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Pathlist

Pathlist-equiv

Basic equivalence relation for pathlist structures.

Definitions and Theorems

Function: pathlist-equiv$inline

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

Theorem: pathlist-equiv-is-an-equivalence

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

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

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

Theorem: pathlist-fix-under-pathlist-equiv

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

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

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

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

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

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

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

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

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