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Integer-ranges

Ushort-integer

Fixtype of ACL2 integers in the range of type unsigned short.

Definitions and Theorems

Function: ushort-integer-equiv$inline

(defun ushort-integer-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (ushort-integerp acl2::x)
                              (ushort-integerp acl2::y))))
  (equal (ushort-integer-fix acl2::x)
         (ushort-integer-fix acl2::y)))

Theorem: ushort-integer-equiv-is-an-equivalence

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

Theorem: ushort-integer-equiv-implies-equal-ushort-integer-fix-1

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

Theorem: ushort-integer-fix-under-ushort-integer-equiv

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

Theorem: equal-of-ushort-integer-fix-1-forward-to-ushort-integer-equiv

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

Theorem: equal-of-ushort-integer-fix-2-forward-to-ushort-integer-equiv

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

Theorem: ushort-integer-equiv-of-ushort-integer-fix-1-forward

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

Theorem: ushort-integer-equiv-of-ushort-integer-fix-2-forward

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

Subtopics

Ushort-integerp: Recognizer for ushort-integer.
Ushort-integer-fix: Fixer for ushort-integer.