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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 "prop")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(dd.open "conjunctions.tex")
(dd.subsection "Conjunction rules")
(dd.write "These builders act as $\\wedge$ introduction and elimination rules.
We typically avoid using conjunctions and instead build both proofs separately
to avoid the conversion overhead.")
(defderiv build.conjoin
:derive (! (v (! A) (! B)))
:from ((proof x A)
(proof y B))
:proof (@derive ((v (! (v (! A) (! B))) (v (! A) (! B))) (build.propositional-schema (@formula (v (! A) (! B)))))
((v (v (! (v (! A) (! B))) (! A)) (! B)) (build.associativity @-))
((v (! B) (v (! (v (! A) (! B))) (! A))) (build.commute-or @-))
(B (@given y))
((v (! (v (! A) (! B))) (! A)) (build.modus-ponens @- @--))
((v (! A) (! (v (! A) (! B)))) (build.commute-or @-))
(A (@given x))
((! (v (! A) (! B))) (build.modus-ponens @- @--))))
(defderiv build.first-conjunct
:derive A
:from ((proof x (! (v (! A) (! B)))))
:proof (@derive ((v (! A) A) (build.propositional-schema (@formula A)))
((v A (! A)) (build.commute-or @-))
((v (! B) (v A (! A))) (build.expansion (@formula (! B)) @-))
((v (v (! B) A) (! A)) (build.associativity @-))
((v (! A) (v (! B) A)) (build.commute-or @-))
((v (v (! A) (! B)) A) (build.associativity @-))
((! (v (! A) (! B))) (@given x))
(A (build.modus-ponens-2 @- @--))))
(defderiv build.second-conjunct
:derive B
:from ((proof x (! (v (! A) (! B)))))
:proof (@derive ((v (! B) B) (build.propositional-schema (@formula B)))
((v (! A) (v (! B) B)) (build.expansion (@formula (! A)) @-))
((v (v (! A) (! B)) B) (build.associativity @-))
((! (v (! A) (! B))) (@given x))
(B (build.modus-ponens-2 @- @--))))
(defderiv build.disjoined-conjoin
:derive (v P (! (v (! A) (! B))))
:from ((proof x (v P A))
(proof y (v P B)))
:proof (@derive ((v (! (v (! A) (! B))) (v (! A) (! B))) (build.propositional-schema (@formula (v (! A) (! B)))))
((v (v (! (v (! A) (! B))) (! A)) (! B)) (build.associativity @-))
((v (! B) (v (! (v (! A) (! B))) (! A))) (build.commute-or @-) *1)
((v P B) (@given y))
((v B P) (build.commute-or @-))
((v P (v (! (v (! A) (! B))) (! A))) (build.cut @- *1))
((v P (v (! A) (! (v (! A) (! B))))) (build.disjoined-commute-or @-))
((v P A) (@given x))
((v P (! (v (! A) (! B)))) (build.disjoined-modus-ponens @- @--))))
(defderiv build.disjoined-first-conjunct
:derive (v P A)
:from ((proof x (v P (! (v (! A) (! B))))))
:proof (@derive ((v (! A) A) (build.propositional-schema (@formula A)))
((v A (! A)) (build.commute-or @-))
((v (! B) (v A (! A))) (build.expansion (@formula (! B)) @-))
((v (v (! B) A) (! A)) (build.associativity @-))
((v (! A) (v (! B) A)) (build.commute-or @-))
((v (v (! A) (! B)) A) (build.associativity @-) *1)
((v P (! (v (! A) (! B)))) (@given x))
((v (! (v (! A) (! B))) P) (build.commute-or @-))
((v A P) (build.cut *1 @-))
((v P A) (build.commute-or @-))))
(defderiv build.disjoined-second-conjunct
:derive (v P B)
:from ((proof x (v P (! (v (! A) (! B))))))
:proof (@derive ((v (! B) B) (build.propositional-schema (@formula B)))
((v (! A) (v (! B) B)) (build.expansion (@formula (! A)) @-))
((v (v (! A) (! B)) B) (build.associativity @-) *1)
((v P (! (v (! A) (! B)))) (@given x))
((v (! (v (! A) (! B))) P) (build.commute-or @-))
((v B P) (build.cut *1 @-))
((v P B) (build.commute-or @-))))
(dd.close)
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