rationalp

(in-package "ACL2")

(local (include-book "predicate"))

(defthm rationalp-product-when-one-arg-is-rational
  (implies (and (rationalp x)
      (case-split (not (equal x 0)))
      (case-split (acl2-numberp y)))
    (and (equal (rationalp (* x y)) (rationalp y))
      (equal (rationalp (* y x)) (rationalp y)))))

(defthm rationalp-sum-when-one-arg-is-rational
  (implies (and (rationalp x) (case-split (acl2-numberp y)))
    (and (equal (rationalp (+ x y)) (rationalp y))
      (equal (rationalp (+ y x)) (rationalp y)))))

(defthm rationalp-unary-divide
  (implies (case-split (acl2-numberp x))
    (equal (rationalp (/ x)) (rationalp x))))

(defthm rationalp-product
  (implies (and (case-split (not (complex-rationalp x)))
      (case-split (not (complex-rationalp y))))
    (rationalp (* x y))))