/usr/share/acl2-6.5/books/cgen/splitnat.lisp is in acl2-books-source 6.5-2.
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(ld ;; Newline to fool ACL2/cert.pl dependency scanner
"portcullis.lsp")
(acl2::begin-book);$ACL2s-Preamble$|#
(in-package "DEFDATA")
(set-verify-guards-eagerness 2)
(set-state-ok t)
(include-book "num-list-fns")
(local (include-book "num-list-thms"))
(local (include-book "rem-and-floor"))
(defun weighted-split-nat-step (weights x old-results)
(declare (xargs :guard (and (2+-listp weights)
(natp x)
(naturals-listp old-results)
(= (len weights) (len old-results)))))
(if (mbe :logic (or (endp weights)
(endp old-results))
:exec (endp weights))
nil
(let ((weight (car weights)))
(cons (+ (* weight (car old-results))
(rem x weight))
(weighted-split-nat-step (cdr weights)
(floor x weight)
(cdr old-results))))))
(local (defthm weighted-split-nat-step--true-listp
(true-listp (weighted-split-nat-step w x r))
:rule-classes (:type-prescription :rewrite)))
(local (defthm weighted-split-nat-step--len
(equal (len (weighted-split-nat-step w x r))
(min (len w) (len r)))))
(local (defthm weighted-split-nat-step--nat-listp
(implies (and (naturals-listp w)
(integerp x)
(<= 0 x)
(naturals-listp r))
(naturals-listp (weighted-split-nat-step w x r)))
:rule-classes (:type-prescription :rewrite)))
(local (defthm weighted-split-nat-step--consp
(implies (and (consp w)
(consp r))
(consp (weighted-split-nat-step w x r)))
:rule-classes (:type-prescription :rewrite)))
(local (defthm weighted-split-nat-step--bound-old
(implies (and (2+-listp w)
(integerp x)
(<= 0 x)
(naturals-listp r)
(equal (len w) (len r)))
(<=-lists (weighted-split-nat-step w x r)
(+-lists
(*-lists r
w)
w)))
:rule-classes (:rewrite)))
(local
(encapsulate nil
(local (include-book "arithmetic-5/top" :dir :system))
(local (SET-DEFAULT-HINTS '((acl2::NONLINEARP-DEFAULT-HINT
acl2::STABLE-UNDER-SIMPLIFICATIONP
acl2::HIST
acl2::PSPV))))
(defthm <=-lists--scale
(implies (and (rationalp v)
(<= 0 v)
(rationalp w)
(<= v w)
(naturals-listp l))
(<=-lists (scale l v)
(scale l w))))
(defthm <=-lists--scale2
(implies (and (<=-lists l1
(scale l2 v))
(rationalp v)
(<= 0 v)
(rationalp w)
(<= v w)
(naturals-listp l2))
(<=-lists l1
(scale l2 w))))
(defthm <=-lists--scale3
(implies (and (rationalp v)
(<= 0 v)
(naturals-listp l1)
(naturals-listp l2)
(<=-lists l1 l2))
(<=-lists (scale l1 v)
(scale l2 v))))
(defthm <=-lists--scale4
(implies (and (rationalp v1)
(<= 0 v1)
(rationalp v2)
(<= 0 v2)
(naturals-listp l1)
(naturals-listp l2)
(<=-lists l1 l2)
(<= v1 v2))
(<=-lists (scale l1 v1)
(scale l2 v2))))
(defthm <=-lists--shift
(implies (and (rationalp v)
(rationalp w)
(<= v w)
(rational-listp l))
(<=-lists (shift l v)
(shift l w))))
(defthm <=-lists--shift2
(implies (and (<=-lists l1
(shift l2 v))
(rationalp v)
(rationalp w)
(<= v w)
(rational-listp l2))
(<=-lists l1
(shift l2 w))))
(defthm <=-lists--shift3
(implies (and (rationalp v)
(rational-listp l1)
(rational-listp l2)
(<=-lists l1 l2))
(<=-lists (shift l1 v)
(shift l2 v))))
(defthm <=-lists--shift4
(implies (and (rationalp v1)
(rationalp v2)
(rational-listp l1)
(rational-listp l2)
(<=-lists l1 l2)
(<= v1 v2))
(<=-lists (shift l1 v1)
(shift l2 v2))))
))
(local (defthm shift--<=
(equal (<=-lists (shift l v1)
(shift l v2))
(or (endp l)
(<= v1 v2)))))
(local (defthm weighted-split-nat-step--bound
(implies (and (2+-listp w)
(integerp x)
(<= 0 x)
(naturals-listp r)
(equal (len w) (len r)))
(<=-lists (weighted-split-nat-step w x r)
(shift
(*-lists r w)
x)))))
(local
(encapsulate nil
(local (include-book "arithmetic-5/top" :dir :system))
(local (SET-DEFAULT-HINTS '((acl2::NONLINEARP-DEFAULT-HINT
acl2::STABLE-UNDER-SIMPLIFICATIONP
acl2::HIST
acl2::PSPV))))
(defthm weighted-split-nat-step--bound2--lemma
(implies (and (naturals-listp r)
(2+-listp w)
(equal (len r) (len w)))
(<=-lists (*-lists r w)
(scale r (product-list w)))))))
(local
(encapsulate nil
(local (defthm <=-lists--transitive--force
(implies (and (<=-lists a b)
(force (<=-lists b c)))
(<=-lists a c))
:rule-classes ((:rewrite :match-free :all))))
(local (defthm <=-lists--scale4--force
(implies (and (rationalp v1)
(<= 0 v1)
(rationalp v2)
(<= 0 v2)
(naturals-listp l1)
(naturals-listp l2)
(force (<=-lists l1 l2))
(force (<= v1 v2)))
(<=-lists (scale l1 v1)
(scale l2 v2)))))
(local (defthm <=-lists--shift4--force
(implies (and (rationalp v1)
(rationalp v2)
(rational-listp l1)
(rational-listp l2)
(force (<=-lists l1 l2))
(force (<= v1 v2)))
(<=-lists (shift l1 v1)
(shift l2 v2)))))
(local (include-book "arithmetic-5/top" :dir :system))
(local (SET-DEFAULT-HINTS '((acl2::NONLINEARP-DEFAULT-HINT
acl2::STABLE-UNDER-SIMPLIFICATIONP
acl2::HIST
acl2::PSPV))))
(defthm weighted-split-nat-step--bound2
(implies (and (2+-listp w)
(integerp x)
(<= 0 x)
(naturals-listp r)
(equal (len w) (len r)))
(<=-lists (weighted-split-nat-step w x r)
(shift
(scale r (product-list w))
x))))))
(local
(defthm get-weights-factor-->=2
(implies (and (2+-listp x)
(consp x))
(<= 2 (product-list x)))
:rule-classes (:linear :rewrite)))
(defun rot-list (x)
(declare (xargs :guard (and (true-listp x)
(consp x))))
(append (cdr x) (list (car x))))
(local
(defthm rot-list--consp
(implies (consp x)
(consp (rot-list x)))
:rule-classes (:type-prescription :rewrite)))
(local (defthm consp-cdr-append
(implies (and (consp x)
(consp y))
(consp (cdr (append x y))))))
(local
(defthm rot-list--consp-cdr
(implies (and (consp x)
(consp (cdr x)))
(consp (cdr (rot-list x))))
:rule-classes (:type-prescription :rewrite)))
(local
(defthm len-append-single
(equal (len (append x (list y)))
(+ 1 (len x)))))
(local
(defthm rot-list--len
(implies (consp x)
(equal (len (rot-list x))
(len x)))))
(local (defthm rot-list--nat-listp
(implies (and (naturals-listp x)
(consp x))
(naturals-listp (rot-list x)))))
(local (defthm rot-list--2+-listp
(implies (and (2+-listp x)
(consp x))
(2+-listp (rot-list x)))))
(local (defthm rot-list--product-list
(implies (consp x)
(equal (product-list (rot-list x))
(product-list x)))))
(local (defthm all-<=--rot-list
(implies (consp l)
(equal (all-<= (rot-list l)
x)
(all-<= l
x)))))
(local (in-theory (disable rot-list)))
(defun weighted-split-nat1 (weights weights-factor x)
(declare (xargs :measure (nfix x)
:verify-guards nil
:guard (and (2+-listp weights)
(equal weights-factor (product-list weights))
(consp weights)
(natp x))))
(if (mbe :logic (or (zp x)
(endp weights)
(not (equal weights-factor (product-list weights)))
(not (2+-listp weights)))
:exec (zp x))
(make-list (len weights) :initial-element 0)
(weighted-split-nat-step weights
(rem x weights-factor)
(rot-list
(weighted-split-nat1
(rot-list weights)
weights-factor
(floor x weights-factor))))))
(local
(defthm weighted-split-nat1--consp
(implies (consp weights)
(consp (weighted-split-nat1 weights weights-factor x)))))
(local
(defthm weighted-split-nat1--len
(equal (len (weighted-split-nat1 weights weights-factor x))
(len weights))))
(local
(defthm weighted-split-nat1--nat-listp
(implies (and (2+-listp weights)
(consp weights)
(equal weights-factor (product-list weights))
(integerp x)
(<= 0 x))
(naturals-listp (weighted-split-nat1 weights weights-factor x)))
:rule-classes ((:rewrite)
(:rewrite :corollary
(implies (and (2+-listp weights)
(consp weights)
(equal weights-factor (product-list weights))
(integerp x)
(<= 0 x))
(true-listp (weighted-split-nat1 weights weights-factor x)))))))
(verify-guards weighted-split-nat1)
(local
(defthm weighted-split-nat1--<=-induction-step1
(implies (and (2+-listp weights)
(consp weights)
(integerp x)
(<= 0 x))
(<=-lists (weighted-split-nat-step
weights
(rem x (product-list weights))
(rot-list (weighted-split-nat1 (rot-list weights)
(product-list weights)
(floor x (product-list weights)))))
(shift
(scale (rot-list (weighted-split-nat1 (rot-list weights)
(product-list weights)
(floor x (product-list weights))))
(product-list weights))
(rem x (product-list weights)))))
:rule-classes nil))
(local
(encapsulate nil
(local (include-book "arithmetic-3/top" :dir :system))
(local (SET-DEFAULT-HINTS '((acl2::NONLINEARP-DEFAULT-HINT
acl2::STABLE-UNDER-SIMPLIFICATIONP
acl2::HIST
acl2::PSPV))))
(local (defthm all-<=--shift
(equal (all-<= (shift l v)
x)
(all-<= l
(- x v)))))
(local (defthm all-<=--scale
(implies (and (rationalp v)
(< 0 v))
(equal (all-<= (scale l v)
x)
(all-<= l
(/ x v))))))
(local (defthm blah
(implies (and (equal n (len l))
(<=-lists l
(make-list-logic x n))
(force (<= x y)))
(all-<= l
y))))
(defthm weighted-split-nat1--<=-induction-step2
(implies (and (2+-listp weights)
(consp weights)
(integerp x)
(<= 0 x)
(all-<= (weighted-split-nat1 (rot-list weights)
(product-list weights)
(floor x (product-list weights)))
(floor x (product-list weights))))
(all-<= (shift
(scale (rot-list (weighted-split-nat1 (rot-list weights)
(product-list weights)
(floor x (product-list weights))))
(product-list weights))
(rem x (product-list weights)))
x))
:rule-classes nil)))
(local
(defthm weighted-split-nat1--<=--consp
(implies (and (2+-listp weights)
(consp weights)
(equal weights-factor (product-list weights))
(integerp x)
(<= 0 x))
(all-<= (weighted-split-nat1 weights weights-factor x) x))
:hints (("Goal" :in-theory (disable len 2+-listp product-list)
:do-not '(eliminate-destructors)
:induct t)
("Subgoal *1/2.1"
:use ((:instance weighted-split-nat1--<=-induction-step1)
(:instance weighted-split-nat1--<=-induction-step2))))))
(local
(defthm weighted-split-nat1--<=--endp
(implies (not (consp weights))
(all-<= (weighted-split-nat1 weights weights-factor x) x))))
(local
(defthm weighted-split-nat1--<=
(implies (and (2+-listp weights)
(equal weights-factor (product-list weights))
(integerp x)
(<= 0 x))
(all-<= (weighted-split-nat1 weights weights-factor x) x))
:hints (("Goal" :cases ((consp weights))))))
#|
(defun pos-list-fix (x)
(if (atom x)
nil
(cons
(if (posp (car x))
(car x)
1)
(pos-list-fix (cdr x)))))
|#
(defthm pos-listp--pos-list-fix
(pos-listp (pos-list-fix x))
:rule-classes (:rewrite :type-prescription))
(defthm len--pos-list-fix
(equal (len (pos-list-fix x))
(len x)))
(defthm pos-list-fix--shortcut
(implies (pos-listp x)
(equal (pos-list-fix x)
x)))
(defun non-empty-pos-list-fix (x)
(if (atom x)
(list 1)
(pos-list-fix x)))
(defthm len--non-empty-pos-list-fix
(equal (len (non-empty-pos-list-fix x))
(max 1 (len x))))
(defthm consp--non-empty-pos-list-fix
(consp (non-empty-pos-list-fix x))
:rule-classes (:rewrite :type-prescription))
(defthm pos-listp--non-empty-pos-list-fix
(pos-listp (non-empty-pos-list-fix x))
:rule-classes (:rewrite :type-prescription))
(defthm non-empty-pos-list-fix--shortcut
(implies (and (pos-listp x)
(consp x))
(equal (non-empty-pos-list-fix x)
x)))
(in-theory (disable non-empty-pos-list-fix))
(defun weighted-split-nat (weights x)
(declare (xargs :measure (nfix x)
:guard (and (pos-listp weights)
(consp weights)
(natp x))))
(mbe :exec
(let ((2+-weights (scale weights 2)))
(weighted-split-nat1 2+-weights (product-list 2+-weights) x))
:logic
(let* ((weights (non-empty-pos-list-fix weights))
(x (nfix x))
(2+-weights (scale weights 2)))
(weighted-split-nat1 2+-weights (product-list 2+-weights) x))))
(local ; weighted-split-nat will later be automatically rewritten, so these
; become useless
(defthm weighted-split-nat--len
(equal (len (weighted-split-nat weights x))
(max 1 (len weights)))))
(local
(defthm weighted-split-nat--consp
(consp (weighted-split-nat weights x))))
(defthm weighted-split-nat--nat-listp
(naturals-listp (weighted-split-nat weights x)))
(local (defthm scale--pos-listp--2+-listp
(implies (pos-listp l)
(2+-listp (scale l 2)))
:rule-classes (:forward-chaining :type-prescription)))
(local
(defthm weighted-split-nat--bound
(implies (and (integerp x)
(<= 0 x))
(all-<= (weighted-split-nat weights x) x))))
(local
(defthm car--nat-list--integer
(implies (and (consp x)
(naturals-listp x))
(integerp (car x)))))
(local
(defthm weighted-split-nat--car-integer
(INTEGERP (CAR (WEIGHTED-SPLIT-NAT WEIGHTS X)))))
(local
(defthm car--nat-list-->=0
(implies (and (consp x)
(naturals-listp x))
(<= 0 (car x)))))
(local
(defthm weighted-split-nat--car->=0
(<= 0 (CAR (WEIGHTED-SPLIT-NAT WEIGHTS X)))))
(local
(defthm not-nat-implies-weighted-split-nat-is-zero
(implies (and (not (natp x))
(pos-listp ws)
(>= (len ws) 1))
(equal (weighted-split-nat ws x)
(make-list (len ws) :initial-element 0)))))
(in-theory (disable weighted-split-nat))
;(set-verify-guards-eagerness 0)
(defun nth-weighted-split-nat (i weights x)
(declare (xargs :guard nil)) ; logic only
;(declare (xargs :verify-guards nil))
(if (and (integerp i)
(<= 0 i)
(< i (len weights)))
(nth i (weighted-split-nat weights x))
(car (weighted-split-nat weights x))))
(defun nthcdr-weighted-split-nat (i weights x)
(declare (xargs :guard nil)) ; logic only
;(declare (xargs :verify-guards nil))
(nthcdr i (weighted-split-nat weights x)))
(defthm nth-weighted-split-nat--bound
(implies (and (integerp x)
(<= 0 x))
(<= (nth-weighted-split-nat i weights x)
x))
;:hints (("Goal" :cases ((all-<= (weighted-split-nat weights x) x))))
:rule-classes (:rewrite :linear))
(defthm nat-listp--nth--integerp
(implies (and (naturals-listp l)
(integerp i)
(<= 0 i)
(< i (len l)))
(integerp (nth i l)))
:rule-classes (:rewrite :type-prescription))
(defthm nth-weighted-split-nat--integerp
(integerp (nth-weighted-split-nat i weights x))
:rule-classes (:rewrite :type-prescription))
(defthm nat-listp--nth-->=0
(implies (and (naturals-listp l)
(integerp i)
(<= 0 i)
(< i (len l)))
(<= 0 (nth i l)))
:rule-classes (:rewrite :linear))
(defthm nth-weighted-split-nat-->=0
(<= 0 (nth-weighted-split-nat i weights x))
:rule-classes (:rewrite :linear))
(local
(defthm nth--nthcdr--decomp
(implies (consp l)
(equal (cons (nth 0 l)
(nthcdr 1 l))
l))))
(local
(defthm len--nthcdr--toobig
(implies (and (integerp i)
(<= (len l) i))
(equal (len (nthcdr i l))
0))))
(local
(defthm len--nthcdr
(implies (and (integerp i)
(<= 0 i)
(<= i (len l)))
(equal (len (nthcdr i l))
(- (len l) i)))))
(defthm nthcdr-weighted-split-nat--len
(equal (len (nthcdr-weighted-split-nat i weights x))
(if (zp i)
(max 1 (len weights))
(if (<= i (len weights))
(- (len weights) i)
0))))
(local
(defthm consp--nthcdr
(equal (consp (nthcdr i l))
(and (consp l)
(implies (integerp i)
(< i (len l)))))))
(defthm nthcdr-weighted-split-nat--consp
(equal (consp (nthcdr-weighted-split-nat i weights x))
(implies (integerp i)
(< i (max 1 (len weights))))))
(local
(defthm naturals-listp--nthcdr
(implies (naturals-listp l)
(naturals-listp (nthcdr i l)))
:rule-classes (:rewrite :type-prescription)))
(defthm nthcdr-weighted-split-nat--naturals-listp
(naturals-listp (nthcdr-weighted-split-nat i weights x)))
(defthm weighted-split-nat--to--nthcdr-weighted-split-nat
(implies (and (pos-listp weights)
(consp weights)
(integerp x)
(<= 0 x))
(equal (weighted-split-nat weights x)
(nthcdr-weighted-split-nat 0 weights x))))
(local
(encapsulate nil
(local (include-book "arithmetic-5/top" :dir :system))
(local
(defthm nthcdr--cdr
(implies (and (integerp i)
(<= 0 i))
(equal (nthcdr i (cdr l))
(nthcdr (+ 1 i) l)))
:hints (("Goal" :expand (nthcdr (+ 1 i) l)))))
(defthm nth--nthcdr--decomp2
(implies (and (integerp i)
(<= 0 i)
(< i (len l)))
(equal (cons (nth i l)
(nthcdr (+ 1 i) l))
(nthcdr i l))))))
(defthm nthcdr-weighted-split-nat--deflike
(implies (and (integerp i)
(<= 0 i)
(< i (len weights)))
(equal (nthcdr-weighted-split-nat i weights x)
(cons (nth-weighted-split-nat i weights x)
(nthcdr-weighted-split-nat (+ 1 i) weights x))))
:hints (("Goal" :in-theory (disable nth nthcdr)))
:rule-classes ((:definition :controller-alist ((nthcdr-weighted-split-nat t nil nil)))))
(in-theory (disable nth-weighted-split-nat nthcdr-weighted-split-nat))
(defthm nthcdr-weighted-split-nat--car
(implies (and (integerp i)
(<= 0 i)
(< i (len weights)))
(equal (car (nthcdr-weighted-split-nat i weights x))
(nth-weighted-split-nat i weights x)))
:hints (("Goal" :expand (nthcdr-weighted-split-nat i weights x))))
(defthm nthcdr-weighted-split-nat--cdr
(implies (and (integerp i)
(<= 0 i)
(< i (len weights)))
(equal (cdr (nthcdr-weighted-split-nat i weights x))
(nthcdr-weighted-split-nat (+ 1 i) weights x)))
:hints (("Goal" :expand (nthcdr-weighted-split-nat i weights x))))
(local
(defthm len-exceeding-return-nil
(implies (and (natp i)
(<= (len l) i))
(equal (nth i l) nil))))
(local
(defthm make-list-logic-len
(implies (natp i)
(equal (len (make-list-logic a i))
i))))
(defun nth-returns-elem-of-make-list-ind-scheme (i x)
(declare (xargs :guard (and (natp i) (natp x))))
(if (or (zp i) (zp x))
0
(nth-returns-elem-of-make-list-ind-scheme (- i 1) (- x 1))))
(local
(defthm nth-returns-elem-of-make-list
(implies (and (natp i)
(natp x)
(< i x))
(equal (nth i (make-list-logic a x)) a))
:hints (("Goal" :induct
(nth-returns-elem-of-make-list-ind-scheme i x)))))
#|
;Above lemmas should help prove this
(defthm nth-i-split-nat-<=
(implies (and (pos-listp ws)
(>= (len ws) 1))
(<= (nth i (weighted-split-nat ws x)) (nfix x)))
:hints (("Goal" :cases ((< i (len ws)))))
:rule-classes (:rewrite :linear))
|#
;ADDED some hack defthms to help termination of recursive records with
;more than 3 fields.
(defthm nth-i-split-nat-3-<=
;(implies (and (integerp x)
; (<= 0 x))
(<= (nth i (weighted-split-nat '(1 1 1) (nfix x))) (nfix x))
:rule-classes (:rewrite :linear))
(defthm nth-i-split-nat-4-<=
;(implies (and (integerp x)
; (<= 0 x))
(<= (nth i (weighted-split-nat '(1 1 1 1) (nfix x))) (nfix x))
:rule-classes (:rewrite :linear))
(defthm nth-i-split-nat-5-<=
;(implies (and (integerp x)
; (<= 0 x))
(<= (nth i (weighted-split-nat '(1 1 1 1 1) (nfix x))) (nfix x))
:rule-classes (:rewrite :linear))
(defthm nth-i-split-nat-6-<=
;(implies (and (integerp x)
; (<= 0 x))
(<= (nth i (weighted-split-nat '(1 1 1 1 1 1) (nfix x))) (nfix x))
:rule-classes (:rewrite :linear))
; testing theorems
#|
(thm
(implies (natp x)
(natp (cadr (weighted-split-nat '(3 2 4) x)))))
(thm
(implies (and (natp n) (natp x))
(<= (nth i (weighted-split-nat '(1 1 1) x)) x)))
;|#
; testing
#|
(defun weighted-split-nat-downfrom (weights n)
(declare (xargs :guard (and (pos-listp weights)
(consp weights)
(natp n))))
(if (zp n)
(list (list 0 '-> (weighted-split-nat weights 0)))
(cons (list n '-> (weighted-split-nat weights n))
(weighted-split-nat-downfrom weights (- n 1)))))
(defun weighted-split-nat-upto (weights n)
(declare (xargs :guard (and (pos-listp weights)
(consp weights)
(natp n))))
(reverse (weighted-split-nat-downfrom weights n)))
;(trace$ weighted-split-nat-step-tail)
(weighted-split-nat-upto '(1 1) 33)
;|#
; alternative interface
(defthm pos-listp--list-expt--2
(implies (naturals-listp l)
(pos-listp (list-expt 2 l)))
:rule-classes (:rewrite :type-prescription))
(defun pow-weighted-split-nat (pow-weights x)
(declare (xargs :guard (and (pos-listp pow-weights)
(consp pow-weights)
(natp x))))
(let ((2**weights (list-expt 2 pow-weights)))
(weighted-split-nat1 2**weights (product-list 2**weights) x)))
(defun split-nat (nways x)
(declare (xargs :guard (and (posp nways)
(natp x))))
(weighted-split-nat (make-list nways :initial-element 1) x))
(defthm split-nat--naturals-listp
(naturals-listp (split-nat nways x))
:rule-classes :type-prescription)
(defthm naturals-listp--true-listp
(implies (naturals-listp x)
(true-listp x))
:rule-classes (:rewrite :forward-chaining))#|ACL2s-ToDo-Line|#
|