1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827
|
;; Copyright (C) 2017, 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 "gcd-body1")
(include-book "gcd-cond")
(include-book "gcd-spec")
(local (include-book "arithmetic-3/top" :dir :system))
(local (in-theory (disable nth)))
;; ======================================================================
;;; Table of Contents:
;;;
;;; 1. DE Module Generator of GCD1
;;; 2. Multi-Step State Lemma
;;; 3. Single-Step-Update Property
;;; 4. Relationship Between the Input and Output Sequences
;; ======================================================================
;; 1. DE Module Generator of GCD1
;;
;; Construct a DE module generator that computes the Greatest Common Divisor
;; (GCD1) of two natural numbers. Prove the value and state lemmas for this
;; module generator. We follow the link-joint model in building this
;; generator.
(defconst *gcd1$go-num* (+ *merge$go-num*
*gcd-cond$go-num*
*gcd-body1$go-num*))
(defun gcd1$data-ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ 2 (* 2 (mbe :logic (nfix data-size)
:exec data-size))))
(defun gcd1$ins-len (data-size)
(declare (xargs :guard (natp data-size)))
(+ (gcd1$data-ins-len data-size)
*gcd1$go-num*))
;; DE module generator of GCD1
(module-generator
gcd1* (data-size)
;; MODULE'S NAME
(si 'gcd1 data-size)
;; INPUTS
;; There are 3 types of inputs for a complex joint:
;; * full-in and empty-out- signals,
;; * input data,
;; * GO signals.
(list* 'full-in 'empty-out- (append (sis 'data-in 0 (* 2 data-size))
(sis 'go 0 *gcd1$go-num*)))
;; OUTPUTS
;; For a complex joint, in addition to outputing the data, we also report the
;; "act" signals from the joints at the module's input and output ports.
(list* 'in-act 'out-act
(sis 'data-out 0 data-size))
;; INTERNAL STATE
'(s l0 l1 l2)
;; OCCURRENCES
;; Our DE description of a self-timed module allows links and joints to appear
;; in any order in the module's occurrence list, except that LINKS MUST BE
;; DECLARED BEFORE JOINTS so that when the module is being evaluated, the "se"
;; function called in the first pass will extract the links' full/empty states
;; and data and provide these values as inputs for the corresponding joints;
;; the "de" function wil make the second pass to update the link's full/empty
;; states and data using the joints' output values calculated from the first
;; pass.
(list
;; LINKS
;; S
'(s (s-status s-out)
link1
(branch-act merge-act done-))
;; L0
(list 'l0
(list* 'l0-status (sis 'd0-out 0 (* 2 data-size)))
(si 'link (* 2 data-size))
(list* 'merge-act 'branch-act (sis 'd0-in 0 (* 2 data-size))))
;; L1
(list 'l1
(list* 'l1-status (sis 'd1-out 0 (* 2 data-size)))
(si 'link (* 2 data-size))
(list* 'branch-act1 'body-act (sis 'd1-in 0 (* 2 data-size))))
;; L2
(list 'l2
(list* 'l2-status (sis 'd2-out 0 (* 2 data-size)))
(si 'link (* 2 data-size))
(list* 'body-act 'merge-act1 (sis 'd2-in 0 (* 2 data-size))))
;; JOINTS
;; Merge-in
'(me-ready-in0 (me-full-in0) b-and (full-in s-status))
'(me-ready-in1 (me-full-in1) b-and (l2-status s-status))
(list 'me
(list* 'merge-act 'in-act 'merge-act1
(sis 'd0-in 0 (* 2 data-size)))
(si 'merge (* 2 data-size))
(list* 'me-full-in0 'me-full-in1 'l0-status 's-out
(append (sis 'data-in 0 (* 2 data-size))
(sis 'd2-out 0 (* 2 data-size))
(sis 'go 0 *merge$go-num*))))
;; Branch-out
'(br-ready-out0 (br-empty-out0-) b-or (empty-out- s-status))
'(br-ready-out1 (br-empty-out1-) b-or (l1-status s-status))
(list 'branch-out
(list* 'branch-act 'out-act 'branch-act1 'done-
(append (sis 'data-out 0 data-size)
(sis 'd1-in 0 (* 2 data-size))))
(si 'gcd-cond data-size)
(list* 'l0-status 'br-empty-out0- 'br-empty-out1-
(append (sis 'd0-out 0 (* 2 data-size))
(sis 'go *merge$go-num* *gcd-cond$go-num*))))
;; Body
(list 'body
(list* 'body-act
(sis 'd2-in 0 (* 2 data-size)))
(si 'gcd-body1 data-size)
(list* 'l1-status 'l2-status
(append (sis 'd1-out 0 (* 2 data-size))
(sis 'go
(+ *merge$go-num*
*gcd-cond$go-num*)
*gcd-body1$go-num*)))))
(declare (xargs :guard (natp data-size))))
(make-event
`(progn
,@(state-accessors-gen 'gcd1 '(s l0 l1 l2) 0)))
;; DE netlist generator. A generated netlist will contain an instance of GCD1.
(defund gcd1$netlist (data-size)
(declare (xargs :guard (and (natp data-size)
(<= 2 data-size))))
(cons (gcd1* data-size)
(union$ (link1$netlist)
(link$netlist (* 2 data-size))
(gcd-cond$netlist data-size)
(gcd-body1$netlist data-size)
:test 'equal)))
;; Recognizer for GCD1
(defund gcd1& (netlist data-size)
(declare (xargs :guard (and (alistp netlist)
(natp data-size)
(<= 2 data-size))))
(b* ((subnetlist (delete-to-eq (si 'gcd1 data-size) netlist)))
(and (equal (assoc (si 'gcd1 data-size) netlist)
(gcd1* data-size))
(link1& subnetlist)
(link& subnetlist (* 2 data-size))
(gcd-cond& subnetlist data-size)
(gcd-body1& subnetlist data-size)
(merge& subnetlist (* 2 data-size)))))
;; Sanity check
(local
(defthmd check-gcd1$netlist-64
(and (net-syntax-okp (gcd1$netlist 64))
(net-arity-okp (gcd1$netlist 64))
(gcd1& (gcd1$netlist 64) 64))))
;; Constraints on the state of GCD1
(defund gcd1$st-format (st data-size)
(b* ((l0 (nth *gcd1$l0* st))
(l1 (nth *gcd1$l1* st))
(l2 (nth *gcd1$l2* st)))
(and (natp data-size)
(<= 3 data-size)
(link$st-format l0 (* 2 data-size))
(link$st-format l1 (* 2 data-size))
(link$st-format l2 (* 2 data-size)))))
(defthm gcd1$st-format=>constraint
(implies (gcd1$st-format st data-size)
(and (natp data-size)
(<= 3 data-size)))
:hints (("Goal" :in-theory (enable gcd1$st-format)))
:rule-classes :forward-chaining)
(defund gcd1$valid-st (st data-size)
(b* ((s (nth *gcd1$s* st))
(l0 (nth *gcd1$l0* st))
(l1 (nth *gcd1$l1* st))
(l2 (nth *gcd1$l2* st)))
(and (gcd1$st-format st data-size)
(link1$valid-st s)
(link$valid-st l0 (* 2 data-size))
(link$valid-st l1 (* 2 data-size))
(link$valid-st l2 (* 2 data-size)))))
(defthmd gcd1$valid-st=>constraint
(implies (gcd1$valid-st st data-size)
(and (natp data-size)
(<= 3 data-size)))
:hints (("Goal" :in-theory (enable gcd1$valid-st)))
:rule-classes :forward-chaining)
(defthmd gcd1$valid-st=>st-format
(implies (gcd1$valid-st st data-size)
(gcd1$st-format st data-size))
:hints (("Goal" :in-theory (e/d (gcd1$valid-st)
()))))
;; Extract the input and output signals for GCD1
(progn
;; Extract the input data
(defun gcd1$data-in (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(take (* 2 (mbe :logic (nfix data-size)
:exec data-size))
(nthcdr 2 inputs)))
(defthm len-gcd1$data-in
(equal (len (gcd1$data-in inputs data-size))
(* 2 (nfix data-size))))
(in-theory (disable gcd1$data-in))
;; Extract the inputs for the merge-in joint
(defund gcd1$me-inputs (inputs st data-size)
(b* ((full-in (nth 0 inputs))
(data-in (gcd1$data-in inputs data-size))
(go-signals (nthcdr (gcd1$data-ins-len data-size) inputs))
(me-go-signals (take *merge$go-num* go-signals))
(s (nth *gcd1$s* st))
(s.s (nth *link1$s* s))
(s.d (nth *link1$d* s))
(l0 (nth *gcd1$l0* st))
(l0.s (nth *link$s* l0))
(l2 (nth *gcd1$l2* st))
(l2.s (nth *link$s* l2))
(l2.d (nth *link$d* l2))
(me-full-in0 (f-and full-in (car s.s)))
(me-full-in1 (f-and (car l2.s) (car s.s))))
(list* me-full-in0 me-full-in1 (car l0.s) (car s.d)
(append data-in
(v-threefix (strip-cars l2.d))
me-go-signals))))
;; Extract the inputs for the branch-out joint
(defund gcd1$br-inputs (inputs st data-size)
(b* ((empty-out- (nth 1 inputs))
(go-signals (nthcdr (gcd1$data-ins-len data-size) inputs))
(br-go-signals (take *gcd-cond$go-num*
(nthcdr *merge$go-num* go-signals)))
(s (nth *gcd1$s* st))
(s.s (nth *link1$s* s))
(l0 (nth *gcd1$l0* st))
(l0.s (nth *link$s* l0))
(l0.d (nth *link$d* l0))
(l1 (nth *gcd1$l1* st))
(l1.s (nth *link$s* l1))
(br-empty-out0- (f-or empty-out- (car s.s)))
(br-empty-out1- (f-or (car l1.s) (car s.s))))
(list* (f-buf (car l0.s)) br-empty-out0- br-empty-out1-
(append (v-threefix (strip-cars l0.d))
br-go-signals))))
;; Extract the inputs for the "body" joint
(defund gcd1$body-inputs (inputs st data-size)
(b* ((go-signals (nthcdr (gcd1$data-ins-len data-size) inputs))
(body-go-signals (take *gcd-body1$go-num*
(nthcdr (+ *merge$go-num*
*gcd-cond$go-num*)
go-signals)))
(l1 (nth *gcd1$l1* st))
(l1.s (nth *link$s* l1))
(l1.d (nth *link$d* l1))
(l2 (nth *gcd1$l2* st))
(l2.s (nth *link$s* l2)))
(list* (f-buf (car l1.s)) (f-buf (car l2.s))
(append (v-threefix (strip-cars l1.d))
body-go-signals))))
;; Extract the "in-act" signal
(defund gcd1$in-act (inputs st data-size)
(merge$act0 (gcd1$me-inputs inputs st data-size)
(* 2 data-size)))
(defthm gcd1$in-act-inactive
(implies (not (nth 0 inputs))
(not (gcd1$in-act inputs st data-size)))
:hints (("Goal" :in-theory (enable gcd1$me-inputs
gcd1$in-act))))
;; Extract the "out-act" signal
(defund gcd1$out-act (inputs st data-size)
(gcd-cond$act0 (gcd1$br-inputs inputs st data-size)
data-size))
(defthm gcd1$out-act-inactive
(implies (equal (nth 1 inputs) t)
(not (gcd1$out-act inputs st data-size)))
:hints (("Goal" :in-theory (enable gcd1$br-inputs
gcd1$out-act))))
;; Extract the output data
(defund gcd1$data-out (inputs st data-size)
(gcd-cond$data0-out (gcd1$br-inputs inputs st data-size)
data-size))
(defthm len-gcd1$data-out-1
(implies (gcd1$st-format st data-size)
(equal (len (gcd1$data-out inputs st data-size))
data-size))
:hints (("Goal" :in-theory (enable gcd1$st-format
gcd1$data-out))))
(defthm len-gcd1$data-out-2
(implies (gcd1$valid-st st data-size)
(equal (len (gcd1$data-out inputs st data-size))
data-size))
:hints (("Goal" :in-theory (enable gcd1$valid-st
gcd1$data-out))))
(defthm bvp-gcd1$data-out
(implies (and (gcd1$valid-st st data-size)
(gcd1$out-act inputs st data-size))
(bvp (gcd1$data-out inputs st data-size)))
:hints (("Goal" :in-theory (enable gcd1$valid-st
gcd1$out-act
gcd1$data-out
gcd1$br-inputs
gcd-cond$br-inputs
gcd-cond$act0
gcd-cond$data-in
branch$act0))))
(defun gcd1$outputs (inputs st data-size)
(list* (gcd1$in-act inputs st data-size)
(gcd1$out-act inputs st data-size)
(gcd1$data-out inputs st data-size)))
)
;; The value lemma for GCD1
(defthm gcd1$value
(b* ((inputs (list* full-in empty-out- (append data-in go-signals))))
(implies (and (gcd1& netlist data-size)
(true-listp data-in)
(equal (len data-in) (* 2 data-size))
(true-listp go-signals)
(equal (len go-signals) *gcd1$go-num*)
(gcd1$st-format st data-size))
(equal (se (si 'gcd1 data-size) inputs st netlist)
(gcd1$outputs inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(se (si 'gcd1 data-size) inputs st netlist))
:in-theory (e/d (de-rules
gcd1&
gcd1*$destructure
gcd1$data-in
merge$act0
gcd1$st-format
gcd1$in-act
gcd1$out-act
gcd1$data-out
gcd1$br-inputs
gcd1$me-inputs)
(de-module-disabled-rules)))))
;; This function specifies the next state of GCD1.
(defun gcd1$step (inputs st data-size)
(b* ((data-in (gcd1$data-in inputs data-size))
(s (nth *gcd1$s* st))
(s.d (nth *link1$d* s))
(l0 (nth *gcd1$l0* st))
(l1 (nth *gcd1$l1* st))
(l2 (nth *gcd1$l2* st))
(l2.d (nth *link$d* l2))
(me-inputs (gcd1$me-inputs inputs st data-size))
(br-inputs (gcd1$br-inputs inputs st data-size))
(body-inputs (gcd1$body-inputs inputs st data-size))
(d1-in (gcd-cond$data1-out br-inputs data-size))
(d2-in (gcd-body1$data-out body-inputs data-size))
(done- (gcd-cond$flag br-inputs data-size))
(merge-act1 (merge$act1 me-inputs (* 2 data-size)))
(merge-act (merge$act me-inputs (* 2 data-size)))
(branch-act1 (gcd-cond$act1 br-inputs data-size))
(branch-act (gcd-cond$act br-inputs data-size))
(body-act (gcd-body1$act body-inputs data-size))
(s-inputs (list branch-act merge-act done-))
(l0-inputs (list* merge-act branch-act
(fv-if (car s.d) (strip-cars l2.d) data-in)))
(l1-inputs (list* branch-act1 body-act d1-in))
(l2-inputs (list* body-act merge-act1 d2-in)))
(list
;; S
(link1$step s-inputs s)
;; L0
(link$step l0-inputs l0 (* 2 data-size))
;; L1
(link$step l1-inputs l1 (* 2 data-size))
;; L2
(link$step l2-inputs l2 (* 2 data-size)))))
;; The state lemma for GCD1
(defthm gcd1$state
(b* ((inputs (list* full-in empty-out- (append data-in go-signals))))
(implies (and (gcd1& netlist data-size)
(true-listp data-in)
(equal (len data-in) (* 2 data-size))
(true-listp go-signals)
(equal (len go-signals) *gcd1$go-num*)
(gcd1$st-format st data-size))
(equal (de (si 'gcd1 data-size) inputs st netlist)
(gcd1$step inputs st data-size))))
:hints (("Goal"
:do-not-induct t
:expand (:free (inputs data-size)
(de (si 'gcd1 data-size) inputs st netlist))
:in-theory (e/d (de-rules
gcd1&
gcd1*$destructure
merge$act
merge$act0
merge$act1
gcd1$st-format
gcd1$data-in
gcd1$data-out
gcd1$br-inputs
gcd1$me-inputs
gcd1$body-inputs)
(de-module-disabled-rules)))))
(in-theory (disable gcd1$step))
;; ======================================================================
;; 2. Multi-Step State Lemma
;; Conditions on the inputs
(defund gcd1$input-format (inputs data-size)
(declare (xargs :guard (and (true-listp inputs)
(natp data-size))))
(b* ((full-in (nth 0 inputs))
(empty-out- (nth 1 inputs))
(data-in (gcd1$data-in inputs data-size))
(go-signals (nthcdr (gcd1$data-ins-len data-size) inputs)))
(and
(booleanp full-in)
(booleanp empty-out-)
(or (not full-in) (bvp data-in))
(true-listp go-signals)
(= (len go-signals) *gcd1$go-num*)
(equal inputs
(list* full-in empty-out- (append data-in go-signals))))))
(defthm booleanp-gcd1$in-act
(implies (and (gcd1$input-format inputs data-size)
(gcd1$valid-st st data-size))
(booleanp (gcd1$in-act inputs st data-size)))
:hints (("Goal" :in-theory (enable merge$act0
gcd1$input-format
gcd1$valid-st
gcd1$in-act
gcd1$me-inputs)))
:rule-classes (:rewrite :type-prescription))
(defthm booleanp-gcd1$out-act
(implies (and (gcd1$input-format inputs data-size)
(gcd1$valid-st st data-size))
(booleanp (gcd1$out-act inputs st data-size)))
:hints (("Goal" :in-theory (enable branch$act0
gcd-cond$act0
gcd-cond$br-inputs
gcd-cond$flag
gcd-cond$data-in
gcd1$input-format
gcd1$valid-st
gcd1$out-act
gcd1$br-inputs)))
:rule-classes (:rewrite :type-prescription))
(simulate-lemma gcd1)
;; ======================================================================
;; 3. Single-Step-Update Property
;; The extraction function for GCD1 that extracts the future output sequence
;; from the current state.
(defund gcd1$extract (st)
(b* ((l0 (nth *gcd1$l0* st))
(l1 (nth *gcd1$l1* st))
(l2 (nth *gcd1$l2* st)))
(gcd$op-map
(extract-valid-data (list l1 l2 l0)))))
(defthm gcd1$extract-not-empty
(implies (and (gcd1$out-act inputs st data-size)
(gcd1$valid-st st data-size))
(< 0 (len (gcd1$extract st))))
:hints (("Goal"
:in-theory (e/d (branch$act0
gcd-cond$br-inputs
gcd-cond$act0
gcd1$valid-st
gcd1$extract
gcd1$br-inputs
gcd1$out-act)
())))
:rule-classes :linear)
;; Specify and prove a state invariant
(progn
(defund gcd1$inv (st)
(b* ((s (nth *gcd1$s* st))
(s.s (nth *link1$s* s))
(s.d (nth *link1$d* s)))
(if (and (fullp s.s) (not (car s.d)))
(= (len (gcd1$extract st))
0)
(= (len (gcd1$extract st))
1))))
(local
(defthm gcd1$input-format-lemma-1
(implies (gcd1$input-format inputs data-size)
(booleanp (nth 0 inputs)))
:hints (("Goal" :in-theory (enable gcd1$input-format)))
:rule-classes (:rewrite :type-prescription)))
(local
(defthm gcd1$input-format-lemma-2
(implies (gcd1$input-format inputs data-size)
(booleanp (nth 1 inputs)))
:hints (("Goal" :in-theory (enable gcd1$input-format)))
:rule-classes (:rewrite :type-prescription)))
(local
(defthm gcd1$input-format-lemma-3
(implies (and (gcd1$input-format inputs data-size)
(nth 0 inputs))
(bvp (gcd1$data-in inputs data-size)))
:hints (("Goal" :in-theory (enable gcd1$input-format)))))
(local
(defthm booleanp-gcd1$body-act
(b* ((body-inputs (gcd1$body-inputs inputs st data-size))
(l1 (nth *gcd1$l1* st))
(l1.d (nth *link$d* l1)))
(implies (or (equal (nth *link$s*
(nth *gcd1$l1* st))
'(nil))
(and (equal (nth *link$s*
(nth *gcd1$l1* st))
'(t))
(equal (nth *link$s*
(nth *gcd1$l2* st))
'(nil))
(equal (len l1.d) (* 2 data-size))
(bvp (strip-cars l1.d))))
(booleanp (gcd-body1$act body-inputs data-size))))
:hints (("Goal" :in-theory (enable merge$act0
merge$act1
merge$act
gcd-body1$data-in
gcd-body1$me-inputs
gcd-body1$act
gcd-body1$a<b
gcd1$body-inputs)))
:rule-classes (:rewrite :type-prescription)))
(local
(defthm gcd1$body-act-inactive
(b* ((body-inputs (gcd1$body-inputs inputs st data-size)))
(implies (or (equal (nth *link$s*
(nth *gcd1$l1* st))
'(nil))
(equal (nth *link$s*
(nth *gcd1$l2* st))
'(t)))
(not (gcd-body1$act body-inputs data-size))))
:hints (("Goal" :in-theory (enable gcd1$body-inputs)))))
(defthm gcd1$inv-preserved
(implies (and (gcd1$input-format inputs data-size)
(gcd1$valid-st st data-size)
(gcd1$inv st))
(gcd1$inv (gcd1$step inputs st data-size)))
:hints (("Goal"
:in-theory (e/d (f-sr
gcd1$valid-st
gcd1$st-format
gcd1$inv
gcd1$step
gcd1$extract
gcd1$br-inputs
gcd1$me-inputs
gcd-cond$data-in
gcd-cond$flag
gcd-cond$act
gcd-cond$act0
gcd-cond$act1
gcd-cond$br-inputs
branch$act0
branch$act1
merge$act
merge$act0
merge$act1)
(b-nor3)))))
)
;; The extracted next-state function for GCD1. Note that this function avoids
;; exploring the internal computation of GCD1.
(defund gcd1$extracted-step (inputs st data-size)
(b* ((data (gcd$op (gcd1$data-in inputs data-size)))
(extracted-st (gcd1$extract st))
(n (1- (len extracted-st))))
(cond
((equal (gcd1$out-act inputs st data-size) t)
(cond
((equal (gcd1$in-act inputs st data-size) t)
(cons data (take n extracted-st)))
(t (take n extracted-st))))
(t (cond
((equal (gcd1$in-act inputs st data-size) t)
(cons data extracted-st))
(t extracted-st))))))
;; The single-step-update property
;; This property characterizes the one-step update on the future output
;; sequence given the current inputs and current state. The trick here is to
;; apply the extraction function gcd1$extract to the step function gcd1$step so
;; that the one-step update on the future output sequence can be expressed in
;; terms of the gcd1$extracted-step function, which abstracts away the internal
;; computation of GCD1.
(progn
(local
(defthm gcd-body1$data-out-expand
(b* ((body-inputs (gcd1$body-inputs inputs st data-size))
(l1 (nth *gcd1$l1* st))
(l1.d (nth *link$d* l1)))
(implies
(and (natp data-size)
(equal (len l1.d) (* 2 data-size))
(bvp (strip-cars l1.d)))
(equal (gcd-body1$data-out body-inputs data-size)
(v-if (v-< nil t
(rev (take data-size (strip-cars l1.d)))
(rev (nthcdr data-size (strip-cars l1.d))))
(append
(v-adder-output
t
(nthcdr data-size (strip-cars l1.d))
(v-not (take data-size (strip-cars l1.d))))
(take data-size (strip-cars l1.d)))
(append
(v-adder-output
t
(take data-size (strip-cars l1.d))
(v-not (nthcdr data-size (strip-cars l1.d))))
(nthcdr data-size (strip-cars l1.d)))))))
:hints (("Goal"
:in-theory (enable gcd-body1$data-in
gcd-body1$a<b
gcd-body1$data-out
gcd-body1$data0-out
gcd-body1$data1-out
gcd1$body-inputs)))))
(defthm gcd1$extracted-step-correct
(b* ((next-st (gcd1$step inputs st data-size)))
(implies (and (gcd1$input-format inputs data-size)
(gcd1$valid-st st data-size)
(gcd1$inv st))
(equal (gcd1$extract next-st)
(gcd1$extracted-step inputs st data-size))))
:hints (("Goal"
:in-theory (e/d (f-sr
joint-act
gcd1$extracted-step
gcd1$valid-st
gcd1$st-format
gcd1$inv
gcd1$step
gcd1$in-act
gcd1$out-act
gcd1$data-out
gcd1$br-inputs
gcd1$me-inputs
gcd1$extract
gcd-cond$data-in
gcd-cond$flag
gcd-cond$act
gcd-cond$act0
gcd-cond$act1
gcd-cond$data1-out
gcd-cond$br-inputs
branch$act0
branch$act1
merge$act
merge$act0
merge$act1)
(b-nor3
v-if-works
v-not-take
v-not-nthcdr)))))
)
;; ======================================================================
;; 4. Relationship Between the Input and Output Sequences
;; Prove that gcd1$valid-st is an invariant.
(defthm gcd1$valid-st-preserved
(implies (and (gcd1$input-format inputs data-size)
(gcd1$valid-st st data-size))
(gcd1$valid-st (gcd1$step inputs st data-size)
data-size))
:hints (("Goal"
:in-theory (e/d (f-sr
joint-act
gcd1$valid-st
gcd1$st-format
gcd1$step
gcd1$br-inputs
gcd1$me-inputs
gcd-cond$data-in
gcd-cond$flag
gcd-cond$act
gcd-cond$act0
gcd-cond$act1
gcd-cond$br-inputs
branch$act0
branch$act1
merge$act
merge$act0
merge$act1)
(b-nor3)))))
(defthm gcd1$extract-lemma
(implies (and (gcd1$valid-st st data-size)
(gcd1$out-act inputs st data-size))
(equal (list (gcd1$data-out inputs st data-size))
(nthcdr (1- (len (gcd1$extract st)))
(gcd1$extract st))))
:hints (("Goal"
:do-not-induct t
:in-theory (e/d (branch$act0
gcd-cond$data-in
gcd-cond$br-inputs
gcd-cond$act0
gcd-cond$flag
gcd-cond$data0-out
gcd1$valid-st
gcd1$extract
gcd$op
gcd1$br-inputs
gcd1$out-act
gcd1$data-out)
(v-if-works)))))
;; Extract the accepted input sequence
(seq-gen gcd1 in in-act 0
(gcd1$data-in inputs data-size))
;; Extract the valid output sequence
(seq-gen gcd1 out out-act 1
(gcd1$data-out inputs st data-size)
:netlist-data (nthcdr 2 outputs))
;; The multi-step input-output relationship
(in-out-stream-lemma gcd1 :op gcd$op :inv t)
|