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