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Dexprefix

Dexprefix-equiv

Basic equivalence relation for dexprefix structures.

Definitions and Theorems

Function: dexprefix-equiv$inline

(defun dexprefix-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (dexprefixp acl2::x)
                              (dexprefixp acl2::y))))
  (equal (dexprefix-fix acl2::x)
         (dexprefix-fix acl2::y)))

Theorem: dexprefix-equiv-is-an-equivalence

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

Theorem: dexprefix-equiv-implies-equal-dexprefix-fix-1

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

Theorem: dexprefix-fix-under-dexprefix-equiv

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

Theorem: equal-of-dexprefix-fix-1-forward-to-dexprefix-equiv

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

Theorem: equal-of-dexprefix-fix-2-forward-to-dexprefix-equiv

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

Theorem: dexprefix-equiv-of-dexprefix-fix-1-forward

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

Theorem: dexprefix-equiv-of-dexprefix-fix-2-forward

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