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• Addnames-indices

Addnames-indices-equiv

Basic equivalence relation for addnames-indices structures.

Definitions and Theorems

Function: addnames-indices-equiv$inline

(defun addnames-indices-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (addnames-indices-p acl2::x)
                              (addnames-indices-p acl2::y))))
  (equal (addnames-indices-fix acl2::x)
         (addnames-indices-fix acl2::y)))

Theorem: addnames-indices-equiv-is-an-equivalence

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

Theorem: addnames-indices-equiv-implies-equal-addnames-indices-fix-1

(defthm addnames-indices-equiv-implies-equal-addnames-indices-fix-1
  (implies (addnames-indices-equiv acl2::x x-equiv)
           (equal (addnames-indices-fix acl2::x)
                  (addnames-indices-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: addnames-indices-fix-under-addnames-indices-equiv

(defthm addnames-indices-fix-under-addnames-indices-equiv
  (addnames-indices-equiv (addnames-indices-fix acl2::x)
                          acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-addnames-indices-fix-1-forward-to-addnames-indices-equiv

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

Theorem: equal-of-addnames-indices-fix-2-forward-to-addnames-indices-equiv

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

Theorem: addnames-indices-equiv-of-addnames-indices-fix-1-forward

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

Theorem: addnames-indices-equiv-of-addnames-indices-fix-2-forward

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