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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 "../link-joint")
(include-book "../vector-module")
(local (in-theory (disable nth)))
;; ======================================================================
;;; Table of Contents:
;;;
;;; 1. DE Module Generator of Q4
;;; 2. Multi-Step State Lemma
;;; 3. Single-Step-Update Property
;;; 4. Relationship Between the Input and Output Sequences
;; ======================================================================
;; 1. DE Module Generator of Q4
;;
;; Construct a DE module generator for a queue of four links, Q4, using the
;; link-joint model. Prove the value and state lemmas for this module
;; generator.
(defconst *queue4$go-num* 5)
(defun queue4$data-ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ 2 (mbe :logic (nfix data-size)
:exec data-size)))
(defun queue4$ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ (queue4$data-ins-len data-size)
*queue4$go-num*))
;; DE module generator of Q4
(module-generator
queue4* (data-size)
(si 'queue4 data-size)
(list* 'full-in 'empty-out- (append (sis 'data-in 0 data-size)
(sis 'go 0 *queue4$go-num*)))
(list* 'in-act 'out-act
(sis 'data-out 0 data-size))
'(l0 l1 l2 l3)
(list
;; LINKS
;; L0
(list 'l0
(list* 'l0-status (sis 'd0-out 0 data-size))
(si 'link data-size)
(list* 'in-act 'trans1-act (sis 'd0-in 0 data-size)))
;; L1
(list 'l1
(list* 'l1-status (sis 'd1-out 0 data-size))
(si 'link data-size)
(list* 'trans1-act 'trans2-act (sis 'd1-in 0 data-size)))
;; L2
(list 'l2
(list* 'l2-status (sis 'd2-out 0 data-size))
(si 'link data-size)
(list* 'trans2-act 'trans3-act (sis 'd2-in 0 data-size)))
;; L3
(list 'l3
(list* 'l3-status (sis 'd3-out 0 data-size))
(si 'link data-size)
(list* 'trans3-act 'out-act (sis 'd3-in 0 data-size)))
;; JOINTS
;; In
(list 'in-cntl
'(in-act)
'joint-cntl
(list 'full-in 'l0-status (si 'go 0)))
(list 'in-op
(sis 'd0-in 0 data-size)
(si 'v-buf data-size)
(sis 'data-in 0 data-size))
;; Transfer data1
(list 'trans1-cntl
'(trans1-act)
'joint-cntl
(list 'l0-status 'l1-status (si 'go 1)))
(list 'trans1-op
(sis 'd1-in 0 data-size)
(si 'v-buf data-size)
(sis 'd0-out 0 data-size))
;; Transfer data2
(list 'trans2-cntl
'(trans2-act)
'joint-cntl
(list 'l1-status 'l2-status (si 'go 2)))
(list 'trans2-op
(sis 'd2-in 0 data-size)
(si 'v-buf data-size)
(sis 'd1-out 0 data-size))
;; Transfer data3
(list 'trans3-cntl
'(trans3-act)
'joint-cntl
(list 'l2-status 'l3-status (si 'go 3)))
(list 'trans3-op
(sis 'd3-in 0 data-size)
(si 'v-buf data-size)
(sis 'd2-out 0 data-size))
;; Out
(list 'out-cntl
'(out-act)
'joint-cntl
(list 'l3-status 'empty-out- (si 'go 4)))
(list 'out-op
(sis 'data-out 0 data-size)
(si 'v-buf data-size)
(sis 'd3-out 0 data-size)))
(declare (xargs :guard (natp data-size))))
(make-event
`(progn
,@(state-accessors-gen 'queue4 '(l0 l1 l2 l3) 0)))
;; DE netlist generator. A generated netlist will contain an instance of Q4.
(defund queue4$netlist (data-size)
(declare (xargs :guard (natp data-size)))
(cons (queue4* data-size)
(union$ (link$netlist data-size)
*joint-cntl*
(v-buf$netlist data-size)
:test 'equal)))
;; Recognizer for Q4
(defund queue4& (netlist data-size)
(declare (xargs :guard (and (alistp netlist)
(natp data-size))))
(b* ((subnetlist (delete-to-eq (si 'queue4 data-size) netlist)))
(and (equal (assoc (si 'queue4 data-size) netlist)
(queue4* data-size))
(link& subnetlist data-size)
(joint-cntl& subnetlist)
(v-buf& subnetlist data-size))))
;; Sanity check
(local
(defthmd check-queue4$netlist-64
(and (net-syntax-okp (queue4$netlist 64))
(net-arity-okp (queue4$netlist 64))
(queue4& (queue4$netlist 64) 64))))
;; Constraints on the state of Q4
(defund queue4$st-format (st data-size)
(b* ((l0 (nth *queue4$l0* st))
(l1 (nth *queue4$l1* st))
(l2 (nth *queue4$l2* st))
(l3 (nth *queue4$l3* st)))
(and (link$st-format l0 data-size)
(link$st-format l1 data-size)
(link$st-format l2 data-size)
(link$st-format l3 data-size))))
(defthm queue4$st-format=>constraint
(implies (queue4$st-format st data-size)
(natp data-size))
:hints (("Goal" :in-theory (enable queue4$st-format)))
:rule-classes :forward-chaining)
(defund queue4$valid-st (st data-size)
(b* ((l0 (nth *queue4$l0* st))
(l1 (nth *queue4$l1* st))
(l2 (nth *queue4$l2* st))
(l3 (nth *queue4$l3* st)))
(and (link$valid-st l0 data-size)
(link$valid-st l1 data-size)
(link$valid-st l2 data-size)
(link$valid-st l3 data-size))))
(defthmd queue4$valid-st=>constraint
(implies (queue4$valid-st st data-size)
(natp data-size))
:hints (("Goal" :in-theory (enable queue4$valid-st)))
:rule-classes :forward-chaining)
(defthmd queue4$valid-st=>st-format
(implies (queue4$valid-st st data-size)
(queue4$st-format st data-size))
:hints (("Goal" :in-theory (e/d (queue4$st-format
queue4$valid-st)
(link$st-format)))))
;; Extract the input and output signals for Q4
(progn
;; Extract the input data
(defun queue4$data-in (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(take (mbe :logic (nfix data-size)
:exec data-size)
(nthcdr 2 inputs)))
(defthm len-queue4$data-in
(equal (len (queue4$data-in inputs data-size))
(nfix data-size)))
(in-theory (disable queue4$data-in))
;; Extract the "in-act" signal
(defund queue4$in-act (inputs st data-size)
(b* ((full-in (nth 0 inputs))
(go-signals (nthcdr (queue4$data-ins-len data-size) inputs))
(go-in (nth 0 go-signals))
(l0 (nth *queue4$l0* st))
(l0.s (nth *link$s* l0)))
(joint-act full-in (car l0.s) go-in)))
(defthm queue4$in-act-inactive
(implies (not (nth 0 inputs))
(not (queue4$in-act inputs st data-size)))
:hints (("Goal" :in-theory (enable queue4$in-act))))
;; Extract the "out-act" signal
(defund queue4$out-act (inputs st data-size)
(b* ((empty-out- (nth 1 inputs))
(go-signals (nthcdr (queue4$data-ins-len data-size) inputs))
(go-out (nth 4 go-signals))
(l3 (nth *queue4$l3* st))
(l3.s (nth *link$s* l3)))
(joint-act (car l3.s) empty-out- go-out)))
(defthm queue4$out-act-inactive
(implies (equal (nth 1 inputs) t)
(not (queue4$out-act inputs st data-size)))
:hints (("Goal" :in-theory (enable queue4$out-act))))
;; Extract the output data
(defund queue4$data-out (st)
(v-threefix (strip-cars (nth *link$d*
(nth *queue4$l3* st)))))
(defthm len-queue4$data-out-1
(implies (queue4$st-format st data-size)
(equal (len (queue4$data-out st))
data-size))
:hints (("Goal" :in-theory (enable queue4$st-format
queue4$data-out))))
(defthm len-queue4$data-out-2
(implies (queue4$valid-st st data-size)
(equal (len (queue4$data-out st))
data-size))
:hints (("Goal" :in-theory (enable queue4$valid-st
queue4$data-out))))
(defthm bvp-queue4$data-out
(implies (and (queue4$valid-st st data-size)
(queue4$out-act inputs st data-size))
(bvp (queue4$data-out st)))
:hints (("Goal" :in-theory (enable queue4$valid-st
queue4$out-act
queue4$data-out))))
(defun queue4$outputs (inputs st data-size)
(list* (queue4$in-act inputs st data-size)
(queue4$out-act inputs st data-size)
(queue4$data-out st)))
)
;; The value lemma for Q4
(defthm queue4$value
(b* ((inputs (list* full-in empty-out- (append data-in go-signals))))
(implies (and (queue4& netlist data-size)
(equal (len data-in) data-size)
(true-listp go-signals)
(equal (len go-signals) *queue4$go-num*)
(queue4$st-format st data-size))
(equal (se (si 'queue4 data-size) inputs st netlist)
(queue4$outputs inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(se (si 'queue4 data-size) inputs st netlist))
:in-theory (e/d (de-rules
queue4&
queue4*$destructure
queue4$st-format
queue4$in-act
queue4$out-act
queue4$data-out)
(de-module-disabled-rules)))))
;; This function specifies the next state of Q4.
(defun queue4$step (inputs st data-size)
(b* ((data-in (queue4$data-in inputs data-size))
(go-signals (nthcdr (queue4$data-ins-len data-size) inputs))
(go-trans1 (nth 1 go-signals))
(go-trans2 (nth 2 go-signals))
(go-trans3 (nth 3 go-signals))
(l0 (nth *queue4$l0* st))
(l0.s (nth *link$s* l0))
(l0.d (nth *link$d* l0))
(l1 (nth *queue4$l1* st))
(l1.s (nth *link$s* l1))
(l1.d (nth *link$d* l1))
(l2 (nth *queue4$l2* st))
(l2.s (nth *link$s* l2))
(l2.d (nth *link$d* l2))
(l3 (nth *queue4$l3* st))
(l3.s (nth *link$s* l3))
(in-act (queue4$in-act inputs st data-size))
(out-act (queue4$out-act inputs st data-size))
(trans1-act (joint-act (car l0.s) (car l1.s) go-trans1))
(trans2-act (joint-act (car l1.s) (car l2.s) go-trans2))
(trans3-act (joint-act (car l2.s) (car l3.s) go-trans3))
(l0-inputs (list* in-act trans1-act data-in))
(l1-inputs (list* trans1-act trans2-act (strip-cars l0.d)))
(l2-inputs (list* trans2-act trans3-act (strip-cars l1.d)))
(l3-inputs (list* trans3-act out-act (strip-cars l2.d))))
(list
;; L0
(link$step l0-inputs l0 data-size)
;; L1
(link$step l1-inputs l1 data-size)
;; L2
(link$step l2-inputs l2 data-size)
;; L3
(link$step l3-inputs l3 data-size))))
;; The state lemma for Q4
(defthm queue4$state
(b* ((inputs (list* full-in empty-out- (append data-in go-signals))))
(implies (and (queue4& netlist data-size)
(true-listp data-in)
(equal (len data-in) data-size)
(true-listp go-signals)
(equal (len go-signals) *queue4$go-num*)
(queue4$st-format st data-size))
(equal (de (si 'queue4 data-size) inputs st netlist)
(queue4$step inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(de (si 'queue4 data-size) inputs st netlist))
:in-theory (e/d (de-rules
queue4&
queue4*$destructure
queue4$st-format
queue4$data-in
queue4$in-act
queue4$out-act)
(de-module-disabled-rules)))))
(in-theory (disable queue4$step))
;; ======================================================================
;; 2. Multi-Step State Lemma
;; Conditions on the inputs
(defund queue4$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 (queue4$data-in inputs data-size))
(go-signals (nthcdr (queue4$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) *queue4$go-num*)
(equal inputs
(list* full-in empty-out- (append data-in go-signals))))))
(defthm booleanp-queue4$in-act
(implies (and (queue4$input-format inputs data-size)
(queue4$valid-st st data-size))
(booleanp (queue4$in-act inputs st data-size)))
:hints (("Goal" :in-theory (enable queue4$input-format
queue4$valid-st
queue4$in-act)))
:rule-classes (:rewrite :type-prescription))
(defthm booleanp-queue4$out-act
(implies (and (queue4$input-format inputs data-size)
(queue4$valid-st st data-size))
(booleanp (queue4$out-act inputs st data-size)))
:hints (("Goal" :in-theory (enable queue4$input-format
queue4$valid-st
queue4$out-act)))
:rule-classes (:rewrite :type-prescription))
(simulate-lemma queue4)
;; ======================================================================
;; 3. Single-Step-Update Property
;; The extraction function for Q4 that extracts the future output sequence from
;; the current state.
(defund queue4$extract (st)
(b* ((l0 (nth *queue4$l0* st))
(l1 (nth *queue4$l1* st))
(l2 (nth *queue4$l2* st))
(l3 (nth *queue4$l3* st)))
(extract-valid-data (list l0 l1 l2 l3))))
(defthm queue4$extract-not-empty
(implies (and (queue4$out-act inputs st data-size)
(queue4$valid-st st data-size))
(< 0 (len (queue4$extract st))))
:hints (("Goal"
:in-theory (e/d (queue4$valid-st
queue4$extract
queue4$out-act)
())))
:rule-classes :linear)
;; The extracted next-state function for Q4. Note that this function avoids
;; exploring the internal computation of Q4.
(defund queue4$extracted-step (inputs st data-size)
(b* ((data (queue4$data-in inputs data-size))
(extracted-st (queue4$extract st))
(n (1- (len extracted-st))))
(cond
((equal (queue4$out-act inputs st data-size) t)
(cond
((equal (queue4$in-act inputs st data-size) t)
(cons data (take n extracted-st)))
(t (take n extracted-st))))
(t (cond
((equal (queue4$in-act inputs st data-size) t)
(cons data extracted-st))
(t extracted-st))))))
;; The single-step-update property
(defthm queue4$extracted-step-correct
(b* ((next-st (queue4$step inputs st data-size)))
(implies (and (queue4$input-format inputs data-size)
(queue4$valid-st st data-size))
(equal (queue4$extract next-st)
(queue4$extracted-step inputs st data-size))))
:hints (("Goal"
:in-theory (enable f-sr
queue4$extracted-step
queue4$input-format
queue4$valid-st
queue4$st-format
queue4$step
queue4$in-act
queue4$out-act
queue4$extract))))
;; ======================================================================
;; 4. Relationship Between the Input and Output Sequences
;; Prove that queue4$valid-st is an invariant.
(defthm queue4$valid-st-preserved
(implies (and (queue4$input-format inputs data-size)
(queue4$valid-st st data-size))
(queue4$valid-st (queue4$step inputs st data-size)
data-size))
:hints (("Goal"
:in-theory (e/d (f-sr
queue4$input-format
queue4$valid-st
queue4$st-format
queue4$step
queue4$in-act
queue4$out-act)
(nfix)))))
(defthm queue4$extract-lemma
(implies (and (queue4$valid-st st data-size)
(queue4$out-act inputs st data-size))
(equal (list (queue4$data-out st))
(nthcdr (1- (len (queue4$extract st)))
(queue4$extract st))))
:hints (("Goal"
:in-theory (enable queue4$valid-st
queue4$st-format
queue4$extract
queue4$out-act
queue4$data-out))))
;; Extract the accepted input sequence
(seq-gen queue4 in in-act 0
(queue4$data-in inputs data-size))
;; Extract the valid output sequence
(seq-gen queue4 out out-act 1
(queue4$data-out st)
:netlist-data (nthcdr 2 outputs))
;; The multi-step input-output relationship
(in-out-stream-lemma queue4)
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