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; Append 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>
;
; append.lisp
; This file was originally part of the Unicode library.
(in-package "ACL2")
(include-book "list-fix")
(local (include-book "std/basic/inductions" :dir :system))
(defsection std/lists/append
:parents (std/lists append)
:short "Lemmas about @(see append) available in the @(see std/lists)
library."
(local (defthm len-when-consp
(implies (consp x)
(< 0 (len x)))
:rule-classes ((:linear) (:rewrite))))
(defthm append-when-not-consp
(implies (not (consp x))
(equal (append x y)
y)))
(defthm append-of-cons
(equal (append (cons a x) y)
(cons a (append x y))))
(defthm true-listp-of-append
(equal (true-listp (append x y))
(true-listp y)))
(defthm consp-of-append
;; Note that data-structures/list-defthms has a similar rule, except that
;; it adds two type-prescription corollaries. I found these corollaries to
;; be expensive, so I don't bother with them.
(equal (consp (append x y))
(or (consp x)
(consp y))))
(defthm append-under-iff
(iff (append x y)
(or (consp x)
y)))
(defthm len-of-append
(equal (len (append x y))
(+ (len x) (len y))))
;; (defthm nth-of-append
;; (equal (nth n (append x y))
;; (if (< (nfix n) (len x))
;; (nth n x)
;; (nth (- n (len x)) y))))
(defthm equal-when-append-same
(equal (equal (append x y1)
(append x y2))
(equal y1 y2)))
(local (defthm append-nonempty-list
(implies (consp x)
(not (equal (append x y) y)))
:hints(("Goal" :use ((:instance len (x (append x y)))
(:instance len (x y)))))))
(defthm equal-of-appends-when-true-listps
(implies (and (true-listp x1)
(true-listp x2))
(equal (equal (append x1 y)
(append x2 y))
(equal x1 x2)))
:hints(("Goal" :induct (cdr-cdr-induct x1 x2))))
(defthm append-of-nil
(equal (append x nil)
(list-fix x)))
;; Disable this built-in ACL2 rule since append-of-nil is stronger.
(in-theory (disable append-to-nil))
(defthm list-fix-of-append
(equal (list-fix (append x y))
(append x (list-fix y))))
(defthm car-of-append
(equal (car (append x y))
(if (consp x)
(car x)
(car y))))
(defthmd car-of-append-when-consp
(implies (consp x)
(equal (car (append x y))
(car x))))
(theory-invariant (incompatible (:rewrite car-of-append-when-consp)
(:rewrite car-of-append))
:key car-of-append-consp-invariant
:error t)
(defthmd cdr-of-append
(equal (cdr (append x y))
(if (consp x)
(append (cdr x) y)
(cdr y))))
(defthm cdr-of-append-when-consp
; We enable the version that requires consp, because it's less likely we want
; to unconditionally open-up (cdr (append ...)) unless we know ... is a consp.
(implies (consp x)
(equal (cdr (append x y))
(append (cdr x) y))))
(theory-invariant (incompatible (:rewrite cdr-of-append-when-consp)
(:rewrite cdr-of-append))
:key cdr-of-append-consp-invariant
:error t)
(defthm associativity-of-append
(equal (append (append a b) c)
(append a (append b c))))
(defcong element-list-equiv element-list-equiv (append a b) 1)
(table listfix-rules 'element-list-equiv-implies-element-list-equiv-append-1 t)
(defcong element-list-equiv element-list-equiv (append a b) 2)
(table listfix-rules 'element-list-equiv-implies-element-list-equiv-append-2 t)
(def-listp-rule element-list-p-of-append-true-list
(equal (element-list-p (append a b))
(and (element-list-p (list-fix a))
(element-list-p b)))
:requirement true-listp
:name element-list-p-of-append))
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