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