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Cgraph

Cgraph-fix

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

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

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

Definitions and Theorems

Function: cgraph-fix$inline

(defun cgraph-fix$inline (x)
 (declare (xargs :guard (cgraph-p x)))
 (let ((__function__ 'cgraph-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (if (atom x)
          nil
        (let ((rest (cgraph-fix (cdr x))))
          (if (and (consp (car x))
                   (fgl-object-p (caar x)))
              (let ((fty::first-key (caar x))
                    (fty::first-val (cgraph-outedges-fix (cdar x))))
                (cons (cons fty::first-key fty::first-val)
                      rest))
            rest)))
      :exec x)))

Theorem: cgraph-p-of-cgraph-fix

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

Theorem: cgraph-fix-when-cgraph-p

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

Function: cgraph-equiv$inline

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

Theorem: cgraph-equiv-is-an-equivalence

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

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

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

Theorem: cgraph-fix-under-cgraph-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-cgraph-outedges-fix-v-under-cgraph-equiv

(defthm cons-of-cgraph-outedges-fix-v-under-cgraph-equiv
  (cgraph-equiv (cons (cons k (cgraph-outedges-fix v))
                      x)
                (cons (cons k v) x)))

Theorem: cons-cgraph-outedges-equiv-congruence-on-v-under-cgraph-equiv

(defthm
      cons-cgraph-outedges-equiv-congruence-on-v-under-cgraph-equiv
  (implies (cgraph-outedges-equiv v v-equiv)
           (cgraph-equiv (cons (cons k v) x)
                         (cons (cons k v-equiv) x)))
  :rule-classes :congruence)

Theorem: cons-of-cgraph-fix-y-under-cgraph-equiv

(defthm cons-of-cgraph-fix-y-under-cgraph-equiv
  (cgraph-equiv (cons x (cgraph-fix y))
                (cons x y)))

Theorem: cons-cgraph-equiv-congruence-on-y-under-cgraph-equiv

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

Theorem: cgraph-fix-of-acons

(defthm cgraph-fix-of-acons
  (equal (cgraph-fix (cons (cons a b) x))
         (let ((rest (cgraph-fix x)))
           (if (and (fgl-object-p a))
               (let ((fty::first-key a)
                     (fty::first-val (cgraph-outedges-fix b)))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest))))

Theorem: hons-assoc-equal-of-cgraph-fix

(defthm hons-assoc-equal-of-cgraph-fix
  (equal (hons-assoc-equal k (cgraph-fix x))
         (let ((fty::pair (hons-assoc-equal k x)))
           (and (fgl-object-p k)
                fty::pair
                (cons k
                      (cgraph-outedges-fix (cdr fty::pair)))))))

Theorem: cgraph-fix-of-append

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

Theorem: consp-car-of-cgraph-fix

(defthm consp-car-of-cgraph-fix
  (equal (consp (car (cgraph-fix x)))
         (consp (cgraph-fix x))))