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