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|
;; Copyright (C) 2018, Regents of the University of Texas
;; Written by Cuong Chau
;; License: A 3-clause BSD license. See the LICENSE file distributed with
;; ACL2.
;; Cuong Chau <ckcuong@cs.utexas.edu>
;; May 2019
(in-package "ADE")
(include-book "queue4")
(include-book "queue5")
(local (include-book "arithmetic/top" :dir :system))
(local (in-theory (disable nth)))
;; ======================================================================
;;; Table of Contents:
;;;
;;; 1. DE Module Generator of Q10
;;; 2. Multi-Step State Lemma
;;; 3. Single-Step-Update Property
;;; 4. Relationship Between the Input and Output Sequences
;; ======================================================================
;; 1. DE Module Generator of Q10
;;
;; Construct a DE module generator for a queue of four links, Q10, using the
;; link-joint model. Prove the value and state lemmas for this module
;; generator.
(defconst *queue10$go-num* (+ *queue4$go-num*
*queue5$go-num*))
(defun queue10$data-ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ 2 (mbe :logic (nfix data-size)
:exec data-size)))
(defun queue10$ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ (queue10$data-ins-len data-size)
*queue10$go-num*))
;; DE module generator of Q10
(module-generator
queue10* (data-size)
(si 'queue10 data-size)
(list* 'full-in 'empty-out- (append (sis 'data-in 0 data-size)
(sis 'go 0 *queue10$go-num*)))
(list* 'in-act 'out-act
(sis 'data-out 0 data-size))
'(l q4 q5)
(list
;; LINK
;; L
(list 'l
(list* 'l-status (sis 'd-out 0 data-size))
(si 'link data-size)
(list* 'q4-out-act 'q5-in-act (sis 'd-in 0 data-size)))
;; JOINTS
;; Q4
(list 'q4
(list* 'in-act 'q4-out-act
(sis 'd-in 0 data-size))
(si 'queue4 data-size)
(list* 'full-in 'l-status (append (sis 'data-in 0 data-size)
(sis 'go 0 *queue4$go-num*))))
;; Q5
(list 'q5
(list* 'q5-in-act 'out-act
(sis 'data-out 0 data-size))
(si 'queue5 data-size)
(list* 'l-status 'empty-out- (append (sis 'd-out 0 data-size)
(sis 'go
*queue4$go-num*
*queue5$go-num*)))))
(declare (xargs :guard (natp data-size))))
(make-event
`(progn
,@(state-accessors-gen 'queue10 '(l q4 q5) 0)))
;; DE netlist generator. A generated netlist will contain an instance of Q10.
(defund queue10$netlist (data-size)
(declare (xargs :guard (natp data-size)))
(cons (queue10* data-size)
(union$ (queue4$netlist data-size)
(queue5$netlist data-size)
:test 'equal)))
;; Recognizer for Q10
(defund queue10& (netlist data-size)
(declare (xargs :guard (and (alistp netlist)
(natp data-size))))
(b* ((subnetlist (delete-to-eq (si 'queue10 data-size) netlist)))
(and (equal (assoc (si 'queue10 data-size) netlist)
(queue10* data-size))
(link& subnetlist data-size)
(queue4& subnetlist data-size)
(queue5& subnetlist data-size))))
;; Sanity check
(local
(defthmd check-queue10$netlist-64
(and (net-syntax-okp (queue10$netlist 64))
(net-arity-okp (queue10$netlist 64))
(queue10& (queue10$netlist 64) 64))))
;; Constraints on the state of Q10
(defund queue10$st-format (st data-size)
(b* ((l (nth *queue10$l* st))
(q4 (nth *queue10$q4* st))
(q5 (nth *queue10$q5* st)))
(and (link$st-format l data-size)
(queue4$st-format q4 data-size)
(queue5$st-format q5 data-size))))
(defthm queue10$st-format=>constraint
(implies (queue10$st-format st data-size)
(natp data-size))
:hints (("Goal" :in-theory (enable queue10$st-format)))
:rule-classes :forward-chaining)
(defund queue10$valid-st (st data-size)
(b* ((l (nth *queue10$l* st))
(q4 (nth *queue10$q4* st))
(q5 (nth *queue10$q5* st)))
(and (link$valid-st l data-size)
(queue4$valid-st q4 data-size)
(queue5$valid-st q5 data-size))))
(defthmd queue10$valid-st=>constraint
(implies (queue10$valid-st st data-size)
(natp data-size))
:hints (("Goal" :in-theory (enable queue10$valid-st)))
:rule-classes :forward-chaining)
(defthmd queue10$valid-st=>st-format
(implies (queue10$valid-st st data-size)
(queue10$st-format st data-size))
:hints (("Goal" :in-theory (e/d (queue4$valid-st=>st-format
queue5$valid-st=>st-format
queue10$st-format
queue10$valid-st)
(link$st-format)))))
;; Extract the input and output signals for Q10
(progn
;; Extract the input data
(defun queue10$data-in (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(take (mbe :logic (nfix data-size)
:exec data-size)
(nthcdr 2 inputs)))
(defthm len-queue10$data-in
(equal (len (queue10$data-in inputs data-size))
(nfix data-size)))
(in-theory (disable queue10$data-in))
;; Extract the inputs for joint Q4
(defund queue10$q4-inputs (inputs st data-size)
(b* ((full-in (nth 0 inputs))
(data-in (queue10$data-in inputs data-size))
(go-signals (nthcdr (queue10$data-ins-len data-size) inputs))
(q4-go-signals (take *queue4$go-num* go-signals))
(l (nth *queue10$l* st))
(l.s (nth *link$s* l)))
(list* full-in (f-buf (car l.s))
(append data-in q4-go-signals))))
;; Extract the inputs for joint Q5
(defund queue10$q5-inputs (inputs st data-size)
(b* ((empty-out- (nth 1 inputs))
(go-signals (nthcdr (queue10$data-ins-len data-size) inputs))
(q5-go-signals (take *queue5$go-num*
(nthcdr *queue4$go-num* go-signals)))
(l (nth *queue10$l* st))
(l.s (nth *link$s* l))
(l.d (nth *link$d* l)))
(list* (f-buf (car l.s)) empty-out-
(append (v-threefix (strip-cars l.d)) q5-go-signals))))
;; Extract the "in-act" signal
(defund queue10$in-act (inputs st data-size)
(b* ((q4-inputs (queue10$q4-inputs inputs st data-size))
(q4 (nth *queue10$q4* st)))
(queue4$in-act q4-inputs q4 data-size)))
(defthm queue10$in-act-inactive
(implies (not (nth 0 inputs))
(not (queue10$in-act inputs st data-size)))
:hints (("Goal" :in-theory (enable queue10$q4-inputs
queue10$in-act))))
;; Extract the "out-act" signal
(defund queue10$out-act (inputs st data-size)
(b* ((q5-inputs (queue10$q5-inputs inputs st data-size))
(q5 (nth *queue10$q5* st)))
(queue5$out-act q5-inputs q5 data-size)))
(defthm queue10$out-act-inactive
(implies (equal (nth 1 inputs) t)
(not (queue10$out-act inputs st data-size)))
:hints (("Goal" :in-theory (enable queue10$q5-inputs
queue10$out-act))))
;; Extract the output data
(defund queue10$data-out (st)
(b* ((q5 (nth *queue10$q5* st)))
(queue5$data-out q5)))
(defthm len-queue10$data-out-1
(implies (queue10$st-format st data-size)
(equal (len (queue10$data-out st))
data-size))
:hints (("Goal" :in-theory (enable queue10$st-format
queue10$data-out))))
(defthm len-queue10$data-out-2
(implies (queue10$valid-st st data-size)
(equal (len (queue10$data-out st))
data-size))
:hints (("Goal" :in-theory (enable queue10$valid-st
queue10$data-out))))
(defthm bvp-queue10$data-out
(implies (and (queue10$valid-st st data-size)
(queue10$out-act inputs st data-size))
(bvp (queue10$data-out st)))
:hints (("Goal" :in-theory (enable queue10$valid-st
queue10$out-act
queue10$data-out))))
(defun queue10$outputs (inputs st data-size)
(list* (queue10$in-act inputs st data-size)
(queue10$out-act inputs st data-size)
(queue10$data-out st)))
)
;; The value lemma for Q10
(defthm queue10$value
(b* ((inputs (list* full-in empty-out- (append data-in go-signals))))
(implies (and (queue10& netlist data-size)
(true-listp data-in)
(equal (len data-in) data-size)
(true-listp go-signals)
(equal (len go-signals) *queue10$go-num*)
(queue10$st-format st data-size))
(equal (se (si 'queue10 data-size) inputs st netlist)
(queue10$outputs inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(se (si 'queue10 data-size) inputs st netlist))
:in-theory (e/d (de-rules
queue10&
queue10*$destructure
queue10$st-format
queue10$data-in
queue10$q4-inputs
queue10$q5-inputs
queue10$in-act
queue10$out-act
queue10$data-out)
(de-module-disabled-rules)))))
;; This function specifies the next state of Q10.
(defun queue10$step (inputs st data-size)
(b* ((l (nth *queue10$l* st))
(q4 (nth *queue10$q4* st))
(q5 (nth *queue10$q5* st))
(q4-inputs (queue10$q4-inputs inputs st data-size))
(q5-inputs (queue10$q5-inputs inputs st data-size))
(q4-out-act (queue4$out-act q4-inputs q4 data-size))
(q5-in-act (queue5$in-act q5-inputs q5 data-size))
(d-in (queue4$data-out q4))
(l-inputs (list* q4-out-act q5-in-act d-in)))
(list
;; L
(link$step l-inputs l data-size)
;; Joint Q4
(queue4$step q4-inputs q4 data-size)
;; Joint Q5
(queue5$step q5-inputs q5 data-size))))
;; The state lemma for Q10
(defthm queue10$state
(b* ((inputs (list* full-in empty-out- (append data-in go-signals))))
(implies (and (queue10& netlist data-size)
(true-listp data-in)
(equal (len data-in) data-size)
(true-listp go-signals)
(equal (len go-signals) *queue10$go-num*)
(queue10$st-format st data-size))
(equal (de (si 'queue10 data-size) inputs st netlist)
(queue10$step inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(de (si 'queue10 data-size) inputs st netlist))
:in-theory (e/d (de-rules
queue10&
queue10*$destructure
queue10$st-format
queue10$data-in
queue10$q4-inputs
queue10$q5-inputs
queue10$in-act
queue10$out-act)
(de-module-disabled-rules)))))
(in-theory (disable queue10$step))
;; ======================================================================
;; 2. Multi-Step State Lemma
;; Conditions on the inputs
(defund queue10$input-format (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(b* ((full-in (nth 0 inputs))
(empty-out- (nth 1 inputs))
(data-in (queue10$data-in inputs data-size))
(go-signals (nthcdr (queue10$data-ins-len data-size) inputs)))
(and
(booleanp full-in)
(booleanp empty-out-)
(or (not full-in) (bvp data-in))
(true-listp go-signals)
(= (len go-signals) *queue10$go-num*)
(equal inputs
(list* full-in empty-out- (append data-in go-signals))))))
(local
(defthm queue10$input-format=>q4$input-format
(implies (and (queue10$input-format inputs data-size)
(queue10$valid-st st data-size))
(queue4$input-format
(queue10$q4-inputs inputs st data-size)
data-size))
:hints (("Goal"
:in-theory (e/d (queue4$input-format
queue4$data-in
queue10$input-format
queue10$valid-st
queue10$q4-inputs)
())))))
(local
(defthm queue10$input-format=>q5$input-format
(implies (and (queue10$input-format inputs data-size)
(queue10$valid-st st data-size))
(queue5$input-format
(queue10$q5-inputs inputs st data-size)
data-size))
:hints (("Goal"
:in-theory (e/d (queue5$input-format
queue5$data-in
queue10$input-format
queue10$valid-st
queue10$q5-inputs)
())))))
(defthm booleanp-queue10$in-act
(implies (and (queue10$input-format inputs data-size)
(queue10$valid-st st data-size))
(booleanp (queue10$in-act inputs st data-size)))
:hints (("Goal"
:in-theory (enable queue10$valid-st
queue10$in-act)))
:rule-classes (:rewrite :type-prescription))
(defthm booleanp-queue10$out-act
(implies (and (queue10$input-format inputs data-size)
(queue10$valid-st st data-size))
(booleanp (queue10$out-act inputs st data-size)))
:hints (("Goal"
:in-theory (enable queue10$valid-st
queue10$out-act)))
:rule-classes (:rewrite :type-prescription))
(simulate-lemma queue10)
;; ======================================================================
;; 3. Single-Step-Update Property
;; The extraction function for Q10 that extracts the future output sequence from
;; the current state.
(defund queue10$extract (st)
(b* ((l (nth *queue10$l* st))
(q4 (nth *queue10$q4* st))
(q5 (nth *queue10$q5* st)))
(append (queue4$extract q4)
(extract-valid-data (list l))
(queue5$extract q5))))
(defthm queue10$extract-not-empty
(implies (and (queue10$out-act inputs st data-size)
(queue10$valid-st st data-size))
(< 0 (len (queue10$extract st))))
:hints (("Goal"
:in-theory (e/d (queue10$valid-st
queue10$extract
queue10$out-act)
())))
:rule-classes :linear)
;; The extracted next-state function for Q10. Note that this function avoids
;; exploring the internal computation of Q10.
(defund queue10$extracted-step (inputs st data-size)
(b* ((data (queue10$data-in inputs data-size))
(extracted-st (queue10$extract st))
(n (1- (len extracted-st))))
(cond
((equal (queue10$out-act inputs st data-size) t)
(cond
((equal (queue10$in-act inputs st data-size) t)
(cons data (take n extracted-st)))
(t (take n extracted-st))))
(t (cond
((equal (queue10$in-act inputs st data-size) t)
(cons data extracted-st))
(t extracted-st))))))
;; The single-step-update property
(progn
(local
(defthm queue10$q4-out-act-inactive
(implies (equal (nth *link$s* (nth *queue10$l* st))
'(t))
(not (queue4$out-act (queue10$q4-inputs inputs st data-size)
(nth *queue10$q4* st)
data-size)))
:hints (("Goal" :in-theory (enable queue10$q4-inputs)))))
(local
(defthm queue10$q5-in-act-inactive
(implies (equal (nth *link$s* (nth *queue10$l* st))
'(nil))
(not (queue5$in-act (queue10$q5-inputs inputs st data-size)
(nth *queue10$q5* st)
data-size)))
:hints (("Goal" :in-theory (enable queue10$q5-inputs)))))
(local
(defthm queue10$q4-data-in-rewrite
(equal (queue4$data-in
(queue10$q4-inputs inputs st data-size)
data-size)
(queue10$data-in inputs data-size))
:hints (("Goal"
:in-theory (enable queue4$data-in
queue10$q4-inputs)))))
(local
(defthm queue10$q5-data-in-rewrite
(b* ((l (nth *queue10$l* st))
(l.d (nth *link$d* l)))
(implies (equal (len l.d) data-size)
(equal (queue5$data-in
(queue10$q5-inputs inputs st data-size)
data-size)
(v-threefix (strip-cars l.d)))))
:hints (("Goal"
:in-theory (enable queue5$data-in
queue10$q5-inputs)))))
(defthm queue10$extracted-step-correct
(b* ((next-st (queue10$step inputs st data-size)))
(implies (and (queue10$input-format inputs data-size)
(queue10$valid-st st data-size))
(equal (queue10$extract next-st)
(queue10$extracted-step inputs st data-size))))
:hints (("Goal"
:use (queue10$input-format=>q4$input-format
queue10$input-format=>q5$input-format)
:in-theory (e/d (f-sr
left-associativity-of-append
queue4$extracted-step
queue5$extracted-step
queue10$extracted-step
queue10$valid-st
queue10$step
queue10$in-act
queue10$out-act
queue10$extract)
(queue10$input-format=>q4$input-format
queue10$input-format=>q5$input-format
associativity-of-append)))))
)
;; ======================================================================
;; 4. Relationship Between the Input and Output Sequences
;; Prove that queue10$valid-st is an invariant.
(defthm queue10$valid-st-preserved
(implies (and (queue10$input-format inputs data-size)
(queue10$valid-st st data-size))
(queue10$valid-st (queue10$step inputs st data-size)
data-size))
:hints (("Goal"
:use (queue10$input-format=>q4$input-format
queue10$input-format=>q5$input-format)
:in-theory (e/d (f-sr
queue10$valid-st
queue10$step)
(queue10$input-format=>q4$input-format
queue10$input-format=>q5$input-format)))))
(defthm queue10$extract-lemma
(implies (and (queue10$valid-st st data-size)
(queue10$out-act inputs st data-size))
(equal (list (queue10$data-out st))
(nthcdr (1- (len (queue10$extract st)))
(queue10$extract st))))
:hints (("Goal"
:in-theory (enable queue10$valid-st
queue10$extract
queue10$out-act
queue10$data-out))))
;; Extract the accepted input sequence
(seq-gen queue10 in in-act 0
(queue10$data-in inputs data-size))
;; Extract the valid output sequence
(seq-gen queue10 out out-act 1
(queue10$data-out st)
:netlist-data (nthcdr 2 outputs))
;; The multi-step input-output relationship
(in-out-stream-lemma queue10)
|
