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Cgraph-derivstates

Cgraph-derivstates-fix

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

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

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

Definitions and Theorems

Function: cgraph-derivstates-fix$inline

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

Theorem: cgraph-derivstates-p-of-cgraph-derivstates-fix

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

Theorem: cgraph-derivstates-fix-when-cgraph-derivstates-p

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

Function: cgraph-derivstates-equiv$inline

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

Theorem: cgraph-derivstates-equiv-is-an-equivalence

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

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

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

Theorem: cgraph-derivstates-fix-under-cgraph-derivstates-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-cgraph-derivstate-fix-v-under-cgraph-derivstates-equiv

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

Theorem: cons-cgraph-derivstate-equiv-congruence-on-v-under-cgraph-derivstates-equiv

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

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

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

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

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

Theorem: cgraph-derivstates-fix-of-acons

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

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

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

Theorem: cgraph-derivstates-fix-of-append

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

Theorem: consp-car-of-cgraph-derivstates-fix

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