		Search-engine friendly clone of the ACL2 documentation.

Cgraph-edgelist

Cgraph-edgelist-fix

(cgraph-edgelist-fix x) is a usual ACL2::fty list fixing function.

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

In the logic, we apply cgraph-edge-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: cgraph-edgelist-fix$inline

(defun cgraph-edgelist-fix$inline (x)
  (declare (xargs :guard (cgraph-edgelist-p x)))
  (let ((__function__ 'cgraph-edgelist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (cgraph-edge-fix (car x))
                 (cgraph-edgelist-fix (cdr x))))
         :exec x)))

Theorem: cgraph-edgelist-p-of-cgraph-edgelist-fix

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

Theorem: cgraph-edgelist-fix-when-cgraph-edgelist-p

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

Function: cgraph-edgelist-equiv$inline

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

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

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

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

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

Theorem: cgraph-edgelist-fix-under-cgraph-edgelist-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-cgraph-edgelist-fix-x-under-cgraph-edge-equiv

(defthm car-of-cgraph-edgelist-fix-x-under-cgraph-edge-equiv
  (cgraph-edge-equiv (car (cgraph-edgelist-fix x))
                     (car x)))

Theorem: car-cgraph-edgelist-equiv-congruence-on-x-under-cgraph-edge-equiv

(defthm
  car-cgraph-edgelist-equiv-congruence-on-x-under-cgraph-edge-equiv
  (implies (cgraph-edgelist-equiv x x-equiv)
           (cgraph-edge-equiv (car x)
                              (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-cgraph-edgelist-fix-x-under-cgraph-edgelist-equiv

(defthm cdr-of-cgraph-edgelist-fix-x-under-cgraph-edgelist-equiv
  (cgraph-edgelist-equiv (cdr (cgraph-edgelist-fix x))
                         (cdr x)))

Theorem: cdr-cgraph-edgelist-equiv-congruence-on-x-under-cgraph-edgelist-equiv

(defthm
 cdr-cgraph-edgelist-equiv-congruence-on-x-under-cgraph-edgelist-equiv
 (implies (cgraph-edgelist-equiv x x-equiv)
          (cgraph-edgelist-equiv (cdr x)
                                 (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-cgraph-edge-fix-x-under-cgraph-edgelist-equiv

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

Theorem: cons-cgraph-edge-equiv-congruence-on-x-under-cgraph-edgelist-equiv

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

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

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

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

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

Theorem: consp-of-cgraph-edgelist-fix

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

Theorem: cgraph-edgelist-fix-under-iff

(defthm cgraph-edgelist-fix-under-iff
  (iff (cgraph-edgelist-fix x) (consp x)))

Theorem: cgraph-edgelist-fix-of-cons

(defthm cgraph-edgelist-fix-of-cons
  (equal (cgraph-edgelist-fix (cons a x))
         (cons (cgraph-edge-fix a)
               (cgraph-edgelist-fix x))))

Theorem: len-of-cgraph-edgelist-fix

(defthm len-of-cgraph-edgelist-fix
  (equal (len (cgraph-edgelist-fix x))
         (len x)))

Theorem: cgraph-edgelist-fix-of-append

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

Theorem: cgraph-edgelist-fix-of-repeat

(defthm cgraph-edgelist-fix-of-repeat
  (equal (cgraph-edgelist-fix (acl2::repeat n x))
         (acl2::repeat n (cgraph-edge-fix x))))

Theorem: list-equiv-refines-cgraph-edgelist-equiv

(defthm list-equiv-refines-cgraph-edgelist-equiv
  (implies (acl2::list-equiv x y)
           (cgraph-edgelist-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-cgraph-edgelist-fix

(defthm nth-of-cgraph-edgelist-fix
  (equal (nth n (cgraph-edgelist-fix x))
         (if (< (nfix n) (len x))
             (cgraph-edge-fix (nth n x))
           nil)))

Theorem: cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-append-1

(defthm cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-append-1
  (implies (cgraph-edgelist-equiv x fty::x-equiv)
           (cgraph-edgelist-equiv (append x y)
                                  (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-append-2

(defthm cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-append-2
  (implies (cgraph-edgelist-equiv y fty::y-equiv)
           (cgraph-edgelist-equiv (append x y)
                                  (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-nthcdr-2

(defthm cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-nthcdr-2
  (implies (cgraph-edgelist-equiv l l-equiv)
           (cgraph-edgelist-equiv (nthcdr n l)
                                  (nthcdr n l-equiv)))
  :rule-classes (:congruence))

Theorem: cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-take-2

(defthm cgraph-edgelist-equiv-implies-cgraph-edgelist-equiv-take-2
  (implies (cgraph-edgelist-equiv l l-equiv)
           (cgraph-edgelist-equiv (take n l)
                                  (take n l-equiv)))
  :rule-classes (:congruence))