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; acl2-count of lists
; Copyright (C) 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: Sol Swords <sswords@centtech.com>
;
; These theorems are proved so often that I don't think there's any sense
; copyrighting them.
; Not sure if it's best to put theorems about acl2-count of other list-related
; stuff here or in the books for the various list functions, e.g., there's some
; stuff about acl2-count of nthcdr in the nthcdr book. For now here are some
; lemmas about car and cdr that are all over the place.
; It looks like there aren't any exact name clashes -- some overlap in
; centaur/vl/util/arithmetic, but in the VL package.
(in-package "ACL2")
;; These are strictly weaker than acl2-count-of-sum, below (at least, together
;; with (:type-prescription acl2-count)).
(defthmd acl2-count-of-car
(and (implies (consp x)
(< (acl2-count (car x))
(acl2-count x)))
(<= (acl2-count (car x))
(acl2-count x)))
:hints(("Goal" :in-theory (enable acl2-count)))
:rule-classes :linear)
(defthmd acl2-count-of-cdr
(and (implies (consp x)
(< (acl2-count (cdr x))
(acl2-count x)))
(<= (acl2-count (cdr x))
(acl2-count x)))
:hints(("Goal" :in-theory (enable acl2-count)))
:rule-classes :linear)
(defthm acl2-count-of-sum
(and (implies (consp x)
(< (+ (acl2-count (car x))
(acl2-count (cdr x)))
(acl2-count x)))
(<= (+ (acl2-count (car x))
(acl2-count (cdr x)))
(acl2-count x)))
:hints(("Goal" :in-theory (enable acl2-count)))
:rule-classes :linear)
(defthm acl2-count-of-consp-positive
(implies (consp x)
(< 0 (acl2-count x)))
:rule-classes (:type-prescription :linear))
(defthm acl2-count-of-cons-greater
;; [Jared] renamed this form acl2-count-of-cons for compatibility with
;; coi/lists/acl2-count
(> (acl2-count (cons a b))
(+ (acl2-count a) (acl2-count b)))
:rule-classes :linear)
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