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(in-package "ACL2")
(local (include-book "bias-proofs"))
(defund bvecp (x k)
(declare (xargs :guard (integerp k)))
(and (integerp x)
(<= 0 x)
(< x (expt 2 k))))
; bias of a q bit exponent field is 2^(q-1)-1
(defund bias (q) (- (expt 2 (- q 1)) 1) )
(defthm bias-non-negative-integerp-type-prescription
(implies (and (case-split (integerp q))
(case-split (< 0 q))
)
(and (integerp (bias q))
(<= 0 (bias q))))
:rule-classes :TYPE-PRESCRIPTION)
;BOZO rename q2 to k?
(defthm bias-bvecp
(implies (and (<= (1- q) q2)
(case-split (< 0 q))
(case-split (integerp q))
(case-split (integerp q2))
)
(bvecp (bias q) q2)))
(defthm bias-integerp-rewrite
(equal (integerp (bias q))
(or (and (acl2-numberp q) (not (integerp q)))
(<= 1 q))))
;where's bias-integerp?
(defthm bias-integerp
(implies (case-split (< 0 k))
(integerp (bias k))))
(defthm bias-with-q-an-acl2-number-but-not-an-integer
(implies (and (not (integerp q))
(case-split (acl2-numberp q)))
(equal (bias q)
0)))
(defthm bias-with-q-not-an-acl2-number
(implies (not (acl2-numberp q))
(equal (bias q)
-1/2)))
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