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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 "eqtrace-okp")
(include-book "../../clauses/basic-bldrs")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(defderiv rw.direct-iff-eqtrace-nhyp-bldr-lemma-1
:derive (v (!= (iff (? a) (? b)) nil) (!= (? nhyp) nil))
:from ((proof x (= (? nhyp) (not (iff (? a) (? b))))))
:proof (@derive
((v (!= x nil) (= (not x) t)) (build.theorem (theorem-not-when-nil)))
((v (!= (iff (? a) (? b)) nil) (= (not (iff (? a) (? b))) t)) (build.instantiation @- (@sigma (x . (iff (? a) (? b))))) *1)
((= (? nhyp) (not (iff (? a) (? b)))) (@given x))
((v (!= (iff (? a) (? b)) nil) (= (? nhyp) (not (iff (? a) (? b))))) (build.expansion (@formula (!= (iff (? a) (? b)) nil)) @-))
((v (!= (iff (? a) (? b)) nil) (= (? nhyp) t)) (build.disjoined-transitivity-of-pequal @- *1))
((v (!= (iff (? a) (? b)) nil) (!= (? nhyp) nil)) (build.disjoined-not-nil-from-t @-))))
(defderiv rw.direct-iff-eqtrace-nhyp-bldr-lemma-2
:derive (v (!= (? nhyp) nil) (= (iff (? a) (? b)) t))
:from ((proof x (= (? nhyp) (not (iff (? a) (? b))))))
:proof (@derive
((= (? nhyp) (not (iff (? a) (? b)))) (@given x))
((v (!= (iff (? a) (? b)) nil) (!= (? nhyp) nil)) (rw.direct-iff-eqtrace-nhyp-bldr-lemma-1 @-))
((v (!= (? nhyp) nil) (!= (iff (? a) (? b)) nil)) (build.commute-or @-))
((v (!= (? nhyp) nil) (= (iff (? a) (? b)) t)) (build.disjoined-iff-t-from-not-nil @-))))
(defund@ rw.direct-iff-eqtrace-nhyp-bldr (nhyp x)
;; Given an nhyp that matches a direct-iff eqtrace, prove:
;; nhyp != nil v (equal lhs rhs) = t
(declare (xargs :guard (and (logic.termp nhyp)
(rw.eqtracep x)
(equal (rw.direct-iff-eqtrace t nhyp) x))
:verify-guards nil))
;; Let nhyp be (not* (equal a b)).
(let* ((guts (clause.negative-term-guts nhyp))
(args (logic.function-args guts))
(a (first args))
(main-proof (@derive
((= nhyp (not (iff a b))) (clause.standardize-negative-term-bldr nhyp))
((v (!= nhyp nil) (= (iff a b) t)) (rw.direct-iff-eqtrace-nhyp-bldr-lemma-2 @-)))))
(if (equal a (rw.eqtrace->lhs x))
main-proof
(build.disjoined-commute-iff main-proof))))
(defobligations rw.direct-iff-eqtrace-nhyp-bldr
(clause.standardize-negative-term-bldr
rw.direct-iff-eqtrace-nhyp-bldr-lemma-2
build.disjoined-commute-iff))
(encapsulate
()
(local (in-theory (enable rw.direct-iff-eqtrace
rw.direct-iff-eqtrace-nhyp-bldr
theorem-not-when-nil
logic.term-formula)))
(local (in-theory (disable forcing-equal-of-logic.pequal-rewrite-two
forcing-equal-of-logic.pequal-rewrite
forcing-equal-of-logic.por-rewrite-two
forcing-equal-of-logic.por-rewrite
forcing-equal-of-logic.pnot-rewrite-two
forcing-equal-of-logic.pnot-rewrite)))
(defthm rw.direct-iff-eqtrace-nhyp-bldr-under-iff
(iff (rw.direct-iff-eqtrace-nhyp-bldr nhyp x)
t))
(defthm forcing-logic.appealp-of-rw.direct-iff-eqtrace-nhyp-bldr
(implies (force (and (logic.termp nhyp)
(rw.eqtracep x)
(equal (rw.direct-iff-eqtrace t nhyp) x)))
(equal (logic.appealp (rw.direct-iff-eqtrace-nhyp-bldr nhyp x))
t)))
(defthm forcing-logic.conclusion-of-rw.direct-iff-eqtrace-nhyp-bldr
(implies (force (and (logic.termp nhyp)
(rw.eqtracep x)
(equal (rw.direct-iff-eqtrace t nhyp) x)))
(equal (logic.conclusion (rw.direct-iff-eqtrace-nhyp-bldr nhyp x))
(logic.por (logic.term-formula nhyp)
(logic.pequal (logic.function 'iff
(list (rw.eqtrace->lhs x)
(rw.eqtrace->rhs x)))
''t))))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-rw.direct-iff-eqtrace-nhyp-bldr
(implies (force (and (logic.termp nhyp)
(rw.eqtracep x)
(equal (rw.direct-iff-eqtrace t nhyp) x)
;; ---
(logic.term-atblp nhyp atbl)
(equal (cdr (lookup 'not atbl)) 1)
(@obligations rw.direct-iff-eqtrace-nhyp-bldr)))
(equal (logic.proofp (rw.direct-iff-eqtrace-nhyp-bldr nhyp x) axioms thms atbl)
t)))
(verify-guards rw.direct-iff-eqtrace-nhyp-bldr))
(defund rw.direct-iff-eqtrace-bldr (x box)
;; Given a direct-iff eqtrace that is box-okp, prove
;; hypbox-formula v (iff lhs rhs) = t
(declare (xargs :guard (and (rw.eqtracep x)
(rw.hypboxp box)
(rw.direct-iff-eqtrace-okp x box))
:verify-guards nil))
(let* ((left (rw.hypbox->left box))
(right (rw.hypbox->right box))
(nhyp-left (rw.find-nhyp-for-direct-iff-eqtracep left x)))
;; First search for a working hyp on the left.
(if nhyp-left
;; 1. nhyp-left v (iff lhs rhs) = t Direct-Iff eqtrace nhyp bldr
;; 2. Left v (iff lhs rhs) = t Multi assoc expansion
(let* ((line-1 (rw.direct-iff-eqtrace-nhyp-bldr nhyp-left x))
(line-2 (build.multi-assoc-expansion line-1 (logic.term-list-formulas left))))
(if right
;; 3. Left v (Right v (iff lhs rhs) = t) DJ Left Expansion
;; 4. (Left v Right) v (iff lhs rhs) = t Associativity
(build.associativity (build.disjoined-left-expansion line-2 (clause.clause-formula right)))
;; Else we're done already
line-2))
;; Else we know there must be a matching hyp on the right, since our guard
;; requires we are a box-okp direct-iff eqtrace.
;;
;; 1. nhyp-right v (iff lhs rhs) = t Direct-Iff eqtrace nhyp bldr
;; 2. Right v (iff lhs rhs) = t Multi assoc expansion.
(let* ((nhyp-right (rw.find-nhyp-for-direct-iff-eqtracep right x))
(line-1 (rw.direct-iff-eqtrace-nhyp-bldr nhyp-right x))
(line-2 (build.multi-assoc-expansion line-1 (logic.term-list-formulas right))))
(if left
;; 3. Left v (Right v (iff lhs rhs) = t) Expansion
;; 4. (Left v Right) v (iff lhs rhs) = t Associativity
(build.associativity
(build.expansion (clause.clause-formula left) line-2))
;; Else we're done already.
line-2)))))
(defobligations rw.direct-iff-eqtrace-bldr
(rw.direct-iff-eqtrace-nhyp-bldr
build.multi-assoc-expansion
build.disjoined-left-expansion))
(encapsulate
()
(local (in-theory (enable rw.direct-iff-eqtrace-bldr
rw.direct-iff-eqtrace-okp
rw.hypbox-formula
rw.eqtrace-formula
)))
(defthm rw.direct-iff-eqtrace-bldr-under-iff
(iff (rw.direct-iff-eqtrace-bldr x box)
t))
(defthm forcing-logic.appealp-of-rw.direct-iff-eqtrace-bldr
(implies (force (and (rw.eqtracep x)
(rw.hypboxp box)
(rw.direct-iff-eqtrace-okp x box)))
(equal (logic.appealp (rw.direct-iff-eqtrace-bldr x box))
t)))
(defthm forcing-logic.conclusion-of-rw.direct-iff-eqtrace-bldr
(implies (force (and (rw.eqtracep x)
(rw.hypboxp box)
(rw.direct-iff-eqtrace-okp x box)))
(equal (logic.conclusion (rw.direct-iff-eqtrace-bldr x box))
(rw.eqtrace-formula x box)))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-rw.direct-iff-eqtrace-bldr
(implies (force (and (rw.eqtracep x)
(rw.hypboxp box)
(rw.direct-iff-eqtrace-okp x box)
;; ---
(equal (cdr (lookup 'not atbl)) 1)
(rw.hypbox-atblp box atbl)
(@obligations rw.direct-iff-eqtrace-bldr)))
(equal (logic.proofp (rw.direct-iff-eqtrace-bldr x box) axioms thms atbl)
t)))
(verify-guards rw.direct-iff-eqtrace-bldr))
|
