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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 "../build/prop")
(include-book "../clauses/prop")
(include-book "../rewrite/prop")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(defund level2.step-okp (x axioms thms atbl)
(declare (xargs :guard (and (logic.appealp x)
(logic.formula-listp axioms)
(logic.formula-listp thms)
(logic.arity-tablep atbl))))
(let ((method (logic.method x)))
(cond ((equal method 'build.commute-or) (build.commute-or-okp x atbl))
((equal method 'build.right-expansion) (build.right-expansion-okp x atbl))
((equal method 'build.modus-ponens) (build.modus-ponens-okp x atbl))
((equal method 'build.modus-ponens-2) (build.modus-ponens-2-okp x atbl))
((equal method 'build.right-associativity) (build.right-associativity-okp x atbl))
((equal method 'build.disjoined-left-expansion) (build.disjoined-left-expansion-okp x atbl))
((equal method 'build.disjoined-right-expansion) (build.disjoined-right-expansion-okp x atbl))
((equal method 'build.disjoined-contraction) (build.disjoined-contraction-okp x atbl))
((equal method 'build.cancel-neg-neg) (build.cancel-neg-neg-okp x atbl))
((equal method 'build.insert-neg-neg) (build.insert-neg-neg-okp x atbl))
((equal method 'build.lhs-insert-neg-neg) (build.lhs-insert-neg-neg-okp x atbl))
((equal method 'build.merge-negatives) (build.merge-negatives-okp x atbl))
((equal method 'build.merge-implications) (build.merge-implications-okp x atbl))
((equal method 'build.disjoined-commute-or) (build.disjoined-commute-or-okp x atbl))
((equal method 'build.disjoined-right-associativity) (build.disjoined-right-associativity-okp x atbl))
((equal method 'build.disjoined-associativity) (build.disjoined-associativity-okp x atbl))
((equal method 'build.disjoined-cut) (build.disjoined-cut-okp x atbl))
((equal method 'build.disjoined-modus-ponens) (build.disjoined-modus-ponens-okp x atbl))
((equal method 'build.disjoined-modus-ponens-2) (build.disjoined-modus-ponens-2-okp x atbl))
((equal method 'build.lhs-commute-or-then-rassoc) (build.lhs-commute-or-then-rassoc-okp x atbl))
((equal method 'rw.crewrite-twiddle-bldr) (rw.crewrite-twiddle-bldr-okp x atbl))
((equal method 'rw.crewrite-twiddle2-bldr) (rw.crewrite-twiddle2-bldr-okp x atbl))
((equal method 'clause.aux-split-twiddle) (clause.aux-split-twiddle-okp x atbl))
((equal method 'clause.aux-split-twiddle2) (clause.aux-split-twiddle2-okp x atbl))
((equal method 'clause.aux-split-default-3-bldr) (clause.aux-split-default-3-bldr-okp x atbl))
((equal method 'clause.aux-limsplit-cutoff-step-bldr) (clause.aux-limsplit-cutoff-step-bldr-okp x atbl))
(t
(logic.appeal-step-okp x axioms thms atbl)))))
(encapsulate
()
(local (in-theory (enable level2.step-okp)))
(defthm soundness-of-level2.step-okp
(implies (and (logic.appealp x)
(level2.step-okp x axioms thms atbl)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t)))
(defthm level2.step-okp-when-logic.appeal-step-okp
;; This shows that our new step checker is "complete" in the sense that all
;; previously acceptable appeals are still acceptable.
(implies (logic.appeal-step-okp x axioms thms atbl)
(level2.step-okp x axioms thms atbl))
:hints(("Goal" :in-theory (enable logic.appeal-step-okp))))
(defthm level2.step-okp-when-not-consp
(implies (not (consp x))
(equal (level2.step-okp x axioms thms atbl)
nil))
:hints(("Goal" :in-theory (enable logic.method)))))
(encapsulate
()
(defund level2.flag-proofp (flag x axioms thms atbl)
(declare (xargs :guard (and (if (equal flag 'proof)
(logic.appealp x)
(and (equal flag 'list)
(logic.appeal-listp x)))
(logic.formula-listp axioms)
(logic.formula-listp thms)
(logic.arity-tablep atbl))
:measure (two-nats-measure (rank x)
(if (equal flag 'proof) 1 0))))
(if (equal flag 'proof)
(and (level2.step-okp x axioms thms atbl)
(level2.flag-proofp 'list (logic.subproofs x) axioms thms atbl))
(if (consp x)
(and (level2.flag-proofp 'proof (car x) axioms thms atbl)
(level2.flag-proofp 'list (cdr x) axioms thms atbl))
t)))
(definlined level2.proofp (x axioms thms atbl)
(declare (xargs :guard (and (logic.appealp x)
(logic.formula-listp axioms)
(logic.formula-listp thms)
(logic.arity-tablep atbl))))
(level2.flag-proofp 'proof x axioms thms atbl))
(definlined level2.proof-listp (x axioms thms atbl)
(declare (xargs :guard (and (logic.appeal-listp x)
(logic.formula-listp axioms)
(logic.formula-listp thms)
(logic.arity-tablep atbl))))
(level2.flag-proofp 'list x axioms thms atbl))
(defthmd definition-of-level2.proofp
(equal (level2.proofp x axioms thms atbl)
(and (level2.step-okp x axioms thms atbl)
(level2.proof-listp (logic.subproofs x) axioms thms atbl)))
:rule-classes :definition
:hints(("Goal" :in-theory (enable level2.proofp level2.proof-listp level2.flag-proofp))))
(defthmd definition-of-level2.proof-listp
(equal (level2.proof-listp x axioms thms atbl)
(if (consp x)
(and (level2.proofp (car x) axioms thms atbl)
(level2.proof-listp (cdr x) axioms thms atbl))
t))
:rule-classes :definition
:hints(("Goal" :in-theory (enable level2.proofp level2.proof-listp level2.flag-proofp))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition level2.proofp))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition demo.proof-list)))))
(defthm level2.proofp-when-not-consp
(implies (not (consp x))
(equal (level2.proofp x axioms thms atbl)
nil))
:hints(("Goal" :in-theory (enable definition-of-level2.proofp))))
(defthm level2.proof-listp-when-not-consp
(implies (not (consp x))
(equal (level2.proof-listp x axioms thms atbl)
t))
:hints (("Goal" :in-theory (enable definition-of-level2.proof-listp))))
(defthm level2.proof-listp-of-cons
(equal (level2.proof-listp (cons a x) axioms thms atbl)
(and (level2.proofp a axioms thms atbl)
(level2.proof-listp x axioms thms atbl)))
:hints (("Goal" :in-theory (enable definition-of-level2.proof-listp))))
(defthms-flag
:thms ((proof booleanp-of-level2.proofp
(equal (booleanp (level2.proofp x axioms thms atbl))
t))
(t booleanp-of-level2.proof-listp
(equal (booleanp (level2.proof-listp x axioms thms atbl))
t)))
:hints(("Goal"
:in-theory (enable definition-of-level2.proofp)
:induct (logic.appeal-induction flag x))))
(deflist level2.proof-listp (x axioms thms atbl)
(level2.proofp x axioms thms atbl)
:already-definedp t)
(defthms-flag
;; We now prove that level2.proofp is sound. I.e., it only accepts appeals
;; whose conclusions are provable in the sense of logic.proofp.
:thms ((proof logic.provablep-when-level2.proofp
(implies (and (logic.appealp x)
(level2.proofp x axioms thms atbl))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t)))
(t logic.provable-listp-when-level2.proof-listp
(implies (and (logic.appeal-listp x)
(level2.proof-listp x axioms thms atbl))
(equal (logic.provable-listp (logic.strip-conclusions x) axioms thms atbl)
t))))
:hints(("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (enable definition-of-level2.proofp))))
(defthms-flag
;; We also show that any proof accepted by logic.proofp is still accepted,
;; i.e., level2.proofp is "strictly more capable" than logic.proofp.
;;
;; WARNING: THESE THEOREMS MUST BE LEFT DISABLED!
;;
;; Suppose this rule is enabled, and we are trying to prove (level2.proofp X
;; ...) Using this rule, we backchain and try to show (logic.proofp X ...),
;; which causes our forcing rules to kick in and assert that the subproofs of
;; X are acceptable using logic.proofp.
;;
;; But this is horrible; if any of the subproofs are derived rules that only
;; level2.proofp understands, we end up stuck in forcing rounds that we cannot
;; relieve. So, we should always be reasoning about some single layer and
;; never about previous layers.
:thms ((proof level2.proofp-when-logic.proofp
(implies (logic.proofp x axioms thms atbl)
(equal (level2.proofp x axioms thms atbl)
t)))
(t level2.proof-listp-when-logic.proof-listp
(implies (logic.proof-listp x axioms thms atbl)
(equal (level2.proof-listp x axioms thms atbl)
t))))
:hints (("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (enable definition-of-level2.proofp
definition-of-logic.proofp))))
(in-theory (disable level2.proofp-when-logic.proofp
level2.proof-listp-when-logic.proof-listp))
(defthm forcing-level2.proofp-of-logic.provable-witness
;; Corollary: Suppose F is any provable formula. Then, the witnessing
;; proof of F is acceptable by level2.proofp.
(implies (force (logic.provablep formula axioms thms atbl))
(equal (level2.proofp (logic.provable-witness formula axioms thms atbl) axioms thms atbl)
t))
:hints(("Goal" :in-theory (enable level2.proofp-when-logic.proofp))))
|
