Basic equivalence relation for id-neg-alist structures.
Function:
(defun id-neg-alist-equiv$inline (x acl2::y) (declare (xargs :guard (and (id-neg-alist-p x) (id-neg-alist-p acl2::y)))) (equal (id-neg-alist-fix x) (id-neg-alist-fix acl2::y)))
Theorem:
(defthm id-neg-alist-equiv-is-an-equivalence (and (booleanp (id-neg-alist-equiv x y)) (id-neg-alist-equiv x x) (implies (id-neg-alist-equiv x y) (id-neg-alist-equiv y x)) (implies (and (id-neg-alist-equiv x y) (id-neg-alist-equiv y z)) (id-neg-alist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm id-neg-alist-equiv-implies-equal-id-neg-alist-fix-1 (implies (id-neg-alist-equiv x x-equiv) (equal (id-neg-alist-fix x) (id-neg-alist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm id-neg-alist-fix-under-id-neg-alist-equiv (id-neg-alist-equiv (id-neg-alist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-id-neg-alist-fix-1-forward-to-id-neg-alist-equiv (implies (equal (id-neg-alist-fix x) acl2::y) (id-neg-alist-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-id-neg-alist-fix-2-forward-to-id-neg-alist-equiv (implies (equal x (id-neg-alist-fix acl2::y)) (id-neg-alist-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm id-neg-alist-equiv-of-id-neg-alist-fix-1-forward (implies (id-neg-alist-equiv (id-neg-alist-fix x) acl2::y) (id-neg-alist-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm id-neg-alist-equiv-of-id-neg-alist-fix-2-forward (implies (id-neg-alist-equiv x (id-neg-alist-fix acl2::y)) (id-neg-alist-equiv x acl2::y)) :rule-classes :forward-chaining)