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Sparseint

Sparseint-binary-minus

Subtract two sparseints.

Signature
(sparseint-binary-minus x y) → minus
Arguments: x — Guard (sparseint-p x).; y — Guard (sparseint-p y).
Returns: minus — Type (sparseint-p minus).

Definitions and Theorems

Function: sparseint-binary-minus

(defun sparseint-binary-minus (x y)
  (declare (xargs :guard (and (sparseint-p x) (sparseint-p y))))
  (let ((__function__ 'sparseint-binary-minus))
    (declare (ignorable __function__))
    (sparseint$-binary-minus (sparseint-fix x)
                             (sparseint-fix y))))

Theorem: sparseint-p-of-sparseint-binary-minus

(defthm sparseint-p-of-sparseint-binary-minus
  (b* ((minus (sparseint-binary-minus x y)))
    (sparseint-p minus))
  :rule-classes :rewrite)

Theorem: sparseint-val-of-sparseint-binary-minus

(defthm sparseint-val-of-sparseint-binary-minus
  (b* ((?minus (sparseint-binary-minus x y)))
    (equal (sparseint-val minus)
           (- (sparseint-val x)
              (sparseint-val y)))))

Theorem: sparseint-binary-minus-of-sparseint-fix-x

(defthm sparseint-binary-minus-of-sparseint-fix-x
  (equal (sparseint-binary-minus (sparseint-fix x)
                                 y)
         (sparseint-binary-minus x y)))

Theorem: sparseint-binary-minus-sparseint-equiv-congruence-on-x

(defthm sparseint-binary-minus-sparseint-equiv-congruence-on-x
  (implies (sparseint-equiv x x-equiv)
           (equal (sparseint-binary-minus x y)
                  (sparseint-binary-minus x-equiv y)))
  :rule-classes :congruence)

Theorem: sparseint-binary-minus-of-sparseint-fix-y

(defthm sparseint-binary-minus-of-sparseint-fix-y
  (equal (sparseint-binary-minus x (sparseint-fix y))
         (sparseint-binary-minus x y)))

Theorem: sparseint-binary-minus-sparseint-equiv-congruence-on-y

(defthm sparseint-binary-minus-sparseint-equiv-congruence-on-y
  (implies (sparseint-equiv y y-equiv)
           (equal (sparseint-binary-minus x y)
                  (sparseint-binary-minus x y-equiv)))
  :rule-classes :congruence)