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S4vecs

S4vec-bit?

Signature
(s4vec-bit? test then else) → res
Arguments: test — Guard (s4vec-p test).; then — Guard (s4vec-p then).; else — Guard (s4vec-p else).
Returns: res — Type (s4vec-p res).

Definitions and Theorems

Function: s4vec-bit?

(defun s4vec-bit? (test then else)
  (declare (xargs :guard (and (s4vec-p test)
                              (s4vec-p then)
                              (s4vec-p else))))
  (let ((__function__ 's4vec-bit?))
    (declare (ignorable __function__))
    (s3vec-bit? (s3vec-fix test)
                then else)))

Theorem: s4vec-p-of-s4vec-bit?

(defthm s4vec-p-of-s4vec-bit?
  (b* ((res (s4vec-bit? test then else)))
    (s4vec-p res))
  :rule-classes :rewrite)

Theorem: s4vec-bit?-correct

(defthm s4vec-bit?-correct
  (b* ((?res (s4vec-bit? test then else)))
    (equal (s4vec->4vec res)
           (4vec-bit? (s4vec->4vec test)
                      (s4vec->4vec then)
                      (s4vec->4vec else)))))

Theorem: s4vec-bit?-of-s4vec-fix-test

(defthm s4vec-bit?-of-s4vec-fix-test
  (equal (s4vec-bit? (s4vec-fix test) then else)
         (s4vec-bit? test then else)))

Theorem: s4vec-bit?-s4vec-equiv-congruence-on-test

(defthm s4vec-bit?-s4vec-equiv-congruence-on-test
  (implies (s4vec-equiv test test-equiv)
           (equal (s4vec-bit? test then else)
                  (s4vec-bit? test-equiv then else)))
  :rule-classes :congruence)

Theorem: s4vec-bit?-of-s4vec-fix-then

(defthm s4vec-bit?-of-s4vec-fix-then
  (equal (s4vec-bit? test (s4vec-fix then) else)
         (s4vec-bit? test then else)))

Theorem: s4vec-bit?-s4vec-equiv-congruence-on-then

(defthm s4vec-bit?-s4vec-equiv-congruence-on-then
  (implies (s4vec-equiv then then-equiv)
           (equal (s4vec-bit? test then else)
                  (s4vec-bit? test then-equiv else)))
  :rule-classes :congruence)

Theorem: s4vec-bit?-of-s4vec-fix-else

(defthm s4vec-bit?-of-s4vec-fix-else
  (equal (s4vec-bit? test then (s4vec-fix else))
         (s4vec-bit? test then else)))

Theorem: s4vec-bit?-s4vec-equiv-congruence-on-else

(defthm s4vec-bit?-s4vec-equiv-congruence-on-else
  (implies (s4vec-equiv else else-equiv)
           (equal (s4vec-bit? test then else)
                  (s4vec-bit? test then else-equiv)))
  :rule-classes :congruence)