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+;;; ------------------------
+;;; Unnormalized polynomials
+;;; ------------------------
+
+(in-package "UPOL")
+
+(encapsulate
+
+  ;;; ----------
+  ;;; Signatures
+  ;;; ----------
+
+  ((polynomialp (p) t)
+   (nf (p) t)
+   (nfp (p) t)
+   (= (p q) t)
+   (+ (p q) t)
+   (* (p q) t)
+   (- (p) t)
+   (null () t)
+   (identity () t))
+
+  ;;; ---------------
+  ;;; Local witnesses
+  ;;; ---------------
+
+  (local
+    (defun polynomialp (p)
+      (acl2-numberp p)))
+
+  (local
+    (defun nf (p)
+      (fix p)))
+
+  (local
+    (defun nfp (p)
+      (equal (nf p) p)))
+
+  (local
+    (defun = (p q)
+      (equal (nf p) (nf q))))
+
+  (local
+    (defun + (p q)
+      (ACL2::+ p q)))
+
+  (local
+    (defun * (p q)
+      (ACL2::* p q)))
+
+  (local
+    (defun - (p)
+      (ACL2::- p)))
+
+  (local
+    (defun null ()
+      0))
+
+  (local
+    (defun identity ()
+      1))
+
+  ;;; ------
+  ;;; Axioms
+  ;;; ------
+
+  ;;; Normal form.
+
+  (defthm |nfp(p) <=> nf(p) = p|
+    (iff (nfp p) (equal (nf p) p)))
+
+  (defthm |nf(0) = 0|
+    (equal (nf (null)) (null)))
+
+  (defthm |nf(1) = 1|
+    (equal (nf (identity)) (identity)))
+
+  ;;; Equivalence.
+
+  (defequiv =)
+
+  ;;; Compatibility.
+
+  (defthm |p + nf(q) = p + q|
+    (= (+ p (nf q)) (+ p q)))
+
+  (defthm |p * nf(q) = p * q|
+    (= (* p (nf q)) (* p q)))
+
+  ;;; Congruences.
+
+  (defcong = equal (nf p) 1)
+
+  (defcong = = (+ p q) 1)
+  (defcong = = (+ p q) 2)
+
+  (defcong = = (* p q) 1)
+  (defcong = = (* p q) 2)
+
+  ;;; Ring.
+
+  (defthm |p + q = q + p|
+    (= (+ p q) (+ q p)))
+
+  (defthm |(p + q) + r = p + (q + r)|
+    (= (+ (+ p q) r) (+ p (+ q r))))
+
+  (defthm |p * q = q * p|
+    (= (* p q) (* q p)))
+
+  (defthm |(p * q) * r = p * (q * r)|
+    (= (* (* p q) r) (* p (* q r))))
+
+  (defthm |p * (q + r) = (p * q) + (p * r)|
+    (= (* p (+ q r)) (+ (* p q) (* p r))))
+
+  (defthm |p + (- p) = 0|
+    (= (+ p (- p)) (null)))
+
+  (defthm |0 + p = p|
+    (implies (polynomialp p)
+	     (= (+ (null) p) p)))
+
+  (defthm |1 * p = p|
+    (implies (polynomialp p)
+	     (= (* (identity) p) p))))
+
+;;; ----------
+;;; Properties
+;;; ----------
+
+;;; Completion of commutativity and associativity rules.
+
+(defthm |p + (q + r) = q + (p + r)|
+  (= (+ p (+ q r)) (+ q (+ p r)))
+  :hints (("Goal"
+	   :in-theory (disable |(p + q) + r = p + (q + r)|)
+	   :use (|(p + q) + r = p + (q + r)|
+		 (:instance |(p + q) + r = p + (q + r)| (p q) (q p))))))
+
+(defthm |p * (q * r) = q * (p * r)|
+  (= (* p (* q r)) (* q (* p r)))
+  :hints (("Goal"
+	   :in-theory (disable |(p * q) * r = p * (q * r)|)
+	   :use (|(p * q) * r = p * (q * r)|
+		 (:instance |(p * q) * r = p * (q * r)| (p q) (q p))))))