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• Octdig/uscore

Octdig/uscore-equiv

Basic equivalence relation for octdig/uscore structures.

Definitions and Theorems

Function: octdig/uscore-equiv$inline

(defun octdig/uscore-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (octdig/uscore-p acl2::x)
                              (octdig/uscore-p acl2::y))))
  (equal (octdig/uscore-fix acl2::x)
         (octdig/uscore-fix acl2::y)))

Theorem: octdig/uscore-equiv-is-an-equivalence

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

Theorem: octdig/uscore-equiv-implies-equal-octdig/uscore-fix-1

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

Theorem: octdig/uscore-fix-under-octdig/uscore-equiv

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

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

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

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

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

Theorem: octdig/uscore-equiv-of-octdig/uscore-fix-1-forward

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

Theorem: octdig/uscore-equiv-of-octdig/uscore-fix-2-forward

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