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