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