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Number-theory

Prime

Fixtype of prime numbers.

Definitions and Theorems

Function: prime-equiv$inline

(defun prime-equiv$inline (x y)
       (declare (xargs :guard (and (dm::primep x) (dm::primep y))))
       (equal (prime-fix x) (prime-fix y)))

Theorem: prime-equiv-is-an-equivalence

(defthm prime-equiv-is-an-equivalence
        (and (booleanp (prime-equiv x y))
             (prime-equiv x x)
             (implies (prime-equiv x y)
                      (prime-equiv y x))
             (implies (and (prime-equiv x y)
                           (prime-equiv y z))
                      (prime-equiv x z)))
        :rule-classes (:equivalence))

Theorem: prime-equiv-implies-equal-prime-fix-1

(defthm prime-equiv-implies-equal-prime-fix-1
        (implies (prime-equiv x x-equiv)
                 (equal (prime-fix x)
                        (prime-fix x-equiv)))
        :rule-classes (:congruence))

Theorem: prime-fix-under-prime-equiv

(defthm prime-fix-under-prime-equiv
        (prime-equiv (prime-fix x) x)
        :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-prime-fix-1-forward-to-prime-equiv

(defthm equal-of-prime-fix-1-forward-to-prime-equiv
        (implies (equal (prime-fix x) y)
                 (prime-equiv x y))
        :rule-classes :forward-chaining)

Theorem: equal-of-prime-fix-2-forward-to-prime-equiv

(defthm equal-of-prime-fix-2-forward-to-prime-equiv
        (implies (equal x (prime-fix y))
                 (prime-equiv x y))
        :rule-classes :forward-chaining)

Theorem: prime-equiv-of-prime-fix-1-forward

(defthm prime-equiv-of-prime-fix-1-forward
        (implies (prime-equiv (prime-fix x) y)
                 (prime-equiv x y))
        :rule-classes :forward-chaining)

Theorem: prime-equiv-of-prime-fix-2-forward

(defthm prime-equiv-of-prime-fix-2-forward
        (implies (prime-equiv x (prime-fix y))
                 (prime-equiv x y))
        :rule-classes :forward-chaining)