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Snippet-mismatch-list

Snippet-mismatch-list-fix

(snippet-mismatch-list-fix x) is a usual ACL2::fty list fixing function.

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

In the logic, we apply snippet-mismatch-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: snippet-mismatch-list-fix$inline

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

Theorem: snippet-mismatch-list-p-of-snippet-mismatch-list-fix

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

Theorem: snippet-mismatch-list-fix-when-snippet-mismatch-list-p

(defthm snippet-mismatch-list-fix-when-snippet-mismatch-list-p
  (implies (snippet-mismatch-list-p x)
           (equal (snippet-mismatch-list-fix x)
                  x)))

Function: snippet-mismatch-list-equiv$inline

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

Theorem: snippet-mismatch-list-equiv-is-an-equivalence

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

Theorem: snippet-mismatch-list-equiv-implies-equal-snippet-mismatch-list-fix-1

(defthm
 snippet-mismatch-list-equiv-implies-equal-snippet-mismatch-list-fix-1
 (implies (snippet-mismatch-list-equiv x x-equiv)
          (equal (snippet-mismatch-list-fix x)
                 (snippet-mismatch-list-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: snippet-mismatch-list-fix-under-snippet-mismatch-list-equiv

(defthm snippet-mismatch-list-fix-under-snippet-mismatch-list-equiv
  (snippet-mismatch-list-equiv (snippet-mismatch-list-fix x)
                               x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-snippet-mismatch-list-fix-1-forward-to-snippet-mismatch-list-equiv

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

Theorem: equal-of-snippet-mismatch-list-fix-2-forward-to-snippet-mismatch-list-equiv

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

Theorem: snippet-mismatch-list-equiv-of-snippet-mismatch-list-fix-1-forward

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

Theorem: snippet-mismatch-list-equiv-of-snippet-mismatch-list-fix-2-forward

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

Theorem: car-of-snippet-mismatch-list-fix-x-under-snippet-mismatch-equiv

(defthm
    car-of-snippet-mismatch-list-fix-x-under-snippet-mismatch-equiv
  (snippet-mismatch-equiv (car (snippet-mismatch-list-fix x))
                          (car x)))

Theorem: car-snippet-mismatch-list-equiv-congruence-on-x-under-snippet-mismatch-equiv

(defthm
 car-snippet-mismatch-list-equiv-congruence-on-x-under-snippet-mismatch-equiv
 (implies (snippet-mismatch-list-equiv x x-equiv)
          (snippet-mismatch-equiv (car x)
                                  (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-snippet-mismatch-list-fix-x-under-snippet-mismatch-list-equiv

(defthm
 cdr-of-snippet-mismatch-list-fix-x-under-snippet-mismatch-list-equiv
 (snippet-mismatch-list-equiv (cdr (snippet-mismatch-list-fix x))
                              (cdr x)))

Theorem: cdr-snippet-mismatch-list-equiv-congruence-on-x-under-snippet-mismatch-list-equiv

(defthm
 cdr-snippet-mismatch-list-equiv-congruence-on-x-under-snippet-mismatch-list-equiv
 (implies (snippet-mismatch-list-equiv x x-equiv)
          (snippet-mismatch-list-equiv (cdr x)
                                       (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-snippet-mismatch-fix-x-under-snippet-mismatch-list-equiv

(defthm
   cons-of-snippet-mismatch-fix-x-under-snippet-mismatch-list-equiv
  (snippet-mismatch-list-equiv (cons (snippet-mismatch-fix x) y)
                               (cons x y)))

Theorem: cons-snippet-mismatch-equiv-congruence-on-x-under-snippet-mismatch-list-equiv

(defthm
 cons-snippet-mismatch-equiv-congruence-on-x-under-snippet-mismatch-list-equiv
 (implies (snippet-mismatch-equiv x x-equiv)
          (snippet-mismatch-list-equiv (cons x y)
                                       (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-snippet-mismatch-list-fix-y-under-snippet-mismatch-list-equiv

(defthm
 cons-of-snippet-mismatch-list-fix-y-under-snippet-mismatch-list-equiv
 (snippet-mismatch-list-equiv (cons x (snippet-mismatch-list-fix y))
                              (cons x y)))

Theorem: cons-snippet-mismatch-list-equiv-congruence-on-y-under-snippet-mismatch-list-equiv

(defthm
 cons-snippet-mismatch-list-equiv-congruence-on-y-under-snippet-mismatch-list-equiv
 (implies (snippet-mismatch-list-equiv y y-equiv)
          (snippet-mismatch-list-equiv (cons x y)
                                       (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-snippet-mismatch-list-fix

(defthm consp-of-snippet-mismatch-list-fix
  (equal (consp (snippet-mismatch-list-fix x))
         (consp x)))

Theorem: snippet-mismatch-list-fix-under-iff

(defthm snippet-mismatch-list-fix-under-iff
  (iff (snippet-mismatch-list-fix x)
       (consp x)))

Theorem: snippet-mismatch-list-fix-of-cons

(defthm snippet-mismatch-list-fix-of-cons
  (equal (snippet-mismatch-list-fix (cons a x))
         (cons (snippet-mismatch-fix a)
               (snippet-mismatch-list-fix x))))

Theorem: len-of-snippet-mismatch-list-fix

(defthm len-of-snippet-mismatch-list-fix
  (equal (len (snippet-mismatch-list-fix x))
         (len x)))

Theorem: snippet-mismatch-list-fix-of-append

(defthm snippet-mismatch-list-fix-of-append
  (equal (snippet-mismatch-list-fix (append std::a std::b))
         (append (snippet-mismatch-list-fix std::a)
                 (snippet-mismatch-list-fix std::b))))

Theorem: snippet-mismatch-list-fix-of-repeat

(defthm snippet-mismatch-list-fix-of-repeat
  (equal (snippet-mismatch-list-fix (acl2::repeat n x))
         (acl2::repeat n (snippet-mismatch-fix x))))

Theorem: list-equiv-refines-snippet-mismatch-list-equiv

(defthm list-equiv-refines-snippet-mismatch-list-equiv
  (implies (acl2::list-equiv x y)
           (snippet-mismatch-list-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-snippet-mismatch-list-fix

(defthm nth-of-snippet-mismatch-list-fix
  (equal (nth n (snippet-mismatch-list-fix x))
         (if (< (nfix n) (len x))
             (snippet-mismatch-fix (nth n x))
           nil)))

Theorem: snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-append-1

(defthm
 snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-append-1
 (implies (snippet-mismatch-list-equiv x fty::x-equiv)
          (snippet-mismatch-list-equiv (append x y)
                                       (append fty::x-equiv y)))
 :rule-classes (:congruence))

Theorem: snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-append-2

(defthm
 snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-append-2
 (implies (snippet-mismatch-list-equiv y fty::y-equiv)
          (snippet-mismatch-list-equiv (append x y)
                                       (append x fty::y-equiv)))
 :rule-classes (:congruence))

Theorem: snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-nthcdr-2

(defthm
 snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-nthcdr-2
 (implies (snippet-mismatch-list-equiv l l-equiv)
          (snippet-mismatch-list-equiv (nthcdr n l)
                                       (nthcdr n l-equiv)))
 :rule-classes (:congruence))

Theorem: snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-take-2

(defthm
 snippet-mismatch-list-equiv-implies-snippet-mismatch-list-equiv-take-2
 (implies (snippet-mismatch-list-equiv l l-equiv)
          (snippet-mismatch-list-equiv (take n l)
                                       (take n l-equiv)))
 :rule-classes (:congruence))