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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 "pequal-list")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(dd.open "lambda-rules.tex")
(dd.section "$\\lambda$ reduction rules")
(defderiv build.dual-substitution-lemma-1
:from ((proof x (= (? a) (? b)))
(proof y (= (? b) (? d)))
(proof z (= (? c) (? d))))
:derive (= (? a) (? c))
:proof (@derive
((= (? a) (? b)) (@given x))
((= (? b) (? d)) (@given y))
((= (? a) (? d)) (build.transitivity-of-pequal @-- @-) *1)
((= (? c) (? d)) (@given z))
((= (? d) (? c)) (build.commute-pequal @-))
((= (? a) (? c)) (build.transitivity-of-pequal *1 @-))))
(defun build.flag-dual-substitution (flag x vars proofs)
(declare (xargs :guard (and (if (equal flag 'term)
(logic.termp x)
(and (equal flag 'list)
(logic.term-listp x)))
(logic.variable-listp vars)
(uniquep vars)
(logic.appeal-listp proofs)
(same-lengthp vars proofs)
(logic.all-atomicp (logic.strip-conclusions proofs)))
:verify-guards nil))
(if (equal flag 'term)
(cond ((logic.constantp x)
;; If x is a constant, then x/[Vi<-Ti] = x and x/[Vi<-Si] = x, so our goal
;; is to prove x = x, which is trivial by reflexivity.
(build.reflexivity x))
((logic.variablep x)
;; If x is a variable, then either it is some vi or not.
(let ((index (first-index x vars)))
(if (equal index (len vars))
;; If it wasn't found among the vi, then as in the constant case we
;; have x/[Vi<-Ti] = x and x/[Vi<-Si] = x, so our goal is to prove x
;; = x, which is again trivial by reflexivity.
(build.reflexivity x)
;; Else x is exactly some vi, and we have x/[Vi<-Ti] = ti and
;; x/[Vi<-Si] = si, so our goal is to prove ti = si, which is trivial
;; by the ith proof.
(logic.appeal-identity (nth index proofs)))))
((logic.functionp x)
;; If x is (f a1 ... an), then
;; x/[Vi<-Ti] = (f a1/[Vi<-Ti] ... an/[Vi<-Ti]), and
;; x/[Vi<-Si] = (f a1/[Vi<-Si] ... an/[Vi<-Si]).
;;
;; For each ai, we can recursively prove ai/[Vi<-Ti] = ai/[Vi<-Si], then
;; by pequal-by-args we can prove our goal.
(build.pequal-by-args (logic.function-name x)
(build.flag-dual-substitution 'list (logic.function-args x) vars proofs)))
((logic.lambdap x)
;; If x is ((lambda (x1...xn) B) a1...an), then by beta reduction we have
;;
;; To begin, we recursively construct proofs of:
;; ai/[Vi<-Ti] = ai/[Vi<-Si] (*1)
;;
;; To make our next steps more clear, let
;; ci = ai/[Vi<-Ti], and
;; di = ai/[Vi<-Si]
;;
;; In other words, the proofs in *1 establish:
;; ci = di
;;
;; Using these proofs, we can recursively prove B/[Xi<-Ci] = B/[Xi<-Di].
;; This is well-founded because we are recurring over only the structure
;; of x, and B is contained in x and hence is smaller than x. If we now
;; expand away the definitions of Ci and Di, we have proven:
;;
;; B/[Xi<-(Ai/[Vi<-Ti])] = B/[Xi<-(Ai/[Vi<-Si])]
;;
;; But since x is a lambda, all of the variables in B must be among the
;; Xi, so the substitutions can be teased apart. We find that the proof
;; above is exactly a proof of:
;;
;; (B/[Xi<-Ai])/[Vi<-Ti] = (B/[Xi<-Ai])/[Vi<-Si] (*2)
;;
;; We can now derive our goal, x/[Vi<-Ti] = x/[Vi<-Si], as follows:
;;
;; 1. x = B/[Xi<-Ai] Beta Reduction
;; 2. x/[Vi<-Ti] = (B/[Xi<-Ai])/[Vi<-Ti] Instantiation; 1 a=b
;; 3. (B/[Xi<-Ai])/[Vi<-Ti] = (B/[Xi<-Ai])/[Vi<-Si] By *2 b=d
;; 4. x/[Vi<-Si] = (B/[Xi<-Ai])/[Vi<-Si] Instantiation; 1 c=d
;; 5. x/[Vi<-Ti] = x/[Vi<-Si] Lemma; 2,3,4
(let* ((ti=si* (logic.strip-conclusions proofs))
(line-1 (build.beta-reduction (logic.lambda-formals x)
(logic.lambda-body x)
(logic.lambda-actuals x))))
(build.dual-substitution-lemma-1
(build.instantiation line-1 (pair-lists vars (logic.=lhses ti=si*)))
(build.flag-dual-substitution 'term (logic.lambda-body x) (logic.lambda-formals x)
(build.flag-dual-substitution 'list (logic.lambda-actuals x) vars proofs))
(build.instantiation line-1 (pair-lists vars (logic.=rhses ti=si*))))))
(t
;; sneaky hack so that it's always iff to t.
t))
;; List case.
(if (consp x)
(cons (build.flag-dual-substitution 'term (car x) vars proofs)
(build.flag-dual-substitution 'list (cdr x) vars proofs))
nil)))
(definlined build.dual-substitution (x vars proofs)
(declare (xargs :guard (and (logic.termp x)
(logic.variable-listp vars)
(uniquep vars)
(logic.appeal-listp proofs)
(same-lengthp vars proofs)
(logic.all-atomicp (logic.strip-conclusions proofs)))
:verify-guards nil))
(build.flag-dual-substitution 'term x vars proofs))
(definlined build.dual-substitution-list (x vars proofs)
(declare (xargs :guard (and (logic.term-listp x)
(logic.variable-listp vars)
(uniquep vars)
(logic.appeal-listp proofs)
(same-lengthp vars proofs)
(logic.all-atomicp (logic.strip-conclusions proofs)))
:verify-guards nil))
(build.flag-dual-substitution 'list x vars proofs))
(defthmd definition-of-build.dual-substitution
(equal (build.dual-substitution x vars proofs)
(cond ((logic.constantp x)
(build.reflexivity x))
((logic.variablep x)
(let ((index (first-index x vars)))
(if (equal index (len vars))
(build.reflexivity x)
(logic.appeal-identity (nth index proofs)))))
((logic.functionp x)
(build.pequal-by-args (logic.function-name x)
(build.dual-substitution-list (logic.function-args x) vars proofs)))
((logic.lambdap x)
(let* ((ti=si* (logic.strip-conclusions proofs))
(line-1 (build.beta-reduction (logic.lambda-formals x)
(logic.lambda-body x)
(logic.lambda-actuals x))))
(build.dual-substitution-lemma-1
(build.instantiation line-1 (pair-lists vars (logic.=lhses ti=si*)))
(build.flag-dual-substitution 'term (logic.lambda-body x) (logic.lambda-formals x)
(build.flag-dual-substitution 'list (logic.lambda-actuals x) vars proofs))
(build.instantiation line-1 (pair-lists vars (logic.=rhses ti=si*))))))
(t t)))
:rule-classes :definition
:hints(("Goal" :in-theory (enable build.dual-substitution
build.dual-substitution-list))))
(defthmd definition-of-build.dual-substitution-list
(equal (build.dual-substitution-list x vars proofs)
(if (consp x)
(cons (build.dual-substitution (car x) vars proofs)
(build.dual-substitution-list (cdr x) vars proofs))
nil))
:rule-classes :definition
:hints(("Goal" :in-theory (enable build.dual-substitution
build.dual-substitution-list))))
(defobligations build.dual-substitution
(build.reflexivity
build.pequal-by-args
build.beta-reduction
build.instantiation
build.dual-substitution-lemma-1))
(defobligations build.dual-substitution-list
(build.dual-substitution))
(defthm build.flag-dual-substitution-of-term-removal
(equal (build.flag-dual-substitution 'term x vars proofs)
(build.dual-substitution x vars proofs))
:hints(("Goal" :in-theory (enable build.dual-substitution))))
(defthm build.flag-dual-substitution-of-list-removal
(equal (build.flag-dual-substitution 'list x vars proofs)
(build.dual-substitution-list x vars proofs))
:hints(("Goal" :in-theory (enable build.dual-substitution-list))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition build.dual-substitution))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition build.dual-substitution-list))))
(defthm build.dual-substitution-under-iff
(iff (build.dual-substitution x vars proofs)
t)
:hints(("Goal" :use ((:instance definition-of-build.dual-substitution)))))
(defthm build.dual-substitution-list-when-not-consp
(implies (not (consp x))
(equal (build.dual-substitution-list x vars proofs)
nil))
:hints(("Goal" :in-theory (enable definition-of-build.dual-substitution-list))))
(defthm build.dual-substitution-list-of-cons
(equal (build.dual-substitution-list (cons a x) vars proofs)
(cons (build.dual-substitution a vars proofs)
(build.dual-substitution-list x vars proofs)))
:hints(("Goal" :in-theory (enable definition-of-build.dual-substitution-list))))
(defthm len-of-build.dual-substitution-list
(equal (len (build.dual-substitution-list x vars proofs))
(len x))
:hints(("Goal" :induct (cdr-induction x))))
(defthms-flag
:shared-hyp (force (and (logic.variable-listp vars)
(uniquep vars)
(logic.appeal-listp proofs)
(equal (len vars) (len proofs))
(logic.all-atomicp (logic.strip-conclusions proofs))))
:thms ((term forcing-logic.appealp-of-build.dual-substitution
(implies (force (logic.termp x))
(equal (logic.appealp (build.dual-substitution x vars proofs))
t)))
(term forcing-logic.conclusion-of-build.dual-substitution
(implies (force (logic.termp x))
(equal (logic.conclusion (build.dual-substitution x vars proofs))
(logic.pequal
(logic.substitute x (pair-lists vars (logic.=lhses (logic.strip-conclusions proofs))))
(logic.substitute x (pair-lists vars (logic.=rhses (logic.strip-conclusions proofs))))))))
(t forcing-logic.appealp-listp-of-build.dual-substitution-list
(implies (force (logic.term-listp x))
(equal (logic.appeal-listp (build.dual-substitution-list x vars proofs))
t)))
(t forcing-logic.strip-conclusions-of-build.dual-substitution-list
(implies (force (logic.term-listp x))
(equal (logic.strip-conclusions (build.dual-substitution-list x vars proofs))
(logic.pequal-list
(logic.substitute-list x (pair-lists vars (logic.=lhses (logic.strip-conclusions proofs))))
(logic.substitute-list x (pair-lists vars (logic.=rhses (logic.strip-conclusions proofs)))))))))
:hints (("Goal"
:restrict ((definition-of-build.dual-substitution ((x x))))
:in-theory (e/d (definition-of-build.dual-substitution)
(forcing-lookup-of-logic.function-name))
:induct (build.flag-dual-substitution flag x vars proofs))))
(verify-guards build.flag-dual-substitution
:hints(("Goal" :in-theory (disable forcing-lookup-of-logic.function-name))))
(verify-guards build.dual-substitution)
(verify-guards build.dual-substitution-list)
(defthms-flag
:@contextp t
:shared-hyp (force (and (logic.variable-listp vars)
(uniquep vars)
(logic.appeal-listp proofs)
(logic.proof-listp proofs axioms thms atbl)
(equal (len vars) (len proofs))
(logic.all-atomicp (logic.strip-conclusions proofs))))
:thms ((term forcing-logic.proofp-of-build.dual-substitution
(implies (force (and (logic.termp x)
(logic.term-atblp x atbl)
(@obligations build.dual-substitution)))
(equal (logic.proofp (build.dual-substitution x vars proofs) axioms thms atbl)
t)))
(t forcing-logic.proof-listp-of-build.dual-substitution-list
(implies (force (and (logic.term-listp x)
(logic.term-list-atblp x atbl)
(@obligations build.dual-substitution-list)))
(equal (logic.proof-listp (build.dual-substitution-list x vars proofs) axioms thms atbl)
t))))
:hints (("Goal"
:restrict ((definition-of-build.dual-substitution ((x x))))
:in-theory (enable definition-of-build.dual-substitution)
:induct (build.flag-dual-substitution flag x vars proofs))))
(defund build.lambda-pequal-by-args (formals body proofs)
;; Derive ((lambda (x1...xn) B) t1...tn) = ((lambda (x1...xn) B) s1...sn) from t1 = s1, ..., tn = sn
(declare (xargs :guard (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.appeal-listp proofs)
(same-lengthp proofs formals)
(logic.all-atomicp (logic.strip-conclusions proofs)))
:verify-guards nil))
(let* ((conclusions (logic.strip-conclusions proofs)) ;; (t1 = s1, ..., tn = sn)
(t* (logic.=lhses conclusions)) ;; (t1, ..., tn)
(s* (logic.=rhses conclusions))) ;; (s1, ..., sn)
(if (equal t* s*)
;; Optimization. As with build.pequal-by-args, if all of the arguments
;; are identical, we will just use reflexivity to build the proof.
(build.reflexivity (logic.lambda formals body t*))
;; Otherwise, we use the following derivation.
;;
;; 1. ((lambda (x1...xn) B) t1...tn) = B/[Xi<-Ti] Beta reduction
;; 2. B/[Xi<-Ti] = B/[Xi<-Si] Dual substitution
;; 3. ((lambda (x1...xn) B) t1...tn) = B/[Xi<-Si] Trans =; 1,2
;; 4. ((lambda (x1...xn) B) s1...sn) = B/[Xi<-Si] Beta reduction
;; 5. B/[Xi<-Si] = ((lambda (x1...xn) B) s1...sn) Commute =; 4
;; 6. ((lambda (x1...xn) B) t1...tn) = ((lambda (x1...xn) B) s1...sn) Trans =; 3,5
;;
;; Q.E.D.
(let* ((line-1 (build.beta-reduction formals body t*))
(line-2 (build.dual-substitution body formals proofs))
(line-3 (build.transitivity-of-pequal line-1 line-2))
(line-4 (build.beta-reduction formals body s*))
(line-5 (build.commute-pequal line-4))
(line-6 (build.transitivity-of-pequal line-3 line-5)))
line-6))))
(defobligations build.lambda-pequal-by-args
(build.reflexivity
build.beta-reduction
build.dual-substitution
build.transitivity-of-pequal
build.commute-pequal
build.transitivity-of-pequal))
(encapsulate
()
(local (in-theory (enable build.lambda-pequal-by-args)))
(defthm build.lambda-pequal-by-args-under-iff
(iff (build.lambda-pequal-by-args formals body proofs)
t))
(defthm forcing-logic.appealp-of-build.lambda-pequal-by-args
(implies (force (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.appeal-listp proofs)
(equal (len proofs) (len formals))
(logic.all-atomicp (logic.strip-conclusions proofs))))
(equal (logic.appealp (build.lambda-pequal-by-args formals body proofs))
t)))
(defthm forcing-logic.conclusion-of-build.lambda-pequal-by-args
(implies (force (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.appeal-listp proofs)
(equal (len proofs) (len formals))
(logic.all-atomicp (logic.strip-conclusions proofs))))
(equal (logic.conclusion (build.lambda-pequal-by-args formals body proofs))
(logic.pequal (logic.lambda formals body (logic.=lhses (logic.strip-conclusions proofs)))
(logic.lambda formals body (logic.=rhses (logic.strip-conclusions proofs))))))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-build.lambda-pequal-by-args
(implies (force (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(logic.term-atblp body atbl)
(subsetp (logic.term-vars body) formals)
(logic.appeal-listp proofs)
(logic.proof-listp proofs axioms thms atbl)
(equal (len proofs) (len formals))
(logic.all-atomicp (logic.strip-conclusions proofs))
(@obligations build.lambda-pequal-by-args)))
(equal (logic.proofp (build.lambda-pequal-by-args formals body proofs) axioms thms atbl)
t)))
(verify-guards build.lambda-pequal-by-args))
(defund build.lambda-pequal-by-args-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.lambda-pequal-by-args)
(tuplep 2 extras)
(let ((formals (first extras))
(body (second extras)))
(and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(logic.term-atblp body atbl)
(subsetp (logic.term-vars body) formals)
(same-lengthp subproofs formals)
(logic.all-atomicp-of-strip-conclusions subproofs)
(equal conclusion
(logic.pequal (logic.lambda formals body (logic.=lhses-of-strip-conclusions subproofs))
(logic.lambda formals body (logic.=rhses-of-strip-conclusions subproofs)))))))))
(defund build.lambda-pequal-by-args-high (formals body proofs)
;; BOZO we could add a true-listp restriction to proofs to avoid the list-fix
(declare (xargs :guard (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.appeal-listp proofs)
(same-lengthp proofs formals)
(logic.all-atomicp (logic.strip-conclusions proofs)))))
(logic.appeal 'build.lambda-pequal-by-args
(logic.pequal (logic.lambda formals body (logic.=lhses-of-strip-conclusions proofs))
(logic.lambda formals body (logic.=rhses-of-strip-conclusions proofs)))
(list-fix proofs)
(list formals body)))
(encapsulate
()
(local (in-theory (enable build.lambda-pequal-by-args-okp)))
(defthm booleanp-of-build.lambda-pequal-by-args-okp
(equal (booleanp (build.lambda-pequal-by-args-okp x atbl))
t)
:hints(("goal" :in-theory (disable (:executable-counterpart ACL2::force)))))
(defthm build.lambda-pequal-by-args-okp-of-logic.appeal-identity
(equal (build.lambda-pequal-by-args-okp (logic.appeal-identity x) atbl)
(build.lambda-pequal-by-args-okp x atbl))
:hints(("goal" :in-theory (disable (:executable-counterpart ACL2::force)))))
(local (in-theory (e/d (backtracking-logic.formula-atblp-rules)
(forcing-logic.formula-atblp-rules
forcing-lookup-of-logic.function-name-free))))
(defthm lemma-1-for-soundness-of-build.lambda-pequal-by-args-okp
(implies (and (build.lambda-pequal-by-args-okp x atbl)
(logic.appealp x)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl))
(equal (logic.conclusion (build.lambda-pequal-by-args
(first (logic.extras x))
(second (logic.extras x))
(logic.provable-list-witness (logic.strip-conclusions (logic.subproofs x))
axioms thms atbl)))
(logic.conclusion x))))
(defthm@ lemma-2-for-soundness-of-build.lambda-pequal-by-args-okp
(implies (and (build.lambda-pequal-by-args-okp x atbl)
(logic.appealp x)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl)
(@obligations build.lambda-pequal-by-args))
(equal (logic.proofp (build.lambda-pequal-by-args
(first (logic.extras x))
(second (logic.extras x))
(logic.provable-list-witness (logic.strip-conclusions (logic.subproofs x))
axioms thms atbl))
axioms thms atbl)
t)))
(defthm@ forcing-soundness-of-build.lambda-pequal-by-args-okp
(implies (and (build.lambda-pequal-by-args-okp x atbl)
(force (and (logic.appealp x)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl)
(@obligations build.lambda-pequal-by-args))))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t))
:hints (("Goal"
:in-theory (enable lemma-1-for-soundness-of-build.lambda-pequal-by-args-okp
lemma-2-for-soundness-of-build.lambda-pequal-by-args-okp)
:use ((:instance forcing-logic.provablep-when-logic.proofp
(x (build.lambda-pequal-by-args
(first (logic.extras x))
(second (logic.extras x))
(logic.provable-list-witness (logic.strip-conclusions (logic.subproofs x))
axioms thms atbl)))))))))
(defderiv build.disjoined-dual-substitution-lemma-1
:from ((proof x (v P (= (? a) (? b))))
(proof y (v P (= (? b) (? d))))
(proof z (v P (= (? c) (? d)))))
:derive (v P (= (? a) (? c)))
:proof (@derive
((v P (= (? a) (? b))) (@given x))
((v P (= (? b) (? d))) (@given y))
((v P (= (? a) (? d))) (build.disjoined-transitivity-of-pequal @-- @-) *1)
((v P (= (? c) (? d))) (@given z))
((v P (= (? d) (? c))) (build.disjoined-commute-pequal @-))
((v P (= (? a) (? c))) (build.disjoined-transitivity-of-pequal *1 @-))))
(defun build.flag-disjoined-dual-substitution (flag x vars p proofs)
;;
;; P v t1 = s1
;; ...
;; P v tn = sn
;; -----------------------------------------------------
;; P v x/[v1<-t1, ..., vn<-tn] = x/[v1<-s1, ..., vn<-sn]
;;
;; - x is the term we are instantiating
;; - vars are the list of variables (v1 ... vn) from above
;; - p is explicitly given; we can't extract it if proofs is empty
;; - proofs are the givens, i.e., proofs of (t1 = s1, ..., tn = sn)
;;
;; Note: we don't comment this proof since it's almost identical to the one
;; in build.flag-dual-substitution. So see the comments there instead.
(declare (xargs :guard (and (if (equal flag 'term)
(logic.termp x)
(and (equal flag 'list)
(logic.term-listp x)))
(logic.variable-listp vars)
(uniquep vars)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len vars) (len proofs))
(let ((conclusions (logic.strip-conclusions proofs)))
(and (logic.all-disjunctionsp conclusions)
(all-equalp p (logic.vlhses conclusions))
(logic.all-atomicp (logic.vrhses conclusions)))))
:verify-guards nil))
(if (equal flag 'term)
(cond ((logic.constantp x)
(build.expansion p (build.reflexivity x)))
((logic.variablep x)
(let ((index (first-index x vars)))
(if (equal index (len vars))
(build.expansion p (build.reflexivity x))
(logic.appeal-identity (nth index proofs)))))
((logic.functionp x)
(let ((name (logic.function-name x))
(args (logic.function-args x)))
(build.disjoined-pequal-by-args name p (build.flag-disjoined-dual-substitution 'list args vars p proofs))))
((logic.lambdap x)
(let* ((ti=si* (logic.vrhses (logic.strip-conclusions proofs)))
(line-1 (build.beta-reduction (logic.lambda-formals x)
(logic.lambda-body x)
(logic.lambda-actuals x))))
(build.disjoined-dual-substitution-lemma-1
(build.expansion p (build.instantiation line-1 (pair-lists vars (logic.=lhses ti=si*))))
(build.flag-disjoined-dual-substitution 'term
(logic.lambda-body x)
(logic.lambda-formals x)
p
(build.flag-disjoined-dual-substitution 'list
(logic.lambda-actuals x)
vars
p
proofs))
(build.expansion p (build.instantiation line-1 (pair-lists vars (logic.=rhses ti=si*)))))))
(t t))
(if (consp x)
(cons (build.flag-disjoined-dual-substitution 'term (car x) vars p proofs)
(build.flag-disjoined-dual-substitution 'list (cdr x) vars p proofs))
nil)))
(definlined build.disjoined-dual-substitution (x vars p proofs)
(declare (xargs :guard (and (logic.termp x)
(logic.variable-listp vars)
(uniquep vars)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len vars) (len proofs))
(let ((conclusions (logic.strip-conclusions proofs)))
(and (logic.all-disjunctionsp conclusions)
(all-equalp p (logic.vlhses conclusions))
(logic.all-atomicp (logic.vrhses conclusions)))))
:verify-guards nil))
(build.flag-disjoined-dual-substitution 'term x vars p proofs))
(definlined build.disjoined-dual-substitution-list (x vars p proofs)
(declare (xargs :guard (and (logic.term-listp x)
(logic.variable-listp vars)
(uniquep vars)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len vars) (len proofs))
(let ((conclusions (logic.strip-conclusions proofs)))
(and (logic.all-disjunctionsp conclusions)
(all-equalp p (logic.vlhses conclusions))
(logic.all-atomicp (logic.vrhses conclusions)))))
:verify-guards nil))
(build.flag-disjoined-dual-substitution 'list x vars p proofs))
(defthmd definition-of-build.disjoined-dual-substitution
(equal (build.disjoined-dual-substitution x vars p proofs)
(cond ((logic.constantp x)
(build.expansion p (build.reflexivity x)))
((logic.variablep x)
(let ((index (first-index x vars)))
(if (equal index (len vars))
(build.expansion p (build.reflexivity x))
(logic.appeal-identity (nth index proofs)))))
((logic.functionp x)
(let ((name (logic.function-name x))
(args (logic.function-args x)))
(build.disjoined-pequal-by-args name p (build.disjoined-dual-substitution-list args vars p proofs))))
((logic.lambdap x)
(let* ((ti=si* (logic.vrhses (logic.strip-conclusions proofs)))
(line-1 (build.beta-reduction (logic.lambda-formals x)
(logic.lambda-body x)
(logic.lambda-actuals x))))
(build.disjoined-dual-substitution-lemma-1
(build.expansion p (build.instantiation line-1 (pair-lists vars (logic.=lhses ti=si*))))
(build.disjoined-dual-substitution (logic.lambda-body x) (logic.lambda-formals x) p
(build.disjoined-dual-substitution-list (logic.lambda-actuals x) vars p proofs))
(build.expansion p (build.instantiation line-1 (pair-lists vars (logic.=rhses ti=si*)))))))
(t t)))
:rule-classes :definition
:hints(("Goal" :in-theory (enable build.disjoined-dual-substitution
build.disjoined-dual-substitution-list))))
(defthmd definition-of-build.disjoined-dual-substitution-list
(equal (build.disjoined-dual-substitution-list x vars p proofs)
(if (consp x)
(cons (build.disjoined-dual-substitution (car x) vars p proofs)
(build.disjoined-dual-substitution-list (cdr x) vars p proofs))
nil))
:rule-classes :definition
:hints(("Goal" :in-theory (enable build.disjoined-dual-substitution
build.disjoined-dual-substitution-list))))
(defobligations build.disjoined-dual-substitution
(build.expansion
build.reflexivity
build.disjoined-pequal-by-args
build.beta-reduction
build.instantiation
build.disjoined-transitivity-of-pequal
build.commute-pequal))
(defobligations build.disjoined-dual-substitution-list
(build.disjoined-dual-substitution))
(defthm build.flag-disjoined-dual-substitution-of-term-removal
(equal (build.flag-disjoined-dual-substitution 'term x vars p proofs)
(build.disjoined-dual-substitution x vars p proofs))
:hints(("Goal" :in-theory (enable build.disjoined-dual-substitution))))
(defthm build.flag-disjoined-dual-substitution-of-list-removal
(equal (build.flag-disjoined-dual-substitution 'list x vars p proofs)
(build.disjoined-dual-substitution-list x vars p proofs))
:hints(("Goal" :in-theory (enable build.disjoined-dual-substitution-list))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition build.disjoined-dual-substitution))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition build.disjoined-dual-substitution-list))))
(defthm forcing-build.disjoined-dual-substitution-under-iff
(iff (build.disjoined-dual-substitution x vars p proofs)
t)
:hints(("Goal" :use ((:instance definition-of-build.disjoined-dual-substitution)))))
(defthm build.disjoined-dual-substitution-list-when-not-consp
(implies (not (consp x))
(equal (build.disjoined-dual-substitution-list x vars p proofs)
nil))
:hints(("Goal" :in-theory (enable definition-of-build.disjoined-dual-substitution-list))))
(defthm build.disjoined-dual-substitution-list-of-cons
(equal (build.disjoined-dual-substitution-list (cons a x) vars p proofs)
(cons (build.disjoined-dual-substitution a vars p proofs)
(build.disjoined-dual-substitution-list x vars p proofs)))
:hints(("Goal" :in-theory (enable definition-of-build.disjoined-dual-substitution-list))))
(defthm len-of-build.disjoined-dual-substitution-list
(equal (len (build.disjoined-dual-substitution-list x vars p proofs))
(len x))
:hints(("Goal" :induct (cdr-induction x))))
(defthms-flag
:shared-hyp (force (and (logic.variable-listp vars)
(uniquep vars)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len vars) (len proofs))
(logic.all-disjunctionsp (logic.strip-conclusions proofs))
(all-equalp p (logic.vlhses (logic.strip-conclusions proofs)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions proofs)))))
:thms ((term forcing-logic.appealp-of-build.disjoined-dual-substitution
(implies (force (logic.termp x))
(equal (logic.appealp (build.disjoined-dual-substitution x vars p proofs))
t)))
(term forcing-logic.conclusion-of-build.disjoined-dual-substitution
(implies (force (logic.termp x))
(equal (logic.conclusion (build.disjoined-dual-substitution x vars p proofs))
(logic.por p (logic.pequal (logic.substitute x (pair-lists vars (logic.=lhses (logic.vrhses (logic.strip-conclusions proofs)))))
(logic.substitute x (pair-lists vars (logic.=rhses (logic.vrhses (logic.strip-conclusions proofs))))))))))
(t forcing-logic.appeal-listp-of-build.disjoined-dual-substitution
(implies (force (logic.term-listp x))
(equal (logic.appeal-listp (build.disjoined-dual-substitution-list x vars p proofs))
t)))
(t forcing-logic.strip-conclusions-of-build.disjoined-dual-substitution
(implies (force (logic.term-listp x))
(equal (logic.strip-conclusions (build.disjoined-dual-substitution-list x vars p proofs))
(logic.por-list (repeat p (len x))
(logic.pequal-list
(logic.substitute-list x (pair-lists vars (logic.=lhses (logic.vrhses (logic.strip-conclusions proofs)))))
(logic.substitute-list x (pair-lists vars (logic.=rhses (logic.vrhses (logic.strip-conclusions proofs)))))))))))
:hints (("Goal"
:in-theory (enable definition-of-build.disjoined-dual-substitution)
:induct (build.flag-disjoined-dual-substitution flag x vars p proofs))))
(verify-guards build.flag-disjoined-dual-substitution)
(verify-guards build.disjoined-dual-substitution)
(verify-guards build.disjoined-dual-substitution-list)
(defthms-flag
:@contextp t
:shared-hyp (force (and (logic.variable-listp vars)
(uniquep vars)
(logic.formulap p)
(logic.formula-atblp p atbl)
(logic.appeal-listp proofs)
(equal (len vars) (len proofs))
(logic.all-disjunctionsp (logic.strip-conclusions proofs))
(all-equalp p (logic.vlhses (logic.strip-conclusions proofs)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions proofs)))
(logic.proof-listp proofs axioms thms atbl)))
:thms ((term forcing-logic.proofp-of-build.disjoined-dual-substitution
(implies (force (and (logic.termp x)
(logic.term-atblp x atbl)
(logic.proof-listp proofs axioms thms atbl)
(@obligations build.disjoined-dual-substitution)))
(equal (logic.proofp (build.disjoined-dual-substitution x vars p proofs) axioms thms atbl)
t)))
(t forcing-logic.proof-listp-of-build.disjoined-dual-substitution
(implies (force (and (logic.term-listp x)
(logic.term-list-atblp x atbl)
(logic.proof-listp proofs axioms thms atbl)
(@obligations build.disjoined-dual-substitution-list)))
(equal (logic.proof-listp (build.disjoined-dual-substitution-list x vars p proofs) axioms thms atbl)
t))))
:hints (("Goal"
:in-theory (enable definition-of-build.disjoined-dual-substitution)
:induct (build.flag-disjoined-dual-substitution flag x vars p proofs))))
(defund build.disjoined-lambda-pequal-by-args (formals body p proofs)
(declare (xargs :guard (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len proofs) (len formals))
(let ((conclusions (logic.strip-conclusions proofs)))
(and (logic.all-disjunctionsp conclusions)
(all-equalp p (logic.vlhses conclusions))
(logic.all-atomicp (logic.vrhses conclusions)))))))
(let* ((ti=si* (logic.vrhses (logic.strip-conclusions proofs)))
(ti* (logic.=lhses ti=si*))
(si* (logic.=rhses ti=si*)))
(if (equal ti* si*)
;; Optimization. If all the args are identical, we can just use
;; reflexivity and expansion to build the proof.
(build.expansion p (build.reflexivity (logic.lambda formals body ti*)))
;; Otherwise, we use the following derivation:
;;
;; 1. P v ((lambda (x1...xn) B) t1...tn) = B/[Xi<-Ti] Beta reduction, expansion
;; 2. P v B/[Xi<-Ti] = B/[Xi<-Si] DJ Dual Substitution
;; 3. P v ((lambda (x1...xn) B) t1...tn) = B/[Xi<-Si] DJ Trans =; 1,2
;; 4. P v B/[Xi<-Si] = ((lambda (x1...xn) B) s1...sn) Beta, commute =, expansion
;; 5. P v ((lambda (x1...xn) B) t1...tn) = ((lambda (x1...xn) B) s1...sn) DJ Trans =; 3,4
;;
;; Q.E.D.
(let* ((line-1 (build.expansion p (build.beta-reduction formals body ti*)))
(line-2 (build.disjoined-dual-substitution body formals p proofs))
(line-3 (build.disjoined-transitivity-of-pequal line-1 line-2))
(line-4 (build.expansion p (build.commute-pequal (build.beta-reduction formals body si*))))
(line-5 (build.disjoined-transitivity-of-pequal line-3 line-4)))
line-5))))
(defobligations build.disjoined-lambda-pequal-by-args
(build.expansion
build.reflexivity
build.disjoined-dual-substitution
build.disjoined-transitivity-of-pequal
build.commute-pequal
build.beta-reduction))
(encapsulate
()
(local (in-theory (enable build.disjoined-lambda-pequal-by-args)))
(defthm forcing-build.disjoined-lambda-pequal-by-args-under-iff
(iff (build.disjoined-lambda-pequal-by-args formals body p proofs)
t))
(defthm forcing-logic.appealp-of-build.disjoined-lambda-pequal-by-args
(implies (force (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len proofs) (len formals))
(logic.all-disjunctionsp (logic.strip-conclusions proofs))
(all-equalp p (logic.vlhses (logic.strip-conclusions proofs)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions proofs)))))
(equal (logic.appealp (build.disjoined-lambda-pequal-by-args formals body p proofs))
t)))
(defthm forcing-logic.conclusion-of-build.disjoined-lambda-pequal-by-args
(implies (force (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len proofs) (len formals))
(logic.all-disjunctionsp (logic.strip-conclusions proofs))
(all-equalp p (logic.vlhses (logic.strip-conclusions proofs)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions proofs)))))
(equal (logic.conclusion (build.disjoined-lambda-pequal-by-args formals body p proofs))
(logic.por p (logic.pequal
(logic.lambda formals body (logic.=lhses (logic.vrhses (logic.strip-conclusions proofs))))
(logic.lambda formals body (logic.=rhses (logic.vrhses (logic.strip-conclusions proofs))))))))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
(defthm@ forcing-logic.proofp-of-build.disjoined-lambda-pequal-by-args
(implies (force (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len proofs) (len formals))
(logic.all-disjunctionsp (logic.strip-conclusions proofs))
(all-equalp p (logic.vlhses (logic.strip-conclusions proofs)))
(logic.all-atomicp (logic.vrhses (logic.strip-conclusions proofs)))
;; ---
(logic.term-atblp body atbl)
(logic.formula-atblp p atbl)
(logic.proof-listp proofs axioms thms atbl)
(@obligations build.disjoined-lambda-pequal-by-args)))
(equal (logic.proofp (build.disjoined-lambda-pequal-by-args formals body p proofs) axioms thms atbl)
t))))
(defund build.disjoined-lambda-pequal-by-args-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.disjoined-lambda-pequal-by-args)
(tuplep 2 extras)
(let ((formals (first extras))
(body (second extras)))
(and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(logic.term-atblp body atbl)
(subsetp (logic.term-vars body) formals)
(same-lengthp subproofs formals)
(logic.all-disjunctionsp-of-strip-conclusions subproofs)
(logic.all-atomicp-of-vrhses-of-strip-conclusions subproofs)
(equal (logic.fmtype conclusion) 'por*)
(logic.formula-atblp (logic.vlhs conclusion) atbl)
;; BOZO efficiency -- add an all-equalp-of-vlhses-of-strip-conclusions
(all-equalp (logic.vlhs conclusion) (logic.vlhses-of-strip-conclusions subproofs))
(equal (logic.vrhs conclusion)
(logic.pequal (logic.lambda formals body (logic.=lhses-of-vrhses-of-strip-conclusions subproofs))
(logic.lambda formals body (logic.=rhses-of-vrhses-of-strip-conclusions subproofs)))))))))
(defund build.disjoined-lambda-pequal-by-args-high (formals body p proofs)
;; BOZO we could add a true-listp restriction to proofs to avoid the list-fix
(declare (xargs :guard (and (true-listp formals)
(logic.variable-listp formals)
(uniquep formals)
(logic.termp body)
(subsetp (logic.term-vars body) formals)
(logic.formulap p)
(logic.appeal-listp proofs)
(equal (len proofs) (len formals))
(let ((conclusions (logic.strip-conclusions proofs)))
(and (logic.all-disjunctionsp conclusions)
(all-equalp p (logic.vlhses conclusions))
(logic.all-atomicp (logic.vrhses conclusions)))))))
(logic.appeal 'build.disjoined-lambda-pequal-by-args
(logic.por p (logic.pequal (logic.lambda formals body (logic.=lhses-of-vrhses-of-strip-conclusions proofs))
(logic.lambda formals body (logic.=rhses-of-vrhses-of-strip-conclusions proofs))))
(list-fix proofs)
(list formals body)))
(encapsulate
()
(local (in-theory (enable build.disjoined-lambda-pequal-by-args-okp)))
(defthm booleanp-of-build.disjoined-lambda-pequal-by-args-okp
(equal (booleanp (build.disjoined-lambda-pequal-by-args-okp x atbl))
t)
:hints(("goal" :in-theory (disable (:executable-counterpart ACL2::force)))))
(defthm build.disjoined-lambda-pequal-by-args-okp-of-logic.appeal-identity
(equal (build.disjoined-lambda-pequal-by-args-okp (logic.appeal-identity x) atbl)
(build.disjoined-lambda-pequal-by-args-okp x atbl))
:hints(("goal" :in-theory (disable (:executable-counterpart ACL2::force)))))
(local (in-theory (e/d (backtracking-logic.formula-atblp-rules)
(forcing-logic.formula-atblp-rules
forcing-lookup-of-logic.function-name-free))))
(defthm lemma-1-for-soundness-of-build.disjoined-lambda-pequal-by-args-okp
(implies (and (build.disjoined-lambda-pequal-by-args-okp x atbl)
(logic.appealp x)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl))
(equal (logic.conclusion (build.disjoined-lambda-pequal-by-args
(first (logic.extras x))
(second (logic.extras x))
(logic.vlhs (logic.conclusion x))
(logic.provable-list-witness (logic.strip-conclusions (logic.subproofs x))
axioms thms atbl)))
(logic.conclusion x))))
(defthm@ lemma-2-for-soundness-of-build.disjoined-lambda-pequal-by-args-okp
(implies (and (build.disjoined-lambda-pequal-by-args-okp x atbl)
(logic.appealp x)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl)
(@obligations build.disjoined-lambda-pequal-by-args))
(equal (logic.proofp (build.disjoined-lambda-pequal-by-args
(first (logic.extras x))
(second (logic.extras x))
(logic.vlhs (logic.conclusion x))
(logic.provable-list-witness (logic.strip-conclusions (logic.subproofs x))
axioms thms atbl))
axioms thms atbl)
t)))
(defthm@ forcing-soundness-of-build.disjoined-lambda-pequal-by-args-okp
(implies (and (build.disjoined-lambda-pequal-by-args-okp x atbl)
(force (and (logic.appealp x)
(logic.provable-listp (logic.strip-conclusions (logic.subproofs x)) axioms thms atbl)
(@obligations build.disjoined-lambda-pequal-by-args))))
(equal (logic.provablep (logic.conclusion x) axioms thms atbl)
t))
:hints (("Goal"
:in-theory (enable lemma-1-for-soundness-of-build.disjoined-lambda-pequal-by-args-okp
lemma-2-for-soundness-of-build.disjoined-lambda-pequal-by-args-okp)
:use ((:instance forcing-logic.provablep-when-logic.proofp
(x (build.disjoined-lambda-pequal-by-args
(first (logic.extras x))
(second (logic.extras x))
(logic.vlhs (logic.conclusion x))
(logic.provable-list-witness (logic.strip-conclusions (logic.subproofs x))
axioms thms atbl)))))))))
(dd.close)
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