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Hex-digit

Hex-digit-equiv

Basic equivalence relation for hex-digit structures.

Definitions and Theorems

Function: hex-digit-equiv$inline

(defun hex-digit-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (hex-digitp acl2::x)
                              (hex-digitp acl2::y))))
  (equal (hex-digit-fix acl2::x)
         (hex-digit-fix acl2::y)))

Theorem: hex-digit-equiv-is-an-equivalence

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

Theorem: hex-digit-equiv-implies-equal-hex-digit-fix-1

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

Theorem: hex-digit-fix-under-hex-digit-equiv

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

Theorem: equal-of-hex-digit-fix-1-forward-to-hex-digit-equiv

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

Theorem: equal-of-hex-digit-fix-2-forward-to-hex-digit-equiv

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

Theorem: hex-digit-equiv-of-hex-digit-fix-1-forward

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

Theorem: hex-digit-equiv-of-hex-digit-fix-2-forward

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