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Renaming

Renaming-fix

Fixing function for renaming structures.

Signature
(renaming-fix x) → new-x
Arguments: x — Guard (renamingp x).
Returns: new-x — Type (renamingp new-x).

Definitions and Theorems

Function: renaming-fix$inline

(defun renaming-fix$inline (x)
 (declare (xargs :guard (renamingp x)))
 (let ((__function__ 'renaming-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b*
    ((list
         (identifier-identifier-alist-fix (cdr (std::da-nth 0 x)))))
    (let ((list (if (and (no-duplicatesp-equal (strip-cars list))
                         (no-duplicatesp-equal (strip-cdrs list)))
                    list
                  nil)))
      (list (cons 'list list))))
   :exec x)))

Theorem: renamingp-of-renaming-fix

(defthm renamingp-of-renaming-fix
  (b* ((new-x (renaming-fix$inline x)))
    (renamingp new-x))
  :rule-classes :rewrite)

Theorem: renaming-fix-when-renamingp

(defthm renaming-fix-when-renamingp
  (implies (renamingp x)
           (equal (renaming-fix x) x)))

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)