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