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