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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 "find-proof")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
; Proof Replacement.
;
; A basic goal of this file is to be able to support reasoning of the form,
;
; Given: BLDR is sound as long as FOO is a theorem.
; Given: A proof of FOO.
; Conclude: BLDR's conclusions are sound even when FOO is not a theorem.
;
; To carry out this sort of thing, we introduce a proof-replacing function,
; which walks through an appeal and replaces any conclusions of some form
; with a new proof. Then, we see that
;
; Replace(BLDR's output, Proof of FOO)
;
; Is a proof of BLDR's output which is true even without having FOO as a
; theorem of the history.
(defund logic.flag-replace-proofs (flag x proofs)
(declare (xargs :guard (and (if (equal flag 'proof)
(logic.appealp x)
(and (equal flag 'list)
(logic.appeal-listp x)))
(logic.appeal-listp proofs))
:measure (two-nats-measure (rank x) (if (equal flag 'proof) 1 0))
:verify-guards nil))
(if (equal flag 'proof)
(or (logic.find-proof (logic.conclusion x) proofs)
(logic.appeal (logic.method x)
(logic.conclusion x)
(logic.flag-replace-proofs 'list (logic.subproofs x) proofs)
(logic.extras x)))
(if (consp x)
(cons (logic.flag-replace-proofs 'proof (car x) proofs)
(logic.flag-replace-proofs 'list (cdr x) proofs))
nil)))
(defund logic.replace-proofs (x proofs)
(declare (xargs :guard (and (logic.appealp x)
(logic.appeal-listp proofs))
:verify-guards nil))
(logic.flag-replace-proofs 'proof x proofs))
(defund logic.replace-proofs-list (x proofs)
(declare (xargs :guard (and (logic.appeal-listp x)
(logic.appeal-listp proofs))
:verify-guards nil))
(logic.flag-replace-proofs 'list x proofs))
(defthmd definition-of-logic.replace-proofs
(equal (logic.replace-proofs x proofs)
(or (logic.find-proof (logic.conclusion x) proofs)
(logic.appeal (logic.method x)
(logic.conclusion x)
(logic.replace-proofs-list (logic.subproofs x) proofs)
(logic.extras x))))
:rule-classes :definition
:hints(("Goal" :in-theory (enable logic.flag-replace-proofs
logic.replace-proofs
logic.replace-proofs-list))))
(defthmd definition-of-logic.replace-proofs-list
(equal (logic.replace-proofs-list x proofs)
(if (consp x)
(cons (logic.replace-proofs (car x) proofs)
(logic.replace-proofs-list (cdr x) proofs))
nil))
:rule-classes :definition
:hints(("Goal"
:in-theory (enable logic.flag-replace-proofs
logic.replace-proofs
logic.replace-proofs-list)
:expand (logic.flag-replace-proofs 'list x proofs))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition logic.replace-proofs))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition logic.replace-proofs-list))))
(defthm logic.replace-proofs-list-when-not-consp
(implies (not (consp x))
(equal (logic.replace-proofs-list x proofs)
nil))
:hints(("Goal" :in-theory (enable definition-of-logic.replace-proofs-list))))
(defthm logic.replace-proofs-list-of-cons
(equal (logic.replace-proofs-list (cons a x) proofs)
(cons (logic.replace-proofs a proofs)
(logic.replace-proofs-list x proofs)))
:hints(("Goal" :in-theory (enable definition-of-logic.replace-proofs-list))))
(defprojection
:list (logic.replace-proofs-list x proofs)
:element (logic.replace-proofs x proofs)
:already-definedp t)
(defthms-flag
:shared-hyp (force (logic.appeal-listp proofs))
:thms ((proof logic.appealp-of-logic.replace-proofs
(implies (force (logic.appealp x))
(equal (logic.appealp (logic.replace-proofs x proofs))
t)))
(t logic.appeal-listp-of-logic.replace-proofs-list
(implies (force (logic.appeal-listp x))
(equal (logic.appeal-listp (logic.replace-proofs-list x proofs))
t))))
:hints(("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (e/d (definition-of-logic.replace-proofs)
((ACL2::force))))))
(defthms-flag
:shared-hyp (force (logic.appeal-listp proofs))
:thms ((proof logic.conclusion-of-logic.replace-proofs
(implies (force (logic.appealp x))
(equal (logic.conclusion (logic.replace-proofs x proofs))
(logic.conclusion x))))
(t logic.strip-conclusions-of-logic.replace-proofs-list
(implies (force (logic.appeal-listp x))
(equal (logic.strip-conclusions (logic.replace-proofs-list x proofs))
(logic.strip-conclusions x)))))
:hints(("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (e/d (definition-of-logic.replace-proofs)
((ACL2::force))))))
(defthmd lemma-1-for-logic.proofp-of-logic.replace-proofs
(implies (and (not (equal (logic.method x) 'axiom))
(not (equal (logic.method x) 'theorem)))
(equal
(logic.appeal-step-okp x
(difference axioms remove)
(difference thms remove)
atbl)
(logic.appeal-step-okp x axioms thms atbl)))
:hints(("Goal" :in-theory (enable logic.appeal-step-okp))))
(defthmd lemma-2-for-logic.proofp-of-logic.replace-proofs
(implies (and (or (equal (logic.method x) 'axiom)
(equal (logic.method x) 'theorem))
(not (memberp (logic.conclusion x) remove)))
(equal
(logic.appeal-step-okp x
(difference axioms remove)
(difference thms remove)
atbl)
(logic.appeal-step-okp x axioms thms atbl)))
:hints(("Goal" :in-theory (enable logic.appeal-step-okp
logic.axiom-okp
logic.theorem-okp))))
(defthmd lemma-3-for-logic.proofp-of-logic.replace-proofs
(implies (not (memberp (logic.conclusion x) remove))
(equal (logic.appeal-step-okp x
(difference axioms remove)
(difference thms remove)
atbl)
(logic.appeal-step-okp x axioms thms atbl)))
:hints(("Goal" :use ((:instance lemma-1-for-logic.proofp-of-logic.replace-proofs)
(:instance lemma-2-for-logic.proofp-of-logic.replace-proofs)))))
(defthmd lemma-4-for-logic.proofp-of-logic.replace-proofs
(implies (and (logic.proof-listp (logic.replace-proofs-list (logic.subproofs x) proofs)
(difference axioms (logic.strip-conclusions proofs))
(difference thms (logic.strip-conclusions proofs))
atbl)
(logic.appeal-listp proofs)
(logic.proof-listp proofs
(difference axioms (logic.strip-conclusions proofs))
(difference thms (logic.strip-conclusions proofs))
atbl)
(logic.appealp x)
(logic.appeal-step-okp x axioms thms atbl)
(logic.proof-listp (logic.subproofs x) axioms thms atbl)
(not (logic.find-proof (logic.conclusion x) proofs)))
(logic.proofp (logic.appeal (logic.method x)
(logic.conclusion x)
(logic.replace-proofs-list (logic.subproofs x)
proofs)
(logic.extras x))
(difference axioms (logic.strip-conclusions proofs))
(difference thms (logic.strip-conclusions proofs))
atbl))
:hints(("Goal"
:in-theory (enable definition-of-logic.proofp)
:use ((:instance lemma-appeal-step-for-forcing-logic.provablep-when-logic.subproofs-provable
(new-subproofs (logic.replace-proofs-list (logic.subproofs x) proofs)))
(:instance lemma-3-for-logic.proofp-of-logic.replace-proofs
(x (logic.appeal (logic.method x)
(logic.conclusion x)
(logic.replace-proofs-list (logic.subproofs x) proofs)
(logic.extras x)))
(remove (logic.strip-conclusions proofs)))))))
(defthms-flag
:shared-hyp (and (force (logic.appeal-listp proofs))
(force (logic.proof-listp proofs
(difference axioms (logic.strip-conclusions proofs))
(difference thms (logic.strip-conclusions proofs))
atbl)))
:thms ((proof logic.proofp-of-logic.replace-proofs
(implies (and (force (logic.appealp x))
(force (logic.proofp x axioms thms atbl)))
(equal (logic.proofp (logic.replace-proofs x proofs)
(difference axioms (logic.strip-conclusions proofs))
(difference thms (logic.strip-conclusions proofs))
atbl)
t)))
(t logic.proof-listp-of-logic.replace-proofs-list
(implies (and (force (logic.appeal-listp x))
(force (logic.proof-listp x axioms thms atbl)))
(equal (logic.proof-listp (logic.replace-proofs-list x proofs)
(difference axioms (logic.strip-conclusions proofs))
(difference thms (logic.strip-conclusions proofs))
atbl)
t))))
:hints(("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (enable definition-of-logic.replace-proofs
definition-of-logic.proofp
lemma-4-for-logic.proofp-of-logic.replace-proofs))))
(defthm logic.appeal-step-okp-in-larger-axiom-set
(implies (and (subsetp axioms more-axioms)
(logic.appeal-step-okp x axioms thms atbl))
(equal (logic.appeal-step-okp x more-axioms thms atbl)
t))
:hints(("Goal" :in-theory (enable logic.appeal-step-okp
logic.axiom-okp))))
(defthms-flag
:shared-hyp (subsetp axioms more-axioms)
:thms ((proof logic.proofp-in-larger-axiom-set
(implies (logic.proofp x axioms thms atbl)
(equal (logic.proofp x more-axioms thms atbl)
t)))
(t logic.proof-listp-in-larger-axiom-set
(implies (logic.proof-listp x axioms thms atbl)
(equal (logic.proof-listp x more-axioms thms atbl)
t))))
:hints(("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (enable definition-of-logic.proofp))))
(defthm logic-provablep-in-larger-axiom-set
(implies (and (subsetp axioms more-axioms)
(logic.provablep x axioms thms atbl))
(equal (logic.provablep x more-axioms thms atbl)
t))
:hints(("Goal" :in-theory (e/d (logic.provablep)
((ACL2::force))))))
(defthm logic.appeal-step-okp-in-larger-theorem-set
(implies (and (subsetp thms more-thms)
(logic.appeal-step-okp x axioms thms atbl))
(equal (logic.appeal-step-okp x axioms more-thms atbl)
t))
:hints(("Goal" :in-theory (enable logic.appeal-step-okp
logic.theorem-okp))))
(defthms-flag
:shared-hyp (subsetp thms more-thms)
:thms ((proof logic.proofp-in-larger-theorem-set
(implies (logic.proofp x axioms thms atbl)
(equal (logic.proofp x axioms more-thms atbl)
t)))
(t logic.proof-listp-in-larger-theorem-set
(implies (logic.proof-listp x axioms thms atbl)
(equal (logic.proof-listp x axioms more-thms atbl)
t))))
:hints(("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (enable definition-of-logic.proofp))))
(defthm logic-provablep-in-larger-theorem-set
(implies (and (subsetp thms more-thms)
(logic.provablep x axioms thms atbl))
(equal (logic.provablep x axioms thms atbl)
t))
:hints(("Goal" :in-theory (e/d (logic.provablep)
((ACL2::force))))))
|
