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