Basic equivalence relation for cuts4-config structures.
Function:
(defun cuts4-config-equiv$inline (x acl2::y) (declare (xargs :guard (and (cuts4-config-p x) (cuts4-config-p acl2::y)))) (equal (cuts4-config-fix x) (cuts4-config-fix acl2::y)))
Theorem:
(defthm cuts4-config-equiv-is-an-equivalence (and (booleanp (cuts4-config-equiv x y)) (cuts4-config-equiv x x) (implies (cuts4-config-equiv x y) (cuts4-config-equiv y x)) (implies (and (cuts4-config-equiv x y) (cuts4-config-equiv y z)) (cuts4-config-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm cuts4-config-equiv-implies-equal-cuts4-config-fix-1 (implies (cuts4-config-equiv x x-equiv) (equal (cuts4-config-fix x) (cuts4-config-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm cuts4-config-fix-under-cuts4-config-equiv (cuts4-config-equiv (cuts4-config-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-cuts4-config-fix-1-forward-to-cuts4-config-equiv (implies (equal (cuts4-config-fix x) acl2::y) (cuts4-config-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-cuts4-config-fix-2-forward-to-cuts4-config-equiv (implies (equal x (cuts4-config-fix acl2::y)) (cuts4-config-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm cuts4-config-equiv-of-cuts4-config-fix-1-forward (implies (cuts4-config-equiv (cuts4-config-fix x) acl2::y) (cuts4-config-equiv x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm cuts4-config-equiv-of-cuts4-config-fix-2-forward (implies (cuts4-config-equiv x (cuts4-config-fix acl2::y)) (cuts4-config-equiv x acl2::y)) :rule-classes :forward-chaining)