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• Schar-array

Schar-array-equiv

Basic equivalence relation for schar-array structures.

Definitions and Theorems

Function: schar-array-equiv$inline

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

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

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

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

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

Theorem: schar-array-fix-under-schar-array-equiv

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

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

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

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

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

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

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

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

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