diff --git a/collects/racket/function.rkt b/collects/racket/function.rkt
index ad34a29b9a..a5deb1a183 100644
--- a/collects/racket/function.rkt
+++ b/collects/racket/function.rkt
@@ -83,8 +83,6 @@
 (define curryr (make-curry #t))
 
 (define (normalized-arity? a)
-  (unless (procedure-arity? a)
-          (raise-argument-error 'normalized-arity? "procedure-arity?" a))
   (or (null? a)
       (arity? a)
       (and (list? a)
@@ -160,7 +158,7 @@
         (for/and {[i (in-range n min-at-least)]}
           (arity-supports-number? arity i))])]))
 
-(define (arity-includes? one two)
+(define (unchecked-arity-includes? one two)
   (cond
     [(exact-nonnegative-integer? two)
      (arity-supports-number? one two)]
@@ -168,7 +166,20 @@
      (arity-supports-at-least? one (arity-at-least-value two))]
     [(list? two)
      (for/and {[elem (in-list two)]}
-       (arity-includes? one elem))]))
+       (unchecked-arity-includes? one elem))]))
+
+(define (arity-includes? one two)
+  (unless (procedure-arity? one)
+    (raise-argument-error 'arity-includes? "procedure-arity?" 0 one two))
+  (unless (procedure-arity? two)
+    (raise-argument-error 'arity-includes? "procedure-arity?" 1 one two))
+  (unchecked-arity-includes? one two))
 
 (define (arity=? one two)
-  (and (arity-includes? one two) (arity-includes? two one)))
+  (unless (procedure-arity? one)
+    (raise-argument-error 'arity=? "procedure-arity?" 0 one two))
+  (unless (procedure-arity? two)
+    (raise-argument-error 'arity=? "procedure-arity?" 1 one two))
+  (and
+    (unchecked-arity-includes? one two)
+    (unchecked-arity-includes? two one)))
diff --git a/collects/racket/private/norm-arity.rkt b/collects/racket/private/norm-arity.rkt
index 8ef8de12a7..246b7f8abd 100644
--- a/collects/racket/private/norm-arity.rkt
+++ b/collects/racket/private/norm-arity.rkt
@@ -37,6 +37,8 @@
   ;;   - if there is only one possibility, it is returned by itself (ie,
   ;;     not in a list)
   (define (normalize-arity arity)
+    (unless (procedure-arity? arity)
+            (raise-argument-error 'normalize-arity "procedure-arity?" arity))
     (if (pair? arity)
         (let* ([reversed (reverse-sort-arity arity)]
                [normalized (normalize-reversed-arity reversed '())]
diff --git a/collects/scribblings/reference/procedures.scrbl b/collects/scribblings/reference/procedures.scrbl
index 69756be16b..aa7692132f 100644
--- a/collects/scribblings/reference/procedures.scrbl
+++ b/collects/scribblings/reference/procedures.scrbl
@@ -126,8 +126,7 @@ the @exnraise[exn:fail:contract].
 (keyword-apply f #:z 7 '(#:y) '(2) '(1))
 ]}
 
-@defproc[(procedure-arity [proc procedure?])
-         (and/c procedure-arity? normalized-arity?)]{
+@defproc[(procedure-arity [proc procedure?]) normalized-arity?]{
 
 Returns information about the number of by-position arguments accepted
 by @racket[proc]. See also @racket[procedure-arity?] and
@@ -612,7 +611,7 @@ arguments, and following steps add arguments to the left of these.
   (map (curryr list 'foo) '(1 2 3))
 ]}
 
-@defproc[(normalized-arity? [arity procedure-arity?]) boolean?]{
+@defproc[(normalized-arity? [arity any/c]) boolean?]{
 
 A normalized arity has one of the following forms:
 @itemize[
@@ -625,8 +624,9 @@ or}
 followed by a single @racket[arity-at-least] instance whose value is greater
 than the preceding integer by at least 2.}
 ]
-Any two normalized arity values that are @racket[arity=?] must also be
-@racket[equal?].
+Every normalized arity is a valid procedure arity and satisfies
+@racket[procedure-arity?].  Any two normalized arity values that are
+@racket[arity=?] must also be @racket[equal?].
 
 @mz-examples[#:eval fun-eval
 (normalized-arity? (arity-at-least 1))
@@ -639,8 +639,7 @@ Any two normalized arity values that are @racket[arity=?] must also be
 }
 
 @defproc[(normalize-arity [arity procedure-arity?])
-         (and/c procedure-arity? normalized-arity?
-                (lambda (x) (arity=? x arity)))]{
+         (and/c normalized-arity? (lambda (x) (arity=? x arity)))]{
 
 Produces a normalized form of @racket[arity].  See also
 @racket[normalized-arity?] and @racket[arity=?].