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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 "level2")
(include-book "../build/prop-list")
(include-book "../build/disjoined-subset")
(include-book "../build/equal")
(include-book "../build/iff")
(include-book "../build/if")
(include-book "../build/not")
(include-book "../clauses/disjoined-update-clause-bldr")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(defund level3.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
;; Propositional rules
((equal method 'build.modus-ponens-list) (build.modus-ponens-list-okp x))
((equal method 'build.disjoined-modus-ponens-list) (build.disjoined-modus-ponens-list-okp x))
((equal method 'build.generic-subset) (build.generic-subset-okp x atbl))
((equal method 'build.multi-assoc-expansion) (build.multi-assoc-expansion-okp x atbl))
((equal method 'clause.aux-disjoined-update-clause-twiddle) (clause.aux-disjoined-update-clause-twiddle-okp x atbl))
;; Pequal rules
((equal method 'build.reflexivity) (build.reflexivity-okp x atbl))
((equal method 'build.commute-pequal) (build.commute-pequal-okp x atbl))
((equal method 'build.disjoined-commute-pequal) (build.disjoined-commute-pequal-okp x atbl))
((equal method 'build.commute-not-pequal) (build.commute-not-pequal-okp x atbl))
((equal method 'build.disjoined-commute-not-pequal) (build.disjoined-commute-not-pequal-okp x atbl))
((equal method 'build.substitute-into-not-pequal) (build.substitute-into-not-pequal-okp x atbl))
((equal method 'build.disjoined-substitute-into-not-pequal) (build.disjoined-substitute-into-not-pequal-okp x atbl))
((equal method 'build.transitivity-of-pequal) (build.transitivity-of-pequal-okp x atbl))
((equal method 'build.disjoined-transitivity-of-pequal) (build.disjoined-transitivity-of-pequal-okp x atbl))
((equal method 'build.not-nil-from-t) (build.not-nil-from-t-okp x atbl))
((equal method 'build.disjoined-not-nil-from-t) (build.disjoined-not-nil-from-t-okp x atbl))
((equal method 'build.not-t-from-nil) (build.not-t-from-nil-okp x atbl))
((equal method 'build.disjoined-not-t-from-nil) (build.disjoined-not-t-from-nil-okp x atbl))
;; Equal rules
((equal method 'build.equal-reflexivity) (build.equal-reflexivity-okp x atbl))
((equal method 'build.equal-t-from-not-nil) (build.equal-t-from-not-nil-okp x atbl))
((equal method 'build.disjoined-equal-t-from-not-nil) (build.disjoined-equal-t-from-not-nil-okp x atbl))
((equal method 'build.equal-nil-from-not-t) (build.equal-nil-from-not-t-okp x atbl))
((equal method 'build.disjoined-equal-nil-from-not-t) (build.disjoined-equal-nil-from-not-t-okp x atbl))
((equal method 'build.pequal-from-equal) (build.pequal-from-equal-okp x atbl))
((equal method 'build.disjoined-pequal-from-equal) (build.disjoined-pequal-from-equal-okp x atbl))
((equal method 'build.not-equal-from-not-pequal) (build.not-equal-from-not-pequal-okp x atbl))
((equal method 'build.disjoined-not-equal-from-not-pequal) (build.disjoined-not-equal-from-not-pequal-okp x atbl))
((equal method 'build.commute-equal) (build.commute-equal-okp x atbl))
((equal method 'build.disjoined-commute-equal) (build.disjoined-commute-equal-okp x atbl))
((equal method 'build.equal-from-pequal) (build.equal-from-pequal-okp x atbl))
((equal method 'build.disjoined-equal-from-pequal) (build.disjoined-equal-from-pequal-okp x atbl))
((equal method 'build.not-pequal-from-not-equal) (build.not-pequal-from-not-equal-okp x atbl))
((equal method 'build.disjoined-not-pequal-from-not-equal) (build.disjoined-not-pequal-from-not-equal-okp x atbl))
((equal method 'build.transitivity-of-equal) (build.transitivity-of-equal-okp x atbl))
((equal method 'build.disjoined-transitivity-of-equal) (build.disjoined-transitivity-of-equal-okp x atbl))
((equal method 'build.not-pequal-constants) (build.not-pequal-constants-okp x atbl))
;; If rules
((equal method 'build.if-when-not-nil) (build.if-when-not-nil-okp x atbl))
((equal method 'build.if-when-nil) (build.if-when-nil-okp x atbl))
;; Iff rules
((equal method 'build.iff-t-from-not-pequal-nil) (build.iff-t-from-not-pequal-nil-okp x atbl))
((equal method 'build.disjoined-iff-t-from-not-pequal-nil) (build.disjoined-iff-t-from-not-pequal-nil-okp x atbl))
((equal method 'build.not-pequal-nil-from-iff-t) (build.not-pequal-nil-from-iff-t-okp x atbl))
((equal method 'build.disjoined-not-pequal-nil-from-iff-t) (build.disjoined-not-pequal-nil-from-iff-t-okp x atbl))
((equal method 'build.iff-t-from-not-nil) (build.iff-t-from-not-nil-okp x atbl))
((equal method 'build.disjoined-iff-t-from-not-nil) (build.disjoined-iff-t-from-not-nil-okp x atbl))
((equal method 'build.iff-reflexivity) (build.iff-reflexivity-okp x atbl))
((equal method 'build.commute-iff) (build.commute-iff-okp x atbl))
((equal method 'build.disjoined-commute-iff) (build.disjoined-commute-iff-okp x atbl))
((equal method 'build.transitivity-of-iff) (build.transitivity-of-iff-okp x atbl))
((equal method 'build.disjoined-transitivity-of-iff) (build.disjoined-transitivity-of-iff-okp x atbl))
((equal method 'build.iff-from-pequal) (build.iff-from-pequal-okp x atbl))
((equal method 'build.disjoined-iff-from-pequal) (build.disjoined-iff-from-pequal-okp x atbl))
((equal method 'build.iff-from-equal) (build.iff-from-equal-okp x atbl))
((equal method 'build.disjoined-iff-from-equal) (build.disjoined-iff-from-equal-okp x atbl))
;; dead rules now, i think
;;((equal method 'build.pequal-nil-from-iff-nil) (build.pequal-nil-from-iff-nil-okp x atbl))
;;((equal method 'build.disjoined-pequal-nil-from-iff-nil) (build.disjoined-pequal-nil-from-iff-nil-okp x atbl))
;;((equal method 'build.not-equal-from-not-iff) (build.not-equal-from-not-iff-okp x atbl))
;;((equal method 'build.iff-nil-from-not-t) (build.iff-nil-from-not-t-okp x atbl))
;;((equal method 'build.disjoined-iff-nil-from-not-t) (build.disjoined-iff-nil-from-not-t-okp x atbl))
;; Not rules
((equal method 'build.disjoined-negative-lit-from-pequal-nil) (build.disjoined-negative-lit-from-pequal-nil-okp x atbl))
((equal method 'build.disjoined-pequal-nil-from-negative-lit) (build.disjoined-pequal-nil-from-negative-lit-okp x atbl))
((equal method 'build.disjoined-iff-when-not-nil) (build.disjoined-iff-when-not-nil-okp x atbl))
;; Extended propositional rules
;; Other rules
(t
(level2.step-okp x axioms thms atbl)))))
(defobligations level3.step-okp
(build.modus-ponens-list
build.disjoined-modus-ponens-list
build.generic-subset
build.multi-assoc-expansion
clause.aux-disjoined-update-clause-twiddle
build.reflexivity
build.commute-pequal
build.disjoined-commute-pequal
build.commute-not-pequal
build.disjoined-commute-not-pequal
build.substitute-into-not-pequal
build.disjoined-substitute-into-not-pequal
build.transitivity-of-pequal
build.disjoined-transitivity-of-pequal
build.not-nil-from-t
build.disjoined-not-nil-from-t
build.not-t-from-nil
build.disjoined-not-t-from-nil
build.equal-reflexivity
build.equal-t-from-not-nil
build.disjoined-equal-t-from-not-nil
build.equal-nil-from-not-t
build.disjoined-equal-nil-from-not-t
build.pequal-from-equal
build.disjoined-pequal-from-equal
build.not-equal-from-not-pequal
build.disjoined-not-equal-from-not-pequal
build.commute-equal
build.disjoined-commute-equal
build.equal-from-pequal
build.disjoined-equal-from-pequal
build.not-pequal-from-not-equal
build.disjoined-not-pequal-from-not-equal
build.transitivity-of-equal
build.disjoined-transitivity-of-equal
build.not-pequal-constants
build.if-when-not-nil
build.if-when-nil
build.iff-t-from-not-pequal-nil
build.disjoined-iff-t-from-not-pequal-nil
build.not-pequal-nil-from-iff-t
build.disjoined-not-pequal-nil-from-iff-t
build.iff-t-from-not-nil
build.disjoined-iff-t-from-not-nil
build.iff-reflexivity
build.commute-iff
build.disjoined-commute-iff
build.transitivity-of-iff
build.disjoined-transitivity-of-iff
build.iff-from-pequal
build.disjoined-iff-from-pequal
build.iff-from-equal
build.disjoined-iff-from-equal
build.disjoined-negative-lit-from-pequal-nil
build.disjoined-pequal-nil-from-negative-lit
build.disjoined-iff-when-not-nil))
(encapsulate
()
(local (in-theory (enable level3.step-okp)))
(defthm@ soundness-of-level3.step-okp
(implies (and (logic.appealp x)
(level3.step-okp x axioms thms atbl)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl)
(equal (cdr (lookup 'not atbl)) 1)
(equal (cdr (lookup 'equal atbl)) 2)
(equal (cdr (lookup 'iff atbl)) 2)
(equal (cdr (lookup 'if atbl)) 3)
(@obligations level3.step-okp))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t)))
(defthm level3.step-okp-when-level2.step-okp
;; This shows that our new step checker is "complete" in the sense that all
;; previously acceptable appeals are still acceptable.
(implies (level2.step-okp x axioms thms atbl)
(level3.step-okp x axioms thms atbl))
:hints(("Goal" :in-theory (enable level2.step-okp logic.appeal-step-okp))))
(defthm level3.step-okp-when-not-consp
(implies (not (consp x))
(equal (level3.step-okp x axioms thms atbl)
nil))
:hints(("Goal" :in-theory (enable logic.method)))))
(encapsulate
()
(defund level3.flag-proofp-aux (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 (level3.step-okp x axioms thms atbl)
(level3.flag-proofp-aux 'list (logic.subproofs x) axioms thms atbl))
(if (consp x)
(and (level3.flag-proofp-aux 'proof (car x) axioms thms atbl)
(level3.flag-proofp-aux 'list (cdr x) axioms thms atbl))
t)))
(definlined level3.proofp-aux (x axioms thms atbl)
(declare (xargs :guard (and (logic.appealp x)
(logic.formula-listp axioms)
(logic.formula-listp thms)
(logic.arity-tablep atbl))))
(level3.flag-proofp-aux 'proof x axioms thms atbl))
(definlined level3.proof-listp-aux (x axioms thms atbl)
(declare (xargs :guard (and (logic.appeal-listp x)
(logic.formula-listp axioms)
(logic.formula-listp thms)
(logic.arity-tablep atbl))))
(level3.flag-proofp-aux 'list x axioms thms atbl))
(defthmd definition-of-level3.proofp-aux
(equal (level3.proofp-aux x axioms thms atbl)
(and (level3.step-okp x axioms thms atbl)
(level3.proof-listp-aux (logic.subproofs x) axioms thms atbl)))
:rule-classes :definition
:hints(("Goal" :in-theory (enable level3.proofp-aux level3.proof-listp-aux level3.flag-proofp-aux))))
(defthmd definition-of-level3.proof-listp-aux
(equal (level3.proof-listp-aux x axioms thms atbl)
(if (consp x)
(and (level3.proofp-aux (car x) axioms thms atbl)
(level3.proof-listp-aux (cdr x) axioms thms atbl))
t))
:rule-classes :definition
:hints(("Goal" :in-theory (enable level3.proofp-aux level3.proof-listp-aux level3.flag-proofp-aux))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition level3.proofp-aux))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition demo.proof-list)))))
(defthm level3.proofp-aux-when-not-consp
(implies (not (consp x))
(equal (level3.proofp-aux x axioms thms atbl)
nil))
:hints(("Goal" :in-theory (enable definition-of-level3.proofp-aux))))
(defthm level3.proof-listp-aux-when-not-consp
(implies (not (consp x))
(equal (level3.proof-listp-aux x axioms thms atbl)
t))
:hints (("Goal" :in-theory (enable definition-of-level3.proof-listp-aux))))
(defthm level3.proof-listp-aux-of-cons
(equal (level3.proof-listp-aux (cons a x) axioms thms atbl)
(and (level3.proofp-aux a axioms thms atbl)
(level3.proof-listp-aux x axioms thms atbl)))
:hints (("Goal" :in-theory (enable definition-of-level3.proof-listp-aux))))
(defthms-flag
:thms ((proof booleanp-of-level3.proofp-aux
(equal (booleanp (level3.proofp-aux x axioms thms atbl))
t))
(t booleanp-of-level3.proof-listp-aux
(equal (booleanp (level3.proof-listp-aux x axioms thms atbl))
t)))
:hints(("Goal"
:in-theory (enable definition-of-level3.proofp-aux)
:induct (logic.appeal-induction flag x))))
(deflist level3.proof-listp-aux (x axioms thms atbl)
(level3.proofp-aux x axioms thms atbl)
:already-definedp t)
(defthms-flag
;; We now prove that level3.proofp-aux is sound. I.e., it only accepts appeals
;; whose conclusions are provable in the sense of logic.proofp.
:@contextp t
:shared-hyp (and (@obligations level3.step-okp)
(equal (cdr (lookup 'not atbl)) 1)
(equal (cdr (lookup 'equal atbl)) 2)
(equal (cdr (lookup 'iff atbl)) 2)
(equal (cdr (lookup 'if atbl)) 3))
:thms ((proof logic.provablep-when-level3.proofp-aux
(implies (and (logic.appealp x)
(level3.proofp-aux x axioms thms atbl))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t)))
(t logic.provable-listp-when-level3.proof-listp-aux
(implies (and (logic.appeal-listp x)
(level3.proof-listp-aux 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-level3.proofp-aux))))
(defthms-flag
;; We also show that any proof accepted by logic.proofp is still accepted,
;; i.e., level3.proofp-aux is "strictly more capable" than logic.proofp.
;; THESE THEOREMS MUST BE LEFT DISABLED!
:thms ((proof level3.proofp-aux-when-logic.proofp
(implies (logic.proofp x axioms thms atbl)
(equal (level3.proofp-aux x axioms thms atbl)
t)))
(t level3.proof-listp-aux-when-logic.proof-listp
(implies (logic.proof-listp x axioms thms atbl)
(equal (level3.proof-listp-aux x axioms thms atbl)
t))))
:hints (("Goal"
:induct (logic.appeal-induction flag x)
:in-theory (enable definition-of-level3.proofp-aux
definition-of-logic.proofp))))
(in-theory (disable level3.proofp-aux-when-logic.proofp
level3.proof-listp-aux-when-logic.proof-listp))
(defthm forcing-level3.proofp-aux-of-logic.provable-witness
;; Corollary: Suppose F is any provable formula. Then, the witnessing
;; proof of F is acceptable by level3.proofp-aux.
(implies (force (logic.provablep formula axioms thms atbl))
(equal (level3.proofp-aux (logic.provable-witness formula axioms thms atbl) axioms thms atbl)
t))
:hints(("Goal" :in-theory (enable level3.proofp-aux-when-logic.proofp))))
(definlined@ level3.proofp (x axioms thms atbl)
(declare (xargs :guard (and (logic.appealp x)
(logic.formula-listp axioms)
(logic.formula-listp thms)
(logic.arity-tablep atbl))))
(and (@obligations level3.step-okp)
(equal (cdr (lookup 'not atbl)) 1)
(equal (cdr (lookup 'equal atbl)) 2)
(equal (cdr (lookup 'iff atbl)) 2)
(equal (cdr (lookup 'if atbl)) 3)
(level3.proofp-aux x axioms thms atbl)))
(defthm booleanp-of-level3.proofp
(equal (booleanp (level3.proofp x axioms thms atbl))
t)
:hints(("Goal" :in-theory (enable level3.proofp))))
(defthm logic.provablep-when-level3.proofp
(implies (and (logic.appealp x)
(level3.proofp x axioms thms atbl))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t))
:hints(("Goal" :in-theory (enable level3.proofp))))
;; The reflective transition to Level 3.
;;
;; This is a particularly interesting transition because the generic subset
;; builder can be used to replace a whole lot of other builders.
(defund build.generic-subset-high (as bs proof)
(declare (xargs :guard (and (logic.formula-listp bs)
(subsetp as bs)
(consp as)
(logic.appealp proof)
(equal (logic.conclusion proof) (logic.disjoin-formulas as)))))
(if (equal as bs)
;; Important optimization since clause cleaning will apply this to each clause,
;; and often clauses are unchanged.
proof
(logic.appeal 'build.generic-subset
(logic.disjoin-formulas bs)
(list proof)
(list as bs))))
(defund@ build.multi-expansion-high (x as)
(declare (xargs :guard (and (logic.appealp x)
(logic.formula-listp as)
(@match (proof x A_i))
(memberp (@formula A_i) as))))
(build.generic-subset-high (@formulas A_i) as x))
(defund@ build.multi-or-expansion-step-high (base as)
(declare (xargs :guard (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v P Ai)))
(memberp (@formula Ai) as))))
(build.generic-subset-high (@formulas P Ai) (cons (@formula P) as) base))
(defund@ build.multi-or-expansion-high (base as)
(declare (xargs :guard (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v Ai Aj)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as))))
(build.generic-subset-high (@formulas Ai Aj) as base))
(defund build.rev-disjunction-high (x proof)
(declare (xargs :guard (and (consp x)
(logic.formula-listp x)
(logic.appealp proof)
(equal (logic.conclusion proof) (logic.disjoin-formulas x)))))
(build.generic-subset-high x (fast-rev x) proof))
(defund build.ordered-subset-high (sub sup proof)
(declare (xargs :guard (and (logic.formula-listp sup)
(logic.appealp proof)
(consp sub)
(ordered-subsetp sub sup)
(equal (logic.conclusion proof) (logic.disjoin-formulas sub)))))
(build.generic-subset-high sub sup proof))
(defund build.disjoined-rev-disjunction-high (x proof)
(declare (xargs :guard (and (consp x)
(logic.formula-listp x)
(logic.appealp proof)
(equal (logic.fmtype (logic.conclusion proof)) 'por*)
(equal (logic.vrhs (logic.conclusion proof))
(logic.disjoin-formulas x)))))
(let ((P (logic.vlhs (logic.conclusion proof))))
(build.generic-subset-high (cons P x) (cons P (fast-rev x)) proof)))
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