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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")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(dd.open "disjoined-subset.tex")
;; We introduce builders to prove B1 v ... v Bm from A1 v ... v An, where
;; [A1,...,An] is a subset of [B1,...,Bm].
;;
;; We actually introduce several such builders. The first of these is called
;; build.generic-subset, and can construct the proof in the general case. But
;; it is not very efficient. To address this, we introduce some variants for
;; the special purposes of proving list reversals, ordered subsets, and the
;; like. In the end, we create an "adaptive" builder which tries to
;; semi-intelligently choose an efficient proof method.
(defund@ build.multi-or-expansion-step (base as)
;; Derive P v A1 v ... v An from P v Ai
;; Note: this is basically like Shankar's function M2-Proof-Step
(declare (xargs :guard (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v P Ai)))
(memberp (@formula Ai) as))
:verify-guards nil))
(@extend ((formula (car as) A1))
(if (and (consp as)
(consp (cdr as)))
(cond ((equal (@formula A1) (@formula Ai))
(@derive ((v P A1) (@given base))
((v P (v A1 (v A2 (v dots An)))) (build.disjoined-right-expansion @- (logic.disjoin-formulas (cdr as))))))
(t
(@derive ((v P (v A2 (v dots An))) (build.multi-or-expansion-step base (cdr as)))
((v P (v A1 (v A2 (v dots An)))) (build.disjoined-left-expansion @- (@formula A1))))))
;; Else there is only one A, so it must be Ai
(@derive ((v P Ai) (logic.appeal-identity base))))))
(encapsulate
()
(local (in-theory (enable build.multi-or-expansion-step)))
(defthm build.multi-or-expansion-step-under-iff
(iff (build.multi-or-expansion-step base as)
t))
(defthm@ lemma-for-forcing-logic.appealp-of-build.multi-or-expansion-step
;; BOZO add to disjoined-subset.lisp
(implies (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v P Ai)))
(memberp (@formula Ai) as))
(and (logic.appealp (build.multi-or-expansion-step base as))
(equal (logic.conclusion (build.multi-or-expansion-step base as))
(logic.por (@formula P) (logic.disjoin-formulas as)))))
:rule-classes nil)
(defthm@ forcing-logic.appealp-of-build.multi-or-expansion-step
(implies (force (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v P Ai)))
(memberp (@formula Ai) as)))
(equal (logic.appealp (build.multi-or-expansion-step base as))
t))
:hints(("Goal" :use ((:instance lemma-for-forcing-logic.appealp-of-build.multi-or-expansion-step)))))
(defthm@ forcing-logic.conclusion-of-build.multi-or-expansion-step
(implies (force (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v P Ai)))
(memberp (@formula Ai) as)))
(equal (logic.conclusion (build.multi-or-expansion-step base as))
(logic.por (@formula P) (logic.disjoin-formulas as))))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :use ((:instance lemma-for-forcing-logic.appealp-of-build.multi-or-expansion-step)))))
(verify-guards build.multi-or-expansion-step)
(defthm@ forcing-logic.proofp-of-build.multi-or-expansion-step
(implies (force (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v P Ai)))
(memberp (@formula Ai) as)
;; ---
(logic.formula-list-atblp as atbl)
(logic.proofp base axioms thms atbl)))
(equal (logic.proofp (build.multi-or-expansion-step base as) axioms thms atbl)
t))))
(defund@ build.multi-or-expansion (base as)
;; Derive A1 v ... v An from Ai v Aj for any 1 <= i,j <= n
;; Note: this is basically like Shankar's function "M2-Proof"
(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))))
(if (consp as)
(@extend ((formula (car as) A1))
(cond ((equal (@formula Ai) (@formula Aj))
(@derive ((v Ai Ai) (@given base))
(Ai (build.contraction @-))
((v A1 (v dots An)) (build.multi-expansion @- as))))
((equal (@formula Ai) (@formula A1))
(@derive ((v A1 Aj) (@given base))
((v A1 (v dots An)) (build.multi-or-expansion-step @- (cdr as)))))
((equal (@formula Aj) (@formula A1))
(@derive ((v Ai A1) (@given base))
((v A1 Ai) (build.commute-or @-))
((v A1 (v dots An)) (build.multi-or-expansion-step @- (cdr as)))))
(t
(@derive ((v A2 (v dots An)) (build.multi-or-expansion base (cdr as)))
((v A1 (v A2 (v dots An))) (build.expansion (@formula A1) @-))))))
;; Degenerate case.
(logic.appeal-identity base)))
(encapsulate
()
(local (in-theory (enable build.multi-or-expansion)))
(defthm build.multi-or-expansion-under-iff
(iff (build.multi-or-expansion base as)
t))
(defthm@ forcing-logic.appealp-of-build.multi-or-expansion
(implies (force (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v Ai Aj)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as)))
(equal (logic.appealp (build.multi-or-expansion base as))
t)))
(defthm@ forcing-logic.conclusion-of-build.multi-or-expansion
(implies (force (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v Ai Aj)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as)))
(equal (logic.conclusion (build.multi-or-expansion base as))
(logic.disjoin-formulas as)))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-build.multi-or-expansion
(implies (force (and (logic.appealp base)
(logic.formula-listp as)
(@match (proof base (v Ai Aj)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as)
;; ---
(force (logic.formula-list-atblp as atbl))
(force (logic.proofp base axioms thms atbl))))
(equal (logic.proofp (build.multi-or-expansion base as) axioms thms atbl)
t))))
(defderiv build.generic-subset-step-lemma-1
:from ((proof x (v (v P A) P)))
:derive (v P A)
:proof (@derive ((v (v P A) P) (@given x))
((v P (v P A)) (build.commute-or @-))
((v (v P P) A) (build.associativity @-))
((v A (v P P)) (build.commute-or @-))
((v A P) (build.disjoined-contraction @-))
((v P A) (build.commute-or @-))))
(defund@ build.generic-subset-step (proof as)
;; Derive A1 v ... v An from (Ai v Aj) v (A1 v ... v An)
;;
;; I originally based this on Shankar's M3-Proof function, but since then
;; I've tweaked it slightly and it no now uses multi-or-expansion-step
;; instead of multi-or-expansion. This seems to save about 4-5% off of
;; proofs of the generic subset builder.
(declare (xargs :guard (and (logic.formula-listp as)
(logic.appealp proof)
(@match (proof proof (v (v Ai Aj) A1-An)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as)
(equal (@formula A1-An) (logic.disjoin-formulas as)))))
(@derive
((v (v Ai Aj) A1-An) (@given proof))
((v A1-An (v Ai Aj)) (build.commute-or @-))
((v (v A1-An Ai) Aj) (build.associativity @-))
((v (v A1-An Ai) A1-An) (build.multi-or-expansion-step @- as))
((v A1-An Ai) (build.generic-subset-step-lemma-1 @-))
((v A1-An A1-An) (build.multi-or-expansion-step @- as))
(A1-An (build.contraction @-))))
(encapsulate
()
(local (in-theory (enable build.generic-subset-step)))
(defthm build.generic-subset-step-under-iff
(iff (build.generic-subset-step proof as)
t))
(defthm@ forcing-logic.appealp-of-build.generic-subset-step
(implies (force (and (logic.formula-listp as)
(logic.appealp proof)
(@match (proof proof (v (v Ai Aj) A1-An)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as)
(equal (@formula A1-An) (logic.disjoin-formulas as))))
(equal (logic.appealp (build.generic-subset-step proof as))
t)))
(verify-guards build.generic-subset-step)
(defthm@ forcing-logic.conclusion-of-build.generic-subset-step
(implies (force (and (logic.formula-listp as)
(logic.appealp proof)
(@match (proof proof (v (v Ai Aj) A1-An)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as)
(equal (@formula A1-An) (logic.disjoin-formulas as))))
(equal (logic.conclusion (build.generic-subset-step proof as))
(logic.disjoin-formulas as)))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-build.generic-subset-step
(implies (force (and (logic.formula-listp as)
(logic.appealp proof)
(@match (proof proof (v (v Ai Aj) A1-An)))
(memberp (@formula Ai) as)
(memberp (@formula Aj) as)
(equal (@formula A1-An) (logic.disjoin-formulas as))
;; ---
(logic.formula-list-atblp as atbl)
(logic.proofp proof axioms thms atbl)))
(equal (logic.proofp (build.generic-subset-step proof as) axioms thms atbl)
t))))
;: build.generic-subset Cost: 12(n^2) + x
;: Derive B1 v ... v Bm
;: from A1 v ... v An,
;: where {A1 ... An} is a subset of {B1 ... Bm}
;:
;; Derivation. (defined by induction on n)
;;
;; Base Case: n = 1 Cost: n + x
;; 1. Ai Given
;; 2. B1 v ... v Bm Multi Expansion
;;
;; Base Case: n = 2 Cost: 6n + x
;; 1. Ai v Aj
;; 2. B1 v ... v Bm Multi Or Expansion
;;
;; Induction Case: n >= 3 Cost: 12n + x
;; Let F = (A1 v A2).
;; F v (A3 v (A4 v ... v An)) is a disjunction of n-1 formulas.
;; Inductively assume we can prove F v (B1 v ... v Bm) from a proof
;; of F v (A3 v ... v An).
;;
;; Derivation.
;; 1. A1 v (A2 v ... v An) Given
;; 2. (A1 v A2) v (A3 v ... v An) Associativity
;; 2'. F v (A3 v ... v An) (By the definition of F)
;; 3. F v (B1 v ... v Bm) Inductive Hypothesis
;; 3'. (A1 v A2) v (B1 v ... v Bm) (By the definition of F)
;; 4. B1 v B2 v ... v Bm Disjoined-Subset-Step
;;
;;
;; Note: This is basically like Shankar's function "M-Proof." We take two
;; lists of formulas, (A1 ... An) and (B1 ... Bm), where each A occurs in the
;; list of B's. We also take, as another input, a proof of A1 v A2 v ... v An.
;; We construct a proof of B1 v B2 v ... v Bm. Note that both n,m must be >=
;; 1; i.e., we must have non-empty lists of formulas.
(defund@ build.generic-subset (as bs proof)
;; Derive B1 v ... v Bm from A1 v ... v An, where as are a subset of bs
;; Note: this is basically like Shankar's function "M-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)))
:measure (len as)
:verify-guards nil))
(@extend ((formula (first as) A1)
(formula (second as) A2)
(formula (third as) A3))
(cond ((not (consp as))
;; Degenerate case -- this isn't even allowed by our guard
(logic.appeal-identity proof))
((not (consp (cdr as)))
;; as = [A1]
(@derive (A1 (@given proof))
((v B1 (v dots Bn)) (build.multi-expansion @- bs))))
((not (consp (cdr (cdr as))))
;; as = [A1 A2]
(@derive ((v A1 A2) (@given proof))
((v B1 (v dots Bn)) (build.multi-or-expansion @- bs))))
(t
;; as = [A1 A2 A3 ...]
(@derive ((v A1 (v A2 (v A3 (v dots An)))) (@given proof))
((v (v A1 A2) (v A3 (v dots An))) (build.associativity @-))
((v (v A1 A2) (v B1 (v dots Bm))) (build.generic-subset (cons (@formula (v A1 A2)) (cdr (cdr as)))
(cons (@formula (v A1 A2)) bs)
@-))
((v B1 (v dots Bm)) (build.generic-subset-step @- bs)))))))
(encapsulate
()
(local (in-theory (enable build.generic-subset)))
(defthm build.generic-subset-under-iff
(iff (build.generic-subset as bs proof)
t))
(defthm lemma-for-forcing-logic.appealp-of-build.generic-subset
(implies (and (logic.formula-listp bs)
(subsetp as bs)
(consp as)
(logic.appealp proof)
(equal (logic.conclusion proof) (logic.disjoin-formulas as)))
(and (logic.appealp (build.generic-subset as bs proof))
(equal (logic.conclusion (build.generic-subset as bs proof))
(logic.disjoin-formulas bs))))
:rule-classes nil)
(defthm forcing-logic.appealp-of-build.generic-subset
(implies (force (and (logic.formula-listp bs)
(logic.appealp proof)
(subsetp as bs)
(consp as)
(equal (logic.conclusion proof) (logic.disjoin-formulas as))))
(equal (logic.appealp (build.generic-subset as bs proof))
t))
:hints(("Goal" :use ((:instance lemma-for-forcing-logic.appealp-of-build.generic-subset)))))
(defthm forcing-logic.conclusion-of-build.generic-subset
(implies (force (and (logic.formula-listp bs)
(logic.appealp proof)
(subsetp as bs)
(consp as)
(equal (logic.conclusion proof) (logic.disjoin-formulas as))))
(equal (logic.conclusion (build.generic-subset as bs proof))
(logic.disjoin-formulas bs)))
:rule-classes ((:rewrite :backchain-limit-lst 0))
:hints(("Goal" :use ((:instance lemma-for-forcing-logic.appealp-of-build.generic-subset)))))
(verify-guards build.generic-subset)
(defthm forcing-logic.proofp-of-build.generic-subset
(implies (force (and (logic.formula-listp bs)
(logic.appealp proof)
(subsetp as bs)
(consp as)
(equal (logic.conclusion proof) (logic.disjoin-formulas as))
;; ---
(logic.formula-list-atblp bs atbl)
(logic.proofp proof axioms thms atbl)))
(equal (logic.proofp (build.generic-subset as bs proof) axioms thms atbl)
t))))
;; We now introduce build.rev-disjunction, which efficiently builds a proof of
;; (an v ... v a1) from a proof of (a1 v ... v an). We could already build
;; such proofs with our generic subset builder, but this builder is much more
;; efficient. For simple tests of the variety shown at the end of this
;; section, we obtain the following savings in terms of "rank":
;;
;; n generic-subset rev-disjunction savings
;; ----------------------------------------------------
;; 1 5 5 0%
;; 2 38 38 0%
;; 3 1,864 771 59%
;; 5 15,194 3,605 76%
;; 10 176,269 18,670 89%
;; 20 2,042,969 83,000 96%
;; 30 8,936,569 192,930 98%
;; ----------------------------------------------------
;;
;; Our proof construction mirrors the "revappend" function, and is implemented
;; as build.revappend-disjunction. Most users should call build.rev-disjunction
;; instead, which hides the accumulator.
(defund@ build.revappend-disjunction (todo done proof)
;; Derive tn v ... v t1 v d1 v ... v dm from (t1 v ... v tn) v (d1 v ... v dm)
(declare (xargs :guard (and (logic.formula-listp todo)
(logic.formula-listp done)
(or (consp todo) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp todo)
(consp done))
(logic.por (logic.disjoin-formulas todo)
(logic.disjoin-formulas done)))
((consp todo)
(logic.disjoin-formulas todo))
(t
(logic.disjoin-formulas done)))))
:verify-guards nil))
(if (and (consp todo)
(consp (cdr todo)))
(if (consp done)
(@derive
((v (v t1 t2-tn) d1-dm) (@given proof))
((v d1-dm (v t1 t2-tn)) (build.commute-or @-))
((v (v d1-dm t1) t2-tn) (build.associativity @-))
((v t2-tn (v d1-dm t1)) (build.commute-or @-))
((v t2-tn (v t1 d1-dm)) (build.disjoined-commute-or @-))
((v tn-t1 dm-d1) (build.revappend-disjunction (cdr todo) (cons (car todo) done) @-)))
(@derive
((v t1 t2-n) (@given proof))
((v t2-n t1) (build.commute-or @-))
(tn-t1 (build.revappend-disjunction (cdr todo) (list (car todo)) @-))))
;; Otherwise, the todo list is only one long, so we already have the proof
;; we were looking for.
(logic.appeal-identity proof)))
(encapsulate
()
(local (in-theory (enable build.revappend-disjunction)))
(defthm build.revappend-disjunction-under-iff
(iff (build.revappend-disjunction todo done proof)
t))
(local (defthm lemma
(implies (and (logic.formula-listp todo)
(logic.formula-listp done)
(or (consp todo) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp todo)
(consp done))
(logic.por (logic.disjoin-formulas todo)
(logic.disjoin-formulas done)))
((consp todo)
(logic.disjoin-formulas todo))
(t
(logic.disjoin-formulas done)))))
(and (logic.appealp (build.revappend-disjunction todo done proof))
(equal (logic.conclusion (build.revappend-disjunction todo done proof))
(logic.disjoin-formulas (app (rev todo) done)))))))
(defthm forcing-logic.appealp-of-build.revappend-disjunction
(implies (force (and (logic.formula-listp todo)
(logic.formula-listp done)
(or (consp todo) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp todo)
(consp done))
(logic.por (logic.disjoin-formulas todo)
(logic.disjoin-formulas done)))
((consp todo)
(logic.disjoin-formulas todo))
(t
(logic.disjoin-formulas done))))))
(equal (logic.appealp (build.revappend-disjunction todo done proof))
t)))
(defthm forcing-logic.conclusion-of-build.revappend-disjunction
(implies (force (and (logic.formula-listp todo)
(logic.formula-listp done)
(or (consp todo) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp todo)
(consp done))
(logic.por (logic.disjoin-formulas todo)
(logic.disjoin-formulas done)))
((consp todo)
(logic.disjoin-formulas todo))
(t
(logic.disjoin-formulas done))))))
(equal (logic.conclusion (build.revappend-disjunction todo done proof))
(logic.disjoin-formulas (app (rev todo) done))))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(verify-guards build.revappend-disjunction)
(defthm forcing-logic.proofp-of-build.revappend-disjunction
(implies (force (and (logic.formula-listp todo)
(logic.formula-listp done)
(or (consp todo) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp todo)
(consp done))
(logic.por (logic.disjoin-formulas todo)
(logic.disjoin-formulas done)))
((consp todo)
(logic.disjoin-formulas todo))
(t
(logic.disjoin-formulas done))))
;; ---
(logic.proofp proof axioms thms atbl)))
(equal (logic.proofp (build.revappend-disjunction todo done proof) axioms thms atbl)
t))))
(defund build.rev-disjunction (x proof)
;; Derive tn v ... v t1 from t1 v ... v tn
(declare (xargs :guard (and (consp x)
(logic.formula-listp x)
(logic.appealp proof)
(equal (logic.conclusion proof) (logic.disjoin-formulas x)))
;; As far as Milawa is concerned, build.rev-disjunction is going to be
;; an alias for build.generic-subset. This way we don't have to write
;; any proofs about it, we can just let it expand.
:export (build.generic-subset x (fast-rev x) proof)))
(build.revappend-disjunction x nil proof))
(encapsulate
()
(local (in-theory (enable build.rev-disjunction)))
(defthm build.rev-disjunction-under-iff
(iff (build.rev-disjunction x proof)
t))
(defthm forcing-logic.appealp-of-build.rev-disjunction
(implies (force (and (consp x)
(logic.formula-listp x)
(logic.appealp proof)
(equal (logic.conclusion proof) (logic.disjoin-formulas x))))
(equal (logic.appealp (build.rev-disjunction x proof))
t)))
(defthm forcing-logic.conclusion-of-build.rev-disjunction
(implies (force (and (consp x)
(logic.formula-listp x)
(logic.appealp proof)
(equal (logic.conclusion proof) (logic.disjoin-formulas x))))
(equal (logic.conclusion (build.rev-disjunction x proof))
(logic.disjoin-formulas (rev x))))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm forcing-logic.proofp-of-build.rev-disjunction
(implies (force (and (consp x)
(logic.formula-listp x)
(logic.appealp proof)
(equal (logic.conclusion proof) (logic.disjoin-formulas x))
;; ---
(logic.proofp proof axioms thms atbl)))
(equal (logic.proofp (build.rev-disjunction x proof) axioms thms atbl)
t))))
;; Our test data above was obtained by just running this let expression and
;; commenting out the appropriate number of lines.
;;
;; (let* ((formulas (list
;; (logic.pequal 'a1 'a1-prime)
;; (logic.pequal 'a2 'a2-prime)
;; (logic.pequal 'a3 'a3-prime)
;; (logic.pequal 'a4 'a4-prime)
;; (logic.pequal 'a5 'a5-prime)
;; (logic.pequal 'a6 'a6-prime)
;; (logic.pequal 'a7 'a7-prime)
;; (logic.pequal 'a8 'a8-prime)
;; (logic.pequal 'a9 'a9-prime)
;; (logic.pequal 'a10 'a10-prime)
;; ;(logic.pequal 'a11 'a11-prime)
;; ;(logic.pequal 'a12 'a12-prime)
;; ;(logic.pequal 'a13 'a13-prime)
;; ;(logic.pequal 'a14 'a14-prime)
;; ;(logic.pequal 'a15 'a15-prime)
;; ;(logic.pequal 'a16 'a16-prime)
;; ;(logic.pequal 'a17 'a17-prime)
;; ;(logic.pequal 'a18 'a18-prime)
;; ;(logic.pequal 'a19 'a19-prime)
;; ;(logic.pequal 'a20 'a20-prime)
;; ;(logic.pequal 'a21 'a21-prime)
;; ;(logic.pequal 'a22 'a22-prime)
;; ;(logic.pequal 'a23 'a23-prime)
;; ;(logic.pequal 'a24 'a24-prime)
;; ;(logic.pequal 'a25 'a25-prime)
;; ;(logic.pequal 'a26 'a26-prime)
;; ;(logic.pequal 'a27 'a27-prime)
;; ;(logic.pequal 'a28 'a28-prime)
;; ;(logic.pequal 'a29 'a29-prime)
;; ;(logic.pequal 'a30 'a30-prime)
;; ))
;; (axiom (build.axiom (logic.disjoin-formulas formulas)))
;; (proof1 (build.generic-subset formulas (rev formulas) axiom))
;; (proof2 (build.rev-disjunction formulas axiom)))
;; (list (list 'OK (equal (logic.conclusion proof1)
;; (logic.conclusion proof2)))
;; (list 'generic-rank (rank proof1))
;; (list 'rev-rank (rank proof2))))
;; We now introduce the ordered subset bldr, which is more efficient than the
;; generic builder when applied to large, ordered subsets (such as are obtained
;; by applying remove-duplicates or remove-all to a list). Note that for small
;; lists, the generic builder is actually better.
;;
;; n generic-subset ordered-subset savings
;; ----------------------------------------------------------------
;; 1 64 818 Lose
;; 2 523 6,626 Lose
;; 3 8,557 17,378 Lose
;; 4 25,380 33,074 Lose
;; 5 53,714 55,555 Lose
;; 6 103,552 79,298 23%
;; 10 579,540 231,074 60%
;; 15 2,306,725 532,034 77%
;; 20 6,263,860 956,594 85%
;;
;; Note that our test data comes from the experiment shown at the end of this
;; section.
(defund@ build.ordered-subset-aux (sub sup done proof)
;; Derive [bm,...,b1,d1,...,dk] from [d1,...,dk] v [a1,...,an]
;; where [a1,...,an] is an ordered subset of [b1,...,bm]
(declare (xargs :guard (and (logic.formula-listp sup)
(ordered-subsetp sub sup)
(logic.formula-listp done)
(or (consp sub) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp sub) (consp done))
(logic.por (logic.disjoin-formulas done)
(logic.disjoin-formulas sub)))
((consp sub)
(logic.disjoin-formulas sub))
(t
(logic.disjoin-formulas done)))))
:verify-guards nil
:measure (+ (rank sub) (rank sup))))
(@extend ((formula (car sub) A1)
(formula (car sup) B1))
(cond ((and (consp sub)
(consp sup))
(if (consp done)
(if (equal (@formula A1) (@formula B1))
(if (consp (cdr sub))
(@derive
;; Case 1: [a1,a2...], [b1=a1,...], [d1,...]
;; Step Goal: [a1,d1,...,dk] v [a2,...,an]
((v D1-Dk (v A1 A2-An)) (@given proof))
((v D1-Dk (v A2-An A1)) (build.disjoined-commute-or @-))
((v (v D1-Dk A2-An) A1) (build.associativity @-))
((v A1 (v D1-Dk A2-An)) (build.commute-or @-))
((v (v A1 D1-Dk) A2-An) (build.associativity @-))
((v Bm-B1 D1-Dk) (build.ordered-subset-aux (cdr sub) (cdr sup) (cons (car sup) done) @-)))
(@derive
;; Case 2: [a1], [b1=a1,...], [d1,...]
;; Step Goal: [a1,d1,...,dk]
((v D1-Dk A1) (@given proof))
((v A1 D1-Dk) (build.commute-or @-))
((v Bm-B1 D1-Dk) (build.ordered-subset-aux (cdr sub) (cdr sup) (cons (car sup) done) @-))))
;; Case 3: [a1,...], [b1!=a1,...], [d1,...]
;; Step Goal: [b1,d1,...,dk] v [a1,...,an]
(@derive
((v D1-Dk A1-An) (@given proof))
((v B1 (v D1-Dk A1-An)) (build.expansion (@formula B1) @-))
((v (v B1 D1-Dk) A1-An) (build.associativity @-))
((v Bm-B1 D1-Dk) (build.ordered-subset-aux sub (cdr sup) (cons (car sup) done) @-))))
(if (equal (@formula A1) (@formula B1))
;; Case 4: [a1,...], [b1=a1,...], done = empty
;; Step Goal: a1..n
(@derive
(A1-An (@given proof))
(Bm-B1 (build.ordered-subset-aux (cdr sub) (cdr sup) (cons (car sup) done) @-)))
;; Case 5: [a1,...], [b1!=a1,...], done = empty
;; Step Goal: b1 v (a1..n)
(@derive
(A1-An (@given proof))
((v B1 A1-An) (build.expansion (@formula B1) @-))
(Bm-B1 (build.ordered-subset-aux sub (cdr sup) (cons (car sup) done) @-))))))
((consp sup)
;; Case 6: sub = empty, [b1,...], done = [d1,...] (done is nonempty via our guard)
;; Step Goal: [b1,d1,...,dk]
(@derive
(D1-Dk (@given proof))
((v B1 D1-Dk) (build.expansion (@formula B1) @-))
((v Bm-B1 D1-Dk) (build.ordered-subset-aux sub (cdr sup) (cons (car sup) done) @-))))
(t
;; Case 7: sup = empty
(logic.appeal-identity proof)))))
(encapsulate
()
(local (in-theory (enable build.ordered-subset-aux)))
(defthm build.ordered-subset-aux-under-iff
(iff (build.ordered-subset-aux sub sup done proof)
t))
(defthm forcing-logic.appealp-of-build.ordered-subset-aux
(implies (force (and (logic.formula-listp sup)
(ordered-subsetp sub sup)
(logic.formula-listp done)
(or (consp sub) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp sub) (consp done))
(logic.por (logic.disjoin-formulas done)
(logic.disjoin-formulas sub)))
((consp sub)
(logic.disjoin-formulas sub))
(t
(logic.disjoin-formulas done))))))
(equal (logic.appealp (build.ordered-subset-aux sub sup done proof))
t)))
(defthm forcing-logic.conclusion-of-build.ordered-subset-aux
(implies (force (and (logic.formula-listp sup)
(ordered-subsetp sub sup)
(logic.formula-listp done)
(or (consp sub) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp sub) (consp done))
(logic.por (logic.disjoin-formulas done)
(logic.disjoin-formulas sub)))
((consp sub)
(logic.disjoin-formulas sub))
(t
(logic.disjoin-formulas done))))))
(equal (logic.conclusion (build.ordered-subset-aux sub sup done proof))
(logic.disjoin-formulas (app (rev sup) done)))))
(verify-guards build.ordered-subset-aux)
(defthm forcing-logic.proofp-of-build.ordered-subset-aux
(implies (force (and (logic.formula-listp sup)
(ordered-subsetp sub sup)
(logic.formula-listp done)
(or (consp sub) (consp done))
(logic.appealp proof)
(equal (logic.conclusion proof)
(cond ((and (consp sub) (consp done))
(logic.por (logic.disjoin-formulas done)
(logic.disjoin-formulas sub)))
((consp sub)
(logic.disjoin-formulas sub))
(t
(logic.disjoin-formulas done))))
;; ---
(logic.formula-list-atblp sup atbl)
(logic.proofp proof axioms thms atbl)))
(equal (logic.proofp (build.ordered-subset-aux sub sup done proof) axioms thms atbl)
t))))
(defund build.ordered-subset (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.rev-disjunction (fast-rev sup)
(build.ordered-subset-aux sub sup nil proof)))
(encapsulate
()
(local (in-theory (enable build.ordered-subset)))
(defthm build.ordered-subset-under-iff
(iff (build.ordered-subset sub sup proof)
t))
(defthm forcing-logic.appealp-of-build.ordered-subset
(implies (force (and (logic.formula-listp sup)
(logic.appealp proof)
(consp sub)
(ordered-subsetp sub sup)
(equal (logic.conclusion proof) (logic.disjoin-formulas sub))))
(equal (logic.appealp (build.ordered-subset sub sup proof))
t)))
(defthm forcing-logic.conclusion-of-build.ordered-subset
(implies (force (and (logic.formula-listp sup)
(logic.appealp proof)
(consp sub)
(ordered-subsetp sub sup)
(equal (logic.conclusion proof) (logic.disjoin-formulas sub))))
(equal (logic.conclusion (build.ordered-subset sub sup proof))
(logic.disjoin-formulas sup))))
(defthm forcing-logic.proofp-of-build.ordered-subset
(implies (force (and (logic.formula-listp sup)
(logic.appealp proof)
(consp sub)
(ordered-subsetp sub sup)
(equal (logic.conclusion proof) (logic.disjoin-formulas sub))
;; ---
(logic.formula-list-atblp sup atbl)
(logic.proofp proof axioms thms atbl)))
(equal (logic.proofp (build.ordered-subset sub sup proof) axioms thms atbl)
t))))
;; Test data for the build.ordered-subset was taken from running this let*
;; expression with the appropriate lines commented out.
;; (let* ((sub (list
;; (logic.pequal 'a1 'a1-prime)
;; (logic.pequal 'a2 'a2-prime)
;; (logic.pequal 'a3 'a3-prime)
;; (logic.pequal 'a4 'a4-prime)
;; (logic.pequal 'a5 'a5-prime)
;; (logic.pequal 'a6 'a6-prime)
;; (logic.pequal 'a7 'a7-prime)
;; (logic.pequal 'a8 'a8-prime)
;; (logic.pequal 'a9 'a9-prime)
;; (logic.pequal 'a10 'a10-prime)
;; (logic.pequal 'a11 'a11-prime)
;; (logic.pequal 'a12 'a12-prime)
;; (logic.pequal 'a13 'a13-prime)
;; (logic.pequal 'a14 'a14-prime)
;; (logic.pequal 'a15 'a15-prime)
;; (logic.pequal 'a16 'a16-prime)
;; (logic.pequal 'a17 'a17-prime)
;; (logic.pequal 'a18 'a18-prime)
;; (logic.pequal 'a19 'a19-prime)
;; (logic.pequal 'a20 'a20-prime)
;; ))
;; (sup (list
;; (logic.pequal 'c1 'c1-prime)
;; (logic.pequal 'a1 'a1-prime)
;; (logic.pequal 'b1 'b1-prime)
;; (logic.pequal 'c2 'c2-prime)
;; (logic.pequal 'a2 'a2-prime)
;; (logic.pequal 'b2 'b2-prime)
;; (logic.pequal 'c3 'c3-prime)
;; (logic.pequal 'a3 'a3-prime)
;; (logic.pequal 'b3 'b3-prime)
;; (logic.pequal 'c4 'c4-prime)
;; (logic.pequal 'a4 'a4-prime)
;; (logic.pequal 'b4 'b4-prime)
;; (logic.pequal 'c5 'c5-prime)
;; (logic.pequal 'a5 'a5-prime)
;; (logic.pequal 'b5 'b5-prime)
;; (logic.pequal 'c6 'c6-prime)
;; (logic.pequal 'a6 'a6-prime)
;; (logic.pequal 'b6 'b6-prime)
;; (logic.pequal 'c7 'c7-prime)
;; (logic.pequal 'a7 'a7-prime)
;; (logic.pequal 'b7 'b7-prime)
;; (logic.pequal 'c8 'c8-prime)
;; (logic.pequal 'a8 'a8-prime)
;; (logic.pequal 'b8 'b8-prime)
;; (logic.pequal 'c9 'c9-prime)
;; (logic.pequal 'a9 'a9-prime)
;; (logic.pequal 'b9 'b9-prime)
;; (logic.pequal 'c10 'c10-prime)
;; (logic.pequal 'a10 'a10-prime)
;; (logic.pequal 'b10 'b10-prime)
;; (logic.pequal 'c11 'c11-prime)
;; (logic.pequal 'a11 'a11-prime)
;; (logic.pequal 'b11 'b11-prime)
;; (logic.pequal 'c12 'c12-prime)
;; (logic.pequal 'a12 'a12-prime)
;; (logic.pequal 'b12 'b12-prime)
;; (logic.pequal 'c13 'c13-prime)
;; (logic.pequal 'a13 'a13-prime)
;; (logic.pequal 'b13 'b13-prime)
;; (logic.pequal 'c14 'c14-prime)
;; (logic.pequal 'a14 'a14-prime)
;; (logic.pequal 'b14 'b14-prime)
;; (logic.pequal 'c15 'c15-prime)
;; (logic.pequal 'a15 'a15-prime)
;; (logic.pequal 'b15 'b15-prime)
;; (logic.pequal 'c16 'c16-prime)
;; (logic.pequal 'a16 'a16-prime)
;; (logic.pequal 'b16 'b16-prime)
;; (logic.pequal 'c17 'c17-prime)
;; (logic.pequal 'a17 'a17-prime)
;; (logic.pequal 'b17 'b17-prime)
;; (logic.pequal 'c18 'c18-prime)
;; (logic.pequal 'a18 'a18-prime)
;; (logic.pequal 'b18 'b18-prime)
;; (logic.pequal 'c19 'c19-prime)
;; (logic.pequal 'a19 'a19-prime)
;; (logic.pequal 'b19 'b19-prime)
;; (logic.pequal 'c20 'c20-prime)
;; (logic.pequal 'a20 'a20-prime)
;; (logic.pequal 'b20 'b20-prime)
;; ))
;; (proof (build.axiom (logic.disjoin-formulas sub)))
;; (oproof (build.ordered-subset sub sup proof))
;; (sproof (build.generic-subset sub sup proof)))
;; (list
;; (list 'ok (equal (logic.conclusion oproof) (logic.conclusion sproof)))
;; (list 'rank-o (rank oproof))
;; (list 'rank-s (rank sproof))))
;; Finally we introduce our "adaptive" builder. We check for certain common
;; cases.
(defund build.disjoined-subset (as bs proof)
(declare (xargs :guard (and (logic.formula-listp bs)
(subsetp as bs)
(logic.appealp proof)
(consp as)
(equal (logic.conclusion proof)
(logic.disjoin-formulas as)))
:export (build.generic-subset as bs proof)))
(cond ((equal bs as)
;; The best case: we can just reuse the proof verbatim.
(logic.appeal-identity proof))
((equal bs (fast-rev as))
;; Another good case, we can use our reversal builder.
(build.rev-disjunction as proof))
((ordered-subsetp as bs)
;; Here we may be able to optimize. As a heuristic, if there are more
;; than 5 members in the subset or more than 10 in the superset, we
;; just use the ordered builder since the generic builder is probably
;; not as efficient. Otherwise, we'll try both builders and just use
;; whichever proof turns out to be smaller.
(if (or (< 5 (len as))
(< 10 (len bs)))
(build.ordered-subset as bs proof)
(let* ((proof1 (build.generic-subset as bs proof))
(proof2 (build.ordered-subset as bs proof))
(rank1 (rank proof1))
(rank2 (rank proof2)))
(ACL2::prog2$
(ACL2::cw "BDJS n,m,generic,ord = ~x0, ~x1, ~x2, ~x3~%"
(len as) (len bs) rank1 rank2)
(if (< (rank proof1) (rank proof2))
proof1
proof2)))))
(t
;; If we get here, we have no special tricks to use. We fall back to
;; using the generic subset builder.
(build.generic-subset as bs proof))))
(encapsulate
()
(local (in-theory (enable build.disjoined-subset)))
(defthm build.disjoined-subset-under-iff
(iff (build.disjoined-subset as bs proof)
t))
(defthm forcing-logic.appealp-of-build.disjoined-subset
(implies (force (and (logic.formula-listp bs)
(subsetp as bs)
(logic.appealp proof)
(consp as)
(equal (logic.conclusion proof)
(logic.disjoin-formulas as))))
(equal (logic.appealp (build.disjoined-subset as bs proof))
t)))
(defthm forcing-logic.conclusion-of-build.disjoined-subset
(implies (force (and (logic.formula-listp bs)
(subsetp as bs)
(logic.appealp proof)
(consp as)
(equal (logic.conclusion proof)
(logic.disjoin-formulas as))))
(equal (logic.conclusion (build.disjoined-subset as bs proof))
(logic.disjoin-formulas bs)))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm forcing-logic.proofp-of-build.disjoined-subset
(implies (force (and (logic.formula-listp bs)
(subsetp as bs)
(logic.appealp proof)
(consp as)
(equal (logic.conclusion proof)
(logic.disjoin-formulas as))
;; ---
(logic.formula-list-atblp bs atbl)
(logic.proofp proof axioms thms atbl)))
(equal (logic.proofp (build.disjoined-subset as bs proof) axioms thms atbl)
t))))
;; We can now also talk about building a lists of proofs X1,...,Xn from another
;; list of proofs Y1,...,Ym, where "each Xi is a superset of some Yi".
(defund build.all-superset-of-some (x y proofs)
(declare (xargs :guard (and (logic.formula-list-listp x)
(logic.formula-list-listp y)
(cons-listp y)
(all-superset-of-somep x y)
(logic.appeal-listp proofs)
(subsetp (logic.disjoin-each-formula-list y)
(logic.strip-conclusions proofs)))))
(if (consp x)
(let* ((subset (find-subset (car x) y))
(proof (logic.find-proof (logic.disjoin-formulas subset) proofs)))
;; Proof is now a proof of subset, which is a subset of (car x).
;; We can just expand the proof to get a proof of (car x).
(cons (build.disjoined-subset subset (car x) proof)
(build.all-superset-of-some (cdr x) y proofs)))
nil))
(encapsulate
()
(local (in-theory (enable build.all-superset-of-some)))
(defthm forcing-logic.appeal-listp-of-build.all-superset-of-some
(implies (force (and (logic.formula-list-listp x)
(logic.formula-list-listp y)
(cons-listp y)
(all-superset-of-somep x y)
(logic.appeal-listp proofs)
(subsetp (logic.disjoin-each-formula-list y)
(logic.strip-conclusions proofs))))
(equal (logic.appeal-listp (build.all-superset-of-some x y proofs))
t)))
(defthm forcing-logic.strip-conclusions-of-build.all-superset-of-some
(implies (force (and (logic.formula-list-listp x)
(logic.formula-list-listp y)
(cons-listp y)
(all-superset-of-somep x y)
(logic.appeal-listp proofs)
(subsetp (logic.disjoin-each-formula-list y)
(logic.strip-conclusions proofs))))
(equal (logic.strip-conclusions (build.all-superset-of-some x y proofs))
(logic.disjoin-each-formula-list x)))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm forcing-logic.proof-listp-of-build.all-superset-of-some
(implies (force (and (logic.formula-list-listp x)
(logic.formula-list-listp y)
(cons-listp y)
(all-superset-of-somep x y)
(logic.appeal-listp proofs)
(subsetp (logic.disjoin-each-formula-list y)
(logic.strip-conclusions proofs))
;; ---
(logic.formula-list-list-atblp x atbl)
(logic.proof-listp proofs axioms thms atbl)))
(equal (logic.proof-listp (build.all-superset-of-some x y proofs) axioms thms atbl)
t))))
(defund build.generic-subset-okp (x atbl)
(declare (xargs :guard (and (logic.appealp x)
(logic.arity-tablep atbl))))
(let ((method (logic.method x))
(conclusion (logic.conclusion x))
(subproofs (logic.subproofs x))
(extras (logic.extras x)))
(and (equal method 'build.generic-subset)
(equal (len subproofs) 1)
(tuplep 2 extras)
(let ((as (first extras))
(bs (second extras)))
(and (logic.formula-listp bs)
(logic.formula-list-atblp bs atbl)
(subsetp as bs)
(consp as)
(equal conclusion (logic.disjoin-formulas bs))
(equal (logic.conclusion (first subproofs))
(logic.disjoin-formulas as)))))))
(encapsulate
()
(local (in-theory (enable build.generic-subset-okp)))
(defthm booleanp-of-build.generic-subset-okp
(equal (booleanp (build.generic-subset-okp x atbl))
t)
:hints(("goal" :in-theory (disable forcing-true-listp-of-logic.subproofs))))
(defthm build.generic-subset-okp-of-logic.appeal-identity
(equal (build.generic-subset-okp (logic.appeal-identity x) atbl)
(build.generic-subset-okp x atbl))
:hints(("goal" :in-theory (disable forcing-true-listp-of-logic.subproofs))))
(defthm forcing-soundness-of-build.generic-subset-okp
(implies (and (build.generic-subset-okp x atbl)
(force (and (logic.appealp x)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl))))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t))
:hints (("Goal"
:use ((:instance forcing-logic.provablep-when-logic.proofp
(x (build.generic-subset
(first (logic.extras x))
(second (logic.extras x))
(logic.provable-witness (logic.conclusion (car (logic.subproofs x)))
axioms thms atbl)))))))))
(dd.close)
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