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Lhprobe

Lhprobe-equiv

Basic equivalence relation for lhprobe structures.

Definitions and Theorems

Function: lhprobe-equiv$inline

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

Theorem: lhprobe-equiv-is-an-equivalence

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

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

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

Theorem: lhprobe-fix-under-lhprobe-equiv

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

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

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

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

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

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

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

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

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