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; Take 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>
; Contributing author: Alessandro Coglio <coglio@kestrel.edu>
;
; take.lisp
; This file was originally part of the Unicode library.
(in-package "ACL2")
(include-book "list-fix")
(include-book "equiv")
(local (include-book "std/basic/inductions" :dir :system))
(defsection std/lists/take
:parents (std/lists take)
:short "Lemmas about @(see take) available in the @(see std/lists) library."
(defthm consp-of-take
(equal (consp (take n xs))
(not (zp n))))
(defthm take-under-iff
(iff (take n xs)
(not (zp n))))
(defthm len-of-take
(equal (len (take n xs))
(nfix n)))
(defthm take-of-cons
(equal (take n (cons a x))
(if (zp n)
nil
(cons a (take (1- n) x)))))
(defthm take-of-append
(equal (take n (append x y))
(if (< (nfix n) (len x))
(take n x)
(append x (take (- n (len x)) y))))
:hints(("Goal" :induct (take n x))))
(defthm take-of-zero
(equal (take 0 x)
nil))
(defthm take-of-1
(equal (take 1 x)
(list (car x))))
(defthm car-of-take
(implies (<= 1 (nfix n))
(equal (car (take n x))
(car x))))
(defthm second-of-take
(implies (<= 2 (nfix n))
(equal (second (take n x))
(second x))))
(defthm third-of-take
(implies (<= 3 (nfix n))
(equal (third (take n x))
(third x))))
(defthm fourth-of-take
(implies (<= 4 (nfix n))
(equal (fourth (take n x))
(fourth x))))
(defthm take-of-len-free
(implies (equal len (len x))
(equal (take len x)
(list-fix x))))
(defthm equal-of-take-and-list-fix
(equal (equal (take n x) (list-fix x))
(equal (len x) (nfix n))))
(defthm take-of-len
(equal (take (len x) x)
(list-fix x)))
(defthm subsetp-of-take
(implies (<= (nfix n) (len x))
(subsetp (take n x) x)))
(defthm take-fewer-of-take-more
;; Note: see also repeat.lisp for related cases and a stronger rule that
;; case-splits.
(implies (<= (nfix a) (nfix b))
(equal (take a (take b x))
(take a x))))
(defthm take-of-take-same
;; Note: see also repeat.lisp for related cases and a stronger rule that
;; case-splits.
(equal (take a (take a x))
(take a x)))
(defcong list-equiv equal (take n x) 2
:hints(("Goal"
:induct (and (take n x)
(cdr-cdr-induct x x-equiv)))))
(defcong list-equiv equal (take n x) 2)
(defcong list-equiv equal (butlast lst n) 1
:hints(("Goal" :induct (cdr-cdr-induct lst lst-equiv))))
(local (defthm element-list-p-of-take-nil
(implies (element-p nil)
(element-list-p (take n nil)))))
(def-listp-rule element-list-p-of-take
(implies (element-list-p (double-rewrite x))
(iff (element-list-p (take n x))
(or (element-p nil)
(<= (nfix n) (len x))))))
(def-projection-rule elementlist-projection-of-take-nil-preserving
(implies (equal nil (element-xformer nil))
(equal (elementlist-projection (take n x))
(take n (elementlist-projection x))))
:hints (("goal" :induct (take n x)))
:name elementlist-projection-of-take
:requirement nil-preservingp
:body (equal (elementlist-projection (take n x))
(take n (elementlist-projection x))))
(def-projection-rule elementlist-projection-of-take
(implies (<= (nfix n) (len x))
(equal (elementlist-projection (take n x))
(take n (elementlist-projection x))))
:hints (("goal" :induct (take n x)))
:name elementlist-projection-of-take
:requirement (not nil-preservingp)))
(defsection first-n
:parents (std/lists take)
:short "@(call first-n) is logically identical to @('(take n x)'), but its
guard does not require @('(true-listp x)')."
:long "<p><b>Reasoning Note.</b> We leave @('first-n') enabled, so it will
just get rewritten into @('take'). You should typically never write a theorem
about @('first-n'): write theorems about @('take') instead.</p>"
(local (defun repeat (n x)
(if (zp n)
nil
(cons x (repeat (- n 1) x)))))
(local (defthm l0
(equal (append (repeat n x) (cons x y))
(cons x (append (repeat n x) y)))))
(local (defthm l1
(equal (make-list-ac n val acc)
(append (repeat n val) acc))
:hints(("Goal"
:induct (make-list-ac n val acc)))))
(local (defthm l2
(implies (atom x)
(equal (take n x)
(make-list n)))))
(defun first-n (n x)
(declare (xargs :guard (natp n)))
(mbe :logic (take n x)
:exec
(cond ((zp n)
nil)
((atom x)
(make-list n))
(t
(cons (car x)
(first-n (- n 1) (cdr x))))))))
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