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Representation-of-integer-operations

Rem-sllong-sllong

Remainder of a value of type signed long long and a value of type signed long long [C:6.5.5].

Definitions and Theorems

Function: rem-sllong-sllong

(defun rem-sllong-sllong (x y)
  (declare (xargs :guard (and (sllongp x)
                              (sllongp y)
                              (rem-sllong-sllong-okp x y))))
  (sllong-from-integer (rem (integer-from-sllong x)
                            (integer-from-sllong y))))

Theorem: sllongp-of-rem-sllong-sllong

(defthm sllongp-of-rem-sllong-sllong
  (sllongp (rem-sllong-sllong x y)))

Theorem: rem-sllong-sllong-of-sllong-fix-x

(defthm rem-sllong-sllong-of-sllong-fix-x
  (equal (rem-sllong-sllong (sllong-fix x) y)
         (rem-sllong-sllong x y)))

Theorem: rem-sllong-sllong-sllong-equiv-congruence-on-x

(defthm rem-sllong-sllong-sllong-equiv-congruence-on-x
  (implies (sllong-equiv x x-equiv)
           (equal (rem-sllong-sllong x y)
                  (rem-sllong-sllong x-equiv y)))
  :rule-classes :congruence)

Theorem: rem-sllong-sllong-of-sllong-fix-y

(defthm rem-sllong-sllong-of-sllong-fix-y
  (equal (rem-sllong-sllong x (sllong-fix y))
         (rem-sllong-sllong x y)))

Theorem: rem-sllong-sllong-sllong-equiv-congruence-on-y

(defthm rem-sllong-sllong-sllong-equiv-congruence-on-y
  (implies (sllong-equiv y y-equiv)
           (equal (rem-sllong-sllong x y)
                  (rem-sllong-sllong x y-equiv)))
  :rule-classes :congruence)