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Integer-operations

Uint-shl

Left shift of unsigned integer values.

Signature
(uint-shl left-operand right-operand) → result
Arguments: left-operand — Guard (uintp left-operand).; right-operand — Guard (uintp right-operand).
Returns: result — Type (uintp result).

We prove that x << y is equivalent to x * 2**y, as mentioned in [SD: Types: Integers: Shifts].

Definitions and Theorems

Function: uint-shl

(defun uint-shl (left-operand right-operand)
  (declare (xargs :guard (and (uintp left-operand)
                              (uintp right-operand))))
  (b* ((size (uint->size left-operand))
       (x (uint->value left-operand))
       (y (uint->value right-operand)))
    (make-uint :size (uint->size left-operand)
               :value (loghead size (ash x y)))))

Theorem: uintp-of-uint-shl

(defthm uintp-of-uint-shl
  (b* ((result (uint-shl left-operand right-operand)))
    (uintp result))
  :rule-classes :rewrite)

Theorem: uint-shl-of-uint-fix-left-operand

(defthm uint-shl-of-uint-fix-left-operand
  (equal (uint-shl (uint-fix left-operand)
                   right-operand)
         (uint-shl left-operand right-operand)))

Theorem: uint-shl-uint-equiv-congruence-on-left-operand

(defthm uint-shl-uint-equiv-congruence-on-left-operand
  (implies (uint-equiv left-operand left-operand-equiv)
           (equal (uint-shl left-operand right-operand)
                  (uint-shl left-operand-equiv right-operand)))
  :rule-classes :congruence)

Theorem: uint-shl-of-uint-fix-right-operand

(defthm uint-shl-of-uint-fix-right-operand
  (equal (uint-shl left-operand (uint-fix right-operand))
         (uint-shl left-operand right-operand)))

Theorem: uint-shl-uint-equiv-congruence-on-right-operand

(defthm uint-shl-uint-equiv-congruence-on-right-operand
  (implies (uint-equiv right-operand right-operand-equiv)
           (equal (uint-shl left-operand right-operand)
                  (uint-shl left-operand right-operand-equiv)))
  :rule-classes :congruence)