diff --git a/collects/redex/private/reduction-semantics.ss b/collects/redex/private/reduction-semantics.ss
index db17a74e1a..979d5fa0b9 100644
--- a/collects/redex/private/reduction-semantics.ss
+++ b/collects/redex/private/reduction-semantics.ss
@@ -1661,13 +1661,18 @@
          (compiled-lang-nt-map lang)))
 
 (define (apply-reduction-relation* reductions exp)
-  (let ([answers (make-hash)])
+  (let ([answers (make-hash)]
+        [cycles (make-hash)])
     (let loop ([exp exp])
-      (let ([nexts (apply-reduction-relation reductions exp)])
-        (cond
-          [(null? nexts) (hash-set! answers exp #t)]
-          [else (for-each loop nexts)])))
-    (hash-map answers (λ (x y) x))))
+      (unless (hash-ref cycles exp #f)
+        (hash-set! cycles exp #t)
+        (let ([nexts (apply-reduction-relation reductions exp)])
+          (cond
+            [(null? nexts) (hash-set! answers exp #t)]
+            [else (for-each loop nexts)]))))
+    (sort (hash-map answers (λ (x y) x))
+          string<=?
+          #:key (λ (x) (format "~s" x)))))
 
 
 ;; map/mt : (a -> b) (listof a) (listof b) -> (listof b)
diff --git a/collects/redex/redex.scrbl b/collects/redex/redex.scrbl
index b05698c823..7ccb954edb 100644
--- a/collects/redex/redex.scrbl
+++ b/collects/redex/redex.scrbl
@@ -819,11 +819,12 @@ names of the reductions that were used.
           [t any?])
          (listof (listof any?))]{
 
-apply-reduction-relation* accepts a list of reductions and a
-term. It returns the results of following every reduction
-path from the term. If there are infinite reduction
-sequences starting at the term, this function will not
-terminate.
+The function @scheme[apply-reduction-relation*] accepts a reduction relation and a
+term. Starting from @scheme[t], it follows every reduction
+path and returns all of the terms that do not reduce further.
+If there are infinite reduction
+sequences that do not repeat, this function will not
+terminate (it does terminate if the only infinite reduction paths are cyclic).
 }
 
 @defidform[-->]{ Recognized specially within
@@ -974,7 +975,8 @@ Tests to see if @scheme[e1] is equal to @scheme[e2].
 @defform[(test--> reduction-relation e1 e2 ...)]{
 
 Tests to see if the value of @scheme[e1] (which should be a term),
-reduces to the @scheme[e2]s under @scheme[reduction-relation].
+reduces to the @scheme[e2]s under @scheme[reduction-relation]
+(using @scheme[apply-reduction-relation*], so it may not terminate).
 }
 
 @defform[(test-predicate p? e)]{