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Alias

Alias-fix

Fixing function for alias structures.

Signature
(alias-fix x) → new-x
Arguments: x — Guard (alias-p x).
Returns: new-x — Type (alias-p new-x).

Definitions and Theorems

Function: alias-fix$inline

(defun alias-fix$inline (x)
  (declare (xargs :guard (alias-p x)))
  (let ((acl2::__function__ 'alias-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((name (sv::svar-fix (cdr (std::da-nth 0 x))))
              (val (svex-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'name name)
                 (cons 'val val)))
         :exec x)))

Theorem: alias-p-of-alias-fix

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

Theorem: alias-fix-when-alias-p

(defthm alias-fix-when-alias-p
  (implies (alias-p x)
           (equal (alias-fix x) x)))

Function: alias-equiv$inline

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

Theorem: alias-equiv-is-an-equivalence

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

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

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

Theorem: alias-fix-under-alias-equiv

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

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

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

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

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

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

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

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

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