement the simply-typed lambda calculus with general recursion (fix)

3 files changed, 87 insertions, 4 deletions

diff --git a/README.md b/README.md
index e525561..7daa864 100644
--- a/README.md
+++ b/README.md
@@ -8,7 +8,7 @@ For an introduction, see [*Explaining Lisp's quoting without getting tangled*](q
 * [~] LC: The untyped lambda calculus.
 * [x] STLC: The simply-typed lambda calculus.
 * [x] STLC-ext: Simple extensions. Sums, products, lists, ascryptions.
-* [ ] STLC-fix: General recursion.
+* [x] STLC-fix: General recursion.
 * [ ] STLC-sub: Subtyping. Structural records, intersective unions, implicit typeclasses, ⊤, ⊥.
 * [ ] STLC-rec: Recursive types. Also sums, products, ascryption.
 * [x] STLC-ref: References.
@@ -16,9 +16,9 @@ For an introduction, see [*Explaining Lisp's quoting without getting tangled*](q
 * [ ] 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 Rust-style ownership.
-* [ ] STLC-eff: Effects.
-* [ ] STLC-class: Typeclasses
+* [ ] STLC-own: References with first-class ownership, Rust-style.
+* [ ] STLC-lent: References with second-class ownership.
+* [ ] STLC-dep: Dependent types with normalization by evaluation.
 * [~] SKI: The SKI combinator calculus.
 * [~] Iota, Jot, Zot: Implementations of Chris Barker's languages.
 * [~] Aviary: Various combinators and constructions of equivalence.
diff --git a/lib.rkt b/lib.rkt
index 9b65256..b35ea21 100644
--- a/lib.rkt
+++ b/lib.rkt
@@ -86,6 +86,9 @@
     [`(let ,id (: ,t) ,e ,in)
       (desugar `((λ ,id (: ,t) ,in) ,e))]
 
+    [`(letrec ,x (: ,t) ,e ,in)
+      (desugar `(let ,x (: ,t) (fix (λ ,x (: ,t) ,e)) ,in))]
+
     [`(λ ,x (: ,t) ,e) `(λ ,x (: ,t) ,(desugar e))]
     [`(,e1 ,e2 ,e3) `(,(desugar e1) ,(desugar e2) ,(desugar e3))]
     [`(,e1 ,e2) `(,(desugar e1) ,(desugar e2))]
diff --git a/stlc-fix.rkt b/stlc-fix.rkt
new file mode 100644
index 0000000..f5ad384
--- /dev/null
+++ b/stlc-fix.rkt
@@ -0,0 +1,80 @@
+#lang racket
+(require "lib.rkt")
+
+;; The Simply-Typed Lambda Calculus, with general recursion
+
+;;      (interpret Expr Table[Sym, Expr]): Value
+(define (interpret expr [ctx #hash()])
+  (interpret- (strip (desugar expr)) ctx))
+(define (interpret- expr ctx)
+  (match expr
+    ['sole 'sole]
+    [n #:when (natural? n) n]
+    [x #:when (dict-has-key? ctx x) (dict-ref ctx x)]
+
+    [`(fix ,e)
+      (match (interpret- e ctx)
+        [`(λ ,x ,e ,env)
+          ; FIXME: unsure what should be ctx and what should be env
+          (interpret- e (dict-set ctx x `(fix (λ ,x ,e ,ctx))))]
+        [e (err (format "applying fix to unknown expression ~a" e))])]
+
+    [`(λ ,id ,body) `(λ ,id ,body ,ctx)]
+    [`(λ ,id ,body ,env) `(λ ,id ,body ,env)]
+    [`(,body ,arg)
+      (match (interpret- body ctx)
+        [`(λ ,id ,body ,env)
+          (interpret- body (dict-set env id (interpret- arg ctx)))]
+        [e (err (format "applying arg ~a to unknown expression ~a" arg e))])]
+
+    [e (err (format "interpreting an unknown expression ~a" e))]))
+
+;;      (check Expr Type Table[Sym, Type]): Bool
+(define (check expr with [Γ #hash()])
+  (check- (desugar expr) with Γ))
+(define (check- expr with Γ)
+  (let ([with (if (dict-has-key? Γ with) (dict-ref Γ with) with)])
+  (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)]
+
+    [(`(fix ,e) with)
+      (check- (infer- e) `(→ ,with ,with) Γ)]
+
+    [(`(λ ,x (: ,t) ,e) `(→ ,t1 ,t2))
+      (and (equal? t t1) (check- e t2 (dict-set Γ x t1)))]
+    [(`(,e1 ,e2) t)
+      (match (infer- e1 Γ)
+        [`(→ ,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)
+  (match expr
+    ['sole 'Unit]
+    [n #:when (natural? n) 'Nat]
+    [x #:when (dict-has-key? Γ x)
+      (dict-ref Γ x)]
+
+    [`(fix ,e)
+      (match (infer- 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- 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))])]
+
+    [e (err (format "inferring an unknown expression ~a" e))]))