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