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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 "if-lemmas")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(dd.open "urewrite-trace-if-lemmas.tex")
(defderiv rw.iff-implies-equal-if-bldr
:derive (= (equal (if (? a1) (? b1) (? c1)) (if (? a2) (? b2) (? c2))) t)
:from ((proof x (= (iff (? a1) (? a2)) t))
(proof y (v (!= (not (? a2)) nil) (= (equal (? b1) (? b2)) t)))
(proof z (v (!= (? a2) nil) (= (equal (? c1) (? c2)) t))))
:proof (@derive
((= (iff (? a1) (? a2)) t) (@given x) *1)
((v (!= (not (? a2)) nil) (= (equal (? b1) (? b2)) t)) (@given y) *2)
((v (!= (? a2) nil) (= (equal (? c1) (? c2)) t)) (@given z) *3)
((v (!= (iff x1 x2) t)
(v (! (v (!= (not x2) nil) (= (equal y1 y2) t)))
(v (! (v (!= x2 nil) (= (equal z1 z2) t)))
(= (equal (if x1 y1 z1) (if x2 y2 z2)) t)))) (build.theorem (rw.theorem-iff-implies-equal-if-combined)))
((v (!= (iff (? a1) (? a2)) t)
(v (! (v (!= (not (? a2)) nil) (= (equal (? b1) (? b2)) t)))
(v (! (v (!= (? a2) nil) (= (equal (? c1) (? c2)) t)))
(= (equal (if (? a1) (? b1) (? c1)) (if (? a2) (? b2) (? c2))) t)))) (build.instantiation @- (@sigma (x1 . (? a1)) (x2 . (? a2)) (y1 . (? b1)) (y2 . (? b2)) (z1 . (? c1)) (z2 . (? c2)))))
((= (equal (if (? a1) (? b1) (? c1)) (if (? a2) (? b2) (? c2))) t) (rw.three-modus-ponens *1 *2 *3 @-)))
:minatbl ((if . 3)))
(defderiv rw.iff-implies-iff-if-bldr
:derive (= (iff (if (? a1) (? b1) (? c1)) (if (? a2) (? b2) (? c2))) t)
:from ((proof x (= (iff (? a1) (? a2)) t))
(proof y (v (!= (not (? a2)) nil) (= (iff (? b1) (? b2)) t)))
(proof z (v (!= (? a2) nil) (= (iff (? c1) (? c2)) t))))
:proof (@derive
((= (iff (? a1) (? a2)) t) (@given x) *1)
((v (!= (not (? a2)) nil) (= (iff (? b1) (? b2)) t)) (@given y) *2)
((v (!= (? a2) nil) (= (iff (? c1) (? c2)) t)) (@given z) *3)
((v (!= (iff x1 x2) t)
(v (! (v (!= (not x2) nil) (= (iff y1 y2) t)))
(v (! (v (!= x2 nil) (= (iff z1 z2) t)))
(= (iff (if x1 y1 z1) (if x2 y2 z2)) t)))) (build.theorem (rw.theorem-iff-implies-iff-if-combined)))
((v (!= (iff (? a1) (? a2)) t)
(v (! (v (!= (not (? a2)) nil) (= (iff (? b1) (? b2)) t)))
(v (! (v (!= (? a2) nil) (= (iff (? c1) (? c2)) t)))
(= (iff (if (? a1) (? b1) (? c1)) (if (? a2) (? b2) (? c2))) t)))) (build.instantiation @- (@sigma (x1 . (? a1)) (x2 . (? a2)) (y1 . (? b1)) (y2 . (? b2)) (z1 . (? c1)) (z2 . (? c2)))))
((= (iff (if (? a1) (? b1) (? c1)) (if (? a2) (? b2) (? c2))) t) (rw.three-modus-ponens *1 *2 *3 @-)))
:minatbl ((if . 3)))
(defderiv rw.equal-of-if-x-y-y-bldr
:derive (= (equal (if (? a1) (? b) (? c)) (? d)) t)
:from ((proof x (= (iff (? a1) (? a2)) t))
(proof y (v (!= (not (? a2)) nil) (= (equal (? b) (? d)) t)))
(proof z (v (!= (? a2) nil) (= (equal (? c) (? d)) t))))
:proof (@derive
((= (iff (? a1) (? a2)) t) (@given x) *1)
((v (!= (not (? a2)) nil) (= (equal (? b) (? d)) t)) (@given y) *2)
((v (!= (? a2) nil) (= (equal (? c) (? d)) t)) (@given z) *3)
((v (!= (iff x1 x2) t)
(v (! (v (!= (not x2) nil) (= (equal y w) t)))
(v (! (v (!= x2 nil) (= (equal z w) t)))
(= (equal (if x1 y z) w) t)))) (build.theorem (rw.theorem-equal-of-if-x-y-y)))
((v (!= (iff (? a1) (? a2)) t)
(v (! (v (!= (not (? a2)) nil) (= (equal (? b) (? d)) t)))
(v (! (v (!= (? a2) nil) (= (equal (? c) (? d)) t)))
(= (equal (if (? a1) (? b) (? c)) (? d)) t)))) (build.instantiation @- (@sigma (x1 . (? a1)) (x2 . (? a2)) (y . (? b)) (z . (? c)) (w . (? d)))))
((= (equal (if (? a1) (? b) (? c)) (? d)) t) (rw.three-modus-ponens *1 *2 *3 @-)))
:minatbl ((if . 3)))
(defderiv rw.iff-of-if-x-y-y-bldr
:derive (= (iff (if (? a1) (? b) (? c)) (? d)) t)
:from ((proof x (= (iff (? a1) (? a2)) t))
(proof y (v (!= (not (? a2)) nil) (= (iff (? b) (? d)) t)))
(proof z (v (!= (? a2) nil) (= (iff (? c) (? d)) t))))
:proof (@derive
((= (iff (? a1) (? a2)) t) (@given x) *1)
((v (!= (not (? a2)) nil) (= (iff (? b) (? d)) t)) (@given y) *2)
((v (!= (? a2) nil) (= (iff (? c) (? d)) t)) (@given z) *3)
((v (!= (iff x1 x2) t)
(v (! (v (!= (not x2) nil) (= (iff y w) t)))
(v (! (v (!= x2 nil) (= (iff z w) t)))
(= (iff (if x1 y z) w) t)))) (build.theorem (rw.theorem-iff-of-if-x-y-y)))
((v (!= (iff (? a1) (? a2)) t)
(v (! (v (!= (not (? a2)) nil) (= (iff (? b) (? d)) t)))
(v (! (v (!= (? a2) nil) (= (iff (? c) (? d)) t)))
(= (iff (if (? a1) (? b) (? c)) (? d)) t)))) (build.instantiation @- (@sigma (x1 . (? a1)) (x2 . (? a2)) (y . (? b)) (z . (? c)) (w . (? d)))))
((= (iff (if (? a1) (? b) (? c)) (? d)) t) (rw.three-modus-ponens *1 *2 *3 @-)))
:minatbl ((if . 3)))
(dd.close)
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