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