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