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• Comp-db

Comp-db-equiv

Basic equivalence relation for comp-db structures.

Definitions and Theorems

Function: comp-db-equiv$inline

(defun comp-db-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (comp-dbp acl2::x)
                              (comp-dbp acl2::y))))
  (equal (comp-db-fix acl2::x)
         (comp-db-fix acl2::y)))

Theorem: comp-db-equiv-is-an-equivalence

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

Theorem: comp-db-equiv-implies-equal-comp-db-fix-1

(defthm comp-db-equiv-implies-equal-comp-db-fix-1
  (implies (comp-db-equiv acl2::x x-equiv)
           (equal (comp-db-fix acl2::x)
                  (comp-db-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: comp-db-fix-under-comp-db-equiv

(defthm comp-db-fix-under-comp-db-equiv
  (comp-db-equiv (comp-db-fix acl2::x)
                 acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-comp-db-fix-1-forward-to-comp-db-equiv

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

Theorem: equal-of-comp-db-fix-2-forward-to-comp-db-equiv

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

Theorem: comp-db-equiv-of-comp-db-fix-1-forward

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

Theorem: comp-db-equiv-of-comp-db-fix-2-forward

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