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• Decdig/uscore-list

Decdig/uscore-list-equiv

Basic equivalence relation for decdig/uscore-list structures.

Definitions and Theorems

Function: decdig/uscore-list-equiv$inline

(defun decdig/uscore-list-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (decdig/uscore-listp acl2::x)
                              (decdig/uscore-listp acl2::y))))
  (equal (decdig/uscore-list-fix acl2::x)
         (decdig/uscore-list-fix acl2::y)))

Theorem: decdig/uscore-list-equiv-is-an-equivalence

(defthm decdig/uscore-list-equiv-is-an-equivalence
  (and (booleanp (decdig/uscore-list-equiv x y))
       (decdig/uscore-list-equiv x x)
       (implies (decdig/uscore-list-equiv x y)
                (decdig/uscore-list-equiv y x))
       (implies (and (decdig/uscore-list-equiv x y)
                     (decdig/uscore-list-equiv y z))
                (decdig/uscore-list-equiv x z)))
  :rule-classes (:equivalence))

Theorem: decdig/uscore-list-equiv-implies-equal-decdig/uscore-list-fix-1

(defthm
    decdig/uscore-list-equiv-implies-equal-decdig/uscore-list-fix-1
  (implies (decdig/uscore-list-equiv acl2::x x-equiv)
           (equal (decdig/uscore-list-fix acl2::x)
                  (decdig/uscore-list-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: decdig/uscore-list-fix-under-decdig/uscore-list-equiv

(defthm decdig/uscore-list-fix-under-decdig/uscore-list-equiv
  (decdig/uscore-list-equiv (decdig/uscore-list-fix acl2::x)
                            acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-decdig/uscore-list-fix-1-forward-to-decdig/uscore-list-equiv

(defthm
 equal-of-decdig/uscore-list-fix-1-forward-to-decdig/uscore-list-equiv
 (implies (equal (decdig/uscore-list-fix acl2::x)
                 acl2::y)
          (decdig/uscore-list-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-decdig/uscore-list-fix-2-forward-to-decdig/uscore-list-equiv

(defthm
 equal-of-decdig/uscore-list-fix-2-forward-to-decdig/uscore-list-equiv
 (implies (equal acl2::x
                 (decdig/uscore-list-fix acl2::y))
          (decdig/uscore-list-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: decdig/uscore-list-equiv-of-decdig/uscore-list-fix-1-forward

(defthm decdig/uscore-list-equiv-of-decdig/uscore-list-fix-1-forward
 (implies (decdig/uscore-list-equiv (decdig/uscore-list-fix acl2::x)
                                    acl2::y)
          (decdig/uscore-list-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: decdig/uscore-list-equiv-of-decdig/uscore-list-fix-2-forward

(defthm decdig/uscore-list-equiv-of-decdig/uscore-list-fix-2-forward
  (implies
       (decdig/uscore-list-equiv acl2::x
                                 (decdig/uscore-list-fix acl2::y))
       (decdig/uscore-list-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)