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• Sshort-array

Sshort-array-equiv

Basic equivalence relation for sshort-array structures.

Definitions and Theorems

Function: sshort-array-equiv$inline

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

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

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

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

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

Theorem: sshort-array-fix-under-sshort-array-equiv

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

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

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

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

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

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

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

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

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