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Candidate-assigns

Candidate-assigns-fix

(candidate-assigns-fix x) is a usual ACL2::fty list fixing function.

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

In the logic, we apply candidate-assign-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: candidate-assigns-fix$inline

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

Theorem: candidate-assigns-p-of-candidate-assigns-fix

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

Theorem: candidate-assigns-fix-when-candidate-assigns-p

(defthm candidate-assigns-fix-when-candidate-assigns-p
  (implies (candidate-assigns-p x)
           (equal (candidate-assigns-fix x) x)))

Function: candidate-assigns-equiv$inline

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

Theorem: candidate-assigns-equiv-is-an-equivalence

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

Theorem: candidate-assigns-equiv-implies-equal-candidate-assigns-fix-1

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

Theorem: candidate-assigns-fix-under-candidate-assigns-equiv

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

Theorem: equal-of-candidate-assigns-fix-1-forward-to-candidate-assigns-equiv

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

Theorem: equal-of-candidate-assigns-fix-2-forward-to-candidate-assigns-equiv

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

Theorem: candidate-assigns-equiv-of-candidate-assigns-fix-1-forward

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

Theorem: candidate-assigns-equiv-of-candidate-assigns-fix-2-forward

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

Theorem: car-of-candidate-assigns-fix-x-under-candidate-assign-equiv

(defthm car-of-candidate-assigns-fix-x-under-candidate-assign-equiv
  (candidate-assign-equiv (car (candidate-assigns-fix x))
                          (car x)))

Theorem: car-candidate-assigns-equiv-congruence-on-x-under-candidate-assign-equiv

(defthm
 car-candidate-assigns-equiv-congruence-on-x-under-candidate-assign-equiv
 (implies (candidate-assigns-equiv x x-equiv)
          (candidate-assign-equiv (car x)
                                  (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-candidate-assigns-fix-x-under-candidate-assigns-equiv

(defthm cdr-of-candidate-assigns-fix-x-under-candidate-assigns-equiv
  (candidate-assigns-equiv (cdr (candidate-assigns-fix x))
                           (cdr x)))

Theorem: cdr-candidate-assigns-equiv-congruence-on-x-under-candidate-assigns-equiv

(defthm
 cdr-candidate-assigns-equiv-congruence-on-x-under-candidate-assigns-equiv
 (implies (candidate-assigns-equiv x x-equiv)
          (candidate-assigns-equiv (cdr x)
                                   (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-candidate-assign-fix-x-under-candidate-assigns-equiv

(defthm cons-of-candidate-assign-fix-x-under-candidate-assigns-equiv
  (candidate-assigns-equiv (cons (candidate-assign-fix x) y)
                           (cons x y)))

Theorem: cons-candidate-assign-equiv-congruence-on-x-under-candidate-assigns-equiv

(defthm
 cons-candidate-assign-equiv-congruence-on-x-under-candidate-assigns-equiv
 (implies (candidate-assign-equiv x x-equiv)
          (candidate-assigns-equiv (cons x y)
                                   (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-candidate-assigns-fix-y-under-candidate-assigns-equiv

(defthm
      cons-of-candidate-assigns-fix-y-under-candidate-assigns-equiv
  (candidate-assigns-equiv (cons x (candidate-assigns-fix y))
                           (cons x y)))

Theorem: cons-candidate-assigns-equiv-congruence-on-y-under-candidate-assigns-equiv

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

Theorem: consp-of-candidate-assigns-fix

(defthm consp-of-candidate-assigns-fix
  (equal (consp (candidate-assigns-fix x))
         (consp x)))

Theorem: candidate-assigns-fix-under-iff

(defthm candidate-assigns-fix-under-iff
  (iff (candidate-assigns-fix x)
       (consp x)))

Theorem: candidate-assigns-fix-of-cons

(defthm candidate-assigns-fix-of-cons
  (equal (candidate-assigns-fix (cons a x))
         (cons (candidate-assign-fix a)
               (candidate-assigns-fix x))))

Theorem: len-of-candidate-assigns-fix

(defthm len-of-candidate-assigns-fix
  (equal (len (candidate-assigns-fix x))
         (len x)))

Theorem: candidate-assigns-fix-of-append

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

Theorem: candidate-assigns-fix-of-repeat

(defthm candidate-assigns-fix-of-repeat
  (equal (candidate-assigns-fix (acl2::repeat n x))
         (acl2::repeat n (candidate-assign-fix x))))

Theorem: list-equiv-refines-candidate-assigns-equiv

(defthm list-equiv-refines-candidate-assigns-equiv
  (implies (acl2::list-equiv x y)
           (candidate-assigns-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-candidate-assigns-fix

(defthm nth-of-candidate-assigns-fix
  (equal (nth n (candidate-assigns-fix x))
         (if (< (nfix n) (len x))
             (candidate-assign-fix (nth n x))
           nil)))

Theorem: candidate-assigns-equiv-implies-candidate-assigns-equiv-append-1

(defthm
   candidate-assigns-equiv-implies-candidate-assigns-equiv-append-1
  (implies (candidate-assigns-equiv x fty::x-equiv)
           (candidate-assigns-equiv (append x y)
                                    (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: candidate-assigns-equiv-implies-candidate-assigns-equiv-append-2

(defthm
   candidate-assigns-equiv-implies-candidate-assigns-equiv-append-2
  (implies (candidate-assigns-equiv y fty::y-equiv)
           (candidate-assigns-equiv (append x y)
                                    (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: candidate-assigns-equiv-implies-candidate-assigns-equiv-nthcdr-2

(defthm
   candidate-assigns-equiv-implies-candidate-assigns-equiv-nthcdr-2
  (implies (candidate-assigns-equiv l l-equiv)
           (candidate-assigns-equiv (nthcdr n l)
                                    (nthcdr n l-equiv)))
  :rule-classes (:congruence))

Theorem: candidate-assigns-equiv-implies-candidate-assigns-equiv-take-2

(defthm
     candidate-assigns-equiv-implies-candidate-assigns-equiv-take-2
  (implies (candidate-assigns-equiv l l-equiv)
           (candidate-assigns-equiv (take n l)
                                    (take n l-equiv)))
  :rule-classes (:congruence))