		Search-engine friendly clone of the ACL2 documentation.

Constraint-db

Constraint-db-fix

(constraint-db-fix x) is an ACL2::fty alist fixing function that follows the drop-keys strategy.

Signature
(constraint-db-fix x) → fty::newx
Arguments: x — Guard (constraint-db-p x).
Returns: fty::newx — Type (constraint-db-p fty::newx).

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: constraint-db-fix$inline

(defun constraint-db-fix$inline (x)
 (declare (xargs :guard (constraint-db-p x)))
 (let ((__function__ 'constraint-db-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (if (atom x)
       nil
    (let ((rest (constraint-db-fix (cdr x))))
     (if (and (consp (car x))
              (pseudo-fnsym-p (caar x)))
         (let ((fty::first-key (caar x))
               (fty::first-val (constraint-tuplelist-fix (cdar x))))
           (cons (cons fty::first-key fty::first-val)
                 rest))
       rest)))
   :exec x)))

Theorem: constraint-db-p-of-constraint-db-fix

(defthm constraint-db-p-of-constraint-db-fix
  (b* ((fty::newx (constraint-db-fix$inline x)))
    (constraint-db-p fty::newx))
  :rule-classes :rewrite)

Theorem: constraint-db-fix-when-constraint-db-p

(defthm constraint-db-fix-when-constraint-db-p
  (implies (constraint-db-p x)
           (equal (constraint-db-fix x) x)))

Function: constraint-db-equiv$inline

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

Theorem: constraint-db-equiv-is-an-equivalence

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

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

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

Theorem: constraint-db-fix-under-constraint-db-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-constraint-tuplelist-fix-v-under-constraint-db-equiv

(defthm cons-of-constraint-tuplelist-fix-v-under-constraint-db-equiv
  (constraint-db-equiv (cons (cons k (constraint-tuplelist-fix v))
                             x)
                       (cons (cons k v) x)))

Theorem: cons-constraint-tuplelist-equiv-congruence-on-v-under-constraint-db-equiv

(defthm
 cons-constraint-tuplelist-equiv-congruence-on-v-under-constraint-db-equiv
 (implies (constraint-tuplelist-equiv v v-equiv)
          (constraint-db-equiv (cons (cons k v) x)
                               (cons (cons k v-equiv) x)))
 :rule-classes :congruence)

Theorem: cons-of-constraint-db-fix-y-under-constraint-db-equiv

(defthm cons-of-constraint-db-fix-y-under-constraint-db-equiv
  (constraint-db-equiv (cons x (constraint-db-fix y))
                       (cons x y)))

Theorem: cons-constraint-db-equiv-congruence-on-y-under-constraint-db-equiv

(defthm
 cons-constraint-db-equiv-congruence-on-y-under-constraint-db-equiv
 (implies (constraint-db-equiv y y-equiv)
          (constraint-db-equiv (cons x y)
                               (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: constraint-db-fix-of-acons

(defthm constraint-db-fix-of-acons
  (equal (constraint-db-fix (cons (cons a b) x))
         (let ((rest (constraint-db-fix x)))
           (if (and (pseudo-fnsym-p a))
               (let ((fty::first-key a)
                     (fty::first-val (constraint-tuplelist-fix b)))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest))))

Theorem: hons-assoc-equal-of-constraint-db-fix

(defthm hons-assoc-equal-of-constraint-db-fix
 (equal (hons-assoc-equal k (constraint-db-fix x))
        (let ((fty::pair (hons-assoc-equal k x)))
          (and (pseudo-fnsym-p k)
               fty::pair
               (cons k
                     (constraint-tuplelist-fix (cdr fty::pair)))))))

Theorem: constraint-db-fix-of-append

(defthm constraint-db-fix-of-append
  (equal (constraint-db-fix (append std::a std::b))
         (append (constraint-db-fix std::a)
                 (constraint-db-fix std::b))))

Theorem: consp-car-of-constraint-db-fix

(defthm consp-car-of-constraint-db-fix
  (equal (consp (car (constraint-db-fix x)))
         (consp (constraint-db-fix x))))