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3col4vecs

3col4vecs-fix

(3col4vecs-fix x) is a usual fty list fixing function.

Signature
(3col4vecs-fix x) → fty::newx
Arguments: x — Guard (3col4vecs-p x).
Returns: fty::newx — Type (3col4vecs-p fty::newx).

In the logic, we apply 3col4vecline-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: 3col4vecs-fix$inline

(defun 3col4vecs-fix$inline (x)
  (declare (xargs :guard (3col4vecs-p x)))
  (let ((__function__ '3col4vecs-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (3col4vecline-fix (car x))
                 (3col4vecs-fix (cdr x))))
         :exec x)))

Theorem: 3col4vecs-p-of-3col4vecs-fix

(defthm 3col4vecs-p-of-3col4vecs-fix
  (b* ((fty::newx (3col4vecs-fix$inline x)))
    (3col4vecs-p fty::newx))
  :rule-classes :rewrite)

Theorem: 3col4vecs-fix-when-3col4vecs-p

(defthm 3col4vecs-fix-when-3col4vecs-p
  (implies (3col4vecs-p x)
           (equal (3col4vecs-fix x) x)))

Function: 3col4vecs-equiv$inline

(defun 3col4vecs-equiv$inline (x y)
  (declare (xargs :guard (and (3col4vecs-p x) (3col4vecs-p y))))
  (equal (3col4vecs-fix x)
         (3col4vecs-fix y)))

Theorem: 3col4vecs-equiv-is-an-equivalence

(defthm 3col4vecs-equiv-is-an-equivalence
  (and (booleanp (3col4vecs-equiv x y))
       (3col4vecs-equiv x x)
       (implies (3col4vecs-equiv x y)
                (3col4vecs-equiv y x))
       (implies (and (3col4vecs-equiv x y)
                     (3col4vecs-equiv y z))
                (3col4vecs-equiv x z)))
  :rule-classes (:equivalence))

Theorem: 3col4vecs-equiv-implies-equal-3col4vecs-fix-1

(defthm 3col4vecs-equiv-implies-equal-3col4vecs-fix-1
  (implies (3col4vecs-equiv x x-equiv)
           (equal (3col4vecs-fix x)
                  (3col4vecs-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: 3col4vecs-fix-under-3col4vecs-equiv

(defthm 3col4vecs-fix-under-3col4vecs-equiv
  (3col4vecs-equiv (3col4vecs-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-3col4vecs-fix-1-forward-to-3col4vecs-equiv

(defthm equal-of-3col4vecs-fix-1-forward-to-3col4vecs-equiv
  (implies (equal (3col4vecs-fix x) y)
           (3col4vecs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-3col4vecs-fix-2-forward-to-3col4vecs-equiv

(defthm equal-of-3col4vecs-fix-2-forward-to-3col4vecs-equiv
  (implies (equal x (3col4vecs-fix y))
           (3col4vecs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 3col4vecs-equiv-of-3col4vecs-fix-1-forward

(defthm 3col4vecs-equiv-of-3col4vecs-fix-1-forward
  (implies (3col4vecs-equiv (3col4vecs-fix x) y)
           (3col4vecs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 3col4vecs-equiv-of-3col4vecs-fix-2-forward

(defthm 3col4vecs-equiv-of-3col4vecs-fix-2-forward
  (implies (3col4vecs-equiv x (3col4vecs-fix y))
           (3col4vecs-equiv x y))
  :rule-classes :forward-chaining)

Theorem: car-of-3col4vecs-fix-x-under-3col4vecline-equiv

(defthm car-of-3col4vecs-fix-x-under-3col4vecline-equiv
  (3col4vecline-equiv (car (3col4vecs-fix x))
                      (car x)))

Theorem: car-3col4vecs-equiv-congruence-on-x-under-3col4vecline-equiv

(defthm car-3col4vecs-equiv-congruence-on-x-under-3col4vecline-equiv
  (implies (3col4vecs-equiv x x-equiv)
           (3col4vecline-equiv (car x)
                               (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-3col4vecs-fix-x-under-3col4vecs-equiv

(defthm cdr-of-3col4vecs-fix-x-under-3col4vecs-equiv
  (3col4vecs-equiv (cdr (3col4vecs-fix x))
                   (cdr x)))

Theorem: cdr-3col4vecs-equiv-congruence-on-x-under-3col4vecs-equiv

(defthm cdr-3col4vecs-equiv-congruence-on-x-under-3col4vecs-equiv
  (implies (3col4vecs-equiv x x-equiv)
           (3col4vecs-equiv (cdr x) (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-3col4vecline-fix-x-under-3col4vecs-equiv

(defthm cons-of-3col4vecline-fix-x-under-3col4vecs-equiv
  (3col4vecs-equiv (cons (3col4vecline-fix x) y)
                   (cons x y)))

Theorem: cons-3col4vecline-equiv-congruence-on-x-under-3col4vecs-equiv

(defthm
      cons-3col4vecline-equiv-congruence-on-x-under-3col4vecs-equiv
  (implies (3col4vecline-equiv x x-equiv)
           (3col4vecs-equiv (cons x y)
                            (cons x-equiv y)))
  :rule-classes :congruence)

Theorem: cons-of-3col4vecs-fix-y-under-3col4vecs-equiv

(defthm cons-of-3col4vecs-fix-y-under-3col4vecs-equiv
  (3col4vecs-equiv (cons x (3col4vecs-fix y))
                   (cons x y)))

Theorem: cons-3col4vecs-equiv-congruence-on-y-under-3col4vecs-equiv

(defthm cons-3col4vecs-equiv-congruence-on-y-under-3col4vecs-equiv
  (implies (3col4vecs-equiv y y-equiv)
           (3col4vecs-equiv (cons x y)
                            (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-3col4vecs-fix

(defthm consp-of-3col4vecs-fix
  (equal (consp (3col4vecs-fix x))
         (consp x)))

Theorem: 3col4vecs-fix-under-iff

(defthm 3col4vecs-fix-under-iff
  (iff (3col4vecs-fix x) (consp x)))

Theorem: 3col4vecs-fix-of-cons

(defthm 3col4vecs-fix-of-cons
  (equal (3col4vecs-fix (cons a x))
         (cons (3col4vecline-fix a)
               (3col4vecs-fix x))))

Theorem: len-of-3col4vecs-fix

(defthm len-of-3col4vecs-fix
  (equal (len (3col4vecs-fix x)) (len x)))

Theorem: 3col4vecs-fix-of-append

(defthm 3col4vecs-fix-of-append
  (equal (3col4vecs-fix (append std::a std::b))
         (append (3col4vecs-fix std::a)
                 (3col4vecs-fix std::b))))

Theorem: 3col4vecs-fix-of-repeat

(defthm 3col4vecs-fix-of-repeat
  (equal (3col4vecs-fix (repeat acl2::n x))
         (repeat acl2::n (3col4vecline-fix x))))

Theorem: list-equiv-refines-3col4vecs-equiv

(defthm list-equiv-refines-3col4vecs-equiv
  (implies (list-equiv x y)
           (3col4vecs-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-3col4vecs-fix

(defthm nth-of-3col4vecs-fix
  (equal (nth acl2::n (3col4vecs-fix x))
         (if (< (nfix acl2::n) (len x))
             (3col4vecline-fix (nth acl2::n x))
           nil)))

Theorem: 3col4vecs-equiv-implies-3col4vecs-equiv-append-1

(defthm 3col4vecs-equiv-implies-3col4vecs-equiv-append-1
  (implies (3col4vecs-equiv x fty::x-equiv)
           (3col4vecs-equiv (append x y)
                            (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: 3col4vecs-equiv-implies-3col4vecs-equiv-append-2

(defthm 3col4vecs-equiv-implies-3col4vecs-equiv-append-2
  (implies (3col4vecs-equiv y fty::y-equiv)
           (3col4vecs-equiv (append x y)
                            (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: 3col4vecs-equiv-implies-3col4vecs-equiv-nthcdr-2

(defthm 3col4vecs-equiv-implies-3col4vecs-equiv-nthcdr-2
  (implies (3col4vecs-equiv acl2::l l-equiv)
           (3col4vecs-equiv (nthcdr acl2::n acl2::l)
                            (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: 3col4vecs-equiv-implies-3col4vecs-equiv-take-2

(defthm 3col4vecs-equiv-implies-3col4vecs-equiv-take-2
  (implies (3col4vecs-equiv acl2::l l-equiv)
           (3col4vecs-equiv (take acl2::n acl2::l)
                            (take acl2::n l-equiv)))
  :rule-classes (:congruence))