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; Centaur AIG Library
; Copyright (C) 2008-2011 Centaur Technology
;
; Contact:
; Centaur Technology Formal Verification Group
; 7600-C N. Capital of Texas Highway, Suite 300, Austin, TX 78731, USA.
; http://www.centtech.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: Sol Swords <sswords@centtech.com>
(in-package "ACL2")
(include-book "eval-restrict")
(local (in-theory (disable append-of-nil)))
;; This book defines check-property, which checks that a safety property
;; on a FSM holds after some number of steps in which provided inputs are
;; applied. The FSM is expressed as a set of AIG update functions and an AIG
;; property.
;; We prove that provided there exists an invariant, an AIG for which three
;; Boolean properties (checkable by SAT) and a side condition hold, then
;; check-property always holds. This can be used to validate the result of a
;; model checking algorithm such as interpolation or property-driven
;; reachability that produces an invariant, given a method for validating UNSAT
;; proofs.
(defun check-property (updates prop curr-st inputs)
(declare (Xargs :guard (consp inputs)))
(b* ((assign (hons-shrink-alist
(car inputs)
(hons-shrink-alist curr-st nil)))
((when (atom (cdr inputs)))
(b* ((res (aig-eval prop assign)))
(fast-alist-free assign)
res))
(next-st (aig-eval-alist updates assign)))
(fast-alist-free assign)
(check-property updates prop next-st (cdr inputs))))
(defcong alist-equiv equal (check-property updates prop curr-st inputs) 3)
(defcong alist-equiv equal (check-property updates prop curr-st inputs) 1)
(defun check-property-strong (updates prop curr-st inputs)
(declare (Xargs :measure (len inputs)
:guard t))
(if (atom inputs)
t
(b* ((assign (hons-shrink-alist
(car inputs)
(hons-shrink-alist curr-st nil)))
((when (not (aig-eval prop assign)))
(fast-alist-free assign)
nil)
(next-st (aig-eval-alist updates assign)))
(fast-alist-free assign)
(check-property-strong updates prop next-st (cdr inputs)))))
(defun-sk check-ag-property (updates prop curr-st)
(forall inputs (check-property updates prop curr-st inputs)))
(defthm check-ag-property-implies-check-property-strong
(implies (check-ag-property updates prop curr-st)
(check-property-strong updates prop curr-st inputs))
:hints (("goal" :in-theory (disable check-ag-property
check-ag-property-necc)
:induct (check-property-strong updates prop curr-st inputs))
(and stable-under-simplificationp
(cond
((member-equal
'(AIG-EVAL PROP (binary-APPEND CURR-ST (CAR INPUTS)))
clause)
'(:use ((:instance check-ag-property-necc
(inputs (list (car inputs)))))))
(t
'(:expand
((CHECK-AG-PROPERTY
UPDATES PROP
(AIG-EVAL-ALIST UPDATES (APPEND CURR-ST (CAR INPUTS)))))
:use ((:instance check-ag-property-necc
(inputs (cons (car inputs)
(let ((rest (check-ag-property-witness
updates prop
(aig-eval-alist
updates
(append curr-st (car inputs))))))
(if (consp rest)
rest
(list nil)))))))))))))
(defun-sk unsat-p (x)
(forall env (not (aig-eval x env))))
(local
(progn
(defthm aig-eval-when-vars-subset-of-first-keys
(implies (subsetp-equal (aig-vars x) (alist-keys a))
(equal (aig-eval x (append a b))
(aig-eval x a)))
:hints(("Goal" :in-theory (enable aig-env-lookup
subsetp-equal))))
(defthm invar-holds-after-apply-updates1
(implies (and (unsat-p (aig-and invar
(aig-not (aig-restrict invar updates))))
(unsat-p (aig-not (aig-partial-eval invar initst)))
(subsetp-equal (aig-vars invar)
(alist-keys updates)))
(aig-eval
invar
(aig-eval-alist updates (append initst inputs))))
:hints(("Goal" :in-theory (disable unsat-p aig-eval aig-eval-alist)
:do-not-induct t
:use ((:instance unsat-p-necc
(x (aig-and invar
(aig-not (aig-restrict invar updates))))
(env (append initst inputs)))
(:instance unsat-p-necc
(x (aig-not (aig-partial-eval invar initst)))
(env inputs)))))
:otf-flg t)
(defthm invar-holds-after-apply-updates
(implies (and (unsat-p (aig-and invar
(aig-not (aig-restrict invar updates))))
(unsat-p (aig-not (aig-partial-eval invar initst)))
(subsetp-equal (aig-vars invar)
(alist-keys updates)))
(unsat-p (aig-not (aig-partial-eval
invar
(aig-eval-alist
updates
(append initst inputs))))))
:hints(("Goal" :in-theory (disable unsat-p aig-eval aig-eval-alist)
:expand ((unsat-p (aig-not (aig-partial-eval
invar
(aig-eval-alist
updates
(append initst inputs))))))
:do-not-induct t))
:otf-flg t)
(defthm prop-holds-when-invar-holds
(implies (and (unsat-p (aig-and invar (aig-not prop)))
(unsat-p (aig-not (aig-partial-eval invar initst))))
(aig-eval prop (append initst input)))
:hints(("Goal" :in-theory (disable unsat-p aig-eval)
:do-not-induct t
:use ((:instance unsat-p-necc
(x (aig-and invar (aig-not prop)))
(env (append initst input)))
(:instance unsat-p-necc
(x (aig-not (aig-partial-eval invar initst)))
(env input))))))))
;; If there exists an invariant that satisfies inducitivity, sufficiency, and
;; initialization, then check-property is always true.
;; One subtlety: here the initial state may be partial, i.e. an alist that does
;; not bind all the state variables. In this case the full initial state
;; applied is determined by the first input vector.
(defthm inductive-invariant-impl-check-property
(implies (and
;; The variables of the invariant must be state variables, not inputs
(subsetp-equal (aig-vars invar)
(alist-keys updates))
;; The invariant is inductive, i.e. it is preserved by the update function
(unsat-p (aig-and
invar
(aig-not
(aig-restrict invar updates))))
;; The invariant implies the property
(unsat-p (aig-and invar (aig-not prop)))
;; The invariant holds under the initial state.
(unsat-p (aig-not (aig-partial-eval invar initst))))
(check-property updates prop initst inputs))
:hints (("goal" :induct (check-property updates prop initst inputs)
:in-theory (disable unsat-p))))
(defthm inductive-invariant-impl-check-ag-property
(implies (and
;; The variables of the invariant must be state variables, not inputs
(subsetp-equal (aig-vars invar)
(alist-keys updates))
;; The invariant is inductive, i.e. it is preserved by the update function
(unsat-p (aig-and
invar
(aig-not
(aig-restrict invar updates))))
;; The invariant implies the property
(unsat-p (aig-and invar (aig-not prop)))
;; The invariant holds under the initial state.
(unsat-p (aig-not (aig-partial-eval invar initst))))
(check-ag-property updates prop initst)))
(defthm inductive-invariant-impl-check-property-strong
(implies (and
;; The variables of the invariant must be state variables, not inputs
(subsetp-equal (aig-vars invar)
(alist-keys updates))
;; The invariant is inductive, i.e. it is preserved by the update function
(unsat-p (aig-and
invar
(aig-not
(aig-restrict invar updates))))
;; The invariant implies the property
(unsat-p (aig-and invar (aig-not prop)))
;; The invariant holds under the initial state.
(unsat-p (aig-not (aig-partial-eval invar initst))))
(check-property-strong updates prop initst inputs)))
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