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• Path

Path-equiv

Basic equivalence relation for path structures.

Definitions and Theorems

Function: path-equiv$inline

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

Theorem: path-equiv-is-an-equivalence

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

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

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

Theorem: path-fix-under-path-equiv

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

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

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

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

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

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

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

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

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