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; Standard Typed Lists Library
; Copyright (C) 2008-2016 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 "std/basic/arith-equiv-defs" :dir :system)
(include-book "std/lists/list-defuns" :dir :system)
(local (include-book "std/lists/equiv" :dir :system))
(local (include-book "std/lists/take" :dir :system))
(defsection nats-equiv
:parents (std/lists nat-equiv)
:short "Recognizer for lists that are the same length and that are pairwise
equivalent up to @(see nfix)."
(defund nats-equiv (x y)
(declare (xargs :measure (+ (len x) (len y))))
(if (and (atom x)
(atom y))
t
(and (nat-equiv (car x) (car y))
(nats-equiv (cdr x) (cdr y)))))
(local (in-theory (enable nats-equiv)))
"<p>This is an @(see equivalence) relation:</p>"
(encapsulate
()
(local (defthm nats-equiv-reflexive
(nats-equiv x x)
:hints(("Goal" :induct (len x)))))
(local (defthm nats-equiv-symmetric
(implies (nats-equiv x y)
(nats-equiv y x))))
(local (defthm nats-equiv-transitive
(implies (and (nats-equiv x y)
(nats-equiv y z))
(nats-equiv x x))))
(defequiv nats-equiv))
"<p>It is also a @(see refinement) of @(see list-equiv):</p>"
(defrefinement list-equiv nats-equiv)
"<p>It also enjoys several basic congruences:</p>"
(defcong nats-equiv nat-equiv (car x) 1)
(defcong nats-equiv nats-equiv (cdr x) 1)
(defcong nats-equiv nats-equiv (cons a x) 2)
(defthm nats-equiv-of-cons
(equal (nats-equiv (cons a x) z)
(and (nat-equiv a (car z))
(nats-equiv x (cdr z)))))
(local (defun my-ind (n x y)
(if (zp n)
(list n x y)
(my-ind (- n 1) (cdr x) (cdr y)))))
(defcong nats-equiv nat-equiv (nth n x) 2
:hints(("Goal"
:in-theory (enable nth)
:induct (my-ind n x x-equiv))))
(defcong nats-equiv nats-equiv (update-nth n v x) 3
:hints(("Goal"
:in-theory (enable update-nth)
:induct (my-ind n x x-equiv))))
(defcong nat-equiv nats-equiv (update-nth n v x) 2
:hints(("Goal"
:in-theory (enable update-nth)
:induct (my-ind n x x-equiv))))
(defcong nats-equiv nats-equiv (append x y) 2)
(defcong nats-equiv nats-equiv (revappend x y) 2)
(defcong nats-equiv nats-equiv (take n x) 2
:hints(("Goal" :in-theory (enable take))))
(defcong nats-equiv nats-equiv (nthcdr n x) 2)
(defcong nats-equiv nats-equiv (make-list-ac n val ac) 3)
(defcong nat-equiv nats-equiv (replicate n x) 2
:hints(("Goal" :in-theory (enable replicate)))))
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