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• Modname

Modname-equiv

Definitions and Theorems

Function: modname-equiv$inline

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

Theorem: modname-equiv-is-an-equivalence

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

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

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

Theorem: modname-fix-under-modname-equiv

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

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

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

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

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

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

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

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

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