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Sparseint

Sparseint-bit

Extract the bit at the given index from a sparseint

Signature
(sparseint-bit n x) → bit
Arguments: n — Guard (natp n).; x — Guard (sparseint-p x).
Returns: bit — Type (bitp bit).

Definitions and Theorems

Function: sparseint-bit$inline

(defun sparseint-bit$inline (n x)
  (declare (xargs :guard (and (natp n) (sparseint-p x))))
  (let ((__function__ 'sparseint-bit))
    (declare (ignorable __function__))
    (sparseint$-bit n (sparseint-fix x))))

Theorem: bitp-of-sparseint-bit

(defthm bitp-of-sparseint-bit
  (b* ((bit (sparseint-bit$inline n x)))
    (bitp bit))
  :rule-classes :type-prescription)

Theorem: sparseint-bit-correct

(defthm sparseint-bit-correct
  (b* nil
    (equal (sparseint-bit n x)
           (logbit n (sparseint-val x)))))

Theorem: sparseint-bit$inline-of-nfix-n

(defthm sparseint-bit$inline-of-nfix-n
  (equal (sparseint-bit$inline (nfix n) x)
         (sparseint-bit$inline n x)))

Theorem: sparseint-bit$inline-nat-equiv-congruence-on-n

(defthm sparseint-bit$inline-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (sparseint-bit$inline n x)
                  (sparseint-bit$inline n-equiv x)))
  :rule-classes :congruence)

Theorem: sparseint-bit$inline-of-sparseint-fix-x

(defthm sparseint-bit$inline-of-sparseint-fix-x
  (equal (sparseint-bit$inline n (sparseint-fix x))
         (sparseint-bit$inline n x)))

Theorem: sparseint-bit$inline-sparseint-equiv-congruence-on-x

(defthm sparseint-bit$inline-sparseint-equiv-congruence-on-x
  (implies (sparseint-equiv x x-equiv)
           (equal (sparseint-bit$inline n x)
                  (sparseint-bit$inline n x-equiv)))
  :rule-classes :congruence)