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Mod-p

Signature
(mod-p x) → *

Definitions and Theorems

Function: mod-p

(defun mod-p (x)
  (declare (xargs :guard t))
  (let ((__function__ 'mod-p))
    (declare (ignorable __function__))
    (or (not x) (eq x :mem) (2bits-p x))))

Function: mod-fix

(defun mod-fix (x)
  (declare (xargs :guard (mod-p x)))
  (let ((__function__ 'mod-fix))
    (declare (ignorable __function__))
    (mbe :logic (if (mod-p x) x 'nil)
         :exec x)))

Function: mod-equiv$inline

(defun mod-equiv$inline (x y)
  (declare (xargs :guard (and (mod-p x) (mod-p y))))
  (equal (mod-fix x) (mod-fix y)))

Theorem: mod-equiv-is-an-equivalence

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

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

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

Theorem: mod-fix-under-mod-equiv

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