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• Arith-equivs

Bit->bool

Coerce a bit into a Boolean.

This is just (equal 1 x). However, using a function for this allows us to use congruences and to control case-splitting. For example, if we have

(equal (equal 1 (foo x))
       (equal 1 (bar y)))

this will case split into:

(if (equal 1 (foo x))
    (equal 1 (bar y))
  (not (equal 1 (bar y))))

whereas

(equal (bit->bool (foo x)) (bit->bool (bar y)))

will not.

Because a bunch of libraries were written under the assumption that (equal 1 x) was the way to tell if a bit was true or false, we leave this enabled by default.

Definitions and Theorems

Function: bit->bool$inline

(defun bit->bool$inline (x)
       (declare (xargs :guard t))
       (equal 1 x))

Theorem: bit-equiv-implies-equal-bit->bool-1

(defthm bit-equiv-implies-equal-bit->bool-1
        (implies (bit-equiv a a-equiv)
                 (equal (bit->bool a)
                        (bit->bool a-equiv)))
        :rule-classes (:congruence))