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|
; RTL - A Formal Theory of Register-Transfer Logic and Computer Arithmetic
; Copyright (C) 1995-2013 Advanced Mirco Devices, Inc.
;
; Contact:
; David Russinoff
; 1106 W 9th St., Austin, TX 78703
; http://www.russsinoff.com/
;
; This program is free software; you can redistribute it and/or modify it under
; the terms of the GNU General Public License as published by the Free Software
; Foundation; either version 2 of the License, or (at your option) any later
; version.
;
; This program is distributed in the hope that it will be useful but WITHOUT ANY
; WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
; PARTICULAR PURPOSE. See the GNU General Public License for more details.
;
; You should have received a copy of the GNU General Public License along with
; this program; see the file "gpl.txt" in this directory. If not, write to the
; Free Software Foundation, Inc., 51 Franklin Street, Suite 500, Boston, MA
; 02110-1335, USA.
;
; Author: David M. Russinoff (david@russinoff.com)
(in-package "ACL2")
(include-book "add-new")
(local (include-book "../lib2/top"))
(local (include-book "../../arithmetic/top"))
(local (include-book "log-support"))
(local
(encapsulate ()
(local (include-book "bits-new-proofs"))
(defthm bits_alt-is-bits
(equal (bits_alt x i j)
(bits x i j)))
(defthm bitn_alt-is-bitn
(equal (bitn_alt x n)
(bitn x n)))
(defthm binary-cat_alt-is-binary-cat
(equal (binary-cat_alt x m y n)
(binary-cat x m y n)))
))
;;;**********************************************************************
;;; Radix-4 Booth Encoding
;;;**********************************************************************
(defun theta_alt (i y)
(+ (bitn_alt y (1- (* 2 i)))
(bitn_alt y (* 2 i))
(* -2 (bitn_alt y (1+ (* 2 i))))))
(local
(defthm theta_alt-is-theta
(equal (theta_alt i y)
(theta i y))))
(defun sum-theta_alt (m y)
(if (zp m)
0
(+ (* (expt 2 (* 2 (1- m))) (theta_alt (1- m) y))
(sum-theta_alt (1- m) y))))
(local
(defthm sum-theta_alt-is-sum-theta
(equal (sum-theta_alt m y)
(sum-theta m y))))
(defthm sum-theta_alt-lemma
(implies (and (not (zp m))
(bvecp y (1- (* 2 m))))
(equal y (sum-theta_alt m y)))
:rule-classes ()
:hints (("Goal" :use ((:instance sum-theta-lemma)))))
(defun bmux4_alt (zeta x n)
(case zeta
(1 x)
(-1 (bits_alt (lognot x) (1- n) 0))
(2 (* 2 x))
(-2 (bits_alt (lognot (* 2 x)) (1- n) 0))
(0 0)))
(local
(defthm bmux4_alt-is-bmux4
(implies (and (natp n)
(integerp x)
(> n 0))
(equal (bmux4_alt zeta x n)
(bmux4 zeta x n)))
:hints (("Goal" :in-theory (e/d (lnot-lognot) ())))))
(defun neg (x) (if (< x 0) 1 0))
(encapsulate ((zeta (i) t))
(local (defun zeta (i) (declare (ignore i)) 0))
(defthm zeta-bnd
(and (integerp (zeta i))
(<= (zeta i) 2)
(>= (zeta i) -2))))
(defun pp4_alt (i x n)
(if (zerop i)
(cat_alt 1 1
(bitn_alt (lognot (neg (zeta i))) 0) 1
(bmux4_alt (zeta i) x n) n)
(cat_alt 1 1
(bitn_alt (lognot (neg (zeta i))) 0) 1
(bmux4_alt (zeta i) x n) n
0 1
(neg (zeta (1- i))) 1
0 (* 2 (1- i)))))
(local
(defthm bvecp-mux4
(implies (and (bvecp x (+ -1 m))
(natp m)
(>= m n))
(BVECP (BMUX4 (ZETA 0) X N) m))
:hints (("Goal" :in-theory (e/d (bvecp bmux4
expt-2-reduce-leading-constant) (ZETA-BND))
:use ((:instance zeta-bnd
(i 0)))))))
(local
(defthm integerp-bmux4
(implies (integerp x)
(integerp (BMUX4 (ZETA i) X N)))
:rule-classes :type-prescription
:hints (("Goal" :in-theory (e/d (bmux4
expt-2-reduce-leading-constant) (ZETA-BND))
:use ((:instance zeta-bnd
(i i)))))))
(local
(defthm bvecp-0
(BVECP 0 n)
:hints (("Goal" :in-theory (e/d (bvecp) ())))))
(local
(defthm pp4_alt-is-pp4
(implies (and (natp n)
(natp i)
(bvecp x (+ -1 n))
(integerp x)
(> n 0))
(equal (pp4_alt i x n)
(pp4 i x n)))
:hints (("Goal" :in-theory (e/d (lnot-lognot
bvecp-monotone
cat)
(bmux4 bmux4_alt))))))
(defun sum-zeta (m)
(if (zp m)
0
(+ (* (expt 2 (* 2 (1- m))) (zeta (1- m)))
(sum-zeta (1- m)))))
(defun sum-pp4_alt (x m n)
(if (zp m)
0
(+ (pp4_alt (1- m) x n)
(sum-pp4_alt x (1- m) n))))
(local
(defthm sum-pp4_alt-is-sum-pp4
(implies (and (natp n)
(natp i)
(bvecp x (+ -1 n))
(integerp x)
(> n 0))
(equal (sum-pp4_alt x m n)
(sum-pp4 x m n)))
:hints (("Goal" :in-theory (e/d () (pp4_alt pp4))))))
(defthm booth4-thm_alt
(implies (and (not (zp n))
(not (zp m))
(bvecp x (1- n)))
(= (+ (expt 2 n)
(sum-pp4_alt x m n))
(+ (expt 2 (+ n (* 2 m)))
(* x (sum-zeta m))
(- (* (expt 2 (* 2 (1- m))) (neg (zeta (1- m))))))))
:rule-classes ()
:hints (("Goal" :use ((:instance booth4-thm))
:in-theory (e/d () (sum-pp4_alt sum-pp4)))))
(defun pp4_alt-theta_alt (i x y n)
(if (zerop i)
(cat_alt 1 1
(bitn_alt (lognot (neg (theta_alt i y))) 0) 1
(bmux4_alt (theta_alt i y) x n) n)
(cat_alt 1 1
(bitn_alt (lognot (neg (theta_alt i y))) 0) 1
(bmux4_alt (theta_alt i y) x n) n
0 1
(neg (theta_alt (1- i) y)) 1
0 (* 2 (1- i)))))
(local
(defthm bvecp-mux4-theta
(implies (and (bvecp x (+ -1 m))
(natp m)
(>= m n))
(BVECP (BMUX4 (theta i y) X N) m))
:hints (("Goal" :in-theory (e/d (bvecp bmux4
expt-2-reduce-leading-constant) (ZETA-BND))
:use ((:instance bitn-0-1
(x y)
(n (1- (* 2 I))))
(:instance bitn-0-1
(x y)
(n (* 2 I)))
(:instance bitn-0-1
(x y)
(n (+ 1 (* 2 I)))))))))
(local
(defthm integerp-bmux4-theta
(implies (integerp x)
(integerp (BMUX4 (theta i y) X N)))
:rule-classes :type-prescription
:hints (("Goal" :in-theory (e/d (bmux4
expt-2-reduce-leading-constant) (ZETA-BND))
:use ((:instance bitn-0-1
(x y)
(n (1- (* 2 I))))
(:instance bitn-0-1
(x y)
(n (* 2 I)))
(:instance bitn-0-1
(x y)
(n (+ 1 (* 2 I)))))))))
(local
(defthm pp4_alt-theta_alt-is-pp4-theta
(implies (and (not (zp n))
(bvecp x (+ -1 n))
(integerp i)
(integerp y)
(integerp x))
(equal (pp4_alt-theta_alt i x y n)
(pp4-theta i x y n)))
:hints (("Goal" :in-theory (e/d ()
(theta_alt
theta
bmux4_alt
bmux4))))))
(defun sum-pp4_alt-theta_alt (x y m n)
(if (zp m)
0
(+ (pp4_alt-theta_alt (1- m) x y n)
(sum-pp4_alt-theta_alt x y (1- m) n))))
(local
(defthm sum-pp4_alt-theta_alt-is-sum-pp4-theta
(implies (and (not (zp n))
(bvecp x (+ -1 n))
(integerp i)
(integerp y)
(integerp x))
(equal (sum-pp4_alt-theta_alt x y m n)
(sum-pp4-theta x y m n)))))
(defthm booth4-corollary-alt
(implies (and (not (zp n))
(not (zp m))
(bvecp x (1- n))
(bvecp y (1- (* 2 m))))
(= (+ (expt 2 n)
(sum-pp4_alt-theta_alt x y m n))
(+ (expt 2 (+ n (* 2 m)))
(* x y))))
:rule-classes ()
:hints (("Goal" :use ((:instance booth4-corollary)))))
;;;**********************************************************************
;;; Statically Encoded Multiplier Arrays
;;;**********************************************************************
(defun m-mu-chi (i mode)
(cond ((equal mode 'mu)
(if (zp i) 1
(cons (cons 1 i) 1)))
((equal mode 'chi)
(if (zp i) 0
(cons (cons 1 i) 0)))))
(mutual-recursion
(defun mu_alt (i y)
(declare (xargs :measure (m-mu-chi i 'mu)))
(+ (bits_alt y (1+ (* 2 i)) (* 2 i)) (chi_alt i y)))
(defun chi_alt (i y)
(declare (xargs :measure (m-mu-chi i 'chi)))
(if (zp i)
0
(if (>= (mu_alt (1- i) y) 3)
1
0))))
(local
(encapsulate ()
(local (encapsulate ()
(defun mu-chi_alt (i y mode)
(declare (xargs :measure (if (and (not (equal mode 'mu))
(not (equal mode 'chi)))
0
(m-mu-chi i mode))))
(if (equal mode 'mu)
(+ (bits_alt y (1+ (* 2 i)) (* 2 i)) (mu-chi_alt i y 'chi))
(if (equal mode 'chi)
(if (zp i)
0
(if (>= (mu-chi_alt (1- i) y 'mu) 3)
1
0))
nil)))
(defthm mu-chi_alt-is
(equal (mu-chi_alt i y mode)
(if (equal mode 'mu)
(mu_alt i y)
(if (equal mode 'chi)
(chi_alt i y)
(mu-chi_alt i y mode))))
:rule-classes nil)
(defthm mu-chi_alt-is-2
(equal (mu-chi_alt i y mode)
(if (equal mode 'mu)
(mu i y)
(if (equal mode 'chi)
(chi i y)
(mu-chi_alt i y mode))))
:rule-classes nil)
))
(defthm mu_alt-is-mu
(equal (mu_alt i y)
(mu i y))
:hints (("Goal" :use ((:instance mu-chi_alt-is
(mode 'mu))
(:instance mu-chi_alt-is-2
(mode 'mu))))))
(defthm chi_alt-is-chi
(equal (chi_alt i y)
(chi i y))
:hints (("Goal" :use ((:instance mu-chi_alt-is
(mode 'chi))
(:instance mu-chi_alt-is-2
(mode 'chi))))))
))
(defun phi_alt (i y)
(if (= (bits_alt (mu_alt i y) 1 0) 3)
-1
(bits_alt (mu_alt i y) 1 0)))
(local
(defthm phi_alt-is-phi
(equal (phi_alt i y)
(phi i y))))
(defthm phi-bnd-alt
(member (phi_alt i y) '(-1 0 1 2))
:hints (("Goal" :in-theory (e/d (phi-bnd) (phi_alt phi)))))
(defun sum-odd-powers-of-2 (m)
(if (zp m)
0
(+ (expt 2 (1- (* 2 m)))
(sum-odd-powers-of-2 (1- m)))))
(defthm chi_alt-m
(implies (and (natp m)
(natp y)
(<= y (sum-odd-powers-of-2 m)))
(equal (chi_alt m y) 0))
:rule-classes()
:hints (("Goal" :use ((:instance chi-m))
:in-theory (e/d () (chi_alt chi)))))
(defthm phi_alt-m-1
(implies (and (natp m)
(natp y)
(<= y (sum-odd-powers-of-2 m)))
(>= (phi_alt (1- m) y) 0))
:rule-classes()
:hints (("Goal" :use ((:instance phi-m-1))
:in-theory (e/d () (phi_alt phi)))))
(defun sum-phi_alt (m y)
(if (zp m)
0
(+ (* (expt 2 (* 2 (1- m))) (phi_alt (1- m) y))
(sum-phi_alt (1- m) y))))
(local
(defthm sum-phi_alt-is-sum-phi
(equal (sum-phi_alt m y)
(sum-phi m y))
:hints (("Goal" :in-theory (e/d () (phi_alt
phi))))))
(defthm sum-phi_alt-lemma
(implies (and (natp m)
(natp y)
(<= y (sum-odd-powers-of-2 m)))
(equal y (sum-phi_alt m y)))
:rule-classes ()
:hints (("Goal" :use ((:instance sum-phi-lemma)))))
(defun pp4_alt-phi_alt (i x y n)
(if (zerop i)
(cat_alt 1 1
(bitn_alt (lognot (neg (phi_alt i y))) 0) 1
(bmux4_alt (phi_alt i y) x n) n)
(cat_alt 1 1
(bitn_alt (lognot (neg (phi_alt i y))) 0) 1
(bmux4_alt (phi_alt i y) x n) n
0 1
(neg (phi_alt (1- i) y)) 1
0 (* 2 (1- i)))))
(local
(defthm pp4_alt-phi_alt-is-pp4-phi
(implies (and (integerp x)
(natp n)
(> n 0))
(equal (pp4_alt-phi_alt i x y n)
(pp4-phi i x y n)))
:hints (("Goal" :in-theory (e/d () (phi_alt
phi
pp4_alt
pp4
bmux4_alt
bmux4))))))
(defun sum-pp4_alt-phi_alt (x y m n)
(if (zp m)
0
(+ (pp4_alt-phi_alt (1- m) x y n)
(sum-pp4_alt-phi_alt x y (1- m) n))))
(local
(defthm sum-pp4_alt-phi_alt-is-sum-pp4-phi
(implies (and (integerp x)
(natp n)
(> n 0))
(equal (sum-pp4_alt-phi_alt x y m n)
(sum-pp4-phi x y m n)))
:hints (("Goal" :in-theory (e/d ()
(pp4-phi
pp4_alt-phi_alt))))))
(defthm static-booth-alt
(implies (and (not (zp n))
(not (zp m))
(bvecp x (1- n))
(natp y)
(<= y (sum-odd-powers-of-2 m)))
(= (+ (expt 2 n)
(sum-pp4_alt-phi_alt x y m n))
(+ (expt 2 (+ n (* 2 m)))
(* x y))))
:rule-classes ()
:hints (("Goal" :use ((:instance static-booth)))))
;;;**********************************************************************
;;; Encoding Redundant Representations
;;;**********************************************************************
(defun gamma_alt (i a b c)
(if (zp i)
(bitn_alt c 0)
(logior (bitn_alt a (+ -1 (* 2 i)))
(bitn_alt b (+ -1 (* 2 i))))))
(local
(defthm gamma_alt-is-gamma
(equal (gamma_alt i a b c)
(gamma i a b c))
:hints (("Goal" :use ((:instance bitn-0-1
(x a)
(n (+ -1 (* 2 I))))
(:instance bitn-0-1
(x b)
(n (+ -1 (* 2 I)))))))))
(defun delta_alt (i a b c d)
(if (zp i)
(bitn_alt d 0)
(logand (logior (logand (bitn_alt a (+ -2 (* 2 i)))
(bitn_alt b (+ -2 (* 2 i))))
(logior (logand (bitn_alt a (+ -2 (* 2 i)))
(gamma_alt (1- i) a b c))
(logand (bitn_alt b (+ -2 (* 2 i)))
(gamma_alt (1- i) a b c))))
(lognot (logxor (bitn_alt a (1- (* 2 i)))
(bitn_alt b (1- (* 2 i))))))))
(local
(DEFTHM LOGAND-BVECP-G_alt
(IMPLIES (AND (NATP N) (BVECP Y N) (INTEGERP X))
(BVECP (LOGAND X Y) N))
:hints (("Goal" :use ((:instance logand-bvecp-g
(x y)
(y x)))
:in-theory (e/d () (logand-bvecp-g))))))
(local
(defthm delta_alt-is-delta
(equal (delta_alt i a b c d)
(delta i a b c d))
:hints (("Goal" :in-theory (e/d (land-logand
logand-bitn-reduce
lnot-lognot
lxor-logxor
lior-logior)
(gamma_alt
gamma))))))
;;;;
;;;;
;;;;
(defun psi_alt (i a b c d)
(if (not (natp i))
0
(+ (bits_alt a (1+ (* 2 i)) (* 2 i))
(bits_alt b (1+ (* 2 i)) (* 2 i))
(gamma_alt i a b c)
(delta_alt i a b c d)
(* -4 (+ (gamma_alt (1+ i) a b c)
(delta_alt (1+ i) a b c d))))))
(local
(defthm psi_alt-is-psi
(equal (psi_alt i a b c d)
(psi i a b c d))
:hints (("Goal" :in-theory (e/d () (delta_alt
delta
gamma_alt
gamma))))))
(defthm psi_alt-m-1
(implies (and (natp m)
(>= m 1)
(bvecp a (- (* 2 m) 2))
(bvecp b (- (* 2 m) 2))
(bvecp c 1)
(bvecp d 1))
(and (equal (gamma_alt m a b c) 0)
(equal (delta_alt m a b c d) 0)
(>= (psi_alt (1- m) a b c d) 0)))
:rule-classes ()
:hints (("Goal" :use ((:instance psi-m-1))
:in-theory (e/d ()
(psi_alt
psi
gamma_alt
gamma
delta_alt
delta)))))
(defun sum-psi_alt (m a b c d)
(if (zp m)
0
(+ (* (expt 2 (* 2 (1- m))) (psi_alt (1- m) a b c d))
(sum-psi_alt (1- m) a b c d))))
(local
(defthm sum-psi_alt-is-sum-psi
(equal (sum-psi_alt m a b c d)
(sum-psi m a b c d))
:hints (("Goal" :in-theory (e/d () (psi_alt psi))))))
(defthm sum-psi_alt-lemma
(implies (and (natp m)
(<= 1 M) ;; add
(bvecp a (- (* 2 m) 2))
(bvecp b (- (* 2 m) 2))
(bvecp c 1)
(bvecp d 1))
(equal (+ a b c d) (sum-psi_alt m a b c d)))
:rule-classes ()
:hints (("Goal" :use ((:instance sum-psi-lemma)))))
(defthmd psi_alt-bnd
(and (integerp (psi_alt i a b c d))
(<= (psi_alt i a b c d) 2)
(>= (psi_alt i a b c d) -2))
:hints (("Goal" :use ((:instance psi-bnd)))))
(defun pp4_alt-psi_alt (i x a b c d n)
(if (zerop i)
(cat_alt 1 1
(bitn_alt (lognot (neg (psi_alt i a b c d))) 0) 1
(bmux4_alt (psi_alt i a b c d) x n) n)
(cat_alt 1 1
(bitn_alt (lognot (neg (psi_alt i a b c d))) 0) 1
(bmux4_alt (psi_alt i a b c d) x n) n
0 1
(neg (psi_alt (1- i) a b c d)) 1
0 (* 2 (1- i)))))
(local
(defthm pp4_alt-psi_alt-is-pp4-psi
(implies (and (natp n)
(> n 0)
(integerp x))
(equal (pp4_alt-psi_alt i x a b c d n)
(pp4-psi i x a b c d n)))
:hints (("Goal" :in-theory (e/d ()
(bmux4_alt
bmux4
psi_alt
psi))))))
(defun sum-pp4_alt-psi_alt (x a b c d m n)
(if (zp m)
0
(+ (pp4_alt-psi_alt (1- m) x a b c d n)
(sum-pp4_alt-psi_alt x a b c d (1- m) n))))
(local
(defthm sum-pp4_alt-psi_alt-is-sum-pp4-psi
(implies (and (natp n)
(> n 0)
(integerp x))
(equal (sum-pp4_alt-psi_alt x a b c d m n)
(sum-pp4-psi x a b c d m n)))
:hints (("Goal" :in-theory (e/d ()
(pp4_alt-psi_alt
pp4-psi))))))
(defthm redundant-booth_alt
(implies (and (natp m)
(<= 1 m)
(not (zp n))
(bvecp x (1- n))
(bvecp a (- (* 2 m) 2))
(bvecp b (- (* 2 m) 2))
(bvecp c 1)
(bvecp d 1)
(= y (+ a b c d)))
(= (+ (expt 2 n)
(sum-pp4_alt-psi_alt x a b c d m n))
(+ (expt 2 (+ n (* 2 m)))
(* x y))))
:rule-classes ()
:hints (("Goal" :use ((:instance redundant-booth)))))
;;;**********************************************************************
;;; Radix-8 Booth Encoding
;;;**********************************************************************
(defun eta_alt (i y)
(+ (bitn_alt y (1- (* 3 i)))
(bitn_alt y (* 3 i))
(* 2 (bitn_alt y (1+ (* 3 i))))
(* -4 (bitn_alt y (+ 2 (* 3 i))))))
(local
(defthm eta_alt-is-eta
(equal (eta_alt i y)
(eta i y))))
(defun sum-eta_alt (m y)
(if (zp m)
0
(+ (* (expt 2 (* 3 (1- m))) (eta_alt (1- m) y))
(sum-eta_alt (1- m) y))))
(local
(defthm sum-eta_alt-is-sum-eta
(equal (sum-eta_alt m y)
(sum-eta m y))))
(defthm sum-eta_alt-lemma
(implies (and (not (zp m))
(bvecp y (1- (* 3 m))))
(equal y (sum-eta_alt m y)))
:rule-classes ()
:hints (("Goal" :use ((:instance sum-eta-lemma)))))
(defun bmux8_alt (zeta_alt x n)
(case zeta_alt
(1 x)
(-1 (bits_alt (lognot x) (1- n) 0))
(2 (* 2 x))
(-2 (bits_alt (lognot (* 2 x)) (1- n) 0))
(3 (* 3 x))
(-3 (bits_alt (lognot (* 3 x)) (1- n) 0))
(4 (* 4 x))
(-4 (bits_alt (lognot (* 4 x)) (1- n) 0))
(0 0)))
(local
(defthm bmux8_alt-is-bmux8
(implies (and (natp n)
(> n 0)
(integerp x))
(equal (bmux8_alt zeta x n)
(bmux8 zeta x n)))
:hints (("Goal" :in-theory (e/d (lnot-lognot) ())))))
(encapsulate ((xi (i) t))
(local (defun xi (i) (declare (ignore i)) 0))
(defthm xi-bnd
(and (integerp (xi i))
(<= (xi i) 4)
(>= (xi i) -4))))
(defun pp8_alt (i x n)
(if (zerop i)
(cat_alt 3 2
(bitn_alt (lognot (neg (xi i))) 0) 1
(bmux8_alt (xi i) x n) n)
(cat_alt 3 2
(bitn_alt (lognot (neg (xi i))) 0) 1
(bmux8_alt (xi i) x n) n
0 2
(neg (xi (1- i))) 1
0 (* 3 (1- i)))))
(local
(defthm pp8_alt-is-pp8
(implies (and (natp n)
(> n 0)
(integerp x))
(equal (pp8_alt i x n)
(pp8 i x n)))
:hints (("Goal" :in-theory (e/d () (bmux8_alt
bmux8))))))
(defun sum-xi (m)
(if (zp m)
0
(+ (* (expt 2 (* 3 (1- m))) (xi (1- m)))
(sum-xi (1- m)))))
(defun sum-pp8_alt (x m n)
(if (zp m)
0
(+ (pp8_alt (1- m) x n)
(sum-pp8_alt x (1- m) n))))
(local
(defthm sum-pp8_alt-sum-pp8
(implies (and (natp n)
(> n 0)
(integerp x))
(equal (sum-pp8_alt x m n)
(sum-pp8 x m n)))
:hints (("Goal" :in-theory (e/d () (pp8_alt pp8))))))
(defthm booth8-thm_alt
(implies (and (not (zp n))
(not (zp m))
(bvecp x (- n 2)))
(= (+ (expt 2 n)
(sum-pp8_alt x m n))
(+ (expt 2 (+ n (* 3 m)))
(* x (sum-xi m))
(- (* (expt 2 (* 3 (1- m))) (neg (xi (1- m))))))))
:rule-classes ()
:hints (("Goal" :use ((:instance booth8-thm))
:in-theory (e/d () (sum-pp8_alt sum-pp8)))))
(defun pp8_alt-eta_alt (i x y n)
(if (zerop i)
(cat_alt 3 2
(bitn_alt (lognot (neg (eta_alt i y))) 0) 1
(bmux8_alt (eta_alt i y) x n) n)
(cat_alt 3 2
(bitn_alt (lognot (neg (eta_alt i y))) 0) 1
(bmux8_alt (eta_alt i y) x n) n
0 2
(neg (eta_alt (1- i) y)) 1
0 (* 3 (1- i)))))
(local
(defthm pp8_alt-eta_alt-is-pp8-eta
(implies (and (natp n)
(> n 0)
(integerp x))
(equal (pp8_alt-eta_alt i x y n)
(pp8-eta i x y n)))
:hints (("Goal" :in-theory (e/d (lnot-lognot)
(bmux8_alt
bmux8
eta_alt
eta))))))
(defun sum-pp8_alt-eta_alt (x y m n)
(if (zp m)
0
(+ (pp8_alt-eta_alt (1- m) x y n)
(sum-pp8_alt-eta_alt x y (1- m) n))))
(local
(defthm sum-pp8_alt-eta_alt-is-sum-pp8-eta
(implies (and (natp n)
(> n 0)
(integerp x))
(equal (sum-pp8_alt-eta_alt x y m n)
(sum-pp8-eta x y m n)))
:hints (("Goal" :in-theory (e/d () (pp8_alt-eta_alt
pp8-eta))))))
(defthm booth8-corollary_alt
(implies (and (not (zp n))
(not (zp m))
(bvecp x (- n 2))
(bvecp y (1- (* 3 m))))
(= (+ (expt 2 n)
(sum-pp8_alt-eta_alt x y m n))
(+ (expt 2 (+ n (* 3 m)))
(* x y))))
:rule-classes ()
:hints (("Goal" :use ((:instance booth8-corollary)))))
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