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|
;; Mayank Manjrekar <mankmonjre@gmail.com>
;; June 2023
(in-package "RTL")
(include-book "rtl/rel11/lib/top-alt" :dir :system)
(local (include-book "arithmetic-5/top" :dir :system))
(local
(in-theory
#!acl2(disable |(mod (+ x y) z) where (<= 0 z)|
|(mod (+ x (- (mod a b))) y)|
|(mod (mod x y) z)|
|(mod (+ x (mod a b)) y)|
mod-cancel-*-const
cancel-mod-+
reduce-additive-constant-<
ash-to-floor
|(floor x 2)|
|(equal x (if a b c))|
|(equal (if a b c) x)|
|(logior 1 x)|
mod-theorem-one-b
|(mod (- x) y)|)))
(defevaluator bits-evl bits-evl-lst
((bits x i j)
(binary-+ x y)
(if x y z)
(integerp x)
(not x)
(< x y)
(unary-- x)
(unary-/ x)
(binary-* x y)
(bvecp x k)
(expt x y)
(mod x y)
(floor x y)
(fl x)))
(set-state-ok t)
(set-irrelevant-formals-ok t)
(defun find-bits-tail-in-sum-fn1 (sum i mfc state)
(declare (xargs :guard (pseudo-termp sum)))
;; Find an addend of the form (bits x k 0) in a summation "sum", such that
;; k>=i. If the there is such a addend, the return value is a multiple value
;; containing the same same summation with x in place of (bits x k 0), x, k,
;; and the sign with which (bits x k 0) appears in the summation (nil for +ve
;; and t for -ve).
(cond ((and (consp sum)
(eql (car sum) 'binary-+))
(b* ((laddend (cadr sum))
(raddend (caddr sum))
((mv lterm x k sgn) (find-bits-tail-in-sum-fn1 laddend i mfc state))
((when x) (mv (list 'binary-+ lterm raddend) x k sgn))
((mv rterm x k sgn) (find-bits-tail-in-sum-fn1 raddend i mfc state))
((when x) (mv (list 'binary-+ laddend rterm) x k sgn)))
(mv sum nil nil nil)))
((and (consp sum)
(eql (car sum) 'unary--))
(mv-let (sum x k sgn)
(find-bits-tail-in-sum-fn1 (cadr sum) i mfc state)
(mv `(unary-- ,sum) x k (not sgn))))
((and (consp sum)
(eql (car sum) 'bits))
(b* ((x (cadr sum))
(k (caddr sum))
(j (cadddr sum)))
(if (and (quotep j)
(eql (unquote j) '0)
(equal (mfc-rw `(if (integerp ,k) (if (integerp ,x) (not (< ,k ,i)) 'nil) 'nil) t t mfc state) acl2::*t*))
(mv x x k nil)
(mv sum nil nil nil))))
(t (mv sum nil nil nil))))
(defun bits-bits-sum-meta-fn (term mfc state)
(declare (xargs :guard (pseudo-termp term)))
;; Check if term is of the form (bits *summation* i j), and if so, simplify
;; the addends of the form (bits x k 0), k>=i.
(if (and (consp term)
(consp (cdr term))
(eql (car term) 'bits)
(consp (cddr term))
(consp (cdddr term))
(not (cddddr term)))
(b* ((sum (cadr term))
(i (caddr term))
(j (cadddr term))
((unless (equal (mfc-rw `(if (integerp ,i)
(if (integerp ,j)
(not (< ,j '0))
'nil)
'nil) t t mfc state) acl2::*t*))
term)
((mv new-sum x k ?sgn) (find-bits-tail-in-sum-fn1 sum i mfc state)))
(if x
`(if (if (integerp ,new-sum)
(if (integerp ,i)
(if (integerp ,j)
(if (not (< ,j '0))
(if (integerp ,k)
(if (integerp ,x)
(not (< ,k ,i))
'nil)
'nil)
'nil)
'nil)
'nil)
'nil)
(bits ,new-sum ,i ,j)
,term)
term))
term))
(local-defthmd bits-bits-sum-bf-aux-1
(b* (((mv term x k sgn) (find-bits-tail-in-sum-fn1 sum i mfc state)))
(implies (acl2-numberp (bits-evl x a))
(equal (bits-evl term a)
(if sgn
(+ (bits-evl sum a)
(- (bits (bits-evl x a) (bits-evl k a) 0) (bits-evl x a)))
(+ (bits-evl sum a)
(- (bits-evl x a) (bits (bits-evl x a) (bits-evl k a) 0)))))))
:hints (("Goal" :in-theory (e/d () ()))))
(local-defthmd mod-minus
(implies (and (integerp x)
(rationalp l))
(equal (mod (- x) l)
(if (equal (mod x l) 0)
0
(- l (mod x l)))))
:hints (("Goal" :in-theory (e/d () ())
:use ((:instance mod-def
(x (- x)) (y l))
(:instance mod-def
(x x) (y l))
(:instance minus-fl
(x (/ x l)))))))
(local-defthmd bits-bits-sum-lemma-1
(implies (and (integerp x)
(integerp y)
(integerp k)
(integerp i)
(integerp j)
(>= j 0)
(>= k i))
(equal (bits (+ y (- x (bits x k 0))) i j)
(bits y i j)))
:hints (("Goal" :in-theory (e/d (bits mod)
((:DEFINITION NOT)
(:REWRITE ACL2::|(* (if a b c) x)|)
(:REWRITE ACL2::|(+ x (if a b c))|)
(:REWRITE ACL2::|(- (if a b c))|)
(:REWRITE ACL2::|(floor (+ x y) z) where (< 0 z)| . 1)
(:REWRITE ACL2::|(floor (- x) y)| . 1))))))
(local-defthmd bits-bits-sum-lemma-2
(implies (and (integerp x)
(integerp y)
(integerp k)
(integerp i)
(integerp j)
(>= j 0)
(>= k i))
(equal (bits (+ y (- (bits x k 0) x)) i j)
(bits y i j)))
:hints (("Goal" :in-theory (e/d (bits mod)
((:DEFINITION NOT)
(:REWRITE ACL2::|(* (if a b c) x)|)
(:REWRITE ACL2::|(+ x (if a b c))|)
(:REWRITE ACL2::|(- (if a b c))|)
(:REWRITE ACL2::|(floor (+ x y) z) where (< 0 z)| . 1)
(:REWRITE ACL2::|(floor (- x) y)| . 1))))))
(local-defthm not-numberp-bits
(implies (not (acl2-numberp x))
(and (equal (bits x i j) 0)
(equal (bits (- x) i j) 0)))
:hints (("Goal" :in-theory (e/d (bits) ()))))
(local-defthm acl2-numberp-new-term
(implies (acl2-numberp (bits-evl (mv-nth 1 (find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(equal (acl2-numberp (bits-evl (mv-nth 0 (find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(acl2-numberp (bits-evl sum a))))
:hints (("Goal" :in-theory (e/d () ()))))
(local-defthm aux
(implies (not x)
(equal (bits-evl x a) nil))
:hints (("Goal" :in-theory (e/d () ()))))
(local-defthm aux-1
(implies (acl2-numberp x)
(equal (integerp (- x))
(integerp x)))
:hints (("Goal" :in-theory (e/d () ())))
:rule-classes nil)
(local-defthm aux-2
(implies (acl2-numberp (- x y))
(equal (integerp (- x y))
(integerp (- y x))))
:hints (("Goal" :use ((:instance aux-1
(x (- x y))))))
:rule-classes nil)
(local-defthmd integerp-new-term
(implies (integerp (bits-evl (mv-nth 1 (find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(integerp (- (bits-evl sum a) (bits-evl (mv-nth 0 (find-bits-tail-in-sum-fn1 sum i mfc state)) a))))
:hints (("Goal" :in-theory (e/d () ()))
("Subgoal *1/24" :use (:instance aux-2
(x (BITS-EVL (MV-NTH 0
(FIND-BITS-TAIL-IN-SUM-FN1 (CADR SUM)
I MFC STATE))
A))
(y (BITS-EVL (CADR SUM) A))))))
(local-defthmd integerp-new-term-alt
(implies (integerp (bits-evl (mv-nth 1 (find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(equal (integerp (bits-evl (mv-nth 0 (find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(integerp (bits-evl sum a))))
:hints (("Goal" :in-theory (e/d () (find-bits-tail-in-sum-fn1
acl2-numberp-new-term))
:use (integerp-new-term
acl2-numberp-new-term))))
(local-defthmd bits-bits-sum-bf-aux
(b* (((mv new-term x k ?sgn) (find-bits-tail-in-sum-fn1 sum i mfc state)))
(implies (and x
(integerp (bits-evl x a))
(integerp (bits-evl new-term a))
(integerp (bits-evl i a))
(integerp (bits-evl j a))
(integerp (bits-evl k a))
(<= 0 (bits-evl j a))
(<= (bits-evl i a)
(bits-evl k a)))
(equal (bits (bits-evl new-term a)
(bits-evl i a)
(bits-evl j a))
(bits (bits-evl sum a)
(bits-evl i a)
(bits-evl j a)))))
:hints (("Goal" :in-theory (e/d (integerp-new-term-alt)
(find-bits-tail-in-sum-fn1
(:REWRITE ACL2::DEFAULT-PLUS-1)
(:REWRITE ACL2::DEFAULT-PLUS-2)
(:REWRITE ACL2::DEFAULT-MINUS)))
:use (bits-bits-sum-bf-aux-1
(:instance bits-bits-sum-lemma-1
(y (bits-evl sum a))
(x (bits-evl (mv-nth 1
(find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(k (bits-evl (mv-nth 2
(find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(i (bits-evl i a))
(j (bits-evl j a)))
(:instance bits-bits-sum-lemma-2
(y (bits-evl sum a))
(x (bits-evl (mv-nth 1
(find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(k (bits-evl (mv-nth 2
(find-bits-tail-in-sum-fn1 sum i mfc state)) a))
(i (bits-evl i a))
(j (bits-evl j a)))))))
;; Meta rule to automatically handle simplifications performed by
;; bits-bits-sum, bits-bits-sum-alt, and bits-bits-diff
(defthmd bits-bits-sum-meta
(implies (and (pseudo-termp x)
(alistp a))
(equal (bits-evl x a)
(bits-evl (bits-bits-sum-meta-fn x mfc state) a)))
:hints (("Goal" :in-theory (e/d (bits-bits-sum-bf-aux) ())
:do-not-induct t))
:rule-classes ((:meta :trigger-fns (bits))))
#|
;; For example:
(thm
(implies (and (integerp x)
(integerp y)
(integerp z)
(integerp n)
(natp i)
(>= n i))
(equal (bits (+ z (- (bits x 3 0) (bits y n 0))) i 0)
(bits (+ z (bits x 3 0) (- y)) i 0)))
:hints (("Goal" :in-theory (e/d (bits-bits-sum-meta) ()))))
|#
;;======================================================================
(defun find-power-of-2-gt-n (prod n mfc state)
(declare (xargs :guard (and (pseudo-termp prod)
(pseudo-termp n))))
(cond ((and (consp prod)
(eql (car prod) 'binary-*))
(b* ((left (cadr prod))
(right (caddr prod))
((mv kl yl) (find-power-of-2-gt-n left n mfc state))
((when kl) (mv kl `(binary-* ,yl ,right)))
((mv kr yr) (find-power-of-2-gt-n right n mfc state))
((when kr) (mv kr `(binary-* ,left ,yr))))
(mv nil prod)))
((and (consp prod)
(eql (car prod) 'expt)
(equal (cadr prod) ''2)
(equal (mfc-rw `(< ,n ,(caddr prod)) t t mfc state) acl2::*t*))
(mv (caddr prod) ''1))
((and (quotep prod)
(rationalp (unquote prod))
(equal (mfc-rw `(not (equal (fl (binary-* ,prod (unary-/ (expt '2 (binary-+ '1 ,n))))) '0))
t t mfc state) acl2::*t*))
(mv `(binary-+ '1 ,n) `(fl (binary-* ,prod (unary-/ (expt '2 (binary-+ '1 ,n)))))))
(t (mv nil prod))))
(defun remove-addend-gt-n (sum n mfc state)
(declare (xargs :guard (and (pseudo-termp sum)
(pseudo-termp n))
:guard-debug t))
(cond ((and (consp sum)
(equal (car sum) 'binary-+))
(b* ((left (cadr sum))
(right (caddr sum))
((mv term pow factor sgn) (remove-addend-gt-n left n mfc state))
((when pow) (mv `(binary-+ ,term ,right) pow factor sgn))
((mv term pow factor sgn) (remove-addend-gt-n right n mfc state))
((when pow) (mv `(binary-+ ,left ,term) pow factor sgn)))
(mv sum nil nil nil)))
((and (consp sum)
(equal (car sum) 'unary--))
(b* (((mv term pow factor sgn)
(remove-addend-gt-n (cadr sum) n mfc state)))
(mv `(unary-- ,term) pow factor (not sgn))))
(t (b* (((mv pow factor) (find-power-of-2-gt-n sum n mfc state)))
(if (and pow
(mfc-rw `(integerp ,factor) t t mfc state))
(mv `(binary-+ ,sum (unary-- (binary-* ,factor (expt '2 ,pow)))) pow factor nil)
(mv sum nil nil nil))))))
(defun bits-plus-mult-2-fn (term mfc state)
(declare (xargs :guard (pseudo-termp term)))
(cond ((and (consp term)
(eql (car term) 'bits))
(b* ((n (caddr term))
(m (cadddr term))
(sum (cadr term))
((mv new-sum k y ?sgn) (remove-addend-gt-n sum n mfc state)))
(if k
`(if (if (< ,n ,k)
(if (integerp ,y)
(integerp ,k)
'nil)
'nil)
(bits ,new-sum ,n ,m)
,term)
term)))
(t term)))
(local-defthm not-consp-sum-power
(implies (not (consp sum))
(not (mv-nth 0 (find-power-of-2-gt-n sum n mfc state)))))
(local-defthmd no-pow-when-sum-not-acl2-numberp
(implies (not (acl2-numberp (bits-evl sum a)))
(not (mv-nth 0 (find-power-of-2-gt-n sum n mfc state)))))
(local-defthmd remove-addend-gt-n-lemma
(b* (((mv new-sum k factor sgn) (remove-addend-gt-n sum n mfc state)))
(implies k
(equal (bits-evl sum a)
(+ (bits-evl new-sum a) (* (if sgn -1 1)
(expt 2 (bits-evl k a))
(bits-evl factor a))))))
:hints (("Goal" :in-theory (e/d (no-pow-when-sum-not-acl2-numberp)
()))))
(local-defthm bits-idx-not-numberp
(implies (or (not (acl2-numberp i))
(not (acl2-numberp j))
(not (acl2-numberp x)))
(equal (bits x i j) 0))
:hints (("Goal" :in-theory (e/d (bits) ()))))
(defthmd bits-plus-mult-2-meta
(implies (and (pseudo-termp x)
(alistp a))
(equal (bits-evl x a)
(bits-evl (bits-plus-mult-2-fn x mfc state) a)))
:hints (("Goal" :do-not-induct t
:use ((:instance remove-addend-gt-n-lemma
(sum (cadr x))
(n (caddr x)))
(:instance bits-plus-mult-2
(n (bits-evl (caddr x) a))
(m (bits-evl (cadddr x) a))
(y (* (if (mv-nth 3 (remove-addend-gt-n (cadr x) (caddr x) mfc state)) -1 1)
(bits-evl (mv-nth 2 (remove-addend-gt-n (cadr x) (caddr x) mfc state)) a)))
(k (bits-evl (mv-nth 1 (remove-addend-gt-n (cadr x) (caddr x) mfc state)) a))
(x (bits-evl (mv-nth 0 (remove-addend-gt-n (cadr x) (caddr x) mfc state)) a))))))
:rule-classes ((:meta :trigger-fns (bits))))
#|
For example:
(thm
(implies (integerp z)
(EQUAL (BITS (+ X (* 513/2 z)) 7 1)
(BITS (+ X (* 1/2 z)) 7 1)))
:hints (("Goal" :in-theory (e/d (bits-plus-mult-2-meta) ()))))
(thm
(implies (and (integerp k)
(integerp y)
(< n k))
(equal (bits (+ x (* (expt 2 k) y) (* (expt 2 l) z)) n m)
(bits (+ x (* (expt 2 l) z)) n m)))
:hints (("Goal" :in-theory (e/d (bits-plus-mult-2-meta) ()))))
|#
;;======================================================================
(defun addend-count (sum)
(declare (xargs :guard (pseudo-termp sum)))
(cond ((and (consp sum)
(equal (car sum) 'binary-+))
(+ (addend-count (cadr sum)) (addend-count (caddr sum))))
((and (consp sum)
(equal (car sum) 'unary--))
(addend-count (cadr sum)))
(t 1)))
(defun find-power-of-2-le-m (sum m gsum mfc state)
(declare (xargs :guard (and (pseudo-termp sum)
(pseudo-termp gsum)
(pseudo-termp m))))
(cond ((and (consp sum)
(or (eql (car sum) 'binary-+)
(eql (car sum) 'binary-*)))
(b* ((left (cadr sum))
(right (caddr sum))
((mv pow rem) (find-power-of-2-le-m left m gsum mfc state))
((when pow) (mv pow rem)))
(find-power-of-2-le-m right m gsum mfc state)))
((and (consp sum)
(eql (car sum) 'unary--))
(find-power-of-2-le-m (cadr sum) m gsum mfc state))
((and (consp sum)
(eql (car sum) 'expt)
(equal (cadr sum) ''2)
(equal (mfc-rw `(not (< ,m ,(caddr sum))) t t mfc state) acl2::*t*))
(b* ((term `(bits ,gsum (binary-+ '-1 ,(caddr sum)) '0))
(rem (mfc-rw term 'acl2::? nil mfc state))
;; subtracting rem and simplifying should result in fewer terms
(simp (mfc-rw `(binary-+ ,gsum (unary-- ,rem)) 'acl2::? nil mfc state)))
(if (and (pseudo-termp simp)
(<= (addend-count simp)
(addend-count gsum)))
(mv (caddr sum) rem)
(mv nil nil))))
((and (quotep sum)
(rationalp (unquote sum))
(equal (mfc-rw `(not (< ,m ,(kwote (expo (unquote sum))))) t t mfc state) acl2::*t*))
(b* ((term `(bits ,gsum (binary-+ '-1 ,(kwote (expo (unquote sum)))) '0))
(rem (mfc-rw term 'acl2::? nil mfc state))
;; subtracting rem and simplifying should result in fewer terms
(simp (mfc-rw `(binary-+ ,gsum (unary-- ,rem)) 'acl2::? nil mfc state)))
(if (and (pseudo-termp simp)
(<= (addend-count simp)
(addend-count gsum)))
(mv (caddr sum) rem)
(mv nil nil))))
(t (mv nil nil))))
(defun bits-plus-mult-1-meta-fn (term mfc state)
(declare (xargs :guard (pseudo-termp term)))
(cond ((and (consp term)
(eql (car term) 'bits))
(b* ((n (caddr term))
(m (cadddr term))
(sum (cadr term))
((mv pow rem) (find-power-of-2-le-m sum m sum mfc state))
(y `(binary-* (expt '2 (unary-- ,pow)) (binary-+ ,sum (unary-- ,rem)))))
(if pow
`(if (if (bvecp ,rem ,pow)
(if (not (< ,m ,pow))
(if (integerp ,y)
(if (integerp ,m)
(if (integerp ,n)
(integerp ,pow)
'nil)
'nil)
'nil)
'nil)
'nil)
(bits ,y
(binary-+ ,n (unary-- ,pow))
(binary-+ ,m (unary-- ,pow)))
,term)
term)))
(t term)))
(defthmd bits-plus-mult-1-meta-aux
(implies (and (pseudo-termp x)
(alistp a))
(equal (bits-evl x a)
(bits-evl (bits-plus-mult-1-meta-fn x mfc state) a)))
:hints (("Goal" :in-theory (e/d ()
((:REWRITE ACL2::DEFAULT-EXPT-2)
(:REWRITE ACL2::DEFAULT-LESS-THAN-1)
(:REWRITE ACL2::DEFAULT-LESS-THAN-2)
(:REWRITE ACL2::DEFAULT-MINUS)
(:REWRITE ACL2::DEFAULT-PLUS-1)
(:REWRITE ACL2::DEFAULT-TIMES-1)))
:use ((:instance bits-plus-mult-1
(y (* (EXPT 2 (- (BITS-EVL (MV-NTH 0 (FIND-POWER-OF-2-LE-M (CADR X) (CADDDR X) (CADR X) MFC STATE)) A)))
(- (BITS-EVL (CADR X) A)
(BITS-EVL (MV-NTH 1 (FIND-POWER-OF-2-LE-M (CADR X) (CADDDR X) (CADR X) MFC STATE)) A))))
(k (BITS-EVL (MV-NTH 0 (FIND-POWER-OF-2-LE-M (CADR X) (CADDDR X) (CADR X) MFC STATE)) A))
(n (BITS-EVL (CADDR X) A))
(m (bits-evl (CADDDR X) A))
(x (BITS-EVL (MV-NTH 1 (FIND-POWER-OF-2-LE-M (CADR X) (CADDDR X) (CADR X) MFC STATE)) A))))
:do-not-induct t))
:rule-classes ((:meta :trigger-fns (bits))))
;; bits-plus-mult-1-meta-aux works when bits-plus-mult-2-meta is enabled. Note:
;; bits-plus-mult-1-meta also simplifies expressions simplified by the the
;; rewrite rule bits-shift-up-1.
(deftheory bits-plus-mult-1-meta
'(bits-plus-mult-2-meta bits-plus-mult-1-meta-aux))
#|
For example:
(thm
(IMPLIES (AND (integerp l)
(<= K M)
(bvecp (+ x (* (expt 2 l) z)) k)
(INTEGERP Y)
(CASE-SPLIT (INTEGERP M))
(CASE-SPLIT (INTEGERP N))
(CASE-SPLIT (natp K)))
(EQUAL (BITS (+ x (* Y (EXPT 2 K)) (* (expt 2 l) z)) N M)
(BITS Y (- N K) (- M K))))
:hints (("Goal" :in-theory (e/d (bits-plus-mult-1-meta) ()))))
|#
;;======================================================================
(mutual-recursion
(defun bits-shatter-contains-term (cls x n m)
(declare (xargs :guard (pseudo-termp cls)))
(if (or (atom cls) (quotep cls))
nil
(if (and (eql (car cls) 'bits)
(quotep (caddr cls))
(quotep (cadddr cls))
(equal (cadr cls) x))
(b* ((h (unquote (caddr cls)))
(l (unquote (cadddr cls))))
(if (and (equal h n)
(integerp l)
(integerp m)
(> l m))
(cons (cons 'p (kwote l)) nil)
(if (and (equal l m)
(integerp n)
(integerp h)
(> n h))
(cons (cons 'p (kwote (1+ h))) nil)
nil)))
(if (and (eql (car cls) 'bitn)
(quotep (caddr cls))
(equal (cadr cls) x))
(b* ((h (unquote (caddr cls)))
(l (unquote (caddr cls))))
(if (and (equal h n)
(integerp l)
(integerp m)
(> l m))
(cons (cons 'p (kwote l)) nil)
(if (and (equal l m)
(integerp n)
(integerp h)
(> n h))
(cons (cons 'p (kwote (1+ h))) nil)
nil)))
(bits-shatter-contains-term-lst (cdr cls) x n m)))))
(defun bits-shatter-contains-term-lst (lst x n m)
(declare (xargs :guard (pseudo-term-listp lst)))
(if (atom lst)
nil
(or (bits-shatter-contains-term (car lst) x n m)
(bits-shatter-contains-term-lst (cdr lst) x n m)))))
(defun bits-shatter-walk-clauses (clss x n m)
(declare (xargs :guard (pseudo-term-listp clss)))
(if (atom clss)
nil
(or (bits-shatter-contains-term (car clss) x n m)
(bits-shatter-walk-clauses (cdr clss) x n m))))
(defun bits-shatter-fn (x n m mfc state)
(declare (xargs :guard t)
(ignore state))
(if (and (quotep n)
(quotep m)
(consp (cdr n))
(consp (cdr m)))
(bits-shatter-walk-clauses (mfc-clause mfc) x (unquote n) (unquote m))
nil))
(defthmd bits-plus-bits-shatter
(implies (and (bind-free (bits-shatter-fn x n m mfc state) (p))
(integerp m)
(integerp n)
(integerp p)
(<= m p)
(<= p n))
(equal (bits x n m)
(+ (bits x (1- p) m)
(* (expt 2 (- p m)) (bits x n p)))))
:hints (("Goal" :use (bits-plus-bits))))
(defthmd bits-bits-shatter
(implies (and (syntaxp (and (quotep c)
(quotep i)
(quotep j)))
(bind-free (bits-shatter-fn x i j mfc state) (p))
(integerp p)
(integerp i)
(integerp j)
(<= j p)
(<= p i))
(equal (equal (bits x i j) c)
(and (bvecp c (1+ (- i j)))
(equal (bits x i p) (bits c (- i j) (- p j)))
(equal (bits x (1- p) j) (bits c (1- (- p j)) 0)))))
:hints (("Goal" :in-theory (e/d () ())
:use ((:instance bits-plus-bits
(x x) (n i) (m j) (p p))
(:instance bits-plus-bits
(x c) (n (- i j)) (m 0) (p (- p j)))
(:instance bits-bits
(x x) (i i) (j j) (k (1- (- p j))) (l 0))
(:instance bits-bits
(x x) (i i) (j j) (k (- i j)) (l (- p j)))))))
#| For example:
(thm
(implies (and (bvecp x 32)
(equal (bits x 30 0) 0))
(zerp x (sp)))
:hints (("Goal" :in-theory (e/d (zerp encodingp expf sp sigf
bits-bits-shatter) ()))))
|#
|
