Basic equivalence relation for path-alist structures.
Function:
(defun path-alist-equiv$inline (x y) (declare (xargs :guard (and (path-alist-p x) (path-alist-p y)))) (equal (path-alist-fix x) (path-alist-fix y)))
Theorem:
(defthm path-alist-equiv-is-an-equivalence (and (booleanp (path-alist-equiv x y)) (path-alist-equiv x x) (implies (path-alist-equiv x y) (path-alist-equiv y x)) (implies (and (path-alist-equiv x y) (path-alist-equiv y z)) (path-alist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm path-alist-equiv-implies-equal-path-alist-fix-1 (implies (path-alist-equiv x x-equiv) (equal (path-alist-fix x) (path-alist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm path-alist-fix-under-path-alist-equiv (path-alist-equiv (path-alist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-path-alist-fix-1-forward-to-path-alist-equiv (implies (equal (path-alist-fix x) y) (path-alist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-path-alist-fix-2-forward-to-path-alist-equiv (implies (equal x (path-alist-fix y)) (path-alist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm path-alist-equiv-of-path-alist-fix-1-forward (implies (path-alist-equiv (path-alist-fix x) y) (path-alist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm path-alist-equiv-of-path-alist-fix-2-forward (implies (path-alist-equiv x (path-alist-fix y)) (path-alist-equiv x y)) :rule-classes :forward-chaining)