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Integer-ranges

Sshort-integer

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

Definitions and Theorems

Function: sshort-integer-equiv$inline

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

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

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

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

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

Theorem: sshort-integer-fix-under-sshort-integer-equiv

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

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

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

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

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

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

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

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

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

Subtopics

Sshort-integerp: Recognizer for sshort-integer.
Sshort-integer-fix: Fixer for sshort-integer.