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S4vecs

S4vec-<

Signature
(s4vec-< x y) → res
Arguments: x — Guard (s4vec-p x).; y — Guard (s4vec-p y).
Returns: res — Type (s4vec-p res).

Definitions and Theorems

Function: s4vec-<

(defun s4vec-< (x y)
  (declare (xargs :guard (and (s4vec-p x) (s4vec-p y))))
  (let ((__function__ 's4vec-<))
    (declare (ignorable __function__))
    (b* (((unless (and (s4vec-2vec-p x) (s4vec-2vec-p y)))
          (s4vec-x)))
      (s2vec (int-to-sparseint
                  (bool->vec (sparseint-< (s4vec->upper x)
                                          (s4vec->upper y))))))))

Theorem: s4vec-p-of-s4vec-<

(defthm s4vec-p-of-s4vec-<
  (b* ((res (s4vec-< x y))) (s4vec-p res))
  :rule-classes :rewrite)

Theorem: s4vec-<-correct

(defthm s4vec-<-correct
  (b* ((?res (s4vec-< x y)))
    (equal (s4vec->4vec res)
           (4vec-< (s4vec->4vec x)
                   (s4vec->4vec y)))))

Theorem: s4vec-<-of-s4vec-fix-x

(defthm s4vec-<-of-s4vec-fix-x
  (equal (s4vec-< (s4vec-fix x) y)
         (s4vec-< x y)))

Theorem: s4vec-<-s4vec-equiv-congruence-on-x

(defthm s4vec-<-s4vec-equiv-congruence-on-x
  (implies (s4vec-equiv x x-equiv)
           (equal (s4vec-< x y)
                  (s4vec-< x-equiv y)))
  :rule-classes :congruence)

Theorem: s4vec-<-of-s4vec-fix-y

(defthm s4vec-<-of-s4vec-fix-y
  (equal (s4vec-< x (s4vec-fix y))
         (s4vec-< x y)))

Theorem: s4vec-<-s4vec-equiv-congruence-on-y

(defthm s4vec-<-s4vec-equiv-congruence-on-y
  (implies (s4vec-equiv y y-equiv)
           (equal (s4vec-< x y)
                  (s4vec-< x y-equiv)))
  :rule-classes :congruence)