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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 "round")
(local (include-book "add-new"))
(local
(defthm bits-is-bits_alt
(equal (bits x i j)
(bits_alt x i j))
:hints (("Goal" :in-theory (e/d (bits_alt bits) ())))))
(local
(defthm bitn-is-bitn_alt
(equal (bitn x n)
(bitn_alt x n))
:hints (("Goal" :in-theory (e/d (bitn_alt bitn) ())))))
(local
(defthm binary-cat_alt-is-binary-cat
(equal (binary-cat x m y n)
(binary-cat_alt x m y n))
:hints (("Goal" :in-theory (e/d (binary-cat_alt binary-cat) ())))))
(local
(defthm mulcat_alt-is-mulcat
(equal (mulcat l n x)
(mulcat_alt l n x))
:hints (("Goal" :in-theory (e/d (mulcat_alt mulcat) ())))))
;;;**********************************************************************
;;; Bit Vector Addition
;;;**********************************************************************
(defthm half-adder
(implies (and (bvecp u 1)
(bvecp v 1))
(equal (+ u v)
(cat (logand u v) 1 (logxor u v) 1)))
:rule-classes ()
:hints (("Goal" :use ((:instance half-adder_alt)))))
(defthm add-2
(implies (and (natp x) (natp y))
(equal (+ x y)
(+ (logxor x y)
(* 2 (logand x y)))))
:rule-classes ()
:hints (("Goal" :use ((:instance add-2_alt)))))
(defthm full-adder
(implies (and (bvecp u 1)
(bvecp v 1)
(bvecp w 1))
(equal (+ u v w)
(cat (logior (logand u v) (logior (logand u w) (logand v w))) 1
(logxor u (logxor v w)) 1)))
:rule-classes ()
:hints (("Goal" :use ((:instance full-adder_alt)))))
(defthm add-3
(implies (and (natp x)
(natp y)
(natp z))
(= (+ x y z)
(+ (logxor x (logxor y z))
(* 2 (logior (logand x y)
(logior (logand x z)
(logand y z)))))))
:rule-classes ()
:hints (("Goal" :use ((:instance add-3_alt)))))
(defun rc-carry (x y k)
(if (zp k)
0
(logior (logand (bitn x (1- k)) (bitn y (1- k)))
(logior (logand (bitn x (1- k)) (rc-carry x y (1- k)))
(logand (bitn y (1- k)) (rc-carry x y (1- k)))))))
(local
(defthm rc-carry-is-rc-carry_alt
(equal (rc-carry x y k)
(rc-carry_alt x y k))
:hints (("Goal" :induct (rc-carry x y k)))))
(defun rc-sum (x y k)
(if (zp k)
0
(cat (logxor (bitn x (1- k))
(logxor (bitn y (1- k)) (rc-carry x y (1- k))))
1
(rc-sum x y (1- k))
(1- k))))
(local
(defthm rc-sum-is-rc-sum_alt
(equal (rc-sum x y k)
(rc-sum_alt x y k))
:hints (("Goal" :induct (rc-sum x y k)))))
(defthm ripple-carry
(implies (and (natp x)
(natp y)
(natp n))
(equal (+ (bits x (1- n) 0) (bits y (1- n) 0))
(cat (rc-carry x y n) 1 (rc-sum x y n) n)))
:rule-classes ()
:hints (("Goal" :use ((:instance ripple-carry_alt)))))
(defun gen (x y i j)
(declare (xargs :measure (nfix (1+ i))))
(if (and (natp i) (natp j) (>= i j))
(if (= (bitn x i) (bitn y i))
(bitn x i)
(gen x y (1- i) j))
0))
(local
(defthm gen-is-gen_alt
(equal (gen x y i j)
(gen_alt x y i j))))
(defun prop (x y i j)
(declare (xargs :measure (nfix (1+ i))))
(if (and (natp i) (natp j) (>= i j))
(if (= (bitn x i) (bitn y i))
0
(prop x y (1- i) j))
1))
(local
(defthm prop-is-prop_alt
(equal (prop x y i j)
(prop_alt x y i j))))
(defthm bvecp-1-gen
(bvecp (gen x y i j) 1)
:rule-classes (:rewrite
(:forward-chaining :trigger-terms ((gen x y i j)))))
(defthm bvecp-1-prop
(bvecp (prop x y i j) 1)
:rule-classes (:rewrite
(:forward-chaining :trigger-terms ((prop x y i j)))))
(defthmd gen-val
(implies (and (natp j) (>= i j))
(equal (gen x y i j)
(if (>= (+ (bits x i j) (bits y i j))
(expt 2 (1+ (- i j))))
1
0)))
:hints (("Goal" :use ((:instance gen_alt-val)))))
(defthmd gen-val-cor1
(implies (natp j)
(equal (gen x y i j)
(bitn (+ (bits x i j) (bits y i j))
(1+ (- i j)))))
:hints (("Goal" :use ((:instance gen_alt-val-cor1)))))
(defthmd gen-val-cor2
(implies (and (natp x)
(natp y)
(natp i))
(equal (+ (bits x i 0) (bits y i 0))
(+ (* (expt 2 (1+ i)) (gen x y i 0))
(bits (+ x y) i 0))))
:hints (("Goal" :use ((:instance gen_alt-val-cor2)))))
(defthm gen-special-case
(implies (and (integerp i)
(integerp j)
(>= i j)
(>= j 0)
(= (bitn (+ (bits x i j) (bits y i j)) (- i j)) 0))
(equal (gen x y i j)
(logior (bitn x i) (bitn y i))))
:rule-classes ()
:hints (("Goal" :use ((:instance gen_alt-special-case)))))
(defthmd prop-val
(implies (and (integerp i) (natp j) (>= i j))
(equal (prop x y i j)
(if (= (+ (bits x i j) (bits y i j))
(1- (expt 2 (1+ (- i j)))))
1
0)))
:hints (("Goal" :use ((:instance prop_alt-val)))))
(defthmd prop-as-logxor
(implies (and (natp i)
(natp j)
(<= j i)
(natp x)
(natp y))
(equal (prop x y i j)
(if (equal (logxor (bits x i j) (bits y i j))
(1- (expt 2 (1+ (- i j)))))
1
0)))
:hints (("Goal" :use ((:instance prop_alt-as-logxor)))))
(defthm gen-extend
(implies (and (integerp i)
(integerp j)
(integerp k)
(> i k)
(>= k j)
(>= j 0))
(equal (gen x y i j)
(logior (gen x y i (1+ k))
(logand (prop x y i (1+ k))
(gen x y k j)))))
:rule-classes ()
:hints (("Goal" :use ((:instance gen_alt-extend)))))
(defthm gen-extend-cor
(implies (and (natp x)
(natp y)
(natp i)
(natp j)
(natp k)
(> i k)
(>= k j))
(equal (gen x y i j)
(bitn (+ (bits x i (1+ k))
(bits y i (1+ k))
(gen x y k j))
(- i k))))
:rule-classes ()
:hints (("Goal" :use ((:instance gen_alt-extend-cor)))))
(defthm prop-extend
(implies (and (integerp i)
(integerp j)
(integerp k)
(> i k)
(>= k j)
(>= j 0))
(equal (prop x y i j)
(logand (prop x y i (1+ k))
(prop x y k j))))
:rule-classes ()
:hints (("Goal" :use ((:instance prop_alt-extend)))))
(defthm bits-sum
(implies (and (integerp x) (integerp y))
(equal (bits (+ x y) i j)
(bits (+ (bits x i j)
(bits y i j)
(gen x y (1- j) 0))
(- i j) 0)))
:rule-classes ()
:hints (("Goal" :use ((:instance bits_alt-sum)))))
(defthmd bits-sum-swallow
(implies (and (equal (bitn x k) 0)
(natp x)
(natp y)
(integerp i)
(integerp j)
(integerp k)
(>= i j)
(> j k)
(>= k 0)
(<= y (expt 2 k)))
(equal (bits (+ x y) i j)
(bits x i j)))
:hints (("Goal" :use ((:instance bits_alt-sum-swallow)))))
(defthmd bits-sum-cor
(implies (and (integerp x)
(integerp y)
(>= i j)
(>= j 0)
(= (gen x y i j) 0)
(= (gen x y (1- j) 0) 0))
(equal (bits (+ x y) i j)
(+ (bits x i j) (bits y i j))))
:hints (("Goal" :use ((:instance bits_alt-sum-cor)))))
(defthm bits-sum-3
(implies (and (integerp x) (integerp y) (integerp z))
(equal (bits (+ x y z) i j)
(bits (+ (bits x i j)
(bits y i j)
(bits z i j)
(gen x y (1- j) 0)
(gen (+ x y) z (1- j) 0))
(- i j) 0)))
:rule-classes ()
:hints (("Goal" :use ((:instance bits_alt-sum-3)))))
(defthm bits-sum-plus-1
(implies (and (integerp x)
(integerp y)
(integerp i)
(integerp j)
(>= i j)
(>= j 0))
(equal (bits (+ 1 x y) i j)
(bits (+ (bits x i j)
(bits y i j)
(logior (prop x y (1- j) 0)
(gen x y (1- j) 0) ))
(- i j) 0)))
:rule-classes ()
:hints (("Goal" :use ((:instance bits_alt-sum-plus-1)))))
(defthmd logand-gen-0
(implies (and (integerp i)
(integerp j)
(>= i j)
(>= j 0)
(= (logand (bits x i j) (bits y i j)) 0))
(equal (gen x y i j) 0))
:hints (("Goal" :use ((:instance logand-gen_alt-0)))))
(defthm logand-gen-0-cor
(implies (and (integerp x)
(integerp y)
(integerp i)
(integerp j)
(>= i j)
(>= j 0)
(= (logand x y) 0))
(equal (bits (+ x y) i j)
(+ (bits x i j) (bits y i j))))
:rule-classes ()
:hints (("Goal" :use ((:instance logand-gen_alt-0-cor)))))
(defthmd gen-plus
(implies (and (natp x)
(natp y)
(natp k)
(bvecp z (1+ k))
(= (logand z y) 0)
(= (gen x y k 0) 1))
(equal (gen (+ x y) z k 0) 0))
:hints (("Goal" :use ((:instance gen_alt-plus)))))
(defthmd gen-extend-3
(implies (and (natp i)
(natp j)
(> i j)
(natp x)
(natp y)
(bvecp z (1+ j))
(= (logand y z) 0))
(equal (gen (+ x y) z i 0)
(logand (prop x y i (1+ j))
(gen (+ x y) z j 0))))
:hints (("Goal" :use ((:instance gen_alt-extend-3)))))
;;;**********************************************************************
;;; Leading One Prediction
;;;**********************************************************************
(defund lop (a b d k)
(let ((c (- (bitn a (1- k)) (bitn b (1- k)))))
(if (and (integerp k) (>= k 0))
(if (= k 0)
0
(if (= d 0)
(lop a b c (1- k))
(if (= d (- c))
(lop a b (- c) (1- k))
k)))
0)))
(local
(defthm lop-is-lop
(equal (lop a b d k)
(lop_alt a b d k))
:hints (("Goal" :in-theory (e/d (lop lop_alt) ())))))
(defthm lop-bnds
(implies (and (integerp a)
(integerp b)
(integerp n)
(>= a 0)
(>= b 0)
(>= n 0)
(not (= a b))
(< a (expt 2 n))
(< b (expt 2 n)))
(or (= (lop a b 0 n) (expo (- a b)))
(= (lop a b 0 n) (1+ (expo (- a b))))))
:rule-classes ()
:hints (("Goal" :use ((:instance lop_alt-bnds)))))
(defthm lop-thm-1
(implies (and (integerp a)
(> a 0)
(integerp b)
(> b 0)
(= e (expo a))
(< (expo b) e)
(= lambda
(logior (* 2 (mod a (expt 2 e)))
(bits (lognot (* 2 b)) e 0))))
(or (= (expo (- a b)) (expo lambda))
(= (expo (- a b)) (1- (expo lambda)))))
:rule-classes ()
:hints (("Goal" :use ((:instance lop_alt-thm-1)))))
;;;
;;;
;;; We need set of theorem about how lxor is equal logxor
;;;
;;; land is logand
;;;
;;
(defun lamt (a b e)
(logxor a (bits (lognot b) e 0)))
(local
(defthm lamt-is-lamt_alt
(equal (lamt a b e)
(lamt_alt a b e))
:hints (("Goal" :in-theory (e/d (lamt lamt_alt) ())))))
(defun lamg (a b e)
(logand a (bits (lognot b) e 0)))
(local
(defthm lamg-is-lamg_alt
(equal (lamg a b e)
(lamg_alt a b e))
:hints (("Goal" :in-theory (e/d (lamg lamg_alt) ())))))
(defun lamz (a b e)
(bits (lognot (logior a (bits (lognot b) e 0))) e 0))
(local
(defthm lamz-is-lamz_alt
(implies (and (natp e)
(integerp a)
(integerp b))
(equal (lamz a b e)
(lamz_alt a b e)))
:hints (("Goal" :in-theory (e/d (lamz lamz_alt) ())))))
(defun lam1 (a b e)
(logand (bits (lamt a b e) e 2)
(logand (bits (lamg a b e) (1- e) 1)
(bits (lognot (lamz a b e)) (- e 2) 0))))
(local
(defthm lam1-is-lam1_alt
(implies (and (integerp a)
(integerp b)
(natp e))
(equal (lam1 a b e)
(lam1_alt a b e)))
:hints (("Goal" :in-theory (e/d (lam1 lam1_alt) ())))))
(defun lam2 (a b e)
(logand (bits (lognot (lamt a b e)) e 2)
(logand (bits (lamz a b e) (1- e) 1)
(bits (lognot (lamz a b e)) (- e 2) 0))))
(local
(defthm lam2-is-lam2_alt
(implies (and (integerp a)
(integerp b)
(natp e))
(equal (lam2 a b e)
(lam2_alt a b e)))
:hints (("Goal" :in-theory (e/d (lam2 lam2_alt) ())))))
(defun lam3 (a b e)
(logand (bits (lamt a b e) e 2)
(logand (bits (lamz a b e) (1- e) 1)
(bits (lognot (lamg a b e)) (- e 2) 0))))
(local
(defthm lam3-is-lam3_alt
(implies (and (integerp a)
(integerp b)
(natp e))
(equal (lam3 a b e)
(lam3_alt a b e)))
:hints (("Goal" :in-theory (e/d (lam3 lam3_alt) ())))))
(defun lam4 (a b e)
(logand (bits (lognot (lamt a b e)) e 2)
(logand (bits (lamg a b e) (1- e) 1)
(bits (lognot (lamg a b e)) (- e 2) 0))))
(local
(defthm lam4-is-lam4_alt
(implies (and (integerp a)
(integerp b)
(natp e))
(equal (lam4 a b e)
(lam4_alt a b e)))
:hints (("Goal" :in-theory (e/d (lam4 lam4_alt) ())))))
(defun lam0 (a b e)
(logior (lam1 a b e)
(logior (lam2 a b e)
(logior (lam3 a b e)
(lam4 a b e)))))
(local
(defthm lam0-is-lam0_alt
(implies (and (integerp a)
(integerp b)
(natp e)
(> e 1))
(equal (lam0 a b e)
(lam0_alt a b e)))
:hints (("Goal" :in-theory (e/d (lam0 lam0_alt)
())))))
(defun lamb (a b e)
(+ (* 2 (lam0 a b e))
(bitn (lognot(lamt a b e)) 0)))
(local
(defthm lamb-is-lamb
(implies (and (integerp a)
(integerp b)
(natp e)
(> e 1))
(equal (lamb a b e)
(lamb_alt a b e)))
:hints (("Goal" :in-theory (e/d (lamb lamb_alt)
())))))
(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 ()
:hints (("Goal" :use ((:instance lop_alt-thm-2))
:in-theory (e/d ()
(lamb
lamb_alt)))))
;;;**********************************************************************
;;; Trailing One Prediction
;;;**********************************************************************
(defthm top-thm-1
(implies (and (natp n)
(natp k)
(< k n)
(integerp a)
(integerp b))
(equal (equal (bits (+ a b 1) k 0) 0)
(equal (bits (lognot (logxor a b)) k 0) 0)))
:rule-classes ()
:hints (("Goal" :use ((:instance top-thm-1-alt)))))
(defthm top-thm-2
(implies (and (natp n)
(integerp a)
(integerp b)
(natp k)
(< k n)
(or (equal c 0) (equal c 1)))
(equal (equal (bits (+ a b c) k 0) 0)
(equal (bits (logxor (logxor a b)
(cat (logior a b) n c 1))
k 0)
0)))
:rule-classes ()
:hints (("Goal" :use ((:instance top-thm-2-alt)))))
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