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Lhbit

Lhbit-equiv

Basic equivalence relation for lhbit structures.

Definitions and Theorems

Function: lhbit-equiv$inline

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

Theorem: lhbit-equiv-is-an-equivalence

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

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

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

Theorem: lhbit-fix-under-lhbit-equiv

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

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

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

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

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

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

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

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

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