Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 36,685 Bytes
4365a98 |
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 |
(* Title: OAWN_Invariants.thy
License: BSD 2-Clause. See LICENSE.
Author: Timothy Bourke
*)
section "Generic open invariants on sequential AWN processes"
theory OAWN_Invariants
imports Invariants OInvariants
AWN_Cterms AWN_Labels AWN_Invariants
OAWN_SOS
begin
subsection "Open invariants via labelled control terms"
lemma oseqp_sos_subterms:
assumes "wellformed \<Gamma>"
and "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
and "((\<sigma>, p), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
shows "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
using assms
proof (induct p)
fix p1 p2
assume IH1: "\<exists>pn. p1 \<in> subterms (\<Gamma> pn) \<Longrightarrow>
((\<sigma>, p1), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i \<Longrightarrow>
\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
and IH2: "\<exists>pn. p2 \<in> subterms (\<Gamma> pn) \<Longrightarrow>
((\<sigma>, p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i \<Longrightarrow>
\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
and "\<exists>pn. p1 \<oplus> p2 \<in> subterms (\<Gamma> pn)"
and "((\<sigma>, p1 \<oplus> p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
from \<open>\<exists>pn. p1 \<oplus> p2 \<in> subterms (\<Gamma> pn)\<close> obtain pn
where "p1 \<in> subterms (\<Gamma> pn)"
and "p2 \<in> subterms (\<Gamma> pn)" by auto
from \<open>((\<sigma>, p1 \<oplus> p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i\<close>
have "((\<sigma>, p1), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i
\<or> ((\<sigma>, p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i" by auto
thus "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
proof
assume "((\<sigma>, p1), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
with \<open>p1 \<in> subterms (\<Gamma> pn)\<close> show ?thesis by (auto intro: IH1)
next
assume "((\<sigma>, p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
with \<open>p2 \<in> subterms (\<Gamma> pn)\<close> show ?thesis by (auto intro: IH2)
qed
qed auto
lemma oreachable_subterms:
assumes "wellformed \<Gamma>"
and "control_within \<Gamma> (init A)"
and "trans A = oseqp_sos \<Gamma> i"
and "(\<sigma>, p) \<in> oreachable A S U"
shows "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
using assms(4)
proof (induct rule: oreachable_pair_induct)
fix \<sigma> p
assume "(\<sigma>, p) \<in> init A"
with \<open>control_within \<Gamma> (init A)\<close> show "\<exists>pn. p \<in> subterms (\<Gamma> pn)" ..
next
fix \<sigma> p a \<sigma>' p'
assume "(\<sigma>, p) \<in> oreachable A S U"
and "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
and 3: "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S \<sigma> \<sigma>' a"
moreover from 3 and \<open>trans A = oseqp_sos \<Gamma> i\<close>
have "((\<sigma>, p), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i" by simp
ultimately show "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
using \<open>wellformed \<Gamma>\<close>
by (auto elim: oseqp_sos_subterms)
qed
lemma onl_oinvariantI [intro]:
assumes init: "\<And>\<sigma> p l. \<lbrakk> (\<sigma>, p) \<in> init A; l \<in> labels \<Gamma> p \<rbrakk> \<Longrightarrow> P (\<sigma>, l)"
and other: "\<And>\<sigma> \<sigma>' p l. \<lbrakk> (\<sigma>, p) \<in> oreachable A S U;
\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l);
U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> \<forall>l\<in>labels \<Gamma> p. P (\<sigma>', l)"
and step: "\<And>\<sigma> p a \<sigma>' p' l'.
\<lbrakk> (\<sigma>, p) \<in> oreachable A S U;
\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l);
((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A;
l' \<in> labels \<Gamma> p';
S \<sigma> \<sigma>' a \<rbrakk> \<Longrightarrow> P (\<sigma>', l')"
shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
proof
fix \<sigma> p
assume "(\<sigma>, p) \<in> init A"
hence "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)" using init by simp
thus "onl \<Gamma> P (\<sigma>, p)" ..
next
fix \<sigma> p a \<sigma>' p'
assume rp: "(\<sigma>, p) \<in> oreachable A S U"
and "onl \<Gamma> P (\<sigma>, p)"
and tr: "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S \<sigma> \<sigma>' a"
from \<open>onl \<Gamma> P (\<sigma>, p)\<close> have "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)" ..
with rp tr \<open>S \<sigma> \<sigma>' a\<close> have "\<forall>l'\<in>labels \<Gamma> p'. P (\<sigma>', l')" by (auto elim: step)
thus "onl \<Gamma> P (\<sigma>', p')" ..
next
fix \<sigma> \<sigma>' p
assume "(\<sigma>, p) \<in> oreachable A S U"
and "onl \<Gamma> P (\<sigma>, p)"
and "U \<sigma> \<sigma>'"
from \<open>onl \<Gamma> P (\<sigma>, p)\<close> have "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)" by auto
with \<open>(\<sigma>, p) \<in> oreachable A S U\<close> have "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>', l)"
using \<open>U \<sigma> \<sigma>'\<close> by (rule other)
thus "onl \<Gamma> P (\<sigma>', p)" by auto
qed
lemma global_oinvariantI [intro]:
assumes init: "\<And>\<sigma> p. (\<sigma>, p) \<in> init A \<Longrightarrow> P \<sigma>"
and other: "\<And>\<sigma> \<sigma>' p l. \<lbrakk> (\<sigma>, p) \<in> oreachable A S U; P \<sigma>; U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> P \<sigma>'"
and step: "\<And>\<sigma> p a \<sigma>' p'.
\<lbrakk> (\<sigma>, p) \<in> oreachable A S U;
P \<sigma>;
((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A;
S \<sigma> \<sigma>' a \<rbrakk> \<Longrightarrow> P \<sigma>'"
shows "A \<Turnstile> (S, U \<rightarrow>) (\<lambda>(\<sigma>, _). P \<sigma>)"
proof
fix \<sigma> p
assume "(\<sigma>, p) \<in> init A"
thus "(\<lambda>(\<sigma>, _). P \<sigma>) (\<sigma>, p)"
by simp (erule init)
next
fix \<sigma> p a \<sigma>' p'
assume rp: "(\<sigma>, p) \<in> oreachable A S U"
and "(\<lambda>(\<sigma>, _). P \<sigma>) (\<sigma>, p)"
and tr: "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S \<sigma> \<sigma>' a"
from \<open>(\<lambda>(\<sigma>, _). P \<sigma>) (\<sigma>, p)\<close> have "P \<sigma>" by simp
with rp have "P \<sigma>'"
using tr \<open>S \<sigma> \<sigma>' a\<close> by (rule step)
thus "(\<lambda>(\<sigma>, _). P \<sigma>) (\<sigma>', p')" by simp
next
fix \<sigma> \<sigma>' p
assume "(\<sigma>, p) \<in> oreachable A S U"
and "(\<lambda>(\<sigma>, _). P \<sigma>) (\<sigma>, p)"
and "U \<sigma> \<sigma>'"
hence "P \<sigma>'" by simp (erule other)
thus "(\<lambda>(\<sigma>, _). P \<sigma>) (\<sigma>', p)" by simp
qed
lemma onl_oinvariantD [dest]:
assumes "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
and "(\<sigma>, p) \<in> oreachable A S U"
and "l \<in> labels \<Gamma> p"
shows "P (\<sigma>, l)"
using assms unfolding onl_def by auto
lemma onl_oinvariant_weakenD [dest]:
assumes "A \<Turnstile> (S', U' \<rightarrow>) onl \<Gamma> P"
and "(\<sigma>, p) \<in> oreachable A S U"
and "l \<in> labels \<Gamma> p"
and weakenS: "\<And>s s' a. S s s' a \<Longrightarrow> S' s s' a"
and weakenU: "\<And>s s'. U s s' \<Longrightarrow> U' s s'"
shows "P (\<sigma>, l)"
proof -
from \<open>(\<sigma>, p) \<in> oreachable A S U\<close> have "(\<sigma>, p) \<in> oreachable A S' U'"
by (rule oreachable_weakenE)
(erule weakenS, erule weakenU)
with \<open>A \<Turnstile> (S', U' \<rightarrow>) onl \<Gamma> P\<close> show "P (\<sigma>, l)"
using \<open>l \<in> labels \<Gamma> p\<close> ..
qed
lemma onl_oinvariant_initD [dest]:
assumes invP: "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
and init: "(\<sigma>, p) \<in> init A"
and pnl: "l \<in> labels \<Gamma> p"
shows "P (\<sigma>, l)"
proof -
from init have "(\<sigma>, p) \<in> oreachable A S U" ..
with invP show ?thesis using pnl ..
qed
lemma onl_oinvariant_sterms:
assumes wf: "wellformed \<Gamma>"
and il: "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
and rp: "(\<sigma>, p) \<in> oreachable A S U"
and "p'\<in>sterms \<Gamma> p"
and "l\<in>labels \<Gamma> p'"
shows "P (\<sigma>, l)"
proof -
from wf \<open>p'\<in>sterms \<Gamma> p\<close> \<open>l\<in>labels \<Gamma> p'\<close> have "l\<in>labels \<Gamma> p"
by (rule labels_sterms_labels)
with il rp show "P (\<sigma>, l)" ..
qed
lemma onl_oinvariant_sterms_weaken:
assumes wf: "wellformed \<Gamma>"
and il: "A \<Turnstile> (S', U' \<rightarrow>) onl \<Gamma> P"
and rp: "(\<sigma>, p) \<in> oreachable A S U"
and "p'\<in>sterms \<Gamma> p"
and "l\<in>labels \<Gamma> p'"
and weakenS: "\<And>\<sigma> \<sigma>' a. S \<sigma> \<sigma>' a \<Longrightarrow> S' \<sigma> \<sigma>' a"
and weakenU: "\<And>\<sigma> \<sigma>'. U \<sigma> \<sigma>' \<Longrightarrow> U' \<sigma> \<sigma>'"
shows "P (\<sigma>, l)"
proof -
from \<open>(\<sigma>, p) \<in> oreachable A S U\<close> have "(\<sigma>, p) \<in> oreachable A S' U'"
by (rule oreachable_weakenE)
(erule weakenS, erule weakenU)
with assms(1-2) show ?thesis using assms(4-5)
by (rule onl_oinvariant_sterms)
qed
lemma otrans_from_sterms:
assumes "((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
and "wellformed \<Gamma>"
shows "\<exists>p'\<in>sterms \<Gamma> p. ((\<sigma>, p'), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
using assms by (induction p rule: sterms_pinduct [OF \<open>wellformed \<Gamma>\<close>]) auto
lemma otrans_from_sterms':
assumes "((\<sigma>, p'), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
and "wellformed \<Gamma>"
and "p' \<in> sterms \<Gamma> p"
shows "((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
using assms by (induction p rule: sterms_pinduct [OF \<open>wellformed \<Gamma>\<close>]) auto
lemma otrans_to_dterms:
assumes "((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
and "wellformed \<Gamma>"
shows "\<forall>r\<in>sterms \<Gamma> q. r \<in> dterms \<Gamma> p"
using assms by (induction q) auto
theorem cterms_includes_sterms_of_oseq_reachable:
assumes "wellformed \<Gamma>"
and "control_within \<Gamma> (init A)"
and "trans A = oseqp_sos \<Gamma> i"
shows "\<Union>(sterms \<Gamma> ` snd ` oreachable A S U) \<subseteq> cterms \<Gamma>"
proof
fix qs
assume "qs \<in> \<Union>(sterms \<Gamma> ` snd ` oreachable A S U)"
then obtain \<xi> and q where *: "(\<xi>, q) \<in> oreachable A S U"
and **: "qs \<in> sterms \<Gamma> q" by auto
from * have "\<And>x. x \<in> sterms \<Gamma> q \<Longrightarrow> x \<in> cterms \<Gamma>"
proof (induction rule: oreachable_pair_induct)
fix \<sigma> p q
assume "(\<sigma>, p) \<in> init A"
and "q \<in> sterms \<Gamma> p"
from \<open>control_within \<Gamma> (init A)\<close> and \<open>(\<sigma>, p) \<in> init A\<close>
obtain pn where "p \<in> subterms (\<Gamma> pn)" by auto
with \<open>wellformed \<Gamma>\<close> show "q \<in> cterms \<Gamma>" using \<open>q\<in>sterms \<Gamma> p\<close>
by (rule subterms_sterms_in_cterms)
next
fix p \<sigma> a \<sigma>' q x
assume "(\<sigma>, p) \<in> oreachable A S U"
and IH: "\<And>x. x \<in> sterms \<Gamma> p \<Longrightarrow> x \<in> cterms \<Gamma>"
and "((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A"
and "x \<in> sterms \<Gamma> q"
from this(3) and \<open>trans A = oseqp_sos \<Gamma> i\<close>
have step: "((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i" by simp
from step \<open>wellformed \<Gamma>\<close> obtain ps
where ps: "ps \<in> sterms \<Gamma> p"
and step': "((\<sigma>, ps), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
by (rule otrans_from_sterms [THEN bexE])
from ps have "ps \<in> cterms \<Gamma>" by (rule IH)
moreover from step' \<open>wellformed \<Gamma>\<close> \<open>x \<in> sterms \<Gamma> q\<close> have "x \<in> dterms \<Gamma> ps"
by (rule otrans_to_dterms [rule_format])
ultimately show "x \<in> cterms \<Gamma>" by (rule ctermsDI)
qed
thus "qs \<in> cterms \<Gamma>" using ** .
qed
corollary oseq_reachable_in_cterms:
assumes "wellformed \<Gamma>"
and "control_within \<Gamma> (init A)"
and "trans A = oseqp_sos \<Gamma> i"
and "(\<sigma>, p) \<in> oreachable A S U"
and "p' \<in> sterms \<Gamma> p"
shows "p' \<in> cterms \<Gamma>"
using assms(1-3)
proof (rule cterms_includes_sterms_of_oseq_reachable [THEN subsetD])
from assms(4-5) show "p' \<in> \<Union>(sterms \<Gamma> ` snd ` oreachable A S U)"
by (auto elim!: rev_bexI)
qed
lemma oseq_invariant_ctermI:
assumes wf: "wellformed \<Gamma>"
and cw: "control_within \<Gamma> (init A)"
and sl: "simple_labels \<Gamma>"
and sp: "trans A = oseqp_sos \<Gamma> i"
and init: "\<And>\<sigma> p l. \<lbrakk>
(\<sigma>, p) \<in> init A;
l\<in>labels \<Gamma> p
\<rbrakk> \<Longrightarrow> P (\<sigma>, l)"
and other: "\<And>\<sigma> \<sigma>' p l. \<lbrakk>
(\<sigma>, p) \<in> oreachable A S U;
l\<in>labels \<Gamma> p;
P (\<sigma>, l);
U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> P (\<sigma>', l)"
and local: "\<And>p l \<sigma> a q l' \<sigma>' pp. \<lbrakk>
p\<in>cterms \<Gamma>;
l\<in>labels \<Gamma> p;
P (\<sigma>, l);
((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i;
((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A;
l'\<in>labels \<Gamma> q;
(\<sigma>, pp)\<in>oreachable A S U;
p\<in>sterms \<Gamma> pp;
(\<sigma>', q)\<in>oreachable A S U;
S \<sigma> \<sigma>' a
\<rbrakk> \<Longrightarrow> P (\<sigma>', l')"
shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
proof
fix \<sigma> p l
assume "(\<sigma>, p) \<in> init A"
and *: "l \<in> labels \<Gamma> p"
with init show "P (\<sigma>, l)" by auto
next
fix \<sigma> p a \<sigma>' q l'
assume sr: "(\<sigma>, p) \<in> oreachable A S U"
and pl: "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)"
and tr: "((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A"
and A6: "l' \<in> labels \<Gamma> q"
and "S \<sigma> \<sigma>' a"
thus "P (\<sigma>', l')"
proof -
from sr and tr and \<open>S \<sigma> \<sigma>' a\<close> have A7: "(\<sigma>', q) \<in> oreachable A S U"
by - (rule oreachable_local')
from tr and sp have tr': "((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i" by simp
then obtain p' where "p' \<in> sterms \<Gamma> p"
and A4: "((\<sigma>, p'), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
by (blast dest: otrans_from_sterms [OF _ wf])
from wf cw sp sr this(1) have A1: "p'\<in>cterms \<Gamma>"
by (rule oseq_reachable_in_cterms)
from labels_not_empty [OF wf] obtain ll where A2: "ll\<in>labels \<Gamma> p'"
by blast
with \<open>p'\<in>sterms \<Gamma> p\<close> have "ll\<in>labels \<Gamma> p"
by (rule labels_sterms_labels [OF wf])
with pl have A3: "P (\<sigma>, ll)" by simp
from sr \<open>p'\<in>sterms \<Gamma> p\<close>
obtain pp where A7: "(\<sigma>, pp)\<in>oreachable A S U"
and A8: "p'\<in>sterms \<Gamma> pp"
by auto
from sr tr \<open>S \<sigma> \<sigma>' a\<close> have A9: "(\<sigma>', q)\<in>oreachable A S U"
by - (rule oreachable_local')
from sp and \<open>((\<sigma>, p'), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i\<close>
have A5: "((\<sigma>, p'), a, (\<sigma>', q)) \<in> trans A" by simp
from A1 A2 A3 A4 A5 A6 A7 A8 A9 \<open>S \<sigma> \<sigma>' a\<close> show ?thesis by (rule local)
qed
next
fix \<sigma> \<sigma>' p l
assume sr: "(\<sigma>, p) \<in> oreachable A S U"
and "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)"
and "U \<sigma> \<sigma>'"
show "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>', l)"
proof
fix l
assume "l\<in>labels \<Gamma> p"
with \<open>\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)\<close> have "P (\<sigma>, l)" ..
with sr and \<open>l\<in>labels \<Gamma> p\<close>
show "P (\<sigma>', l)" using \<open>U \<sigma> \<sigma>'\<close> by (rule other)
qed
qed
lemma oseq_invariant_ctermsI:
assumes wf: "wellformed \<Gamma>"
and cw: "control_within \<Gamma> (init A)"
and sl: "simple_labels \<Gamma>"
and sp: "trans A = oseqp_sos \<Gamma> i"
and init: "\<And>\<sigma> p l. \<lbrakk>
(\<sigma>, p) \<in> init A;
l\<in>labels \<Gamma> p
\<rbrakk> \<Longrightarrow> P (\<sigma>, l)"
and other: "\<And>\<sigma> \<sigma>' p l. \<lbrakk>
wellformed \<Gamma>;
(\<sigma>, p) \<in> oreachable A S U;
l\<in>labels \<Gamma> p;
P (\<sigma>, l);
U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> P (\<sigma>', l)"
and local: "\<And>p l \<sigma> a q l' \<sigma>' pp pn. \<lbrakk>
wellformed \<Gamma>;
p\<in>ctermsl (\<Gamma> pn);
not_call p;
l\<in>labels \<Gamma> p;
P (\<sigma>, l);
((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i;
((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A;
l'\<in>labels \<Gamma> q;
(\<sigma>, pp)\<in>oreachable A S U;
p\<in>sterms \<Gamma> pp;
(\<sigma>', q)\<in>oreachable A S U;
S \<sigma> \<sigma>' a
\<rbrakk> \<Longrightarrow> P (\<sigma>', l')"
shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
proof (rule oseq_invariant_ctermI [OF wf cw sl sp])
fix \<sigma> p l
assume "(\<sigma>, p) \<in> init A"
and "l \<in> labels \<Gamma> p"
thus "P (\<sigma>, l)" by (rule init)
next
fix \<sigma> \<sigma>' p l
assume "(\<sigma>, p) \<in> oreachable A S U"
and "l \<in> labels \<Gamma> p"
and "P (\<sigma>, l)"
and "U \<sigma> \<sigma>'"
with wf show "P (\<sigma>', l)" by (rule other)
next
fix p l \<sigma> a q l' \<sigma>' pp
assume "p \<in> cterms \<Gamma>"
and otherassms: "l \<in> labels \<Gamma> p"
"P (\<sigma>, l)"
"((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
"((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A"
"l' \<in> labels \<Gamma> q"
"(\<sigma>, pp) \<in> oreachable A S U"
"p \<in> sterms \<Gamma> pp"
"(\<sigma>', q) \<in> oreachable A S U"
"S \<sigma> \<sigma>' a"
from this(1) obtain pn where "p \<in> ctermsl(\<Gamma> pn)"
and "not_call p"
unfolding cterms_def' [OF wf] by auto
with wf show "P (\<sigma>', l')"
using otherassms by (rule local)
qed
subsection "Open step invariants via labelled control terms"
lemma onll_ostep_invariantI [intro]:
assumes *: "\<And>\<sigma> p l a \<sigma>' p' l'. \<lbrakk> (\<sigma>, p)\<in>oreachable A S U;
((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A;
S \<sigma> \<sigma>' a;
l \<in>labels \<Gamma> p;
l'\<in>labels \<Gamma> p' \<rbrakk>
\<Longrightarrow> P ((\<sigma>, l), a, (\<sigma>', l'))"
shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
proof
fix \<sigma> p \<sigma>' p' a
assume "(\<sigma>, p) \<in> oreachable A S U"
and "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S \<sigma> \<sigma>' a"
hence "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, (\<sigma>', l'))" by (auto elim!: *)
thus "onll \<Gamma> P ((\<sigma>, p), a, (\<sigma>', p'))" ..
qed
lemma onll_ostep_invariantE [elim]:
assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
and "(\<sigma>, p) \<in> oreachable A S U"
and "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S \<sigma> \<sigma>' a"
and lp: "l \<in>labels \<Gamma> p"
and lp': "l'\<in>labels \<Gamma> p'"
shows "P ((\<sigma>, l), a, (\<sigma>', l'))"
proof -
from assms(1-4) have "onll \<Gamma> P ((\<sigma>, p), a, (\<sigma>', p'))" ..
with lp lp' show "P ((\<sigma>, l), a, (\<sigma>', l'))" by auto
qed
lemma onll_ostep_invariantD [dest]:
assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
and "(\<sigma>, p) \<in> oreachable A S U"
and "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S \<sigma> \<sigma>' a"
shows "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, (\<sigma>', l'))"
using assms by auto
lemma onll_ostep_invariant_weakenD [dest]:
assumes "A \<Turnstile>\<^sub>A (S', U' \<rightarrow>) onll \<Gamma> P"
and "(\<sigma>, p) \<in> oreachable A S U"
and "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S' \<sigma> \<sigma>' a"
and weakenS: "\<And>s s' a. S s s' a \<Longrightarrow> S' s s' a"
and weakenU: "\<And>s s'. U s s' \<Longrightarrow> U' s s'"
shows "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, (\<sigma>', l'))"
proof -
from \<open>(\<sigma>, p) \<in> oreachable A S U\<close> have "(\<sigma>, p) \<in> oreachable A S' U'"
by (rule oreachable_weakenE)
(erule weakenS, erule weakenU)
with \<open>A \<Turnstile>\<^sub>A (S', U' \<rightarrow>) onll \<Gamma> P\<close> show ?thesis
using \<open>((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A\<close> and \<open>S' \<sigma> \<sigma>' a\<close> ..
qed
lemma onll_ostep_to_invariantI [intro]:
assumes sinv: "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> Q"
and wf: "wellformed \<Gamma>"
and init: "\<And>\<sigma> l p. \<lbrakk> (\<sigma>, p) \<in> init A; l\<in>labels \<Gamma> p \<rbrakk> \<Longrightarrow> P (\<sigma>, l)"
and other: "\<And>\<sigma> \<sigma>' p l.
\<lbrakk> (\<sigma>, p) \<in> oreachable A S U;
l\<in>labels \<Gamma> p;
P (\<sigma>, l);
U \<sigma> \<sigma>' \<rbrakk> \<Longrightarrow> P (\<sigma>', l)"
and local: "\<And>\<sigma> p l \<sigma>' l' a.
\<lbrakk> (\<sigma>, p) \<in> oreachable A S U;
l\<in>labels \<Gamma> p;
P (\<sigma>, l);
Q ((\<sigma>, l), a, (\<sigma>', l'));
S \<sigma> \<sigma>' a\<rbrakk> \<Longrightarrow> P (\<sigma>', l')"
shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
proof
fix \<sigma> p l
assume "(\<sigma>, p) \<in> init A" and "l\<in>labels \<Gamma> p"
thus "P (\<sigma>, l)" by (rule init)
next
fix \<sigma> p a \<sigma>' p' l'
assume sr: "(\<sigma>, p) \<in> oreachable A S U"
and lp: "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)"
and tr: "((\<sigma>, p), a, (\<sigma>', p')) \<in> trans A"
and "S \<sigma> \<sigma>' a"
and lp': "l' \<in> labels \<Gamma> p'"
show "P (\<sigma>', l')"
proof -
from lp obtain l where "l\<in>labels \<Gamma> p" and "P (\<sigma>, l)"
using labels_not_empty [OF wf] by auto
from sinv sr tr \<open>S \<sigma> \<sigma>' a\<close> this(1) lp' have "Q ((\<sigma>, l), a, (\<sigma>', l'))" ..
with sr \<open>l\<in>labels \<Gamma> p\<close> \<open>P (\<sigma>, l)\<close> show "P (\<sigma>', l')" using \<open>S \<sigma> \<sigma>' a\<close> by (rule local)
qed
next
fix \<sigma> \<sigma>' p l
assume "(\<sigma>, p) \<in> oreachable A S U"
and "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)"
and "U \<sigma> \<sigma>'"
show "\<forall>l\<in>labels \<Gamma> p. P (\<sigma>', l)"
proof
fix l
assume "l\<in>labels \<Gamma> p"
with \<open>\<forall>l\<in>labels \<Gamma> p. P (\<sigma>, l)\<close> have "P (\<sigma>, l)" ..
with \<open>(\<sigma>, p) \<in> oreachable A S U\<close> and \<open>l\<in>labels \<Gamma> p\<close>
show "P (\<sigma>', l)" using \<open>U \<sigma> \<sigma>'\<close> by (rule other)
qed
qed
lemma onll_ostep_invariant_sterms:
assumes wf: "wellformed \<Gamma>"
and si: "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
and sr: "(\<sigma>, p) \<in> oreachable A S U"
and sos: "((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A"
and "S \<sigma> \<sigma>' a"
and "l'\<in>labels \<Gamma> q"
and "p'\<in>sterms \<Gamma> p"
and "l\<in>labels \<Gamma> p'"
shows "P ((\<sigma>, l), a, (\<sigma>', l'))"
proof -
from wf \<open>p'\<in>sterms \<Gamma> p\<close> \<open>l\<in>labels \<Gamma> p'\<close> have "l\<in>labels \<Gamma> p"
by (rule labels_sterms_labels)
with si sr sos \<open>S \<sigma> \<sigma>' a\<close> show "P ((\<sigma>, l), a, (\<sigma>', l'))" using \<open>l'\<in>labels \<Gamma> q\<close> ..
qed
lemma oseq_step_invariant_sterms:
assumes inv: "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
and wf: "wellformed \<Gamma>"
and sp: "trans A = oseqp_sos \<Gamma> i"
and "l'\<in>labels \<Gamma> q"
and sr: "(\<sigma>, p) \<in> oreachable A S U"
and tr: "((\<sigma>, p'), a, (\<sigma>', q)) \<in> trans A"
and "S \<sigma> \<sigma>' a"
and "p'\<in>sterms \<Gamma> p"
shows "\<forall>l\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, (\<sigma>', l'))"
proof
from assms(3, 6) have "((\<sigma>, p'), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i" by simp
hence "((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
using wf \<open>p'\<in>sterms \<Gamma> p\<close> by (rule otrans_from_sterms')
with assms(3) have trp: "((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A" by simp
fix l assume "l \<in> labels \<Gamma> p'"
with wf inv sr trp \<open>S \<sigma> \<sigma>' a\<close> \<open>l'\<in>labels \<Gamma> q\<close> \<open>p'\<in>sterms \<Gamma> p\<close>
show "P ((\<sigma>, l), a, (\<sigma>', l'))"
by - (erule(7) onll_ostep_invariant_sterms)
qed
lemma oseq_step_invariant_sterms_weaken:
assumes inv: "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
and wf: "wellformed \<Gamma>"
and sp: "trans A = oseqp_sos \<Gamma> i"
and "l'\<in>labels \<Gamma> q"
and sr: "(\<sigma>, p) \<in> oreachable A S' U'"
and tr: "((\<sigma>, p'), a, (\<sigma>', q)) \<in> trans A"
and "S' \<sigma> \<sigma>' a"
and "p'\<in>sterms \<Gamma> p"
and weakenS: "\<And>\<sigma> \<sigma>' a. S' \<sigma> \<sigma>' a \<Longrightarrow> S \<sigma> \<sigma>' a"
and weakenU: "\<And>\<sigma> \<sigma>'. U' \<sigma> \<sigma>' \<Longrightarrow> U \<sigma> \<sigma>'"
shows "\<forall>l\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, (\<sigma>', l'))"
proof -
from \<open>S' \<sigma> \<sigma>' a\<close> have "S \<sigma> \<sigma>' a" by (rule weakenS)
from \<open>(\<sigma>, p) \<in> oreachable A S' U'\<close>
have Ir: "(\<sigma>, p) \<in> oreachable A S U"
by (rule oreachable_weakenE)
(erule weakenS, erule weakenU)
with assms(1-4) show ?thesis
using tr \<open>S \<sigma> \<sigma>' a\<close> \<open>p'\<in>sterms \<Gamma> p\<close>
by (rule oseq_step_invariant_sterms)
qed
lemma onll_ostep_invariant_any_sterms:
assumes wf: "wellformed \<Gamma>"
and si: "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
and sr: "(\<sigma>, p) \<in> oreachable A S U"
and sos: "((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A"
and "S \<sigma> \<sigma>' a"
and "l'\<in>labels \<Gamma> q"
shows "\<forall>p'\<in>sterms \<Gamma> p. \<forall>l\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, (\<sigma>', l'))"
by (intro ballI) (rule onll_ostep_invariant_sterms [OF assms])
lemma oseq_step_invariant_ctermI [intro]:
assumes wf: "wellformed \<Gamma>"
and cw: "control_within \<Gamma> (init A)"
and sl: "simple_labels \<Gamma>"
and sp: "trans A = oseqp_sos \<Gamma> i"
and local: "\<And>p l \<sigma> a q l' \<sigma>' pp. \<lbrakk>
p\<in>cterms \<Gamma>;
l\<in>labels \<Gamma> p;
((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i;
((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A;
l'\<in>labels \<Gamma> q;
(\<sigma>, pp) \<in> oreachable A S U;
p\<in>sterms \<Gamma> pp;
(\<sigma>', q) \<in> oreachable A S U;
S \<sigma> \<sigma>' a
\<rbrakk> \<Longrightarrow> P ((\<sigma>, l), a, (\<sigma>', l'))"
shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
proof
fix \<sigma> p l a \<sigma>' q l'
assume sr: "(\<sigma>, p) \<in> oreachable A S U"
and tr: "((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A"
and "S \<sigma> \<sigma>' a"
and pl: "l \<in> labels \<Gamma> p"
and A5: "l' \<in> labels \<Gamma> q"
from this(2) and sp have "((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i" by simp
then obtain p' where "p' \<in> sterms \<Gamma> p"
and A3: "((\<sigma>, p'), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
by (blast dest: otrans_from_sterms [OF _ wf])
from this(2) and sp have A4: "((\<sigma>, p'), a, (\<sigma>', q)) \<in> trans A" by simp
from wf cw sp sr \<open>p'\<in>sterms \<Gamma> p\<close> have A1: "p'\<in>cterms \<Gamma>"
by (rule oseq_reachable_in_cterms)
from sr \<open>p'\<in>sterms \<Gamma> p\<close>
obtain pp where A6: "(\<sigma>, pp)\<in>oreachable A S U"
and A7: "p'\<in>sterms \<Gamma> pp"
by auto
from sr tr \<open>S \<sigma> \<sigma>' a\<close> have A8: "(\<sigma>', q)\<in>oreachable A S U"
by - (erule(2) oreachable_local')
from wf cw sp sr have "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
by (rule oreachable_subterms)
with sl wf have "\<forall>p'\<in>sterms \<Gamma> p. l \<in> labels \<Gamma> p'"
using pl by (rule simple_labels_in_sterms)
with \<open>p' \<in> sterms \<Gamma> p\<close> have "l \<in> labels \<Gamma> p'" by simp
with A1 show "P ((\<sigma>, l), a, (\<sigma>', l'))" using A3 A4 A5 A6 A7 A8 \<open>S \<sigma> \<sigma>' a\<close>
by (rule local)
qed
lemma oseq_step_invariant_ctermsI [intro]:
assumes wf: "wellformed \<Gamma>"
and "control_within \<Gamma> (init A)"
and "simple_labels \<Gamma>"
and "trans A = oseqp_sos \<Gamma> i"
and local: "\<And>p l \<sigma> a q l' \<sigma>' pp pn. \<lbrakk>
wellformed \<Gamma>;
p\<in>ctermsl (\<Gamma> pn);
not_call p;
l\<in>labels \<Gamma> p;
((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i;
((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A;
l'\<in>labels \<Gamma> q;
(\<sigma>, pp) \<in> oreachable A S U;
p\<in>sterms \<Gamma> pp;
(\<sigma>', q) \<in> oreachable A S U;
S \<sigma> \<sigma>' a
\<rbrakk> \<Longrightarrow> P ((\<sigma>, l), a, (\<sigma>', l'))"
shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"
using assms(1-4) proof (rule oseq_step_invariant_ctermI)
fix p l \<sigma> a q l' \<sigma>' pp
assume "p \<in> cterms \<Gamma>"
and otherassms: "l \<in> labels \<Gamma> p"
"((\<sigma>, p), a, (\<sigma>', q)) \<in> oseqp_sos \<Gamma> i"
"((\<sigma>, p), a, (\<sigma>', q)) \<in> trans A"
"l' \<in> labels \<Gamma> q"
"(\<sigma>, pp) \<in> oreachable A S U"
"p \<in> sterms \<Gamma> pp"
"(\<sigma>', q) \<in> oreachable A S U"
"S \<sigma> \<sigma>' a"
from this(1) obtain pn where "p \<in> ctermsl(\<Gamma> pn)"
and "not_call p"
unfolding cterms_def' [OF wf] by auto
with wf show "P ((\<sigma>, l), a, (\<sigma>', l'))"
using otherassms by (rule local)
qed
lemma open_seqp_action [elim]:
assumes "wellformed \<Gamma>"
and "((\<sigma> i, p), a, (\<sigma>' i, p')) \<in> seqp_sos \<Gamma>"
shows "((\<sigma>, p), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
proof -
from assms obtain ps where "ps\<in>sterms \<Gamma> p"
and "((\<sigma> i, ps), a, (\<sigma>' i, p')) \<in> seqp_sos \<Gamma>"
by - (drule trans_from_sterms, auto)
thus ?thesis
proof (induction p)
fix p1 p2
assume "\<lbrakk> ps \<in> sterms \<Gamma> p1; ((\<sigma> i, ps), a, \<sigma>' i, p') \<in> seqp_sos \<Gamma> \<rbrakk>
\<Longrightarrow> ((\<sigma>, p1), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
and "\<lbrakk> ps \<in> sterms \<Gamma> p2; ((\<sigma> i, ps), a, \<sigma>' i, p') \<in> seqp_sos \<Gamma> \<rbrakk>
\<Longrightarrow> ((\<sigma>, p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
and "ps \<in> sterms \<Gamma> (p1 \<oplus> p2)"
and "((\<sigma> i, ps), a, (\<sigma>' i, p')) \<in> seqp_sos \<Gamma>"
with assms(1) show "((\<sigma>, p1 \<oplus> p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
by simp (metis oseqp_sos.ochoiceT1 oseqp_sos.ochoiceT2)
next
fix l fip fmsg p1 p2
assume IH1: "\<lbrakk> ps \<in> sterms \<Gamma> p1; ((\<sigma> i, ps), a, \<sigma>' i, p') \<in> seqp_sos \<Gamma> \<rbrakk>
\<Longrightarrow> ((\<sigma>, p1), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
and IH2: "\<lbrakk> ps \<in> sterms \<Gamma> p2; ((\<sigma> i, ps), a, \<sigma>' i, p') \<in> seqp_sos \<Gamma> \<rbrakk>
\<Longrightarrow> ((\<sigma>, p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
and "ps \<in> sterms \<Gamma> ({l}unicast(fip, fmsg). p1 \<triangleright> p2)"
and "((\<sigma> i, ps), a, (\<sigma>' i, p')) \<in> seqp_sos \<Gamma>"
from this(3-4) have "((\<sigma> i, {l}unicast(fip, fmsg). p1 \<triangleright> p2), a, (\<sigma>' i, p')) \<in> seqp_sos \<Gamma>"
by simp
thus "((\<sigma>, {l}unicast(fip, fmsg). p1 \<triangleright> p2), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
proof (rule seqp_unicastTE)
assume "a = unicast (fip (\<sigma> i)) (fmsg (\<sigma> i))"
and "\<sigma>' i = \<sigma> i"
and "p' = p1"
thus ?thesis by auto
next
assume "a = \<not>unicast (fip (\<sigma> i))"
and "\<sigma>' i = \<sigma> i"
and "p' = p2"
thus ?thesis by auto
qed
next
fix p
assume "ps \<in> sterms \<Gamma> (call(p))"
and "((\<sigma> i, ps), a, (\<sigma>' i, p')) \<in> seqp_sos \<Gamma>"
with assms(1) have "((\<sigma>, ps), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
by (cases ps) auto
with assms(1) \<open>ps \<in> sterms \<Gamma> (call(p))\<close> have "((\<sigma>, \<Gamma> p), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i"
by - (rule otrans_from_sterms', simp_all)
thus "((\<sigma>, call(p)), a, (\<sigma>', p')) \<in> oseqp_sos \<Gamma> i" by auto
qed auto
qed
end
|