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Sint-array-equiv

Basic equivalence relation for sint-array structures.

Definitions and Theorems

Function: sint-array-equiv$inline

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

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

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

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

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

Theorem: sint-array-fix-under-sint-array-equiv

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

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

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

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

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

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

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

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

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