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Boolean-if

Boolean-if-spec

Specification of the circuit.

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

We use an if, obviously.

Definitions and Theorems

Function: boolean-if-spec

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

Theorem: booleanp-of-boolean-if-spec

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

Theorem: bitp-w-when-boolean-if-spec

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