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(in-package "ACL2")
(include-book "lop2") ;BOZO yuck!
(local (include-book "lop3-proofs"))
(defund OLAMT (a b e)
(logxor a (lnot b (1+ e))))
(defund OLAMG (a b e)
(logand a (lnot b (1+ e))))
(defund OLAMZ (a b e)
(lnot (logior a (lnot b (1+ e))) (1+ e)))
(defund OLAM1 (a b e)
(logand (bits (olamt a b e) e 2)
(logand (bits (olamg a b e) (1- e) 1)
(lnot (bits (olamz a b e) (- e 2) 0) (1- e)))))
(defund OLAM2 (a b e)
(logand (lnot (bits (olamt a b e) e 2) (1- e))
(logand (bits (olamz a b e) (1- e) 1)
(lnot (bits (olamz a b e) (- e 2) 0) (1- e)))))
(defund OLAM3 (a b e)
(logand (bits (olamt a b e) e 2)
(logand (bits (olamz a b e) (1- e) 1)
(lnot (bits (olamg a b e) (- e 2) 0) (1- e)))))
(defund OLAM4 (a b e)
(logand (lnot (bits (olamt a b e) e 2) (1- e))
(logand (bits (olamg a b e) (1- e) 1)
(lnot (bits (olamg a b e) (- e 2) 0) (1- e)))))
(defund OLAM0 (a b e)
(logior (olam1 a b e)
(logior (olam2 a b e)
(logior (olam3 a b e)
(olam4 a b e)))))
(defund OLAMB (a b e)
(+ (* 2 (olam0 a b e))
(lnot (bitn (olamt a b e) 0) 1)))
(defthm OLAMT-NAT
(implies (and (integerp a)
(>= a 0)
)
(and (integerp (olamt a b e))
(>= (olamt a b e) 0))))
(defthm OLAMG-NAT
(implies (and (integerp a)
(> a 0)
)
(and (integerp (olamg a b e))
(>= (olamg a b e) 0))))
(defthm OLAMZ-NAT
(implies (and (integerp a)
(> a 0)
)
(and (integerp (olamz a b e))
(>= (olamz a b e) 0))))
(defthm OLAM1-NAT
(implies (and (integerp a)
(> a 0)
)
(and (integerp (olam1 a b e))
(>= (olam1 a b e) 0))))
(defthm OLAM3-NAT
(implies (and (integerp a)
(> a 0)
)
(and (integerp (olam3 a b e))
(>= (olam3 a b e) 0)))
:rule-classes ())
(defthm OLAM2-NAT
(implies (and (integerp a)
(> a 0)
(integerp b)
)
(and (integerp (olam2 a b e))
(>= (olam2 a b e) 0))))
(defthm OLAM4-NAT
(implies (and (integerp a)
(> a 0)
)
(and (integerp (olam4 a b e))
(>= (olam4 a b e) 0))))
(defthm OLAM0-NAT
(implies (and (integerp a)
(> a 0)
)
(and (integerp (olam0 a b e))
(>= (olam0 a b e) 0))))
(defthm OLAMB-NAT
(implies (and (integerp a)
(> a 0)
)
(and (integerp (olamb a b e))
(>= (olamb a b e) 0))))
(defthm OLAM1-BND
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1)
(integerp k)
(<= k (- e 2))
(>= k 0))
(< (olam1 a b e) (expt 2 (- e 1))))
:rule-classes ())
(defthm OLAM3-BND
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1)
(integerp k)
(<= k (- e 2))
(>= k 0))
(< (olam3 a b e) (expt 2 (- e 1))))
:rule-classes ())
(defthm OLAM2-BND
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1)
(integerp k)
(<= k (- e 2))
(>= k 0))
(< (olam2 a b e) (expt 2 (- e 1))))
:rule-classes ())
(defthm OLAM4-BND
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1)
(integerp k)
(<= k (- e 2))
(>= k 0))
(< (olam4 a b e) (expt 2 (- e 1))))
:rule-classes ())
(defthm OLAM0-BND
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1))
(< (olam0 a b e) (expt 2 (- e 1))))
:rule-classes ())
(defthm OLAMB-BND
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1))
(< (olamb a b e) (expt 2 e)))
:rule-classes ())
(defthm olop-thm-2
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1))
(and (not (= (olamb a b e) 0))
(or (= (expo (- a b)) (expo (olamb a b e)))
(= (expo (- a b)) (1- (expo (olamb a b e)))))))
:rule-classes ())
(include-book "land")
(include-book "lior")
(include-book "lxor")
(defun lamt (a b e)
(lxor a (lnot b (1+ e)) (1+ e)))
(defun lamg (a b e)
(land a (lnot b (1+ e)) (1+ e)))
(defun lamz (a b e)
(lnot (lior a (lnot b (1+ e)) (1+ e)) (1+ e)))
(defun lam1 (a b e)
(land (bits (lamt a b e) e 2)
(land (bits (lamg a b e) (1- e) 1)
(lnot (bits (lamz a b e) (- e 2) 0) (1- e))
(1- e))
(1- e)))
(defun lam2 (a b e)
(land (lnot (bits (lamt a b e) e 2) (1- e))
(land (bits (lamz a b e) (1- e) 1)
(lnot (bits (lamz a b e) (- e 2) 0) (1- e))
(1- e))
(1- e)))
(defun lam3 (a b e)
(land (bits (lamt a b e) e 2)
(land (bits (lamz a b e) (1- e) 1)
(lnot (bits (lamg a b e) (- e 2) 0) (1- e))
(1- e))
(1- e)))
(defun lam4 (a b e)
(land (lnot (bits (lamt a b e) e 2) (1- e))
(land (bits (lamg a b e) (1- e) 1)
(lnot (bits (lamg a b e) (- e 2) 0) (1- e))
(1- e))
(1- e)))
(defun lam0 (a b e)
(lior (lam1 a b e)
(lior (lam2 a b e)
(lior (lam3 a b e)
(lam4 a b e)
(1- e))
(1- e))
(1- e)))
(defun lamb (a b e)
(+ (* 2 (lam0 a b e))
(lnot (bitn (lamt a b e) 0) 1)))
(in-theory (enable bits-tail bvecp-expo)) ;BOZO yuck!
(defthm lop-thm-2
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(not (= a b))
(= e (expo a))
(= e (expo b))
(> e 1))
(and (not (= (lamb a b e) 0))
(or (= (expo (- a b)) (expo (lamb a b e)))
(= (expo (- a b)) (1- (expo (lamb a b e)))))))
:rule-classes ())
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