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• Renaming

Renaming-equiv

Basic equivalence relation for renaming structures.

Definitions and Theorems

Function: renaming-equiv$inline

(defun renaming-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (renamingp acl2::x)
                              (renamingp acl2::y))))
  (equal (renaming-fix acl2::x)
         (renaming-fix acl2::y)))

Theorem: renaming-equiv-is-an-equivalence

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

Theorem: renaming-equiv-implies-equal-renaming-fix-1

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

Theorem: renaming-fix-under-renaming-equiv

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

Theorem: equal-of-renaming-fix-1-forward-to-renaming-equiv

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

Theorem: equal-of-renaming-fix-2-forward-to-renaming-equiv

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

Theorem: renaming-equiv-of-renaming-fix-1-forward

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

Theorem: renaming-equiv-of-renaming-fix-2-forward

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