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Representation-of-integer-operations

Shl-sllong-okp

Check if the left shift of a value of type signed long long by an integer is well-defined.

Definitions and Theorems

Function: shl-sllong-okp

(defun shl-sllong-okp (x y)
  (declare (xargs :guard (and (sllongp x) (integerp y))))
  (and (integer-range-p 0 (llong-bits)
                        (ifix y))
       (>= (integer-from-sllong x) 0)
       (sllong-integerp (* (integer-from-sllong x)
                           (expt 2 (ifix y))))))

Theorem: booleanp-of-shl-sllong-okp

(defthm booleanp-of-shl-sllong-okp
  (booleanp (shl-sllong-okp x y)))

Theorem: shl-sllong-okp-of-sllong-fix-x

(defthm shl-sllong-okp-of-sllong-fix-x
  (equal (shl-sllong-okp (sllong-fix x) y)
         (shl-sllong-okp x y)))

Theorem: shl-sllong-okp-sllong-equiv-congruence-on-x

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

Theorem: shl-sllong-okp-of-ifix-y

(defthm shl-sllong-okp-of-ifix-y
  (equal (shl-sllong-okp x (ifix y))
         (shl-sllong-okp x y)))

Theorem: shl-sllong-okp-int-equiv-congruence-on-y

(defthm shl-sllong-okp-int-equiv-congruence-on-y
  (implies (acl2::int-equiv y y-equiv)
           (equal (shl-sllong-okp x y)
                  (shl-sllong-okp x y-equiv)))
  :rule-classes :congruence)