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; Repeat function and lemmas
; Copyright (C) 2005-2013 Kookamara LLC
;
; Contact:
;
; Kookamara LLC
; 11410 Windermere Meadows
; Austin, TX 78759, USA
; http://www.kookamara.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Jared Davis <jared@kookamara.com>
;
; repeat.lisp
; This file was originally part of the Unicode library.
(in-package "ACL2")
(include-book "rev")
(local (include-book "take"))
(local (include-book "nthcdr"))
(local (defthm commutativity-2-of-+
(equal (+ x (+ y z))
(+ y (+ x z)))))
(local (defthm fold-consts-in-+
(implies (and (syntaxp (quotep x))
(syntaxp (quotep y)))
(equal (+ x (+ y z)) (+ (+ x y) z)))))
(local (defthm distributivity-of-minus-over-+
(equal (- (+ x y)) (+ (- x) (- y)))))
(defsection repeat
:parents (std/lists make-list)
:short "@(call repeat) creates a list of @('x')es with length @('n'); it
is a simpler alternative to @(see make-list)."
(defund repeat (n x)
(declare (xargs :guard (natp n)
:verify-guards nil))
(mbe :logic (if (zp n)
nil
(cons x (repeat (- n 1) x)))
; On CCL, a simple loop of (loop for i from 1 to 10000 do (repeat 10000 6))
; finished in 2.74 seconds when we use make-list, versus 3.69 seconds when we
; use make-list-ac. So lets use make-list.
:exec (make-list n :initial-element x)))
(local (in-theory (enable repeat)))
(defthm repeat-when-zp
(implies (zp n)
(equal (repeat n a)
nil)))
(defthm |(repeat 0 x)|
(equal (repeat 0 x)
nil))
(defthm repeat-under-iff
(iff (repeat n x)
(not (zp n))))
(defthm consp-of-repeat
(equal (consp (repeat n a))
(not (zp n))))
(defthm repeat-1
(equal (repeat 1 a)
(list a)))
(defthm take-when-atom
(implies (atom x)
(equal (take n x)
(repeat n nil))))
(defthm len-of-repeat
(equal (len (repeat n x))
(nfix n)))
(defthm repeat-of-nfix
(equal (repeat (nfix n) x)
(repeat n x)))
(defthm car-of-repeat-increment
;; Goofy rule that helps when recurring when repeat is involved.
;; BOZO there's a better rule than this in str/arithmetic, but it case-splits.
(implies (natp n)
(equal (car (repeat (+ 1 n) x))
x)))
(defthm cdr-of-repeat-increment
;; Goofy rule that helps when recurring when repeat is involved.
(implies (natp n)
(equal (cdr (repeat (+ 1 n) x))
(repeat n x))))
(defthm member-of-repeat
(equal (member a (repeat n b))
(if (equal a b)
(repeat n b)
nil)))
(encapsulate
()
(local (defun dec-dec-induct (k n)
(if (zp k)
nil
(if (zp n)
nil
(dec-dec-induct (- k 1) (- n 1))))))
(defthm take-of-repeat
(equal (take n (repeat k a))
(if (<= (nfix n) (nfix k))
(repeat n a)
(append (repeat k a)
(repeat (- (nfix n) (nfix k)) nil))))
:hints(("Goal" :induct (dec-dec-induct k n))))
(defthm nthcdr-of-repeat
(equal (nthcdr n (repeat k a))
(if (<= (nfix n) (nfix k))
(repeat (- (nfix k) (nfix n)) a)
nil))
:hints(("Goal" :induct (dec-dec-induct k n)))))
(defthm append-of-repeat-to-cons-of-same
(equal (append (repeat n a) (cons a x))
(cons a (append (repeat n a) x))))
(encapsulate
()
(local (defthm l0
(implies (equal (append (repeat n a) x) y)
(and (equal (repeat n a) (take n y))
(equal (nthcdr n y) x)))))
(local (defthm l1
(implies (not (<= (nfix n) (len y)))
(not (equal (append (repeat n a) x) y)))))
(local (defthm l2
(implies (and (<= n (len y))
(equal (repeat n a) (take n y))
(equal x (nthcdr n y)))
(equal (append (repeat n a) x) y))
:hints(("Goal"
:in-theory (disable append-of-take-and-nthcdr)
:use ((:instance append-of-take-and-nthcdr
(n n)
(x y)))))))
(defthm equal-of-append-repeat
(implies (case-split (<= n (len y)))
(equal (equal (append (repeat n a) x) y)
(and (equal (repeat n a) (take n y))
(equal x (nthcdr n y)))))
:hints(("Goal"
:use ((:instance l0)
(:instance l2))))))
(defthm rev-of-repeat
(equal (rev (repeat n a))
(repeat n a)))
(def-listp-rule element-list-p-of-repeat
(iff (element-list-p (repeat n x))
(or (element-p x)
(zp n)))))
(local (in-theory (enable repeat)))
(defsection make-list-ac-removal
:parents (repeat make-list)
:short "Rewrite rule that eliminates @('make-list-ac') (and hence @(see
make-list)) in favor of @(see repeat)."
(local (defun silly-repeat (n x acc)
(if (zp n)
acc
(cons x (silly-repeat (- n 1) x acc)))))
(local (defthm lemma1
(equal (make-list-ac n x acc)
(silly-repeat n x acc))))
(local (defthm lemma2
(equal (silly-repeat n x acc)
(append (repeat n x) acc))))
(defthm make-list-ac-removal
(equal (make-list-ac n x acc)
(append (repeat n x)
acc))))
(verify-guards repeat)
(defsection take-of-take-split
:parents (std/lists/take)
:short "Aggressive case splitting rule to reduce @('(take a (take b x))')."
:long "@(def take-of-take-split)
<p>This rule may sometimes cause too much case splitting. If you disable it,
nests of @('take') can still be reduced when ACL2 can determine the
relationship between @('a') and @('b'), using the following related rules:</p>
@(def take-of-take-same)
@(def take-more-of-take-fewer)
@(def take-fewer-of-take-more)"
:autodoc nil
(local (defun my-induct (a b x)
(if (or (zp a)
(zp b))
(list a b x)
(my-induct (- a 1) (- b 1) (cdr x)))))
(defthm take-more-of-take-fewer
(implies (< (nfix b) (nfix a))
(equal (take a (take b x))
(append (take b x) (repeat (- (nfix a) (nfix b)) nil))))
:hints(("Goal" :induct (my-induct a b x))))
(defthm take-of-take-split
;; This has a very aggressive case split.
(equal (take a (take b x))
(if (<= (nfix a) (nfix b))
(take a x)
(append (take b x) (repeat (- (nfix a) (nfix b)) nil))))))
(defsection take-of-too-many
:parents (std/lists/take repeat)
:short "Rewrite @('(take n x)') when @('n') is more than @('(len x)')."
:long "<p>This rule may sometimes lead your proof in a bad direction. If you
see unwanted @('repeat') terms, you may want to disable it.</p>"
(defthm take-of-too-many
(implies (<= (len x) (nfix n))
(equal (take n x)
(append x (repeat (- (nfix n) (len x)) nil))))))
(defsection replicate
:parents (repeat)
:short "Alias for @(see repeat)."
(defmacro replicate (n x)
`(repeat ,n ,x))
(add-macro-alias replicate repeat))
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