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