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Boolean-xor

Boolean-xor-spec

Specification of the circuit.

Signature
(boolean-xor-spec x y z prime) → yes/no
Arguments: x — Guard (pfield::fep x prime).; y — Guard (pfield::fep y prime).; z — Guard (pfield::fep z prime).; prime — Guard (primep prime).
Returns: yes/no — Type (booleanp yes/no).

We use an if. An alternative is to use logxor.

Definitions and Theorems

Function: boolean-xor-spec

(defun boolean-xor-spec (x y z prime)
  (declare (xargs :guard (and (primep prime)
                              (pfield::fep x prime)
                              (pfield::fep y prime)
                              (pfield::fep z prime))))
  (declare (ignore prime))
  (declare (xargs :guard (and (bitp x) (bitp y))))
  (let ((__function__ 'boolean-xor-spec))
    (declare (ignorable __function__))
    (equal z (if (= x y) 0 1))))

Theorem: booleanp-of-boolean-xor-spec

(defthm booleanp-of-boolean-xor-spec
  (b* ((yes/no (boolean-xor-spec x y z prime)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: bitp-z-when-boolean-xor-spec

(defthm bitp-z-when-boolean-xor-spec
  (implies (and (boolean-xor-spec x y z prime)
                (bitp x)
                (bitp y))
           (bitp z)))