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A4veclist-nth

Signature
(a4veclist-nth n x) → elt
Arguments: n — Guard (natp n).; x — Guard (a4veclist-p x).
Returns: elt — Type (a4vec-p elt).

Definitions and Theorems

Function: a4veclist-nth

(defun a4veclist-nth (n x)
  (declare (xargs :guard (and (natp n) (a4veclist-p x))))
  (let ((__function__ 'a4veclist-nth))
    (declare (ignorable __function__))
    (mbe :logic
         (if (< (nfix n) (len x))
             (a4vec-fix (nth n x))
           (a4vec-x))
         :exec (or (nth n x) (a4vec-x)))))

Theorem: a4vec-p-of-a4veclist-nth

(defthm a4vec-p-of-a4veclist-nth
  (b* ((elt (a4veclist-nth n x)))
    (a4vec-p elt))
  :rule-classes :rewrite)

Theorem: a4veclist-nth-out-of-bounds

(defthm a4veclist-nth-out-of-bounds
  (implies (<= (len x) (nfix n))
           (equal (a4veclist-nth n x) (a4vec-x))))

Theorem: a4veclist-nth-in-of-bounds

(defthm a4veclist-nth-in-of-bounds
  (implies (< (nfix n) (len x))
           (equal (a4veclist-nth n x)
                  (a4vec-fix (nth n x)))))

Theorem: a4veclist-nth-of-nfix-n

(defthm a4veclist-nth-of-nfix-n
  (equal (a4veclist-nth (nfix n) x)
         (a4veclist-nth n x)))

Theorem: a4veclist-nth-nat-equiv-congruence-on-n

(defthm a4veclist-nth-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (a4veclist-nth n x)
                  (a4veclist-nth n-equiv x)))
  :rule-classes :congruence)

Theorem: a4veclist-nth-of-a4veclist-fix-x

(defthm a4veclist-nth-of-a4veclist-fix-x
  (equal (a4veclist-nth n (a4veclist-fix x))
         (a4veclist-nth n x)))

Theorem: a4veclist-nth-a4veclist-equiv-congruence-on-x

(defthm a4veclist-nth-a4veclist-equiv-congruence-on-x
  (implies (a4veclist-equiv x x-equiv)
           (equal (a4veclist-nth n x)
                  (a4veclist-nth n x-equiv)))
  :rule-classes :congruence)