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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-list")
(include-book "pequal")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(dd.open "pequal-list.tex")
(defund build.reflexivity-list (x)
;; BOZO update the defderiv table with the axiom?
(declare (xargs :guard (logic.term-listp x)))
(if (consp x)
(cons (build.reflexivity (car x))
(build.reflexivity-list (cdr x)))
nil))
(defobligations build.reflexivity-list
(build.reflexivity))
(encapsulate
()
(local (in-theory (enable build.reflexivity-list)))
(defthm forcing-logic.appeal-listp-of-build.reflexivity-list
(implies (force (logic.term-listp x))
(equal (logic.appeal-listp (build.reflexivity-list x))
t)))
(defthm forcing-logic.strip-conclusions-of-build.reflexivity-list
(implies (force (logic.term-listp x))
(equal (logic.strip-conclusions (build.reflexivity-list x))
(logic.pequal-list x x))))
(defthm@ forcing-logic.proof-listp-of-build.reflexivity-list
(implies (force (and (logic.term-listp x)
(logic.term-list-atblp x atbl)
(@obligations build.reflexivity-list)))
(equal (logic.proof-listp (build.reflexivity-list x) axioms thms atbl)
t))))
(defund build.pequal-by-args (f as)
;; Derive (f t1 ... tn) = (f s1 ... sn) from t1 = s1, ..., tn = sn.
(declare (xargs :guard (and (logic.function-namep f)
(logic.appeal-listp as)
(logic.all-atomicp (logic.strip-conclusions as)))
:guard-hints (("Goal" :in-theory (enable logic.functional-axiom)))))
(let* ((conclusions (logic.strip-conclusions as)) ;; (t1 = s1, ..., tn = sn)
(t* (logic.=lhses conclusions)) ;; (t1, ..., tn)
(s* (logic.=rhses conclusions))) ;; (s1, ..., sn)
(cond ((equal t* s*)
;; Optimization. We can just use reflexivity.
(build.reflexivity (logic.function f t*)))
(t
;; Otherwise, take the functional equality axiom,
;; t1 = s1 -> ... -> tn = sn -> (f t1 ... tn) = (f s1 ... sn),
;; and apply repeated modus ponens.
(build.modus-ponens-list (logic.pequal (logic.function f t*) (logic.function f s*))
as
(build.functional-equality f t* s*))))))
(defobligations build.pequal-by-args
(build.reflexivity build.modus-ponens-list build.functional-equality))
(encapsulate
()
(local (in-theory (enable logic.functional-axiom build.pequal-by-args)))
(defthm forcing-build.pequal-by-args-under-iff
(iff (build.pequal-by-args f as)
t))
(defthm forcing-logic.appealp-of-build.pequal-by-args
(implies (force (and (logic.function-namep f)
(logic.appeal-listp as)
(logic.all-atomicp (logic.strip-conclusions as))))
(equal (logic.appealp (build.pequal-by-args f as))
t)))
(defthm forcing-logic.conclusion-of-build.pequal-by-args
(implies (force (and (logic.function-namep f)
(logic.appeal-listp as)
(logic.all-atomicp (logic.strip-conclusions as))))
(equal (logic.conclusion (build.pequal-by-args f as))
(logic.pequal (logic.function f (logic.=lhses (logic.strip-conclusions as)))
(logic.function f (logic.=rhses (logic.strip-conclusions as))))))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-build.pequal-by-args
(implies (force (and (logic.function-namep f)
(logic.appeal-listp as)
(logic.all-atomicp (logic.strip-conclusions as))
(logic.proof-listp as axioms thms atbl)
(equal (len as) (cdr (lookup f atbl)))
(@obligations build.pequal-by-args)))
(equal (logic.proofp (build.pequal-by-args f as) axioms thms atbl)
t))))
(defund build.disjoined-pequal-by-args (f p as)
;; Derive P v (f t1 ... tn) = (f s1 ... sn) from P v t1 = s1, ..., P v tn = sn
(declare (xargs :guard (and (logic.function-namep f)
(logic.formulap p)
(logic.appeal-listp as)
(let ((aconcs (logic.strip-conclusions as)))
(and (logic.all-disjunctionsp aconcs)
(all-equalp p (logic.vlhses aconcs))
(logic.all-atomicp (logic.vrhses aconcs)))))
:guard-hints(("Goal" :in-theory (enable logic.functional-axiom)))))
(let* ((ti=si* (logic.vrhses (logic.strip-conclusions as))) ;; (t1 = s1, ..., tn = sn)
(t* (logic.=lhses ti=si*)) ;; (t1, ..., tn)
(s* (logic.=rhses ti=si*))) ;; (s1, ..., sn)
(cond ((equal t* s*)
;; Optimization. We can just use reflexivity and expansion.
(build.expansion P (build.reflexivity (logic.function f t*))))
(t
;; Otherwise, take the functional equality axiom,
;; t1 = s1 -> ... -> tn = sn -> (f t1 ... tn) = (f s1 ... sn),
;; expand it with P,
;; P v t1 = s1 -> ... -> tn = sn -> (f t1 ... tn) = (f s1 ... sn),
;; and apply repeated disjoined modus ponens.
(build.disjoined-modus-ponens-list (logic.pequal (logic.function f t*) (logic.function f s*))
as
(build.expansion p (build.functional-equality f t* s*)))))))
(defobligations build.disjoined-pequal-by-args
(build.expansion build.reflexivity build.disjoined-modus-ponens-list build.functional-equality))
(encapsulate
()
(local (in-theory (enable logic.functional-axiom build.disjoined-pequal-by-args)))
(defthm forcing-build.disjoined-pequal-by-args-under-iff
(iff (build.disjoined-pequal-by-args f p as)
t))
(defthm forcing-logic.appealp-of-build.disjoined-pequal-by-args
(implies (force (and (logic.function-namep f)
(logic.formulap p)
(logic.appeal-listp as)
(logic.all-disjunctionsp (logic.strip-conclusions as))
(all-equalp p (logic.vlhses (logic.strip-conclusions as)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions as)))))
(equal (logic.appealp (build.disjoined-pequal-by-args f p as))
t)))
(defthm forcing-logic.conclusion-of-build.disjoined-pequal-by-args
(implies (force (and (logic.function-namep f)
(logic.formulap p)
(logic.appeal-listp as)
(logic.all-disjunctionsp (logic.strip-conclusions as))
(all-equalp p (logic.vlhses (logic.strip-conclusions as)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions as)))))
(equal (logic.conclusion (build.disjoined-pequal-by-args f p as))
(logic.por p (logic.pequal (logic.function f (logic.=lhses (logic.vrhses (logic.strip-conclusions as))))
(logic.function f (logic.=rhses (logic.vrhses (logic.strip-conclusions as))))))))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-build.disjoined-pequal-by-args
(implies (force (and (logic.function-namep f)
(logic.formulap p)
(logic.appeal-listp as)
(logic.all-disjunctionsp (logic.strip-conclusions as))
(all-equalp p (logic.vlhses (logic.strip-conclusions as)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions as)))
(logic.formula-atblp p atbl)
(logic.proof-listp as axioms thms atbl)
(equal (cdr (lookup f atbl)) (len as))
(@obligations build.disjoined-pequal-by-args)))
(equal (logic.proofp (build.disjoined-pequal-by-args f p as) axioms thms atbl)
t))))
(dd.close)
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