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;;
;; Copyright (C) 2018, Rockwell Collins
;; All rights reserved.
;;
;; This software may be modified and distributed under the terms
;; of the 3-clause BSD license. See the LICENSE file for details.
;;
;;
(in-package "ACL2")
(include-book "poly-proofs")
(include-book "ordinals/lexicographic-ordering-without-arithmetic" :dir :system)
(include-book "coi/defung/defung" :dir :system)
(defthm set-upper-bound-equiv-subset-p-rewrite
(implies
(and
(subset-p target result)
(syntaxp (not (equal target result))))
(set-upper-bound-equiv target result)))
(defthm tru-dat
(implies
(and
(subset-p list zed)
(not (list::memberp a zed)))
(not (list::memberp a list))))
(defthm try-this
(implies
(and
(subset-p x y)
(force (>-all varid y)))
(>-all varid x))
:hints (("Goal" :in-theory (enable ADVISER::>-ALL-BY-MULTIPLICITY))))
(defthm chaining-on-varid
(implies
(and
(varid-p varid1)
(>-all varid2 list)
(varid-p varid2)
(> varid1 varid2))
(>-ALL varid1 list))
:hints (("Goal" :in-theory (enable ADVISER::>-ALL-BY-MULTIPLICITY))))
(defthm empty-trapezoid-p
(implies
(not (consp zoid))
(iff (trapezoid-p zoid)
(null zoid))))
(defthm open-trapezoid-p
(implies
(consp zoid)
(equal (trapezoid-p zoid)
(let ((list zoid))
(LET ((BV (CAR LIST)))
(AND (NORMALIZED-VARIABLEBOUND-P BV)
(VARIABLEBOUND-LISTP (CDR LIST))
(>-ALL (BOUND-VARID BV)
(ALL-BOUND-LIST-VARIABLES (CDR LIST)))
(TRAPEZOID-P (CDR LIST))))))))
(in-theory
(disable
trapezoid-p
;;variableBound-implication
;;variableRelation-implication
<=-O-FIRST-EXPT-<=
))
(encapsulate
()
(local
(defthm >-all-from->->=all
(IMPLIES (AND (>=ALL X LIST)
(< X VARID)
(varid-p x)
(varid-p varid)
(varid-listp list))
(>-ALL VARID LIST))))
(defthm <-largest-varid-to-<-all
(implies
(and
(consp list)
(varid-p varid)
(varid-listp list))
(iff (< (largest-varid list) varid)
(>-all varid list)))
:hints (("Goal" ;; :induct (>-all varid list)
:do-not-induct t
:do-not '(generalize)
:use largest-varid-is-largest
:in-theory (e/d () (OPEN->=ALL-ON-MEMBERP
OPEN-LARGEST-VARID-ON-MEMBERP
largest-varid-is-largest)))))
(defthm <=largest-varid-to-<=all
(implies
(and
(consp list)
(varid-p varid)
(varid-listp list))
(iff (< varid (largest-varid list))
(not (>=all varid list))))
:hints (("Goal" ;; :induct (>-all varid list)
:do-not-induct t
:do-not '(generalize)
:use largest-varid-is-largest
:in-theory (e/d () (OPEN->=ALL-ON-MEMBERP
OPEN-LARGEST-VARID-ON-MEMBERP
largest-varid-is-largest)))))
)
(defthm consp-bound-varid-list
(iff (consp (bound-varid-list list))
(consp list)))
(in-theory (disable largest-varid))
(defthm equal-if->=all
(implies
(and
(varid-p varid)
(varid-listp list)
(list::memberp varid list)
(>=all varid list))
(iff (equal varid (largest-varid list)) t))
:hints (("Goal" :induct (largest-varid list)
:expand (varid-listp list)
:in-theory (e/d (largest-varid) (OPEN->=ALL-ON-MEMBERP)))))
;; ---------------------------------------------------------------------------------
(defmacro intersect-var-macro (var res cex)
`(intersect :var ,var ,res ,cex))
(defmacro intersect-list-macro (list res cex)
`(intersect :list ,list ,res ,cex))
(defmacro intersect-var-body (var res cex)
`(let ((var ,var)
(res ,res)
(cex ,cex))
(if (not (consp res)) (list var)
(if (< (bound-varid (car res)) (bound-varid var)) (cons var res)
(if (< (bound-varid var) (bound-varid (car res)))
(cons (car res) (intersect-var-macro var (cdr res) cex))
(metlist ((z zres) (andTrue-variableBound-variableBound var (car res) cex))
(let ((res (intersect-list-macro zres (cdr res) cex)))
(cons z res))))))))
(defmacro intersect-list-body (list res cex)
`(let ((list ,list)
(res ,res)
(cex ,cex))
(if (not (consp list)) res
(let ((res (intersect-var-macro (car list) res cex)))
(intersect-list-macro (cdr list) res cex)))))
;; ---------------------------------------------------------------------------------
(def::ung intersect (key arg res cex)
(declare (xargs :default-value nil
:signature ((t (lambda (x) (if (equal key :var) (variableBound-p x) (variableBound-listp x))) variableBound-listp env-p) variableBound-listp)
:guard-hints (("Goal" :do-not-induct t)
(and stable-under-simplificationp
'(:expand ((INTERSECT-0-DOMAIN KEY ARG RES CEX)
(INTERSECT-0-DOMAIN :VAR ARG RES CEX)))))))
(if (equal key :var)
(intersect-var-body arg res cex)
(intersect-list-body arg res cex)))
(defthm trapezoid-p-intersect
(implies
(and
(if (equal key :var) (normalized-variableBound-p arg) (normalized-variableBound-listp arg))
(trapezoid-p res)
(env-p cex))
(and
(trapezoid-p (intersect key arg res cex))
(subset-p (all-bound-list-variables (intersect key arg res cex))
(append
(if (equal key :var) (all-bound-variables arg) (all-bound-list-variables arg))
(all-bound-list-variables res)))))
:hints (("Goal" :do-not-induct t
:do-not '(generalize eliminate-destructors)
:induct (intersect key arg res cex))))
(defthm set-upper-bound-equiv-all-bound-list-variables-intersect
(implies
(and
(if (equal key :var) (normalized-variableBound-p arg) (normalized-variableBound-listp arg))
(trapezoid-p res)
(env-p cex))
(set-upper-bound-equiv (all-bound-list-variables (intersect key arg res cex))
(append
(if (equal key :var) (all-bound-variables arg) (all-bound-list-variables arg))
(all-bound-list-variables res))))
:hints (("Goal" :in-theory (disable trapezoid-p-intersect)
:use trapezoid-p-intersect)))
;; ---------------------------------------------------------------------------------
(defun intersection-measure (key arg res cex)
(declare (ignore cex))
(llist (if (equal key :var) (bound-varid arg) (if (consp arg) (largest-varid (bound-varid-list arg)) 0))
(if (equal key :var) 0 (len arg))
(len res)))
(def::total intersect (key arg res cex)
(declare (xargs :measure (intersection-measure key arg res cex)
:hints (("Goal" :do-not-induct t))
:well-founded-relation l<))
(and
(if (equal key :var) (normalized-variableBound-p arg) (normalized-variableBound-listp arg))
(trapezoid-p res)
(env-p cex)))
(encapsulate
()
(local
(defthm eval-ineq-list-from-conjoinResults-helper
(iff (eval-ineq (conjoinResults x list) env)
(and (eval-ineq-list list env)
(eval-ineq x env)))))
(defthm eval-ineq-from-conjoinResults
(implies
(eval-ineq (conjoinResults (val 0 fn) (val 1 fn)) env)
(and (eval-ineq-list (val 1 fn) env)
(eval-ineq (val 0 fn) env))))
(defthm not-eval-ineq-from-conjoinResults
(implies
(and
(syntaxp (equal any 'any))
(eval-ineq-list (val 1 fn) any)
(not (eval-ineq (conjoinResults (val 0 fn) (val 1 fn)) any))
)
(not (eval-ineq (val 0 fn) any))))
(defthm not-eval-ineq-list-from-conjoinResults
(implies
(and
(syntaxp (equal any 'any))
(eval-ineq (val 0 fn) any)
(not (eval-ineq (conjoinResults (val 0 fn) (val 1 fn)) any)))
(not (eval-ineq-list (val 1 fn) any))))
)
(defthm inv1-intersect
(implies
(and
(if (equal key :var) (normalized-variableBound-p arg) (normalized-variableBound-listp arg))
(trapezoid-p res)
(env-p cex)
(eval-ineq-list res cex)
(if (equal key :var) (eval-ineq arg cex) (eval-ineq-list arg cex)))
(eval-ineq-list (intersect key arg res cex) cex)))
(defthm inv2-intersect
(implies
(and
(if (equal key :var) (normalized-variableBound-p arg) (normalized-variableBound-listp arg))
(trapezoid-p res)
(env-p cex)
(eval-ineq-list res cex)
(if (equal key :var) (eval-ineq arg cex) (eval-ineq-list arg cex))
(not (and (if (equal key :var) (eval-ineq arg any) (eval-ineq-list arg any))
(eval-ineq-list res any))))
(not (eval-ineq-list (intersect key arg res cex) any))))
;; ---------------------------------------------------------------------------------
(def::un intersect-var (var res cex)
(declare (xargs :signature ((variableBound-p variableBound-listp env-p) variableBound-listp)))
(intersect-var-macro var res cex))
(def::signature intersect-var (normalized-variableBound-p trapezoid-p env-p) trapezoid-p)
(def::un intersect-list (list res cex)
(declare (xargs :signature ((variableBound-listp variableBound-listp env-p) variableBound-listp)))
(intersect-list-macro list res cex))
(def::signature intersect-list (normalized-variableBound-listp trapezoid-p env-p) trapezoid-p)
(defthm set-upper-bound-equiv-all-bound-list-variables-intersect-list
(implies
(and
(normalized-variableBound-listp arg)
(trapezoid-p res)
(env-p cex))
(set-upper-bound-equiv (all-bound-list-variables (intersect-list arg res cex))
(double-rewrite (append
(all-bound-list-variables arg)
(all-bound-list-variables res))))))
(defthm inv1-intersect-list
(implies
(and
(normalized-variableBound-listp arg)
(trapezoid-p res)
(env-p cex)
(eval-ineq-list res cex)
(eval-ineq-list arg cex))
(eval-ineq-list (intersect-list arg res cex) cex)))
(defthm inv2-intersect-list
(implies
(and
(normalized-variableBound-listp arg)
(trapezoid-p res)
(env-p cex)
(eval-ineq-list res cex)
(eval-ineq-list arg cex)
(not (and (eval-ineq-list arg any)
(eval-ineq-list res any))))
(not (eval-ineq-list (intersect-list arg res cex) any))))
(in-theory (disable intersect-var intersect-list))
;; ---------------------------------------------------------------------------------
(defmacro and-zoid-macro (xlist ylist env)
`(let ((xlist ,xlist)
(ylist ,ylist)
(env ,env))
(if (and (consp xlist) (consp ylist))
(and-zoid-rec (car xlist) (cdr xlist) (car ylist) (cdr ylist) env)
(if (not (consp xlist)) ylist
xlist))))
(def::ung and-zoid-rec (x xlist y ylist env)
(declare (xargs :default-value nil
:signature ((variableBound-p variableBound-listp variableBound-p variableBound-listp env-p) variableBound-listp)
:guard-hints (("Goal" :do-not-induct t)
(and stable-under-simplificationp
'(:expand ((AND-ZOID-REC-0-DOMAIN X NIL Y YLIST ENV)
(AND-ZOID-REC-0-DOMAIN X XLIST Y NIL ENV)))))))
(if (equal (bound-varid x) (bound-varid y))
(let ((res (and-zoid-macro xlist ylist env)))
(metlist ((z zres) (andTrue-variableBound-variableBound x y env))
(cons z (intersect-list zres res env))))
(if (< (bound-varid x) (bound-varid y))
(cons y (if (not (consp ylist)) (cons x xlist)
(and-zoid-rec x xlist (car ylist) (cdr ylist) env)))
(cons x (if (not (consp xlist)) (cons y ylist)
(and-zoid-rec (car xlist) (cdr xlist) y ylist env))))))
(defthm trapezoid-p-and-zoid-rec
(implies
(and
(normalized-variableBound-p x)
(trapezoid-p xlist)
(>-all (bound-varid x) (all-bound-list-variables xlist))
(normalized-variableBound-p y)
(trapezoid-p ylist)
(>-all (bound-varid y) (all-bound-list-variables ylist))
(env-p env))
(and
(trapezoid-p (and-zoid-rec x xlist y ylist env))
(subset-p (all-bound-list-variables (and-zoid-rec x xlist y ylist env))
(append
(all-bound-variables x)
(all-bound-variables y)
(all-bound-list-variables xlist)
(all-bound-list-variables ylist)))))
:hints (("Goal" :do-not-induct t
:do-not '(generalize eliminate-destructors)
:induct (and-zoid-rec x xlist y ylist env))))
(defthm set-upper-bound-equiv-all-bound-list-variables-and-zoid-rec
(implies
(and
(normalized-variableBound-p x)
(trapezoid-p xlist)
(>-all (bound-varid x) (all-bound-list-variables xlist))
(normalized-variableBound-p y)
(trapezoid-p ylist)
(>-all (bound-varid y) (all-bound-list-variables ylist))
(env-p env))
(set-upper-bound-equiv (all-bound-list-variables (and-zoid-rec x xlist y ylist env))
(append
(all-bound-variables x)
(all-bound-variables y)
(all-bound-list-variables xlist)
(all-bound-list-variables ylist))))
:hints (("Goal" :in-theory (disable trapezoid-p-and-zoid-rec)
:use trapezoid-p-and-zoid-rec)))
(def::total and-zoid-rec (x xlist y ylist env)
(declare (xargs :measure (max (bound-varid x) (bound-varid y))
:hints (("Goal" :do-not-induct t))))
(and
(normalized-variableBound-p x)
(trapezoid-p xlist)
(>-all (bound-varid x) (all-bound-list-variables xlist))
(normalized-variableBound-p y)
(trapezoid-p ylist)
(>-all (bound-varid y) (all-bound-list-variables ylist))
(env-p env)))
(defthm inv1-and-zoid-rec
(implies
(and
(and
(normalized-variableBound-p x)
(trapezoid-p xlist)
(>-all (bound-varid x) (all-bound-list-variables xlist))
(normalized-variableBound-p y)
(trapezoid-p ylist)
(>-all (bound-varid y) (all-bound-list-variables ylist))
(env-p cex))
(and
(eval-ineq x cex)
(eval-ineq y cex)
(eval-ineq-list xlist cex)
(eval-ineq-list ylist cex)))
(eval-ineq-list (and-zoid-rec x xlist y ylist cex) cex)))
(defthm inv2-and-zoid-rec
(implies
(and
(and
(normalized-variableBound-p x)
(trapezoid-p xlist)
(>-all (bound-varid x) (all-bound-list-variables xlist))
(normalized-variableBound-p y)
(trapezoid-p ylist)
(>-all (bound-varid y) (all-bound-list-variables ylist))
(env-p cex))
(and
(eval-ineq x cex)
(eval-ineq y cex)
(eval-ineq-list xlist cex)
(eval-ineq-list ylist cex))
(not
(and
(eval-ineq x any)
(eval-ineq y any)
(eval-ineq-list xlist any)
(eval-ineq-list ylist any)))
)
(not (eval-ineq-list (and-zoid-rec x xlist y ylist cex) any))))
;; ---------------------------------------------------------------------------------
(def::un and-zoid (xlist ylist env)
(declare (xargs :signature ((trapezoid-p trapezoid-p env-p) trapezoid-p)))
(and-zoid-macro xlist ylist env))
(defthm inv1-and-zoid
(implies
(and
(and
(trapezoid-p xlist)
(trapezoid-p ylist)
(env-p cex))
(and
(eval-ineq-list xlist cex)
(eval-ineq-list ylist cex)))
(eval-ineq-list (and-zoid xlist ylist cex) cex)))
(defthm inv2-and-zoid
(implies
(and
(and
(trapezoid-p xlist)
(trapezoid-p ylist)
(env-p cex))
(and
(eval-ineq-list xlist cex)
(eval-ineq-list ylist cex))
(not
(and
(eval-ineq-list xlist any)
(eval-ineq-list ylist any))))
(not (eval-ineq-list (and-zoid xlist ylist cex) any))))
(in-theory (disable and-zoid))
|
