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S4vecs

S4vec-?

Signature
(s4vec-? 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-?

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

Theorem: s4vec-p-of-s4vec-?

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

Theorem: s4vec-?-correct

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

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

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

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

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

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

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

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

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

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

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

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

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