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• Utilities

Sum-nats

Sum a list of natural numbers.

Signature

(sum-nats x) → *

Arguments

x — Guard (nat-listp x).

Definitions and Theorems

Function: tr-sum-nats

(defun tr-sum-nats (x acc)
  (declare (xargs :guard (and (nat-listp x) (natp acc)))
           (type integer acc))
  (if (consp x)
      (tr-sum-nats (cdr x)
                   (the integer (+ (car x) acc)))
    acc))

Function: sum-nats

(defun sum-nats (x)
  (declare (xargs :guard (nat-listp x)))
  (let ((__function__ 'sum-nats))
    (declare (ignorable __function__))
    (mbe :logic
         (if (consp x)
             (+ (nfix (car x)) (sum-nats (cdr x)))
           0)
         :exec (tr-sum-nats x 0))))

Theorem: sum-nats-when-atom

(defthm sum-nats-when-atom
  (implies (atom x)
           (equal (sum-nats x) 0)))

Theorem: sum-nats-of-cons

(defthm sum-nats-of-cons
  (equal (sum-nats (cons a x))
         (+ (nfix a) (sum-nats x))))

Theorem: sum-nats-of-list-fix

(defthm sum-nats-of-list-fix
  (equal (sum-nats (list-fix x))
         (sum-nats x)))

Theorem: sum-nats-of-append

(defthm sum-nats-of-append
  (equal (sum-nats (append x y))
         (+ (sum-nats x) (sum-nats y))))

Theorem: sum-nats-of-rev

(defthm sum-nats-of-rev
  (equal (sum-nats (rev x)) (sum-nats x)))

Theorem: sum-nats-of-revappend

(defthm sum-nats-of-revappend
  (equal (sum-nats (revappend x y))
         (+ (sum-nats x) (sum-nats y))))

Theorem: sum-nats-of-reverse

(defthm sum-nats-of-reverse
  (implies (true-listp x)
           (equal (sum-nats (reverse x))
                  (sum-nats x))))

Theorem: sum-nats-when-all-equalp

(defthm sum-nats-when-all-equalp
  (implies (all-equalp n x)
           (equal (sum-nats x)
                  (* (nfix n) (len x)))))

Theorem: sum-nats-when-all-equalp-1

(defthm sum-nats-when-all-equalp-1
  (implies (all-equalp 1 x)
           (equal (sum-nats x) (len x))))