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; Len lemmas
; Copyright (C) 2008-2013 Centaur Technology
;
; Contact:
; Centaur Technology Formal Verification Group
; 7600-C N. Capital of Texas Highway, Suite 300, Austin, TX 78731, USA.
; http://www.centtech.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@centtech.com>
(in-package "ACL2")
(include-book "abstract")
(defsection std/lists/len
:parents (std/lists len)
:short "Lemmas about @(see len) available in the @(see std/lists) library."
(defthm len-when-atom
(implies (atom x)
(equal (len x)
0)))
(defthm len-of-cons
(equal (len (cons a x))
(+ 1 (len x))))
(defthm |(equal 0 (len x))|
(equal (equal 0 (len x))
(atom x)))
(defthm |(< 0 (len x))|
(equal (< 0 (len x))
(consp x)))
(defthm consp-by-len
(implies (and (equal (len x) n)
(syntaxp (quotep n)))
(equal (consp x)
(< 0 n))))
(defthm consp-of-cdr-by-len
(implies (and (equal (len x) n)
(syntaxp (quotep n)))
(equal (consp (cdr x))
(< 1 n)))
:hints(("Goal" :expand ((len x)
(len (cdr x))))))
(defthm consp-of-cddr-by-len
(implies (and (equal (len x) n)
(syntaxp (quotep n)))
(equal (consp (cddr x))
(< 2 n)))
:hints(("Goal" :expand ((len x)
(len (cdr x))
(len (cddr x))))))
(defthm consp-of-cdddr-by-len
(implies (and (equal (len x) n)
(syntaxp (quotep n)))
(equal (consp (cdddr x))
(< 3 n)))
:hints(("Goal" :expand ((len x)
(len (cdr x))
(len (cddr x))
(len (cdddr x))))))
(def-projection-rule len-of-elementlist-projection
(equal (len (elementlist-projection x))
(len x))))
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