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Uninterpreted

Uninterpreted-fix

Fixing function for uninterpreted structures.

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

Definitions and Theorems

Function: uninterpreted-fix$inline

(defun uninterpreted-fix$inline (x)
 (declare (xargs :guard (uninterpreted-p x)))
 (let ((acl2::__function__ 'uninterpreted-fix))
  (declare (ignorable acl2::__function__))
  (mbe
   :logic
   (b* ((returns (hint-pair-list-fix (cdr (std::da-nth 0 x))))
        (more-returns (hint-pair-list-fix (cdr (std::da-nth 1 x)))))
     (list (cons 'returns returns)
           (cons 'more-returns more-returns)))
   :exec x)))

Theorem: uninterpreted-p-of-uninterpreted-fix

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

Theorem: uninterpreted-fix-when-uninterpreted-p

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

Function: uninterpreted-equiv$inline

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

Theorem: uninterpreted-equiv-is-an-equivalence

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

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

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

Theorem: uninterpreted-fix-under-uninterpreted-equiv

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

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

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

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

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

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

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

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

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