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Constraint

Constraint-fix

Fixing function for constraint structures.

Signature
(constraint-fix x) → new-x
Arguments: x — Guard (constraintp x).
Returns: new-x — Type (constraintp new-x).

Definitions and Theorems

Function: constraint-fix$inline

(defun constraint-fix$inline (x)
 (declare (xargs :guard (constraintp x)))
 (let ((__function__ 'constraint-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (constraint-kind x)
     (:equal (b* ((left (expression-fix (std::da-nth 0 (cdr x))))
                  (right (expression-fix (std::da-nth 1 (cdr x)))))
               (cons :equal (list left right))))
     (:relation
          (b* ((name (str-fix (std::da-nth 0 (cdr x))))
               (args (expression-list-fix (std::da-nth 1 (cdr x)))))
            (cons :relation (list name args)))))
   :exec x)))

Theorem: constraintp-of-constraint-fix

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

Theorem: constraint-fix-when-constraintp

(defthm constraint-fix-when-constraintp
  (implies (constraintp x)
           (equal (constraint-fix x) x)))

Function: constraint-equiv$inline

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

Theorem: constraint-equiv-is-an-equivalence

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

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

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

Theorem: constraint-fix-under-constraint-equiv

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

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

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

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

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

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

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

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

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

Theorem: constraint-kind$inline-of-constraint-fix-x

(defthm constraint-kind$inline-of-constraint-fix-x
  (equal (constraint-kind$inline (constraint-fix x))
         (constraint-kind$inline x)))

Theorem: constraint-kind$inline-constraint-equiv-congruence-on-x

(defthm constraint-kind$inline-constraint-equiv-congruence-on-x
  (implies (constraint-equiv x x-equiv)
           (equal (constraint-kind$inline x)
                  (constraint-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-constraint-fix

(defthm consp-of-constraint-fix
  (consp (constraint-fix x))
  :rule-classes :type-prescription)