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Rewrite

Rewrite-equiv

Basic equivalence relation for rewrite structures.

Definitions and Theorems

Function: rewrite-equiv$inline

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

Theorem: rewrite-equiv-is-an-equivalence

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

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

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

Theorem: rewrite-fix-under-rewrite-equiv

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

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

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

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

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

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

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

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

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