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• Dimb-kind

Dimb-kind-equiv

Basic equivalence relation for dimb-kind structures.

Definitions and Theorems

Function: dimb-kind-equiv$inline

(defun dimb-kind-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (dimb-kindp acl2::x)
                              (dimb-kindp acl2::y))))
  (equal (dimb-kind-fix acl2::x)
         (dimb-kind-fix acl2::y)))

Theorem: dimb-kind-equiv-is-an-equivalence

(defthm dimb-kind-equiv-is-an-equivalence
  (and (booleanp (dimb-kind-equiv x y))
       (dimb-kind-equiv x x)
       (implies (dimb-kind-equiv x y)
                (dimb-kind-equiv y x))
       (implies (and (dimb-kind-equiv x y)
                     (dimb-kind-equiv y z))
                (dimb-kind-equiv x z)))
  :rule-classes (:equivalence))

Theorem: dimb-kind-equiv-implies-equal-dimb-kind-fix-1

(defthm dimb-kind-equiv-implies-equal-dimb-kind-fix-1
  (implies (dimb-kind-equiv acl2::x x-equiv)
           (equal (dimb-kind-fix acl2::x)
                  (dimb-kind-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: dimb-kind-fix-under-dimb-kind-equiv

(defthm dimb-kind-fix-under-dimb-kind-equiv
  (dimb-kind-equiv (dimb-kind-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-dimb-kind-fix-1-forward-to-dimb-kind-equiv

(defthm equal-of-dimb-kind-fix-1-forward-to-dimb-kind-equiv
  (implies (equal (dimb-kind-fix acl2::x) acl2::y)
           (dimb-kind-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-dimb-kind-fix-2-forward-to-dimb-kind-equiv

(defthm equal-of-dimb-kind-fix-2-forward-to-dimb-kind-equiv
  (implies (equal acl2::x (dimb-kind-fix acl2::y))
           (dimb-kind-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: dimb-kind-equiv-of-dimb-kind-fix-1-forward

(defthm dimb-kind-equiv-of-dimb-kind-fix-1-forward
  (implies (dimb-kind-equiv (dimb-kind-fix acl2::x)
                            acl2::y)
           (dimb-kind-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: dimb-kind-equiv-of-dimb-kind-fix-2-forward

(defthm dimb-kind-equiv-of-dimb-kind-fix-2-forward
  (implies (dimb-kind-equiv acl2::x (dimb-kind-fix acl2::y))
           (dimb-kind-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)