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; Milawa - A Reflective Theorem Prover
; Copyright (C) 2005-2009 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>
(in-package "MILAWA")
(include-book "pequal-list")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(dd.open "cons.tex")
(deftheorem theorem-cons-is-not-nil
:derive (!= (cons x y) nil)
:proof (@derive
((v (= (symbolp x) nil) (= (consp x) nil)) (build.axiom (axiom-disjoint-symbols-and-conses)))
((v (= (symbolp (cons x y)) nil) (= (consp (cons x y)) nil)) (build.instantiation @- (@sigma (x . (cons x y)))))
((v (= (consp (cons x y)) nil) (= (symbolp (cons x y)) nil)) (build.commute-or @-) *1)
((= (consp (cons x y)) t) (build.axiom (axiom-consp-of-cons)))
((!= (consp (cons x y)) nil) (build.not-nil-from-t @-))
((= (symbolp (cons x y)) nil) (build.modus-ponens-2 @- *1))
((!= (symbolp (cons x y)) t) (build.not-t-from-nil @-))
((!= t (symbolp (cons x y))) (build.commute-not-pequal @-))
((= (symbolp nil) t) (build.base-eval '(symbolp 'nil)))
((!= (symbolp nil) (symbolp (cons x y))) (build.substitute-into-not-pequal @-- @-))
((!= (symbolp (cons x y)) (symbolp nil)) (build.commute-not-pequal @-) *2)
((v (!= (cons x y) nil) (= (cons x y) nil)) (build.propositional-schema (@formula (= (cons x y) nil))))
((v (!= (cons x y) nil) (= (symbolp (cons x y)) (symbolp nil))) (build.disjoined-pequal-by-args 'symbolp (@formula (!= (cons x y) nil)) (list @-)))
((v (= (symbolp (cons x y)) (symbolp nil)) (!= (cons x y) nil)) (build.commute-or @-))
((!= (cons x y) nil) (build.modus-ponens-2 *2 @-)))
:minatbl ((cons . 2)
(consp . 1)
(symbolp . 1)))
(dd.close)
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